**Preface to "Nonlinear Functional Analysis and Its Applications"**

Originally, functional analysis was that branch of mathematics capable of investigating in an abstract way a series of linear mathematical models from science. The study of these linear models—in fact, only first approximations of real models—proved insufficient, so the theory had to be extended to be able to describe the nonlinear phenomena themselves. In this way nonlinear functional analysis was born and continues to develop, becoming a vast and fascinating field of mathematics, with deep applications to increasingly complex problems in physics, biology, chemistry, and economics.

This book consists of nine papers covering a number of basic ideas, concepts, and methods of nonlinear analysis, as well as some current research problems. Thus, the reader is introduced to the fascinating theory around Brouwer's fixed point theorem, which is the basis of important extensions to infinitely dimensional spaces with numerous applications to boundary value problems for various classes of ordinary and partial differential equations. New results for nonstandard elliptic equations obtained with methods such as the technique of upper and lower solutions, advanced methods of critical point theory, and minimax techniques are then presented. The reader is also introduced to Granas' theory of topological transversality, an alternative to the theory of topological degree. Several contributions address current research issues, such as the problem of discontinuous term equations, results of metric fixed point theory, robust tracker design problems for various classes of nonlinear systems, or the problem of periodic solutions in computer virus propagation models.

I would like to particularly thank Professor Jean Mawhin, Professor Dumitru Motreanu, Professor Donal O'Regan, and Professor Biagio Ricceri, who have positively answered our invitation to contribute a paper to this Special Issue. I am sure that their extremely valuable papers will interest the readers and will stimulate new research work in this direction. I would also like to thank the other contributors for their articles that open new perspectives over some specific problems and applications.

Finally, I would like to thank the editors of the journal *Mathematics*, particularly Assistant Editor Grace Du and Marketing Assistant Rainy Han, for their great support throughout the editing process of the Special Issue for Mathematics and its present MDPI Reprint Book.

> **Radu Precup** *Editor*

## *Review* **Variations on the Brouwer Fixed Point Theorem: A Survey**

#### **Jean Mawhin**

Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, chemin du Cyclotron, 2, 1348 Louvain-la-Neuve, Belgium; jean.mawhin@uclouvain.be

Received: 25 February 2020; Accepted: 21 March 2020; Published: 2 April 2020

**Abstract:** This paper surveys some recent simple proofs of various fixed point and existence theorems for continuous mappings in R*n*. The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is considered, based only on the Cauchy integral theorem.

**Keywords:** Brouwer fixed point theorem; Hamadard theorem; Poincaré–miranda theorem

**MSC:** 55M20; 54C15; 30C15

#### **1. Introduction**

The Bolzano theorem for continuous functions *<sup>f</sup>* : [*a*, *<sup>b</sup>*] <sup>⊂</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup>, which states that *<sup>f</sup> has a zero in* [*a*, *b*] *if f*(*a*)*f*(*b*) ≤ 0, was first proved in 1817 by Bolzano [1] and, independently and differently in 1821 by Cauchy [2]. Its various proofs are not very long, and depend only upon the order and completeness properties of <sup>R</sup>. A consequence of the Bolzano theorem applied to *<sup>I</sup>* <sup>−</sup> *<sup>T</sup>* is that *<sup>T</sup>* : [−*R*, *<sup>R</sup>*] <sup>→</sup> <sup>R</sup>*, continuous, has a fixed point in* [−*R*, *R*] *if T*(−*R*) ∈ [−*R*, *R*] *and T*(*R*) ∈ [−*R*, *R*]. This is the case if *T* : [−*R*, *R*] → [−*R*, *R*].

As [−*R*, *<sup>R</sup>*] is the closed ball of center 0 and radius *<sup>R</sup>* in <sup>R</sup>, a natural question is to know if, *BR* denoting the closed ball *BR* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* of center 0 and radius *<sup>R</sup>* <sup>&</sup>gt; 0, *any continuous mapping <sup>T</sup>* : *BR* <sup>→</sup> <sup>R</sup>*<sup>n</sup> such that T*(*∂BR*) ⊂ *BR has a fixed point*, and, in particular, if *any continuous mapping T* : *BR* → *BR has a fixed point*. The answer is yes, and the first result, usually called the Rothe fixed point theorem (FPT), is more correctly referred as the Birkhoff–Kellog FPT, and the second one as the Brouwer FPT.

Many different proofs of those results have been given since the first published one of the Brouwer FPT by Hadamard in 1910 [3]. Brouwer's original proof [4], published in 1912, was topological and based on some fixed point theorems on spheres proved with the help of the topological degree introduced in the same paper. The Birkhoff–Kellogg FPT was first proved by Birkhoff and Kellogg in 1922 [5]. Its standard name Rothe FPT refers to its extension to Banach spaces by Rothe [6] in 1937.

The existing proofs use ideas from various areas of mathematics such as algebraic topology, combinatorics, differential topology, analysis, algebraic geometry, and even mathematical economics. A survey and a bibliography can be found in [7]. Even for *n* = 2, they cease to be elementary and/or can be technically complicated. The aim of this paper is to survey recent results on some elementary approaches to the Birkhoff–Kellogg and Brouwer FPT, and on how to deduce from them in a simple and systematic way other fixed point and existence theorems for mappings in R*n*. Recall that these results, combined with basic facts of functional analysis, are fundamental in obtaining useful extensions to some classes of mappings in infinite-dimensional normed spaces.

After recalling the simple concept of curvilinear integral in R2, we first propose in Section 2 an elementary proof of the Birkhoff–Kellogg FPT for *n* = 2, based upon such integrals. As the extension to arbitrary *<sup>n</sup>*, using differential (*<sup>n</sup>* <sup>−</sup> <sup>1</sup>) forms in <sup>R</sup>*n*, leads to very cumbersome computations, we adopt in Section 3 a variant given in [8], using differential *n*-forms, which in dimension *n* happens to be significantly simpler than the direct extension of the approach of Section 2.

The generalizations of the Birkhoff–Kellogg and Brouwer FPT to a closed ball in R*<sup>n</sup>* and their homeomorphic images are stated in Section 4. After the concepts of retract and retraction are introduced, the Leray–Schauder–Schaefer FPT on a closed ball is deduced from the Brouwer FPT, whose statement is also extended to retracts of a closed ball in R*n*. Finally, the equivalence of the Birkhoff–Kellogg and Brouwer FPT on a closed ball is established.

The Brouwer FPT and retractions are then used in Section 5 to prove, in a very simple and unified way inspired by the approach of [9], several conditions for the existence of zeros continuous mappings in R*n*, namely the Poincaré–Bohl theorem on a closed ball, the Hadamard theorem on a compact convex set, the Poincaré–Miranda theorem on a closed *n*-interval, and the Hartman–Stampacchia theorem on variational inequalities.

Finally, in Section 6, following the method introduced in [10], simple versions of the Cauchy integral theorem provide criterions for the existence of zeros of a holomorphic function in same spirit of the approach in Section 2. They allow very simple proofs of the Hadamard and Poincaré–Miranda theorems and of the Birkhoff–Kellogg and Brouwer FPT for holomorphic functions.

#### **2. A Proof the Birkhoff–Kellogg Theorem on a Closed Disc Based on Curvilinear Integrals**

Let *<sup>D</sup>* <sup>⊂</sup> <sup>R</sup><sup>2</sup> be open and nonempty and let ·, · denote the usual inner product in <sup>R</sup>2. Given *<sup>f</sup>* = (*f*1, *<sup>f</sup>*2) : *<sup>D</sup>* <sup>→</sup> <sup>R</sup>2, *<sup>x</sup>* → *<sup>f</sup>*(*x*) and *<sup>ϕ</sup>* = (*ϕ*1, *<sup>ϕ</sup>*2) : [*a*, *<sup>b</sup>*] <sup>→</sup> *<sup>D</sup>*, *<sup>t</sup>* → *<sup>ϕ</sup>*(*t*) of class *<sup>C</sup>*1, we consider the corresponding **curvilinear integral** defined by *b <sup>a</sup> f*(*ϕ*(*t*), *ϕ* (*t*)*dt* where denotes the derivative with respect to *t*.

The following result is fundamental for our proof of the Birkhoff–Kellogg FPT on a closed disc.

**Lemma 1.** *If <sup>f</sup>* = (*f*1, *<sup>f</sup>*2) : *<sup>D</sup>* <sup>→</sup> <sup>R</sup><sup>2</sup> *is of class <sup>C</sup>*<sup>1</sup> *and such that <sup>∂</sup>*<sup>1</sup> *<sup>f</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*<sup>2</sup> *<sup>f</sup>*<sup>1</sup> *and if* <sup>Φ</sup> : [*a*, *<sup>b</sup>*] <sup>×</sup> [0, 1] <sup>→</sup> *<sup>D</sup> is of class <sup>C</sup>*<sup>2</sup> *and such that* <sup>Φ</sup>(*b*, *<sup>λ</sup>*) = <sup>Φ</sup>(*a*, *<sup>λ</sup>*) *for all <sup>λ</sup>* <sup>∈</sup> [0, 1]*, then <sup>λ</sup>* <sup>→</sup> *b <sup>a</sup> f*(Φ(*t*, *λ*), *∂t*Φ(*t*, *λ*)*dt is constant on* [0, 1]*.*

**Proof.** It suffices to prove that *∂λ b <sup>a</sup> f*(Φ(*t*, *λ*), *∂t*Φ(*t*, *λ*)*dt* = 0 for all *λ* ∈ [0, 1]. We have, with differentiation under integral sign easily justified and the use of assumptions, the Schwarz theorem and the fundamental theorem of calculus, and omitting the arguments (*t*, *λ*) for the sake of brevity

$$
\begin{split}
&\partial\_{\lambda}\int\_{a}^{b}\langle f(\Phi),\partial\_{t}\Phi\rangle dt = \int\_{a}^{b}\partial\_{\lambda}[\langle f(\Phi),\partial\_{t}\Phi\rangle] dt \\
&= \int\_{a}^{b}\left\langle \left(\partial\_{\lambda}[f(\Phi)],\partial\_{t}\Phi\right) + \langle f(\Phi),\partial\_{\lambda}\partial\_{t}\Phi\rangle\right\rangle dt \\
&= \int\_{a}^{b}\left[\left\langle \sum\_{j=1}^{2}\partial\_{j}f(\Phi)\partial\_{\lambda}\Phi\_{j},\partial\_{t}\Phi\right\rangle + \langle f(\Phi),\partial\_{t}\partial\_{t}\Phi\rangle\right] dt \\
&= \int\_{a}^{b}\left[\sum\_{k=1}^{2}\sum\_{j=1}^{2}\partial\_{j}f\_{k}(\Phi)\partial\_{\lambda}\Phi\_{j}\partial\_{t}\Phi\_{k} + \langle f(\Phi),\partial\_{t}\partial\_{t}\Phi\rangle\right] dt \\
&= \int\_{a}^{b}\left[\sum\_{k=1}^{2}\sum\_{j=1}^{2}\partial\_{k}f\_{j}(\Phi)\partial\_{t}\Phi\_{k}\partial\_{\lambda}\Phi\_{j} + \langle f(\Phi),\partial\_{t}\partial\_{t}\Phi\rangle\right] dt \\
&= \int\_{a}^{b}\left[\sum\_{j=1}^{2}\partial\_{t}f\_{j}(\Phi)\partial\_{\lambda}\Phi\_{j} + \langle f(\Phi),\partial\_{t}\partial\_{t}\Phi\rangle\right] dt \\
&= \int\_{a}^{b}\left\{\langle\partial\_{t}[f(\Phi)],\partial\_{t}\Phi\rangle + \langle f(\Phi),\partial\_{t}\partial\_{t}\Phi\rangle\right\} dt \\
&= \int\_{a}^{b}\partial\_{t}[\langle f(\Phi),\partial\_{t}\Phi\rangle] dt = f(\Phi(b,\lambda)) - f(\Phi(a,\lambda)) = 0.
\end{split}
$$

Let *BR* :<sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> <sup>R</sup><sup>2</sup> : <sup>|</sup>*x*| ≤ *<sup>R</sup>*}, with <sup>|</sup>*x*<sup>|</sup> the Euclidian norm. We prove the **Birkhoff–Kellogg FPT on a closed disc**.

**Theorem 1.** *Any continuous mapping T* : *BR* <sup>→</sup> <sup>R</sup><sup>2</sup> *such that T*(*∂BR*) <sup>⊂</sup> *BR has a fixed point in BR.*

**Proof.** Assume that *T* has no fixed point in *BR*. Then, |*y* − *T*(*y*)| > 0 for all *y* ∈ *∂BR*, and, as *T*(*∂BR*) ⊂ *BR*, |*y* − *λT*(*y*)|≥|*y*| − *λ*|*T*(*y*)| ≥ 1 − *λ* > 0, for all (*y*, *λ*) ∈ *∂BR* × [0, 1). Similarly, *λy* − *T*(*λy*) = 0 for all (*y*, *λ*) ∈ *∂BR* × [0, 1]. As *T* is continuous, there exists *δ* > 0 such that |*y* − *λT*(*y*)| ≥ *δ* and |*λy* − *T*(*λy*)| ≥ *δ* for all (*y*, *λ*) ∈ *∂BR* × [0, 1]. From the Weierstrass approximation theorem, there is a polynomial *<sup>P</sup>* : <sup>R</sup><sup>2</sup> <sup>→</sup> <sup>R</sup><sup>2</sup> such that <sup>|</sup>*T*(*y*) <sup>−</sup> *<sup>P</sup>*(*y*)| ≤ *<sup>δ</sup>* <sup>2</sup> for all *y* ∈ *BR*. Consequently, letting *F*(*y*, *λ*) := *<sup>y</sup>* <sup>−</sup> *<sup>λ</sup>P*(*y*) and *<sup>G</sup>*(*y*, *<sup>λ</sup>*) :<sup>=</sup> *<sup>λ</sup><sup>y</sup>* <sup>−</sup> *<sup>P</sup>*(*λy*), we have, for all (*y*, *<sup>λ</sup>*) <sup>∈</sup> *<sup>∂</sup>BR* <sup>×</sup> [0, 1], <sup>|</sup>*F*(*y*, *<sup>λ</sup>*)| ≥ *<sup>δ</sup>* <sup>2</sup> and <sup>|</sup>*G*(*y*, *<sup>λ</sup>*)| ≥ *<sup>δ</sup>* <sup>2</sup> . Hence, there exists an open neighborhood Δ of *∂BR* such that *F*(*y*, *λ*) = 0 and *G*(*y*, *λ*) = 0 for all (*y*, *λ*) ∈ Δ × [0, 1]. If

$$f\_1: \mathbb{R}^2 \backslash \{0\} \to \mathbb{R}, \; \mathbf{x} \mapsto -|\mathbf{x}|^{-2}\mathbf{x}\_2, \; f\_2: \mathbb{R}^2 \backslash \{0\} \to \mathbb{R}, \mathbf{x} \mapsto |\mathbf{x}|^{-2}\mathbf{x}\_1, \; f\_1$$

then *∂*<sup>2</sup> *f*1(*x*) = |*x*| <sup>−</sup>4(*x*<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> <sup>1</sup>) = *<sup>∂</sup>*<sup>1</sup> *<sup>f</sup>*2(*x*). If *<sup>γ</sup><sup>R</sup>* : [0, 2*π*] <sup>→</sup> <sup>R</sup>2, *<sup>t</sup>* → *<sup>R</sup>*(cos *<sup>t</sup>*, sin *<sup>t</sup>*) is a parametric representation of *∂BR*, so that *γR*(0) = *γR*(2*π*), it follows from Lemma 1 that the integrals

$$\int\_0^{2\pi} \langle f[F(\gamma\_R(t), \lambda)], \partial\_t F(\gamma\_R(t), \lambda) \rangle dt \text{ and } \int\_0^{2\pi} \langle f[G(\gamma\_R(t), \lambda)], \partial\_t G(\gamma\_R(t), \lambda) \rangle dt$$

are constant for *λ* ∈ [0, 1]. Hence, noticing that *F*(·, 1) = *G*(·, 1) = *I* − *P*,

$$\begin{split} &\int\_{0}^{2\pi} \langle f[F(\gamma\_{\mathbb{R}}(t),0)], \partial\_{t}F(\gamma\_{\mathbb{R}}(t),0) \rangle dt = \int\_{0}^{2\pi} \langle f[F(\gamma\_{\mathbb{R}}(t),1)], \partial\_{t}F(\gamma\_{\mathbb{R}}(t),1) \rangle dt \\ &= \int\_{0}^{2\pi} \langle f[G(\gamma\_{\mathbb{R}}(t),1)], \partial\_{t}G(\gamma\_{\mathbb{R}}(t),1) \rangle dt = \int\_{0}^{2\pi} \langle f[G(\gamma\_{\mathbb{R}}(t),0)], \partial\_{t}G(\gamma\_{\mathbb{R}}(t),0) \rangle dt. \end{split}$$

However, as *G*(·, 0) = −*P*(0) is constant and *F*(·, 0) = *I*,

$$\begin{aligned} 0 &= \int\_0^{2\pi} \langle f[G(\gamma\_R(t), 0)], \partial\_t G(\gamma\_R(t), 0) \rangle dt \\ &= \int\_0^{2\pi} \langle f[F(\gamma\_R(t), 0)], \partial\_t F(\gamma\_R(t), 0) \rangle dt \\ &= \int\_0^{2\pi} \langle f(\gamma\_R(t)), \gamma\_R'(t) \rangle dt = \int\_0^{2\pi} (\sin^2 t + \cos^2 t) \, dt = 2\pi, \end{aligned}$$

a contradiction.

A direct consequence is the **Brouwer FPT on a closed disc**.

**Corollary 1.** *Any continuous mapping T* : *BR* → *BR has a fixed point in BR.*

#### **3. A Proof of the Birkhoff–Kellogg Theorem on a Closed** *n***-Ball Based on Differential** *n***-Forms**

The argument used in Section 2 for mappings in R<sup>2</sup> can be extended to mappings in R*n*, using the basic properties of differential *k*-forms in R*n*. For *n* = 2, the differential 1-forms and differential (*n* − 1)-forms coincide, and it is the last ones that are requested for extending the proof of Theorem 1 to arbitrary *n*. We leave to the motivated reader the work to write down this extension of the first approach and to realize that this generalization to dimension *n* of Lemma 1 is very cumbersome and lengthy. Fortunately a similar approach based on differential *n*-forms instead of (*n* − 1)-forms has been

introduced in [8], which, for *n* = 2, has the same length and technicality as the one used in Section 2, but keeps its simplicity for arbitrary *n*. We describe it in this section.

For *<sup>D</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* open, bounded and nonempty, we need the concept of differential (*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)-forms and *n*-forms and suppose that the reader is familiar with the notions, notations and properties of differential *k*-forms (1 ≤ *k* ≤ *n*) on *D*, wedge products, pull backs, exterior differentials and the Stokes–Cartan theorem for differential forms with compact support [11]. All the functions involved in differential forms are supposed to be of class *<sup>C</sup>*2. We associate to the functions *fj* : *<sup>D</sup>* <sup>→</sup> <sup>R</sup> (*<sup>j</sup>* <sup>=</sup> 1, ... , *<sup>n</sup>*) the **differential** 1**-form** *ω<sup>f</sup>* := ∑*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *fj dxj* **in** *D*, and the **differential** (*n* − 1)**-form**

$$\nu\_f = \sum\_{j=1}^n (-1)^{j-1} f\_j \, d\mathfrak{x}\_1 \wedge \dots \wedge \widehat{d\mathfrak{x}\_j} \wedge \dots \wedge d\mathfrak{x}\_{n'} $$

where *dx<sup>j</sup>* means that the corresponding term is missing. We associate also to *<sup>g</sup>* : *<sup>D</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* the **differential** *<sup>n</sup>***-form** *<sup>μ</sup><sup>g</sup>* <sup>=</sup> *g dx*<sup>1</sup> <sup>∧</sup> ... <sup>∧</sup> *dxn*. For example, given the function *<sup>w</sup>* : *<sup>D</sup>* <sup>→</sup> <sup>R</sup> with partial derivatives *∂jw*, its **differential** *dw* := ∑*<sup>n</sup> <sup>j</sup>*=1(*∂jw*) *dxj* is the differential 1-form *<sup>ω</sup>*∇*w*.

Let <sup>Δ</sup> <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* be open, bounded and nonempty, *<sup>F</sup>* : <sup>Δ</sup> <sup>×</sup> [0, 1] <sup>→</sup> *<sup>D</sup>*, (*y*, *<sup>λ</sup>*) → *<sup>F</sup>*(*y*, *<sup>λ</sup>*). For each fixed *λ* ∈ [0, 1],

$$\begin{aligned} \left[F^\*(\cdot,\lambda)\omega\_f\right] &= \sum\_{j=1}^n [f\_j \circ F(\cdot,\lambda)] \, dF\_{\vec{j}}(\cdot,\lambda) \\ &= \sum\_{k=1}^n \left[\sum\_{j=1}^n [f\_j \circ F(\cdot,\lambda)] \partial\_k F\_{\vec{j}}(\cdot,\lambda)\right] dy\_k \quad (j=1,\ldots,n) \end{aligned}$$

is well defined. To shorten the notations, we write *Fj* for *Fj*(·, *λ*). We define the **derivative with respect to** *λ* of *F*∗*ω<sup>f</sup>* by

$$\partial\_{\lambda}(F^\*\omega\_f) := \sum\_{k=1}^n \partial\_{\lambda} \left[ \sum\_{j=1}^n (f\_j \circ F) \partial\_k F\_j \right] dy\_k.$$

so that

$$\begin{aligned} \left(\partial\_{\lambda}(F^\*\omega\_f)\right)\_- &= \sum\_{k=1}^n \sum\_{j=1}^n \left[\partial\_{\lambda}(f\_{\hat{\jmath}}\circ F)\partial\_k F\_{\hat{\jmath}} + (f\_{\hat{\jmath}}\circ F)\partial\_{\lambda}\partial\_k F\_{\hat{\jmath}}\right] dy\_k \\ &= \sum\_{j=1}^n \left[\partial\_{\lambda}(f\_{\hat{\jmath}}\circ F)\,dF\_{\hat{\jmath}} + (f\_{\hat{\jmath}}\circ F)\,\partial\_{\lambda}(dF\_{\hat{\jmath}})\right]. \end{aligned}$$

Furthermore,

$$\partial\_{\lambda}(dF\_{\vec{j}}) = \sum\_{k=1}^{n} (\partial\_{\lambda}\partial\_{k}F\_{\vec{j}}) \, dy\_{k} = \sum\_{k=1}^{n} (\partial\_{k}\partial\_{\lambda}F\_{\vec{j}}) \, dy\_{k} = d(\partial\_{\lambda}F\_{\vec{j}}) \quad (j = 1, \dots, n).$$

On the other hand,

$$dF\_1 \wedge \ldots \wedge dF\_n = f\_F \, dy\_1 \wedge \ldots \, dy\_n$$

where *JF*(·,*<sup>λ</sup>*)(*y*, *<sup>λ</sup>*) denotes the Jacobian of *<sup>F</sup>*(·, *<sup>λ</sup>*) at (*y*, *<sup>λ</sup>*) ∈ <sup>Δ</sup> × [0, 1], and

$$
\partial\_{\lambda} \left[ dF\_1 \wedge \dots \wedge dF\_n \right] = \sum\_{j=1}^n dF\_1 \wedge \dots \wedge \partial\_{\lambda} dF\_j \wedge \dots \wedge dF\_n.
$$

The following two results replace Lemma 1 in Section 2. The first one shows that the differential *n*-form *∂λ*(*F*∗*μg*) is exact in Δ, i.e., is the exterior differential of a (*n* − 1)-differential form in Δ.

*Mathematics* **2020**, *8*, 501

**Lemma 2.** *For each λ* ∈ [0, 1], *we have*

$$\partial\_{\lambda}(F^\*\mu\_{\mathcal{S}}) = d\left[ (\mathcal{g}\circ F) \left( \sum\_{j=1}^n (-1)^{j-1} \partial\_{\lambda} F\_{\bar{j}} dF\_1 \wedge \dots \wedge \widehat{dF\_{\bar{j}}} \wedge \dots \wedge dF\_n \right) \right].$$

**Proof.** We have

*∂λ*(*F*∗*μg*) = *∂λ*(*g* ◦ *F*) *dF*<sup>1</sup> ∧ ... ∧ *dFn* + (*g* ◦ *F*) *∂λ* (*dF*<sup>1</sup> ∧ ... ∧ *dFn*) = *n* ∑ *j*=1 (*∂jg* ◦ *F*)*∂λFj dF*<sup>1</sup> ∧ ... ∧ *dFn* + (*g* ◦ *F*) *n* ∑ *j*=1 *dF*<sup>1</sup> ∧ ... ∧ *∂λdFj* ∧ ... ∧ *dFn* = *n* ∑ *j*=1 (−1)*j*−1(*∂jg* ◦ *<sup>F</sup>*) *dFj* <sup>∧</sup> *∂λFj dF*<sup>1</sup> <sup>∧</sup> ... <sup>∧</sup> *dF<sup>j</sup>* <sup>∧</sup> ... <sup>∧</sup> *dFn* + (*g* ◦ *F*) *n* ∑ *j*=1 (−1)*j*−1*<sup>d</sup> ∂λFj* ∧ *dF*<sup>1</sup> ∧ ... ∧ *dF<sup>j</sup>* ∧ ... ∧ *dFn* = *n* ∑ *j*=1 (−1)*j*−<sup>1</sup> *n* ∑ *k*=1 (*∂kg* ◦ *F*) *dFk* ∧ *∂λFj dF*<sup>1</sup> ∧ ... ∧ *dF<sup>j</sup>* ∧ ... ∧ *dFn* + (*g* ◦ *F*) *n* ∑ *j*=1 (−1)*j*−1*<sup>d</sup> ∂λFj dF*<sup>1</sup> ∧ ... ∧ *dF<sup>j</sup>* ∧ ... ∧ *dFn* = *d*(*g* ◦ *F*) ∧ *n* ∑ *j*=1 (−1)*j*−<sup>1</sup>*∂λFj dF*<sup>1</sup> <sup>∧</sup> ... <sup>∧</sup> *dF<sup>j</sup>* <sup>∧</sup> ... <sup>∧</sup> *dFn* + (*g* ◦ *F*) *d n* ∑ *j*=1 (−1)*j*−<sup>1</sup>*∂λFj dF*<sup>1</sup> <sup>∧</sup> ... <sup>∧</sup> *dF<sup>j</sup>* <sup>∧</sup> ... <sup>∧</sup> *dFn* = *d* (*g* ◦ *F*) *n* ∑ *j*=1 (−1)*j*−<sup>1</sup>*∂λFj dF*<sup>1</sup> <sup>∧</sup> ... <sup>∧</sup> *dF<sup>j</sup>* <sup>∧</sup> ... <sup>∧</sup> *dFn* := *dνg*,*F*.

**Corollary 2.** *If <sup>w</sup>* <sup>∈</sup> *<sup>C</sup>*2(R*n*, <sup>R</sup>)*,* <sup>Δ</sup> *is open, bounded and <sup>F</sup>* <sup>∈</sup> *<sup>C</sup>*2(<sup>Δ</sup> <sup>×</sup> [0, 1],R*n*) *verify <sup>F</sup>*(*∂*<sup>Δ</sup> <sup>×</sup> [0, 1]) <sup>∩</sup> *supp w* = ∅*, then* - <sup>Δ</sup> *F*∗*μ<sup>w</sup> is independent of λ on* [0, 1].

**Proof.** Using Lemma 2, the assumption and Stokes–Cartan theorem, we get

$$\partial\_{\lambda} \int\_{\Delta} F^\* \mu\_w = \int\_{\Delta} \partial\_{\lambda} (F^\* \mu\_w) = \int\_{\Delta} d\nu\_{w,F} = \int\_{\partial \Delta} \nu\_{w,F} = 0.$$

Let *BR* :<sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* : <sup>|</sup>*x*| ≤ *<sup>R</sup>*} with <sup>|</sup>*x*<sup>|</sup> the Euclidian norm. We now show that Proposition <sup>2</sup> allows a simple proof of the **Birkhoff–Kellogg FPT on a closed** *n***-ball**, quite similar to that of Theorem 1.

**Theorem 2.** *Any continuous mapping T* : *BR* <sup>→</sup> <sup>R</sup>*<sup>n</sup> such that T*(*∂BR*) <sup>⊂</sup> *BR has a fixed point in BR.*

**Proof.** Assume that *T* has no fixed point in *BR*. Then, *x* − *T*(*x*) = 0 for *x* ∈ *∂BR*, and for (*x*, *λ*) ∈ *∂BR* × [0, 1), we have |*x* − *λT*(*x*)| ≥ *R* − *λ*|*T*(*x*)| ≥ (1 − *λ*)*R* > 0. Thus, |*x* − *λT*(*x*)| > 0 for all (*x*, *λ*) ∈ *∂BR* × [0, 1]. On the other hand, for (*x*, *λ*) ∈ *∂BR* × [0, 1], we have *λx* ∈ *BR*, *λx* − *T*(*λx*) = 0, and hence |*λx* − *T*(*λx*)| > 0 for all (*x*, *λ*) ∈ *∂B*(*R*) × [0, 1]. By continuity, there exists *δ* > 0 such that <sup>|</sup>*<sup>x</sup>* <sup>−</sup> *<sup>λ</sup>T*(*x*)<sup>|</sup> <sup>&</sup>gt; *<sup>δ</sup>* for all (*x*, *<sup>λ</sup>*) <sup>∈</sup> *<sup>∂</sup>BR* <sup>×</sup> [0, 1]. Let *<sup>P</sup>* : <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* be a polynomial such that max*BR* <sup>|</sup>*<sup>P</sup>* <sup>−</sup> *<sup>T</sup>*<sup>|</sup> <sup>≤</sup> *<sup>δ</sup>*/2, and define *<sup>F</sup>* <sup>∈</sup> *<sup>C</sup>*∞(R*<sup>n</sup>* <sup>×</sup> [0, 1], <sup>R</sup>*n*) and *<sup>G</sup>* <sup>∈</sup> *<sup>C</sup>*∞(R*<sup>n</sup>* <sup>×</sup> [0, 1], <sup>R</sup>*n*) by *<sup>F</sup>*(*x*, *<sup>λ</sup>*) = *<sup>λ</sup><sup>x</sup>* <sup>−</sup> *<sup>P</sup>*(*λx*) and *G*(*x*, *λ*) = *x* − *λP*(*x*), so that |*F*(*x*, *λ*)| ≥ *δ*/2 and |*G*(*x*, *λ*)| ≥ *δ*/2 for all (*x*, *λ*) ∈ *∂BR* × [0, 1]. Let *w* ∈ *<sup>C</sup>*2(R*n*, <sup>R</sup>) with supp *<sup>w</sup>* <sup>⊂</sup> *<sup>B</sup>*(*δ*/2), the open ball of center 0 and radius *<sup>δ</sup>*/2, and - *BR <sup>w</sup>*(*y*) *dy* = 1. Then, by Proposition 2 with Δ = *BR*, we get

$$0 \quad = \int\_{B\_R} F^\*(\cdot, 0) \mu\_w = \int\_{B\_R} F^\*(\cdot, 1) \mu\_w = \int\_{B\_R} (I - P)^\* \mu\_{W'} $$

and

$$\begin{aligned} \int\_{B\_R} (I - P)^\* \mu\_w &= \int\_{B\_R} G^\*(\cdot, 1) \mu\_w = \int\_{B\_R} G^\*(\cdot, 0) \mu\_w = \int\_{B\_R} \mu\_w \\ &= \int\_{B\_R} w(y) \, dy = 1, \end{aligned}$$

a contradiction.

The **Brouwer FPT on a closed** *n***-ball** is a special case.

**Corollary 3.** *Any continuous mapping T* : *BR* → *BR has a fixed point in BR.*

#### **4. Fixed Points, Homeomorphisms and Retractions in** R*<sup>n</sup>*

Now, if *<sup>K</sup>* <sup>⊂</sup> <sup>R</sup>*n*, if there exists a homeomorphism *<sup>h</sup>* : *<sup>B</sup><sup>n</sup>* <sup>→</sup> *<sup>K</sup>*, and if *<sup>T</sup>* : *<sup>K</sup>* <sup>→</sup> *<sup>K</sup>* is continuous, *<sup>h</sup>*−<sup>1</sup> ◦ *<sup>T</sup>* ◦ *<sup>h</sup>* : *<sup>B</sup><sup>n</sup>* <sup>→</sup> *<sup>B</sup><sup>n</sup>* is continuous, has a fixed point *<sup>x</sup>*<sup>∗</sup> by Theorem 3, and *<sup>h</sup>*(*x*∗) <sup>∈</sup> *<sup>K</sup>* is a fixed point of *T*. Consequently, we have a **Brouwer FPT for homeomorphic images of a closed** *n***-ball**.

**Theorem 3.** *If K* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup> is homeomorphic to BR, any continuous mapping T* : *<sup>K</sup>* <sup>→</sup> *K has a fixed point in K.*

For example, *<sup>K</sup>* can be any **closed** *<sup>n</sup>***-interval** [*a*1, *<sup>b</sup>*1] <sup>×</sup> ... <sup>×</sup> [*an*, *bn*], or an *<sup>n</sup>***-simplex** <sup>R</sup>*<sup>n</sup>* <sup>+</sup> := {*x* = ∑*n <sup>j</sup>*=<sup>1</sup> *xje<sup>j</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* : *xj* <sup>≥</sup> 0, <sup>∑</sup>*<sup>n</sup> <sup>j</sup>*=<sup>1</sup> *xj* ≤ 1}.

**Remark 1.** *In Theorem 3, the boundedness assumption on K cannot be omitted: a translation x* → *x* + *a in* <sup>R</sup>*<sup>n</sup> with <sup>a</sup>* <sup>=</sup> <sup>0</sup> *has no fixed point. The closedness assumption on <sup>K</sup> cannot be omitted as well: <sup>T</sup>* : (0, 1) <sup>→</sup> (0, 1), *<sup>x</sup>* → *<sup>x</sup>*<sup>2</sup> *has no fixed point in* (0, 1)*. Theorem <sup>3</sup> does not hold for any closed bounded set: a nontrivial rotation of the closed annulus A* <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> <sup>R</sup><sup>2</sup> : *<sup>r</sup>*<sup>1</sup> ≤ |*x*| ≤ *<sup>r</sup>*2} *has no fixed point in A.*

We now introduce concepts and results due to Borsuk [12] which provide another class of sets on which the Brouwer FPT holds and simple proofs of various equivalent formulations of this theorem. We say that *<sup>U</sup>* <sup>⊂</sup> *<sup>V</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* is a **retract** of *<sup>V</sup>* if there exists a continuous mapping *<sup>r</sup>* : *<sup>V</sup>* <sup>→</sup> *<sup>U</sup>* such that *r* = *I* on *U* (**retraction of** *V* **in** *U*). For example, *BR* is a retract of R*n*, with a retraction *r* given by

$$r(\mathbf{x}) = \begin{cases} \begin{array}{c} \mathbf{x} \\ \end{array} \begin{array}{c} \text{if} \\ \frac{\mathbf{x}}{\|\mathbf{x}\|} \end{array} \begin{array}{c} \|\mathbf{x}\| \le R \\ \end{array} \\ \end{array} \tag{1}$$

Similarly, *for any* 0 < *R*<sup>1</sup> ≤ *R*2*, BR*<sup>1</sup> *is a retract of BR*<sup>2</sup> *.*

**Remark 2.** *The Brouwer FPT on BR implies the Birkhoff–Kellogg FPT on BR. Indeed, if <sup>T</sup>* : *BR* <sup>→</sup> <sup>R</sup>*<sup>n</sup> is continuous, T*(*∂BR*) ⊂ *BR, and r is given by (1), then r* ◦ *T* : *BR* → *BR is continuous and, by the Brouwer FPT 3, has a fixed point x*<sup>∗</sup> ∈ *BR. If* |*T*(*x*∗)| > *R,* |*x*∗| = |*r*(*T*(*x*∗))| = *R and* |*T*(*x*∗)| ≤ *R, a contradiction. Thus,* |*T*(*x*∗)| ≤ *R and x*<sup>∗</sup> = *T*(*x*∗)*. Thus, the two statements are equivalent.*

**Remark 3.** *The Brouwer FPT has for immediate topological consequence the well-known* **no-retraction theorem***, stating that ∂BR is not a retract of BR in* R*n. We do not repeat here the simple proof of this result and the proof of Brouwer FPT from the no-retraction theorem.*

An easy consequence of Theorem 3 is the **Leray–Schauder–Schaefer fixed point theorem**, a special case of a more general result obtained in 1934 by Leray and Schauder [13]. The proof given here is due to Schaefer [14].

**Theorem 4.** *Any continuous mapping <sup>T</sup>* : *BR* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup> such that <sup>x</sup>* <sup>=</sup> *<sup>λ</sup>T*(*x*) *for all* (*x*, *<sup>λ</sup>*) <sup>∈</sup> *<sup>∂</sup>BR* <sup>×</sup> (0, 1) *has a fixed point in BR.*

**Proof.** Let *<sup>r</sup>* : <sup>R</sup>*<sup>n</sup>* <sup>→</sup> *BR* be the retraction of <sup>R</sup>*<sup>n</sup>* onto *BR* defined in Equation (1). Theorem <sup>3</sup> implies the existence of *<sup>x</sup>*<sup>∗</sup> <sup>∈</sup> *BR* such that *<sup>x</sup>*<sup>∗</sup> <sup>=</sup> *<sup>r</sup>*(*T*(*x*∗)). If <sup>|</sup>*T*(*x*∗)<sup>|</sup> <sup>&</sup>gt; *<sup>R</sup>*, then *<sup>x</sup>*<sup>∗</sup> <sup>=</sup> *<sup>R</sup>* |*T*(*x*∗)| *T*(*x*∗), so that |*x*∗| = *R* and *x*<sup>∗</sup> = *λ*∗*T*(*x*∗) with *λ*<sup>∗</sup> = *<sup>R</sup>* <sup>|</sup>*T*(*x*∗)<sup>|</sup> <sup>&</sup>lt; 1, a contradiction with the assumption. Hence, <sup>|</sup>*T*(*x*∗)| ≤ *<sup>R</sup>* and *x*∗ = *T*(*x*∗).

**Remark 4.** If *T* : *∂BR* → *BR*, it is clear that the assumption of Theorem 4 is satisfied. Thus *the Leray–Schauder–Schaefer FPT implies the Birkhoff–Kellogg FPT,* and hence *the two statements are quivalent.*

The Brouwer FPT holds for retracts of a closed ball.

**Theorem 5.** *If U* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup> is a retract of BR, any continuous mapping T* : *<sup>U</sup>* <sup>→</sup> *U has a fixed point.*

**Proof.** Let *U* = *r*(*BR*) for some retraction *r* : *BR* → *U*. Then, *T* ◦ *r* : *BR* → *U* ⊂ *BR* has a fixed point *x*<sup>∗</sup> ∈ *U*. Hence, *x*<sup>∗</sup> = *r*(*x*∗), and *x*<sup>∗</sup> = *T*(*x*∗).

If *<sup>C</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* is non- empty, closed and convex, the **orthogonal projection** *pC*(*x*) **on** *<sup>C</sup>* of *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*n*, defined by <sup>|</sup>*pC*(*x*) <sup>−</sup> *<sup>x</sup>*<sup>|</sup> <sup>=</sup> min*y*∈*<sup>C</sup>* <sup>|</sup>*<sup>y</sup>* <sup>−</sup> *<sup>x</sup>*|, is a retraction of <sup>R</sup>*<sup>n</sup>* onto *<sup>C</sup>* [15]. Consequently, *<sup>C</sup> is a retract of any BR* ⊃ *C*, giving a **Brouwer FPT on compact convex sets**.

**Corollary 4.** *If C* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup> is compact and convex, any continuous mapping T* : *<sup>C</sup>* <sup>→</sup> *C has a fixed point in C.*

#### **5. Zeros of Continuous Mappings in** R*<sup>n</sup>*

The first theorem on the existence of a zero for a mapping from *BR* into R*<sup>n</sup>* was first stated and proved for *C*<sup>1</sup> mappings by Bohl [16] in 1904, and extended to continuous mappings by Hadamard in 1910 [3], under the name **Poincaré–Bohl theorem**. It is a reformulation of the Leray–Schauder–Schaefer FPT Theorem 4.

**Theorem 6.** *Any continuous mapping <sup>f</sup>* : *BR* <sup>→</sup> <sup>R</sup>*<sup>n</sup> such that <sup>f</sup>*(*x*) <sup>=</sup> *<sup>μ</sup><sup>x</sup> for all <sup>x</sup>* <sup>∈</sup> *<sup>∂</sup>BR and for all <sup>μ</sup>* <sup>&</sup>lt; <sup>0</sup> *has a zero in BR.*

**Proof.** Define the continuous mapping *<sup>T</sup>* : *BR* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* by *<sup>T</sup>*(*x*) = *<sup>x</sup>* <sup>−</sup> *<sup>f</sup>*(*x*). For (*x*, *<sup>λ</sup>*) <sup>∈</sup> *<sup>∂</sup>BR* <sup>×</sup> (0, 1), we have, by assumption,

$$
\lambda \mathbf{x} - \lambda T(\mathbf{x}) = (1 - \lambda)\mathbf{x} + \lambda f(\mathbf{x}) = \lambda \left[ f(\mathbf{x}) - \frac{\lambda - 1}{\lambda} \mathbf{x} \right] \neq 0.
$$

By Theorem 4, *T* has a fixed point *x*∗ in *BR*, which is a zero of *f* .

In 1910, two years before the publication of [4], Hadamard, informed by a letter from Brouwer of the statement of his fixed point theorem, published a simple proof based on the Kronecker index (a forerunner of the Brouwer topological degree) in an appendix to an introductory analysis book of Tannery [3]. Hadamard's proof consisted in showing that Brouwer's assumption implies that the condition *x*, *<sup>x</sup>* <sup>−</sup> *<sup>T</sup>*(*x*) ≥ 0 holds for all *<sup>x</sup>* <sup>∈</sup> *<sup>∂</sup>BR*, where ·, · denotes the usual inner product in <sup>R</sup>*n*. This condition implies the existence of a zero of *I* − *T*, because the assumption of the Poincaré–Bohl theorem 6 is satisfied. Hadamard's reasoning using the Kronecker index does not depend upon the special structure *I* − *T* of the mapping in the inner product. Hence, it is natural (although not usual) to call **Hadamard theorem** the statement of existence of a zero for a continuous mapping *<sup>f</sup>* : *BR* <sup>→</sup> <sup>R</sup>*n*, when *x* − *T*(*x*) is replaced by *f*(*x*) in the inequality above, a statement which became in the year 1960 a key ingredient in the theory of monotone operators in reflexive Banach spaces. Using convex analysis, we give an extension to compact convex sets.

Let *<sup>C</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* be compact and convex and *pC* : <sup>R</sup>*<sup>n</sup>* <sup>→</sup> *<sup>C</sup>* be the orthogonal projection of *<sup>x</sup>* on *<sup>C</sup>* [15]. Recall that *pC*(*x*) is characterized by the condition

$$
\langle \mathbf{x} - p\_{\mathbb{C}}(\mathbf{x}), \mathbf{y} - p\_{\mathbb{C}}(\mathbf{x}) \rangle \le 0 \quad \text{for all } \mathbf{y} \in \mathbb{C}. \tag{2}
$$

For *x* ∈ *∂C*, the set

$$N\_{\mathbf{x}} := \{ \boldsymbol{\nu} \in \mathbb{R}^n : \langle \boldsymbol{\nu}, \boldsymbol{y} - \boldsymbol{x} \rangle \le 0 \text{ for all } \boldsymbol{y} \in \mathbb{C} \}$$

is nonempty and called the **normal cone** to *C* at *x*, and its elements *ν* are called the **outer normals** to *C* at *x*. The relation in Equation (2) shows that, for each *x* ∈ *C*, *x* − *p*(*x*) ∈ *Np*(*x*) \ {0}. It can also be shown that each *<sup>x</sup>* <sup>∈</sup> *<sup>∂</sup><sup>C</sup>* is the orthogonal projection of some *<sup>z</sup>* <sup>∈</sup> *<sup>C</sup>*, so that *Nx* <sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* \ *<sup>C</sup>* : *<sup>p</sup>*(*z*) = *<sup>x</sup>*}. The **Hadamard theorem on a convex compact set** follows in a similar way as Theorem 6 from the Brouwer FPT 3.

**Theorem 7.** *If <sup>C</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup> is a compact and convex, any continuous <sup>f</sup>* : *<sup>C</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup> such that ν*, *<sup>f</sup>*(*x*) ≥ <sup>0</sup> *for all x* ∈ *∂C and all ν* ∈ *Nx has a zero in C.*

**Proof.** Let *<sup>T</sup>* : <sup>R</sup>*<sup>n</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup>* be defined by *<sup>T</sup>* <sup>=</sup> *pC* <sup>−</sup> *<sup>f</sup>* ◦ *pC*. Then, for all *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>*n*,

$$|T(\mathbf{x})| \le |p\_{\mathcal{C}}(\mathbf{x})| + |f(p\_{\mathcal{C}}(\mathbf{x}))| \le \max\_{\mathbf{x} \in \mathcal{C}} |\mathbf{x}| + \max\_{y \in \mathcal{C}} |f(y)| := \mathcal{R}\_{\mathbf{x}}$$

and *T* maps *BR* into itself. By Theorem 3, there exists *x*<sup>∗</sup> ∈ *BR* such that *x*<sup>∗</sup> = *pC*(*x*∗) − *f*(*pC*(*x*∗)). If *x*<sup>∗</sup> ∈ *C*, the assumption implies that

$$0 < |\mathbf{x}^\* - p\_\mathcal{C}(\mathbf{x}^\*)|^2 = -\langle \mathbf{x}^\* - p\_\mathcal{C}(\mathbf{x}^\*), f(p\_\mathcal{C}(\mathbf{x}^\*)) \rangle \le 0,$$

a contradiction. Thus, *x*<sup>∗</sup> ∈ *C*, *x*<sup>∗</sup> = *pC*(*x*∗) and *f*(*x*∗) = 0.

**Corollary 5.** *Any continuous mapping f* : *BR* <sup>→</sup> <sup>R</sup>*<sup>n</sup> such that x*, *<sup>f</sup>*(*x*) ≥ <sup>0</sup> *for all x* <sup>∈</sup> *<sup>∂</sup>BR has a zero in BR.*

**Proof.** For each *x* ∈ *∂BR*, *Nx* = {*λx* : *λ* > 0}, and we apply Theorem 7.

**Remark 5.** *As shown when mentioning Hadamard's contribution, Theorem 5 implies the Brouwer FPT, and even the Birkhoff–Kellopg FPT, on BR. Consequently, those statements are equivalent.*

Some twenty years before the publication of Brouwer's paper [4], Poincaré [17] stated in 1883 a theorem about the existence of a zero of a continuous mapping *f* : *P* = [−*R*1, *R*1] ×···× [−*Rn*, *Rn*] → R*<sup>n</sup>* when, for each *<sup>i</sup>* = 1, . . . , *<sup>n</sup>*, *fi* takes opposite signs on the opposite faces of *<sup>P</sup>*

$$P\_i^- := \{ \mathbf{x} \in P : \mathbf{x}\_i = -\mathcal{R}\_i \}, \ P\_i^+ := \{ \mathbf{x} \in P : \mathbf{x}\_i = \mathcal{R}\_i \} \quad (i = 1, \dots, n).$$

Poincaré's proof just told that the result was a consequence of the Kronecker index, which is correct but sketchy. The statement, forgotten for a while, was rediscovered by Cinquini [18] in 1940 with an inconclusive proof, and shown to be equivalent to the Brouwer FPT on *P* one year later by Miranda [19]. Many other proofs have been given since, and we again refer to [7,20] for a more complete history, variations and references, and to [21–23] for useful generalizations to more complicated sets than closed *n*-intervals. Here, we obtain the **Poincaré–Miranda theorem on a closed** *n***-interval** as a special case of Theorem 7.

**Corollary 6.** *Any continuous mapping <sup>f</sup>* : *<sup>P</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup> such that fi*(*x*) <sup>≤</sup> <sup>0</sup> *for all <sup>x</sup>* <sup>∈</sup> *<sup>P</sup>*<sup>−</sup> *<sup>i</sup> and fi*(*x*) ≥ 0 *for all <sup>x</sup>* <sup>∈</sup> *<sup>P</sup>*<sup>+</sup> *<sup>i</sup>* (*i* = 1, . . . , *n*) *has a zero in P.*

**Proof.** If *x* is in the (relative) interior of the face *P*− *<sup>i</sup>* , then *Nx* = {−*λei* : *λ* > 0}, where (*e*1,*e*2, ... ,*en*) is the orthonormal basis in <sup>R</sup>*n*, and the assumption of Theorem <sup>7</sup> becomes <sup>−</sup>*fi*(*x*) <sup>≥</sup> 0, i.e., *fi*(*x*) <sup>≤</sup> 0. Similarly, if *x* is in the (relative) interior to the face *P*<sup>+</sup> *<sup>i</sup>* , then *Nx* = {*λei* : *λ* > 0}, and the assumption of Theorem 7 becomes *fi*(*x*) ≥ 0. Of course, −*λei* and *λei* (*λ* > 0) also belong to the respective normal cones for *x* ∈ *P*<sup>−</sup> *<sup>i</sup>* and *<sup>P</sup>*<sup>+</sup> *<sup>i</sup>* respectively, and if, say, *x* ∈ *P*<sup>−</sup> *<sup>i</sup>* <sup>∩</sup> *<sup>P</sup>*<sup>+</sup> *<sup>j</sup>* then *ν* = −*λei* + *μej* ∈ *Nx* for all *λ*, *μ* > 0, and *ν*, *f*(*x*) = −*λ fi*(*x*) + *μ fk*(*x*) ≥ 0. In general, when *x* belongs to the intersection of several faces of *P*, *Nx* will be made of the linear combination of the *ei* corresponding to the indices of the faces, with a negative coefficient for a face having symbol − and positive coefficient for a face having symbol +, so that, using the assumption, *ν*, *f*(*x*) ≥ 0 for all *x* ∈ *∂P* and all *ν* ∈ *Nx*. The result follows from Theorem 7.

**Remark 6.** *Corollary 6 implies the Brouwer FPT on P. Indeed, if T* : *P* → *P is continuous, and if we set f* = *I* − *T, then, as* −*Ri* ≤ *Ti*(*x*) ≤ *Ri for all x* ∈ *∂P, we have, for x* ∈ *P such that xi* = −*Ri, fi*(*x*) = *xi* − *Ti*(*x*) = −*Ri* − *Ti*(*x*) ≤ 0, *and, for x* ∈ *P such that xi* = *Ri, fi*(*x*) = *xi* − *Ti*(*x*) = *Ri* − *Ti*(*x*) ≥ 0*. Thus f has at least one zero in P, which is a fixed point of T. Consequently, the two statements are equivalent.*

**Remark 7.** *Both the Hadamard theorem on BR and the Poincaré–Miranda theorem can be seen as distinct n-dimensional generalizations of the Bolzano theorem to closed ball and n-intervals respectively.*

**Remark 8.** *Using the Brouwer degree, it is easy to obtain the conclusion of the Hadamard Theorem 7 for a compact convex neighborhood of* 0 *under the weaker condition that for each x* ∈ *∂C*, *there exists ν* ∈ *Nx such that ν*, *f*(*x*) ≥ 0*. No proof based only upon the Brouwer FPT seems to be known.*

If *<sup>C</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup>* is a compact convex set and *<sup>g</sup>* : *<sup>C</sup>* <sup>→</sup> <sup>R</sup> is of class *<sup>C</sup>*1, then *<sup>g</sup>* reaches its minimum on *<sup>C</sup>* at some *x*<sup>∗</sup> ∈ *C* for which

*g*(*x*<sup>∗</sup> + *λ*(*v* − *x*∗)) − *g*(*x*∗) ≥ 0 for all *v* ∈ *C* and for all *λ* ∈ [0, 1],

so that, dividing both members by *λ* and letting *λ* → 0+, we obtain ∇*g*(*x*∗), *v* − *x*∗ ≥ 0 for all *v* ∈ *C*, where <sup>∇</sup>*<sup>g</sup>* denotes the gradient of *<sup>g</sup>*. For example, if *<sup>u</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is fixed and *<sup>g</sup>* : *<sup>C</sup>* <sup>→</sup> <sup>R</sup> is defined by *g*(*x*)=(1/2)|*x* − *u*| 2, the minimization problem corresponds to the definition of *pC*(*u*), and, as ∇*g*(*x*) = *x* − *u*, the inequality above is just Equation (2). In 1966, Hartman and Stampacchia [24] proved that the existence of such a *x*<sup>∗</sup> still holds when ∇*g* is replaced by an arbitrary continuous function *<sup>f</sup>* : *<sup>C</sup>* <sup>→</sup> <sup>R</sup>*n*. When *<sup>C</sup>* is a simplex, the same result was proved independently the same year by Karamardian [25]. We give here a proof, due to Brezis (see [26]) and based upon Brouwer's FPT, of the **Hartman–Stampacchia theorem on variational inequalities**.

**Theorem 8.** *If <sup>C</sup>* <sup>⊂</sup> <sup>R</sup>*<sup>n</sup> is compact, convex and <sup>f</sup>* : *<sup>C</sup>* <sup>→</sup> <sup>R</sup>*<sup>n</sup> continuous, there exists <sup>x</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>C</sup> such that f*(*x*∗), *v* − *x*∗ ≥ 0 *for all v* ∈ *C.*

**Proof.** The Brouwer FPT on *C* (Corollary 4) applied to the continuous mapping *pC* ◦ (*I* − *f*) : *C* → *C* implies the existence of *x*<sup>∗</sup> ∈ *C* such that

$$\mathbf{x}^\* = p\_\mathbb{C}(\mathbf{x}^\* - f(\mathbf{x}^\*)). \tag{3}$$

Taking *x* = *x*<sup>∗</sup> − *f*(*x*∗) in Equation (2) and using Equation (3), one gets

$$\left< \mathbf{x}^\* - f(\mathbf{x}^\*) - \mathbf{x}^\*, \upsilon - \mathbf{x}^\* \right> \le 0 \text{ for all } \upsilon \in \mathbb{C}\_{\prime}.$$

which is the requested inequality.

**Remark 9.** *The conclusion of Theorem 8 is called a* **variational inequality***. In the terminology of the theory of convex sets [15], the conclusion of Theorem 8 means that there exists x*<sup>∗</sup> ∈ *C such that either f*(*x*∗) = 0 *or <sup>f</sup>*(*x*∗) <sup>=</sup> <sup>0</sup> *and <sup>H</sup>* :<sup>=</sup> {*<sup>y</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* : *f*(*x*∗), *<sup>y</sup>* <sup>−</sup> *<sup>x</sup>*∗ <sup>=</sup> <sup>0</sup>} *is a* **supporting hyperplane** *for <sup>C</sup> passing through x*∗ *i.e., C is entirely contained in one of the two closed half-spaces determined by H.*

**Remark 10.** *The Brouwer FPT on C (Corollary 4) also follows from the Hartman–Stampacchia theorem. Indeed, if T* : *C* → *C is continuous and x*<sup>∗</sup> *is given by Theorem 8 applied to f* = *I* − *T, then, taking v* = *T*(*x*∗) ∈ *C in the variational inequality, we obtain* 0 ≤ *x*<sup>∗</sup> − *T*(*x*∗), *T*(*x*∗) − *x*∗ = −|*x*<sup>∗</sup> − *T*(*x*∗)| <sup>2</sup> <sup>≤</sup> 0, *so that x*∗ = *T*(*x*∗)*. Hence, the two statements are equivalent.*

**Remark 11.** *If x*<sup>∗</sup> *and x*# *are two distinct solutions of the variational inequality, then*

$$
\langle \langle f(\mathbf{x}^\*), \mathbf{x}^\# - \mathbf{x}^\* \rangle \rangle \ge 0, \quad \langle f(\mathbf{x}^\#), \mathbf{x}^\* - \mathbf{x}^\# \rangle \ge 0,
$$

*and hence f*(*x*∗) <sup>−</sup> *<sup>f</sup>*(*x*#), *<sup>x</sup>*<sup>∗</sup> <sup>−</sup> *<sup>x</sup>*# ≤ <sup>0</sup>*. Consequently, the variational inequality has a unique solution if <sup>f</sup> satisfies the condition f*(*x*) − *f*(*y*), *x* − *y* > 0 *for all x* = *y* ∈ *C,* i.e., if *f* is **strictly monotone** on *C*.

#### **6. A Direct Approach for Holomorphic Functions in** C

The assumption of the Bolzano theorem for a continuous function *<sup>f</sup>* : [−*R*, *<sup>R</sup>*] <sup>→</sup> <sup>R</sup> can be, without loss of generality, be written *f*(−*R*) ≤ 0 ≤ *f*(*R*) or, equivalently, *x f*(*x*) ≥ 0 for |*x*| = *R*. In 1982, Shih [27] proposed a version of the Bolzano theorem for a complex function *f* holomorphic on a suitable bounded open neighborhood <sup>Ω</sup> <sup>⊂</sup> <sup>C</sup> of 0 and continuous on <sup>Ω</sup>. He showed that *<sup>f</sup>* has a unique zero in Ω when [*z f*(*z*)] > 0 on *∂*Ω, using the Rouché theorem applied to *f*(*z*) and *g*(*z*) = *αz* for a suitable real *α*. As [*z f*(*z*)] = *z* · *f*(*z*) + *z* · *f*(*z*), Shih's condition is just Hadamard's one in Theorem 5 with strict inequality sign. Following the approach introduced in [10], we show in this section that, when the (non strict) Hadamard condition holds on the boundary of a ball, the existence of a zero of a holomorphic function results in a very simple way from an immediate consequence of the Cauchy integral theorem. The same is true for a Poincaré–Miranda theorem on a rectangle, giving another extension of the Bolzano theorem to complex functions. The Brouwer's FPT for holomorphic functions on a closed ball or a closed rectangle follow immediately.

We suppose the reader familiar with the concepts of **holomorphic function** *f* , **piecewise** *C<sup>k</sup>* **cycle** *γ*, and **integral** - *<sup>γ</sup> f*(*z*) *dz* **of** *f* **along** *γ* [28]. We denote by *B*(*R*) the open disc of center 0 and radius *<sup>R</sup>* <sup>&</sup>gt; 0 in <sup>C</sup>, and by *BR* the corresponding closed disc. Let *<sup>γ</sup><sup>R</sup>* : [0, 2*π*] <sup>→</sup> *<sup>∂</sup>BR*, *<sup>t</sup>* → *Reit* be the standard *C*∞-cycle whose image is *∂BR*. The **Cauchy integral theorem on a circle** is proved here in a simple way, reminiscent of Cauchy's proof in 1825 [29], reworked by Falk in 1883 [30], and similar in spirit to the proof of Lemma 2.

**Proposition 1.** *If f* : *BR* <sup>→</sup> <sup>C</sup> *is continuous on BR and holomorphic on B*(*R*)*, then* - *<sup>γ</sup><sup>R</sup> <sup>f</sup>*(*z*) *dz* = <sup>0</sup>*.* **Proof.** Define Γ : [0, 1] × [0, 2*π*] → *BR* by Γ(*λ*, *t*) = *λγR*(*t*), so that Γ(1, ·) = *γ<sup>R</sup>* and Γ(0, ·) is the constant zero mapping. To show that *<sup>λ</sup>* → - <sup>Γ</sup>(*λ*,·) *<sup>f</sup>*(*z*) *dz* is constant in (0, 1), we have (with differentiation under the integral sign easily justified and denoting the derivative with respect to *z*)

$$\begin{split} & \partial\_{\lambda} \quad \int\_{\Gamma(\lambda,\cdot)} f(z) \, dz = \partial\_{\lambda} \int\_{0}^{2\pi} f(\Gamma(\lambda,t)) \partial\_{t} \Gamma(\lambda,t) \, dt \\ & \quad = \int\_{0}^{2\pi} \left[ f'(\Gamma(\lambda,t)) \partial\_{\lambda} \Gamma(s,t) \partial\_{t} \Gamma(\lambda,t) + f(\Gamma(\lambda,t)) \partial\_{\lambda} \partial\_{t} \Gamma(\lambda,t) \right] \, dt \\ & \quad = \int\_{0}^{2\pi} \left[ \partial\_{t} \{ f(\Gamma(\lambda,t)) \} \partial\_{\lambda} \Gamma(\lambda,t) + f(\Gamma(\lambda,t)) \partial\_{t} \partial\_{\lambda} \Gamma(\lambda,t) \right] \, dt \\ & \quad = \int\_{0}^{2\pi} \partial\_{t} \left[ f(\Gamma(\lambda,t)) \partial\_{\lambda} \Gamma(\lambda,t) \right] \, dt \\ & \quad = \int\_{\Gamma} (\Gamma(\lambda,2\pi)) \partial\_{\lambda} \Gamma(\lambda,2\pi) - f(\Gamma(\lambda,0)) \partial\_{\lambda} \Gamma(\lambda,0) = 0. \end{split}$$

By continuity, *<sup>λ</sup>* → - <sup>Γ</sup>(*λ*,·) *<sup>f</sup>*(*z*) *dz* is constant in [0, 1], and hence

$$\int\_{\gamma\_{\mathbb{R}}} f(z) \, dz = \int\_{\Gamma(1, \cdot)} f(z) \, dz = \int\_{\Gamma(0, \cdot)} f(z) \, dz = 0.1$$

Let *<sup>a</sup>* <sup>&</sup>gt; 0, *<sup>b</sup>* <sup>&</sup>gt; 0, *<sup>P</sup>* <sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> <sup>C</sup> : <sup>−</sup>*<sup>a</sup>* ≤ *<sup>z</sup>* <sup>≤</sup> *<sup>a</sup>*, <sup>−</sup>*<sup>b</sup>* ≤ *<sup>z</sup>* <sup>≤</sup> *<sup>b</sup>*} be the corresponding closed rectangle in <sup>C</sup>, and let us introduce the continuous mapping *<sup>ρ</sup>* : [0, 4] <sup>→</sup> *<sup>∂</sup><sup>P</sup>* of class *<sup>C</sup>*<sup>∞</sup> on (0, 1) ∪ (1, 2) ∪ (2, 3) ∪ (3, 4) defined by

$$\rho(t) = \begin{cases} -a + 2ta - ib & \text{if } \quad t \in [0, 1] \\ a + i[-b + 2(t - 1)b] & \text{if } \quad t \in [1, 2] \\\ a - 2(t - 2)a + ib & \text{if } \quad t \in [2, 3] \\\ -a + i[b - 2(t - 3)b] & \text{if } \quad t \in [3, 4] \end{cases} \tag{4}$$

whose image *ρ*([0, 4]) = *∂P*. We state and prove the **Cauchy's integral theorem on the boundary of a rectangle**.

**Proposition 2.** *If f* : *<sup>P</sup>* <sup>→</sup> <sup>C</sup> *is continuous on P and holomorphic on int P, then* - *<sup>ρ</sup> f*(*z*) *dz* = 0*.*

**Proof.** It is entirely similar to that of Proposition 1. If we define *R* : [0, 1] × [0, 4] → *P* by *R*(*λ*, *t*) = *λρ*(*t*), the integral - *<sup>R</sup>*(*λ*,·) *<sup>f</sup>*(*z*) *dz* has to be decomposed into four integrals over [0, 1], [1, 2], [2, 3], and [3, 4], respectively, of *f*(*R*(*λ*, *t*))*∂tR*(*λ*, *t*), and each integral has to be differentiated with respect to *λ* separately. The details are left to the reader.

Propositions 1 and 2 immediately imply the following simple **theorem for the existence of a zero of** *f* .

**Proposition 3.** *Any function <sup>f</sup>* : *BR* <sup>→</sup> <sup>C</sup> *(respectively, <sup>f</sup>* : *<sup>P</sup>* <sup>→</sup> <sup>C</sup>*) holomorphic on <sup>B</sup>*(*R*) *(respectively, int P), continuous on BR (respectively, P), different from zero on ∂BR (respectively, ∂P) and such that*

$$\int\_{\gamma\mathbb{R}} \frac{dz}{f(z)} \neq 0 \quad \left(\text{resp. } \int\_{\rho} \frac{dz}{f(z)} \neq 0\right).$$

*has a zero in B*(*R*) *(respectively, P).*

**Proof.** It is entirely similar in both cases and we prove it for *BR*. If *<sup>f</sup>* has no zero in *<sup>B</sup>*(*R*), then *<sup>z</sup>* → <sup>1</sup> *f*(*z*) is holomorphic on *B*(*R*) and continuous on *BR*. By Proposition 1, - *γ<sup>R</sup> dz <sup>f</sup>*(*z*) = 0, a contradiction to the assumption.

Proposition 3 provides a very simple proof of the **Hadamard theorem for a holomorphic function on** *BR*.

**Theorem 9.** *Any function <sup>f</sup>* : *BR* <sup>→</sup> <sup>C</sup> *holomorphic on <sup>B</sup>*(*R*)*, continuous on BR and such that* [*z f*(*z*)] <sup>≥</sup> <sup>0</sup> *for all z* ∈ *∂BR, has a zero in BR.*

**Proof.** For each integer *<sup>k</sup>* <sup>≥</sup> 1, define *fk* : *BR* <sup>→</sup> <sup>C</sup> by *fk*(*z*) = *<sup>k</sup>*−1*<sup>z</sup>* <sup>+</sup> *<sup>f</sup>*(*z*). Each *fk* has the regularity properties of *<sup>f</sup>* and is such that, for any *<sup>z</sup>* <sup>∈</sup> *<sup>∂</sup>BR*, [*z fk*(*z*)] = *<sup>k</sup>*−1*R*<sup>2</sup> <sup>+</sup> [*z f*(*z*)] <sup>&</sup>gt; 0, so that *fk*(*z*) <sup>=</sup> <sup>0</sup> for all *z* ∈ *∂BR*, and

$$\begin{split} \odot \left[ \int\_{\gamma\_{R}} \frac{dz}{f\_{k}(z)} \right] &=& \odot \left[ \int\_{\gamma\_{R}} \frac{z \overline{z}}{\overline{z} f\_{k}(z)} \frac{dz}{z} \right] \\ &=& \odot \left[ \int\_{\gamma\_{R}} \frac{|z|^{2} \{\Re[\overline{z} f\_{k}(z)] - i \Im[\overline{z} f\_{k}(z)]\}}{|\overline{z} f\_{k}(z)|^{2}} \frac{dz}{z} \right] \\ &=& \odot \left[ \int\_{0}^{2\pi} \frac{i \Re[Re^{-it} f\_{k}(Re^{it})] + \Im[Re^{-it} f(Re^{it})]}{|f\_{k}(Re^{it})|^{2}} dt \right] \\ &=& \int\_{0}^{2\pi} \frac{\Re[Re^{-it} f\_{k}(Re^{it})]}{|f\_{k}(Re^{it})|^{2}} dt > 0. \end{split}$$

By Proposition 3, for each *k* ≥ 1, *fk* has a zero *zk* in *B*(*R*), and, by the Bolzano–Weierstrass theorem, a subsequence (*zkn* )*n*≥<sup>1</sup> of (*zk*)*k*≥<sup>1</sup> converges to some *<sup>z</sup>*<sup>∗</sup> <sup>∈</sup> *BR* such that 0 <sup>=</sup> lim*n*→∞[*k*−<sup>1</sup> *<sup>n</sup> zkn* <sup>+</sup> *<sup>f</sup>*(*zkn* )] = *f*(*z*∗).

The **Birkhoff–Kellog FPT for a holomorphic function on a disc** is a direct consequence of Theorem 9.

**Corollary 7.** *Any function <sup>T</sup>* : *BR* <sup>→</sup> <sup>C</sup> *continuous on BR, holomorphic on <sup>B</sup>*(*R*) *and such that <sup>T</sup>*(*∂BR*) <sup>⊂</sup> *BR has a fixed point in BR.*

**Proof.** For each *<sup>z</sup>* <sup>∈</sup> *<sup>∂</sup>BR*, one has {*z*[*<sup>z</sup>* <sup>−</sup> *<sup>T</sup>*(*z*)]} ≥ *<sup>R</sup>*<sup>2</sup> − |*z*||*T*(*z*)| ≥ 0.

**Example 1.** *For any integer <sup>m</sup>* <sup>≥</sup> <sup>1</sup>*, the mapping <sup>T</sup> defined by <sup>T</sup>*(*z*) = *<sup>z</sup>* <sup>2</sup> (*z<sup>m</sup>* <sup>+</sup> <sup>1</sup>) *is such that for* <sup>|</sup>*z*<sup>|</sup> <sup>=</sup> <sup>1</sup>*,* <sup>|</sup>*T*(*z*)| ≤ <sup>|</sup>*z*<sup>|</sup> <sup>2</sup> (|*z*| *<sup>m</sup>* <sup>+</sup> <sup>1</sup>) <sup>≤</sup> <sup>1</sup>*. There is no uniqueness as T has the fixed points* <sup>0</sup> *and* <sup>1</sup> *in B*1*.*

Let *P*− <sup>1</sup> = {−*a* + *iy* : *y* ∈ [−*b*, *b*]}, *P*−1+ = {*a* + *iy* : *y* ∈ [−*b*, *b*]}, *P*<sup>−</sup> <sup>2</sup> = {*x* − *ib* : *x* ∈ [−*a*, *a*]} and *P*<sup>+</sup> <sup>2</sup> = {*x* + *ib* : *x* ∈ [−*a*, *a*]} be the opposite vertical and horizontal sides of *P*, respectively. Proposition 3 provides a **Poincaré–Miranda theorem for a holomorphic function on a rectangle**.

**Theorem 10.** *Any function f* : *<sup>P</sup>* <sup>→</sup> <sup>C</sup> *continuous on P, holomorphic on int P and such that <sup>f</sup>*(*z*) <sup>≤</sup> <sup>0</sup> *for all z* ∈ *P*<sup>−</sup> *, <sup>f</sup>*(*z*) <sup>≥</sup> <sup>0</sup> *for all z* <sup>∈</sup> *<sup>P</sup>*<sup>+</sup> *, f*(*z*) ≤ 0 *for all z* ∈ *P*<sup>−</sup> *and <sup>f</sup>*(*z*) <sup>≥</sup> <sup>0</sup> *for all z* <sup>∈</sup> *<sup>P</sup>*<sup>+</sup> *has a zero in P.*

**Proof.** For each integer *<sup>k</sup>* <sup>≥</sup> 1, the function *fk* defined on *<sup>P</sup>* by *fk*(*z*) = *<sup>k</sup>*−1*<sup>z</sup>* <sup>+</sup> *<sup>f</sup>*(*z*) is such that *fk*(*z*) < 0 for *z* ∈ *P*<sup>−</sup> <sup>1</sup> , *fk*(*z*) <sup>&</sup>gt; 0 for *<sup>z</sup>* <sup>∈</sup> *<sup>P</sup>*<sup>+</sup> <sup>1</sup> , *fk*(*z*) < 0 for *z* ∈ *P*<sup>−</sup> <sup>2</sup> , and *fk*(*z*) < 0 for

*<sup>z</sup>* <sup>∈</sup> *<sup>P</sup>*<sup>+</sup> <sup>2</sup> . Hence, *fk*(*z*) = 0 for each *z* ∈ *∂P*. Let *ρ* : [0, 4] → Ω be the cycle defined by Equation (4). By the assumptions,

 *ρ dz fk*(*z*) = *ρ* | *fk*(*z*)| <sup>−</sup>2[ *fk*(*z*) <sup>−</sup> *<sup>i</sup> fk*(*z*)] *dz* = *ρ* | *fk*(*z*)| <sup>−</sup>2[− *fk*(*z*) *dx* <sup>+</sup> *fk*(*z*) *dy*] = − 1 0 | *fk*(*ρ*(*t*))| <sup>−</sup>2 *fk*[*ρ*(*t*)]2*a dt* <sup>+</sup> 2 1 | *fk*(*ρ*(*t*))| <sup>−</sup>2 *fk*[*ρ*(*t*)]2*b dt* − 3 2 | *fk*(*ρ*(*t*))| <sup>−</sup>2 *fk*[*ρ*(*t*)]2*a dt* <sup>+</sup> 4 3 | *fk*(*ρ*(*t*))| <sup>−</sup>2 *fk*[*ρ*(*t*)]2*b dt* = − *a* −*a* | *fk*(*s* − *ib*)| <sup>−</sup>2 *fk*(*<sup>s</sup>* <sup>−</sup> *ib*) *ds* <sup>+</sup> *b* −*b* | *fk*(*a* + *it*)| <sup>−</sup>2 *fk*(*<sup>a</sup>* <sup>+</sup> *it*) *ds* + *a* −*a* | *fk*(*s* + *ib*)| <sup>−</sup>2 *fk*(*<sup>s</sup>* <sup>+</sup> *ib*) *dt* <sup>−</sup> *b* −*b* | *fk*(−*a* + *is*)| <sup>−</sup>2 *fk*(−*<sup>a</sup>* <sup>+</sup> *is*) *ds* = *a* −*a* −| *fk*(*s* − *ib*)| <sup>−</sup>2 *fk*(*<sup>s</sup>* <sup>−</sup> *ib*) + <sup>|</sup> *fk*(*<sup>s</sup>* <sup>+</sup> *ib*)<sup>|</sup> <sup>2</sup> *fk*(*<sup>s</sup>* <sup>+</sup> *ib*) *ds* + *b* −*b* | *fk*(*a* + *is*)| <sup>−</sup>2 *fk*(*<sup>a</sup>* <sup>+</sup> *is*) − | *fk*(−*<sup>a</sup>* <sup>+</sup> *is*)<sup>|</sup> <sup>−</sup>2 *fk*(−*<sup>a</sup>* <sup>+</sup> *is*) *ds* > 0,

For *<sup>k</sup>* <sup>≥</sup> 1, Proposition <sup>3</sup> implies the existence of *zk* <sup>∈</sup> int *<sup>P</sup>* such that *<sup>k</sup>*−<sup>1</sup>*zk* <sup>+</sup> *<sup>f</sup>*(*zk*) = 0. Using the Bolzano–Weierstrass theorem, a subsequence (*zkn* )*n*≥<sup>1</sup> converges to some *z*<sup>∗</sup> ∈ *P* such that 0 = lim*n*→<sup>∞</sup> *k*−<sup>1</sup> *<sup>n</sup> zkn* + *f*(*zkn* ) = *f*(*z*∗).

**Example 2.** *Let the holomorphic function <sup>f</sup>* : <sup>C</sup> <sup>→</sup> <sup>C</sup> *be defined by <sup>f</sup>*(*z*) = *<sup>z</sup>*<sup>3</sup> <sup>+</sup> <sup>4</sup>*<sup>z</sup>* <sup>+</sup> <sup>1</sup> <sup>+</sup> *i. Taking P* <sup>=</sup> {*<sup>z</sup>* <sup>∈</sup> <sup>C</sup> : *<sup>z</sup>* <sup>∈</sup> [−1, 1] *and <sup>z</sup>* <sup>∈</sup> [−1, 1]}*, one has*

$$\begin{aligned} z \in P\_1^- &\Rightarrow \Re f(z) = -4 + 3y^2 < 0, \ z \in P\_1^+ &\Rightarrow \Re f(z) = 6 - 3y^2 > 0\\ z \in P\_2^- &\Rightarrow \Im f(z) = -3x^2 - 2 < 0, \ z \in P\_2^+ &\Rightarrow \Im f(z) = 3x^2 + 4 > 0, \ \end{aligned}$$

*and f has a zero in* [−1, 1] × [−1, 1]*.*

A direct consequence of Theorem 10 is the **Birkhoff–Kellogg FPT for a holomorphic function on a rectangle**.

**Corollary 8.** *Any function <sup>T</sup>* : *<sup>P</sup>* <sup>→</sup> <sup>C</sup> *continuous on P, holomorphic on int P, and such that <sup>T</sup>*(*∂P*) <sup>⊂</sup> *<sup>P</sup> has a fixed point in P.*

**Proof.** Define *<sup>f</sup>* : *<sup>P</sup>* <sup>→</sup> <sup>C</sup> by *<sup>f</sup>*(*z*) = *<sup>z</sup>* <sup>−</sup> *<sup>T</sup>*(*z*) for all *<sup>z</sup>* <sup>∈</sup> *<sup>P</sup>*. The assumption *<sup>T</sup>*(*∂P*) <sup>⊂</sup> *<sup>P</sup>* is equivalent to −*a* ≤ *T*(*z*) ≤ *a* and −*b* ≤ *T*(*z*) ≤ *b* for all *z* ∈ *∂P*, and, hence, if *z* ∈ *P*<sup>−</sup> <sup>1</sup> , *f*(*z*) = −*a* − *T*(*z*) ≤ 0, if *<sup>z</sup>* <sup>∈</sup> *<sup>P</sup>*<sup>+</sup> <sup>1</sup> , *f*(*z*) = *a* − *T*(*z*) ≥ 0, if *z* ∈ *P*<sup>−</sup> <sup>2</sup> , *<sup>f</sup>*(*z*) = <sup>−</sup>*<sup>b</sup>* − *T*(*z*) <sup>≤</sup> 0, and if *<sup>z</sup>* <sup>∈</sup> *<sup>P</sup>*<sup>+</sup> <sup>2</sup> , *f*(*z*) = *b* − *T*(*z*) ≥ 0. Thus, by Theorem 10, *f* has a zero in *P* and *T* a fixed point in *P*.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **A Sub-Supersolution Approach for Robin Boundary Value Problems with Full Gradient Dependence**

#### **Dumitru Motreanu 1,\*, Angela Sciammetta <sup>2</sup> and Elisabetta Tornatore <sup>2</sup>**


Received: 19 March 2020; Accepted: 19 April 2020; Published: 27 April 2020

**Abstract:** The paper investigates a nonlinear elliptic problem with a Robin boundary condition, which exhibits a convection term with full dependence on the solution and its gradient. A subsupersolution approach is developed for this type of problems. The main result establishes the existence of a solution enclosed in the ordered interval formed by a sub-supersolution. The result is applied to find positive solutions.

**Keywords:** nonlinear elliptic problem; Robin boundary condition; gradient dependence; sub-supersolution; positive solution

#### **1. Introduction**

In this paper we study the following nonlinear elliptic boundary value problem

$$\begin{cases} -\text{div}(A(\mathbf{x}, \nabla u)) + a(\mathbf{x})|u|^{p-2}u = f(\mathbf{x}, u, \nabla u) & \text{in } \Omega\\ A(\mathbf{x}, \nabla u) \cdot \nu(\mathbf{x}) + \beta(\mathbf{x})|u|^{p-2}u = 0 & \text{on } \partial\Omega \end{cases} \tag{1}$$

on a bounded domain <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup>*<sup>N</sup>* with *<sup>N</sup>* <sup>≥</sup> 3 and with a boundary *<sup>∂</sup>*<sup>Ω</sup> of class *<sup>C</sup>*1. The notation *<sup>ν</sup>*(*x*) stands for the unit exterior normal at any *x* ∈ *∂*Ω and *p* is a real number with 1 < *p* < +∞. We note that, in the stated problem, the boundary condition is of Robin type.

We describe the data entering our problem. The leading differential part of the equation in (1) is the term div(*A*(*x*, <sup>∇</sup>*u*)) in divergence form driven by the map *<sup>A</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>*<sup>N</sup>* which is composed with the (weak) gradient <sup>∇</sup>*<sup>u</sup>* of the solution *<sup>u</sup>* : <sup>Ω</sup> <sup>→</sup> <sup>R</sup>. No homogeneity condition is required for the map *<sup>A</sup>*. Precisely, we assume that *<sup>A</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>*<sup>N</sup>* is continuous and fulfills the conditions:

(A1) There exist constants *c*<sup>1</sup> and *c*<sup>2</sup> with 0 < *c*<sup>1</sup> ≤ *c*<sup>2</sup> such that

$$|A(\mathbf{x},\boldsymbol{\xi}) \cdot \boldsymbol{\xi}| \ge c\_1 |\boldsymbol{\xi}|^p \text{ and } |A(\mathbf{x},\boldsymbol{\xi})| \le c\_2 (|\boldsymbol{\xi}|^{p-1} + 1) \text{ for all } (\mathbf{x},\boldsymbol{\xi}) \in \overline{\Omega} \times \mathbb{R}^N.$$

(A2) For all *x* ∈ Ω, *A*(*x*, *ξ*) is strictly monotone in *ξ*.

Here and subsequently we denote by |·| and · the standard Euclidean norm and scalar product on R*N*, respectively.

As important examples of operators div(*A*(*x*, ∇*u*)) complying with the preceding hypotheses we mention: the *p*-Laplacian Δ*pu* := div(|∇*u*| *<sup>p</sup>*−2∇*u*) where *<sup>A</sup>*(*x*, *<sup>ξ</sup>*) = <sup>|</sup>*ξ*<sup>|</sup> *<sup>p</sup>*−2*ξ*, the (*p*, *q*)-Laplacian Δ*pu* + Δ*qu* := div((|∇*u*| *<sup>p</sup>*−<sup>2</sup> <sup>+</sup> |∇*u*<sup>|</sup> *<sup>q</sup>*−2)∇*u*) where 1 <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; <sup>+</sup><sup>∞</sup> and *<sup>A</sup>*(*x*, *<sup>ξ</sup>*) = <sup>|</sup>*ξ*<sup>|</sup> *<sup>p</sup>*−2*<sup>ξ</sup>* <sup>+</sup> <sup>|</sup>*ξ*<sup>|</sup> *<sup>q</sup>*−2*ξ*, the generalized *p*-mean curvature operator div((1 + |∇*u*| 2) *p*−2 <sup>2</sup> ∇*u*) where *A*(*x*, *ξ*)=(1 + |*ξ*| 2) *p*−2 <sup>2</sup> as well as numerous weighted versions.

The values of *u* on *∂*Ω in the boundary condition of (1) are in the trace sense, whereas *A*(*x*, ∇*u*) · *ν*(*x*) represents the co-normal derivative of *u* associated with *A*. For more details we refer to ([1], pages 7–9) and ([2], Section 2). In the statement of problem (1) we fix the functions *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω) and *<sup>β</sup>* <sup>∈</sup> *<sup>L</sup>*∞(*∂*Ω) satisfying *<sup>α</sup>*(*x*) <sup>≥</sup> 0 for almost everywhere (in short a.e.) *<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup> and *<sup>β</sup>*(*x*) <sup>≥</sup> 0 for a.e. *x* ∈ *∂*Ω, *β* ≡ 0, where *∂*Ω is endowed with the (*N* − 1)-dimensional Hausdorff measure. Contrary to the Neumann problem, here it is allowed to have *<sup>α</sup>* <sup>=</sup> 0. Recall that if *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω) with *<sup>α</sup>* <sup>≥</sup> 0, *<sup>α</sup>* <sup>≡</sup> 0, the term *α*(*x*)|*u*| *<sup>p</sup>*−2*u* was essential to develop the method of sub-supersolution under Neumann boundary condition (see [3]). Actually, in the Robin problem, the hypothesis *β*(*x*) ≥ 0 for a.e. *x* ∈ *∂*Ω, *β* ≡ 0, is a substitute for the condition *α*(*x*) ≥ 0 for a.e. *x* ∈ Ω, *α* ≡ 0, assumed for the Neumann problem.

The reaction term *f*(*x*, *u*, ∇*u*) in the equation (1) is determined by a Carathéodory function *<sup>f</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>, i.e., *<sup>f</sup>*(·,*s*, *<sup>ξ</sup>*) is measurable for all (*s*, *<sup>ξ</sup>*) <sup>∈</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* and *<sup>f</sup>*(*x*, ·, ·) is continuous for a.e. *x* ∈ Ω. This term, depending not only on the solution *u* but also on its gradient ∇*u*, is called convection. It prevents to have a variational structure for problem (1) and thus the variational methods are not applicable, which creates a serious difficulty for handling (1).

The Robin problems exhibiting convection term as is the case in (1) have only recently been studied. We refer to [4–8] for results on the existence of solutions to such problems, where the approach is based on fixed point theorems or on surjectivity criteria for monotone-type operators. We also mention that a singular Robin problem involving convection has recently been treated in [9]. There are many results for Robin problems with variational structure, thus without a convection term. In this direction, we cite, e.g., [10–15]. The aim of the present work is to study the Robin problem (1) with general gradient dependence through the method of sub-supersolution. Due to the lack of variational structure, one cannot handle such a problem by variational methods. We recall that in the study of non-variational elliptic problems one develops arguments as, for instance, the lower and upper solution method with monotone iterations, approximation approach of Galerkin-type, surjectivity theorems for monotone-type operators, fixed point theorems, topological degree theory, bifurcation theory examining phenomena as branches of solutions and blow-up. It is beyond the scope of our paper to review this huge amount of work. We only illustrate certain of these topics with a few recent references: a comparison principle and approximation process relying on a Schauder basis in [16], a fixed point approach using minimal solutions in [17], estimates based on Trudinger-Moser inequality for problems with exponential nonlinearities in [18]. We also mention the classical monographs [19,20], which are fundamental references for general elliptic equations.

According to our knowledge, this is the first time when the method of sub-supersolution is systematically implemented for nonlinear Robin problems with convection. We prove a general existence and location result for a solution to be enclosed in the ordered interval determined by a sub-supersolution. Specifically, given a subsolution *u* and a supersolution *u* for problem (1) with *u* ≤ *u* a.e. in Ω (see Section 2 for the relevant definitions), our main abstract result provides the existence of a solution *u* to problem (1) satisfying *u* ≤ *u* ≤ *u* a.e. in Ω. This is an important qualitative property of the solution *u* offering a priori estimates. The growth condition that we suppose in the variable *s* for the nonlinearity *f*(*x*,*s*, *ξ*) concerns only the real interval [*u*(*x*), *u*(*x*)]. We emphasize that our abstract result can be applied provided we know sub-supersolutions, i.e., ordered pairs of a subsolution *u* and a supersolution *u* for problem (1) with *u* ≤ *u* i.e., in Ω, so the task to find such ordered pairs becomes the primary task in applying the method. In this sense, we provide an application of our main result to get positive solutions for a class of nonlinear Robin problem with convection term by showing explicitly how one can effectively determine sub-supersolutions. Results, as are given here, have recently been established in [21] for nonlinear Dirichlet problems with convection and in [3] for nonlinear Neumann problems with convection. General ideas regarding the method of sub-supersolution can be found in [1,22].

The rest of the paper is organized as follows. Section 2 discusses the background needed in the sequel. Section 3 focuses on a related operator equation, which is of independent interest. Section 4 sets forth our main result. Section 5 contains our application to produce positive solutions.

#### **2. Prerequisites of Sub-Supersolution Method**

This section contains preliminaries that will be used in the sequel. First, we fix some notation. For any *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>, we set *<sup>r</sup>*<sup>+</sup> <sup>=</sup> max{*r*, 0} (the positive part of *<sup>r</sup>*). If *<sup>r</sup>* <sup>&</sup>gt; 1, we also set *<sup>r</sup>* <sup>=</sup> *<sup>r</sup> <sup>r</sup>*−<sup>1</sup> (the Hölder conjugate of *<sup>r</sup>*). In particular, for *<sup>p</sup>* <sup>∈</sup> (1, <sup>+</sup>∞) we have *<sup>p</sup>* <sup>=</sup> *<sup>p</sup> <sup>p</sup>*−<sup>1</sup> .

As indicated in Section 1, <sup>Ω</sup> is a bounded domain in <sup>R</sup>*<sup>N</sup>* with *<sup>N</sup>* <sup>≥</sup> 3 whose boundary *<sup>∂</sup>*<sup>Ω</sup> is of class *C*1. In order to avoid repetitive arguments, we suppose that *N* > *p*. The complementary case *N* ≤ *p* can be treated along the same lines and actually is easier. By ·*Lr*(Ω) we denote the usual norm on the Banach space *Lr*(Ω).

We seek the solutions to problem (1) in the Sobolebv space *W*1,*p*(Ω), which is a Banach space equipped with the norm

$$\|\|u\|\|\_{1,p} := \left(\|\|u\|\|\_{L^p(\Omega)}^p + \|\|\nabla u\|\|\_{L^p(\Omega)}^p\right)^{\frac{1}{p}}.$$

For our study of problem (1) it is convenient to use the following equivalent norm on *W*1,*p*(Ω) (see, e.g., ([23], Lemma 2.7) or ([15], Proposition 2.8))

$$\|\|u\|\|\_{\mathfrak{H},1,p} := \left(\int\_{\partial\Omega} \beta(\sigma) |u(\sigma)|^p d\sigma + \|\nabla u\|\_{L^p(\Omega)}^p \right)^{\frac{1}{p}}.\tag{2}$$

The dual space of *<sup>W</sup>*1,*p*(Ω) is denoted (*W*1,*p*(Ω))∗, while the notation ·, · designates the duality pairing between *<sup>W</sup>*1,*p*(Ω) and (*W*1,*p*(Ω))∗, we denote by <sup>→</sup> the strong convergence and by the weak convergence. The Sobolev embedding theorem ensures that the space *W*1,*p*(Ω) is continuously embedded in *Lp*<sup>∗</sup> (Ω), where *p*<sup>∗</sup> is the Sobolev critical exponent *p*<sup>∗</sup> = *N p <sup>N</sup>*−*<sup>p</sup>* (we have supposed *<sup>N</sup>* <sup>&</sup>gt; *<sup>p</sup>*). Moreover, by the Rellich–Kondrachov theorem, *W*1,*p*(Ω) is compactly embedded in *Lr*(Ω) for every *r* ∈ [1, *p*∗).

Corresponding to the map *<sup>A</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>*<sup>N</sup>* describing the principal part of the equation in problem (1), we introduce the operator *<sup>A</sup>*˜ : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> defined by

$$\langle \bar{A}(u), v \rangle = \int\_{\Omega} A(\mathbf{x}, \nabla u) \cdot \nabla v \, d\mathbf{x} \text{ for all } u,\tag{3}$$

which is well defined thanks to assumption (*A*1). It turns out from assumption (*A*2) and the continuity of *A* that *A*(*x*, *ξ*) is maximal monotone in the variable *ξ* for all *x* ∈ Ω. This allows us to invoke ([2], Proposition 10), which yields:

**Proposition 1.** *Assume that the continuous map <sup>A</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>*<sup>N</sup> satisfies the conditions* (*A*1) *and* (*A*2)*. Then the map <sup>A</sup>*˜ : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> *in* (3) *has the (S*+*)-property, that is, any sequence* {*un*} ⊂ *<sup>W</sup>*1,*p*(Ω) *with un u in W*1,*p*(Ω) *and* lim sup *n*→+∞ *A*˜(*un*), *un* <sup>−</sup> *<sup>u</sup>* ≤ <sup>0</sup> *fulfills un* <sup>→</sup> *u in W*1,*p*(Ω)*.*

There exists a unique continuous linear map *<sup>γ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> *<sup>L</sup>p*(*∂*Ω) called the trace map such that

$$\gamma(\mu) = \mathfrak{u}\_{|\_{\partial\Omega}} \text{ for all } \mathfrak{u} \in \mathcal{W}^{1,p}(\Omega) \cap \mathcal{C}(\overline{\Omega})\dots$$

The kernel of *<sup>γ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> *<sup>L</sup>p*(*∂*Ω) is *<sup>W</sup>*1,*<sup>p</sup>* <sup>0</sup> (Ω). Recalling that *N* > *p*, the trace map *γ* is compact from *<sup>W</sup>*1,*p*(Ω) into *<sup>L</sup>η*(*∂*Ω) for all *<sup>η</sup>* <sup>∈</sup> [1, (*N*−1)*<sup>p</sup> <sup>N</sup>*−*<sup>p</sup>* ) (see, e.g., ([22], Theorem 2.79)). As usual, we drop the notation of the trace map *γ* writing simply *u* in place of *γ*(*u*). The co-normal derivative *A*(*x*, ∇*u*) · *ν*(*x*), appearing in the boundary condition in problem (1), is obtained by extending the map *<sup>u</sup>*(·) → *<sup>A</sup>*(·, <sup>∇</sup>*u*(·)) · *<sup>ν</sup>*(·), from *<sup>C</sup>*1(Ω) to *<sup>W</sup>*1,*p*(Ω).

By a (weak) solution to problem (1) we mean a function *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) such that *<sup>f</sup>*(*x*, *<sup>u</sup>*, <sup>∇</sup>*u*) <sup>∈</sup> *L*(*p*∗) (Ω) and

$$\int\_{\Omega} A(\mathbf{x}, \nabla u) \cdot \nabla v \, d\mathbf{x} + \int\_{\Omega} a(\mathbf{x}) |u|^{p-2} uv \, d\mathbf{x} + \int\_{\partial \Omega} \beta(\mathbf{x}) |u|^{p-2} uv d\sigma = \int\_{\Omega} f(\mathbf{x}, u, \nabla u) v \, d\mathbf{x} \tag{4}$$

for all *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω).

A function *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) is called a subsolution for problem (1) if *<sup>f</sup>*(·, *<sup>u</sup>*(·), <sup>∇</sup>*u*(·)) <sup>∈</sup> *<sup>L</sup>*(*p*∗) (Ω) and

$$\int\_{\Omega} \left( A(\mathbf{x}, \nabla \underline{\mathbf{u}}) \cdot \nabla \mathbf{v} + a(\mathbf{x}) |\underline{\mathbf{u}}|^{p-2} \underline{\mathbf{u}} \mathbf{v} \right) d\mathbf{x} + \int\_{\partial \Omega} \beta(\mathbf{x}) |\underline{\mathbf{u}}|^{p-2} \underline{\mathbf{u}} \mathbf{v} d\sigma \le \int\_{\Omega} f(\mathbf{x}, \underline{\mathbf{u}}, \nabla \underline{\mathbf{u}}) \mathbf{v} d\mathbf{x}, \tag{5}$$

for all *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω), *<sup>v</sup>* <sup>≥</sup> 0 a.e. in <sup>Ω</sup>.

Symmetrically, a function *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) is called a supersolution for problem (1) if *<sup>f</sup>*(·, *<sup>u</sup>*(·), <sup>∇</sup>*u*(·)) <sup>∈</sup> *<sup>L</sup>*(*p*∗) (Ω) and

$$\int\_{\Omega} \left( A(\mathbf{x}, \nabla \overline{\boldsymbol{u}}) \cdot \nabla \boldsymbol{v} + a(\mathbf{x}) |\overline{\boldsymbol{u}}|^{p-2} \overline{\boldsymbol{u}} \boldsymbol{v} \right) d\mathbf{x} + \int\_{\partial \Omega} \beta(\mathbf{x}) |\overline{\boldsymbol{u}}|^{p-2} \overline{\boldsymbol{u}} \boldsymbol{v} d\mathbf{v} \geq \int\_{\Omega} f(\mathbf{x}, \overline{\boldsymbol{u}}, \nabla \overline{\boldsymbol{u}}) \boldsymbol{v} \, d\mathbf{x}, \tag{6}$$

for all *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω), *<sup>v</sup>* <sup>≥</sup> 0 a.e. in <sup>Ω</sup>.

Due to assumption (*A*1), the integrals in the above definitions exist. We notice that *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) is a solution of (1) if and only if *u* is simultaneously a subsolution and a supersolution.

We are going to argue with a sub-supersolution for problem (1), that is, an ordered pair of a subsolution *u* and a supersolution *u* such that *u* ≤ *u*, which means the pointwise inequality *u*(*x*) ≤ *u*(*x*) for a.e. *x* ∈ Ω. Then we can associate the ordered interval

$$[\underline{u}, \overline{u}] = \{ w \in \mathcal{W}^{1,p}(\Omega) : \underline{u} \le w \le \overline{u} \}.$$

Our goal is to obtain a solution *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) of problem (1) with the location property *<sup>u</sup>* <sup>∈</sup> [*u*, *<sup>u</sup>*], which will be achieved through comparison by means of a truncation operator that we now describe. Corresponding to a subsolution *u* and a supersolution *u* satisfying *u* ≤ *u* a.e. in Ω, we define the truncation operator *<sup>T</sup>* <sup>=</sup> *<sup>T</sup>*(*u*, *<sup>u</sup>*) : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> *<sup>W</sup>*1,*p*(Ω) by

$$T(u)(\mathbf{x}) = \begin{cases} \underline{u}(\mathbf{x}) & \text{if } \quad u(\mathbf{x}) < \underline{u}(\mathbf{x}) \\ u(\mathbf{x}) & \text{if } \quad \underline{u}(\mathbf{x}) \le u(\mathbf{x}) \le \overline{u}(\mathbf{x}) \\ \overline{u}(\mathbf{x}) & \text{if } \quad u(\mathbf{x}) > \overline{u}(\mathbf{x}) \end{cases} \tag{7}$$

for all *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) and a.e. *<sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>. It readily follows that *<sup>T</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> *<sup>W</sup>*1,*p*(Ω) is continuous and bounded (in the sense that it maps bounded sets into bounded sets).

We shall also need the (negative) Dirichlet *<sup>p</sup>*-Laplacian, which is the operator <sup>−</sup>Δ*<sup>p</sup>* : *<sup>W</sup>*1,*<sup>p</sup>* <sup>0</sup> (Ω) → *W*−1,*<sup>p</sup>* (Ω)=(*W*1,*<sup>p</sup>* <sup>0</sup> (Ω))<sup>∗</sup> given by

$$\langle -\Delta\_p \mu, v \rangle = \int\_{\Omega} |\nabla u|^{p-2} \nabla u \cdot \nabla v d\mathfrak{x} \text{ for all } \mu, v \in \mathcal{W}\_0^{1,p}(\Omega).$$

It is well-known (see, e.g., ([1], Proposition 9.47)) that there exists a least positive number *λ*<sup>1</sup> > 0 (called the first eigenvalue of −Δ*p*) for which the Dirichlet problem

$$\begin{cases} -\Delta\_p \varphi\_1 = \lambda\_1 |\varphi\_1|^{p-2} \varphi\_1 & \text{in } \Omega\\ \varphi\_1 = 0 & \text{on } \partial\Omega \end{cases} \tag{8}$$

has a nontrivial solution *<sup>ϕ</sup>*<sup>1</sup> <sup>∈</sup> *<sup>W</sup>*1,*<sup>p</sup>* <sup>0</sup> (Ω). By the regularity theory we have *<sup>ϕ</sup>*<sup>1</sup> <sup>∈</sup> *<sup>C</sup>*1(Ω). Moreover, we can choose *ϕ*<sup>1</sup> to satisfy *ϕ*<sup>1</sup> > 0 in Ω.

Finally, we mention a few things about the pseudomonotone operators. Let *X* be a Banach space with the norm · and its dual *X*∗. We denote by ·, · the duality pairing between *X* and *X*∗. A map A : *X* → *X*<sup>∗</sup> is called bounded if it maps bounded sets into bounded sets. The map A : *X* → *X*<sup>∗</sup> is said to be coercive if

$$\lim\_{||u||\to+\infty} \frac{\langle \mathcal{A}(u), u\rangle}{||u||} = +\infty.$$

The map A : *X* → *X*<sup>∗</sup> is called pseudomonotone if for each sequence (*un*) ⊂ *X* satisfying *un u* in *<sup>X</sup>* and lim sup*n*→<sup>∞</sup> A(*un*), *un* <sup>−</sup> *<sup>u</sup>* ≤ 0, it holds

$$\langle \mathcal{A}(\upsilon), \mu - \upsilon \rangle \le \liminf\_{n \to \infty} \langle \mathcal{A}(\mu\_n), \mu\_n - \upsilon \rangle \text{ for all } \upsilon \in X.$$

The main theorem for pseudomonotone operators reads as follows (see, e.g., ([22], Theorem 2.99)).

**Theorem 1.** *Let X be a reflexive Banach space. If* A : *X* → *X*<sup>∗</sup> *is a pseudomonotone, bounded and coercive map, then* A *is surjective.*

#### **3. The Associated Operator Equation**

Assume that a subsolution *u* and a supersolution *u* for problem (1) with *u* ≤ *u* are given and that *<sup>f</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup> satisfies the following growth condition adapted to the ordered interval [*u*, *<sup>u</sup>*]:

(*H*) There exist a function *<sup>σ</sup>* <sup>∈</sup> *<sup>L</sup><sup>r</sup>* (Ω) with *<sup>r</sup>* <sup>∈</sup> (1, *<sup>p</sup>*∗) and constants *<sup>a</sup>* <sup>&</sup>gt; 0 and *<sup>r</sup>*<sup>1</sup> <sup>∈</sup> (0, *<sup>p</sup>* (*p*∗) ) such that

$$|f(\mathbf{x}, \mathbf{s}, \boldsymbol{\xi})| \le \sigma(\mathbf{x}) + a|\boldsymbol{\xi}|^{r\_1} \text{ for a.e.} \mathbf{x} \in \Omega, \text{ all } \mathbf{s} \in [\underline{\mathbf{u}}(\mathbf{x}), \overline{\boldsymbol{\pi}}(\mathbf{x})], \boldsymbol{\xi} \in \mathbb{R}^N.$$

We introduce the cut-off function *<sup>π</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> defined by

$$
\pi(\mathbf{x}, \mathbf{s}) = \begin{cases}
0 & \text{if} \quad \underline{\boldsymbol{u}}(\mathbf{x}) \le \underline{\boldsymbol{s}} \le \overline{\boldsymbol{u}}(\mathbf{x}), \\
(\underline{\mathbf{s}} - \overline{\boldsymbol{u}}(\mathbf{x}))^{\frac{r\_1}{p - r\_1}} & \text{if} \quad \mathbf{s} > \overline{\boldsymbol{u}}(\mathbf{x}),
\end{cases} \tag{9}
$$

where *<sup>r</sup>*<sup>1</sup> <sup>&</sup>gt; 0 is the constant postulated in hypothesis (*H*). From (9) and the fact that *<sup>u</sup>*, *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*<sup>∗</sup> (Ω) we infer that *π* verifies the growth condition

$$|\pi(\mathbf{x}, \mathbf{s})| \le c|\mathbf{s}|^{\frac{r\_1}{p-r\_1}} + \varrho(\mathbf{x}) \text{ for a.e. } \mathbf{x} \in \Omega \text{, all } \mathbf{s} \in \mathbb{R},\tag{10}$$

with a constant *c* > 0 and a function ∈ *L <sup>p</sup>*∗(*p*−*r*1) *<sup>r</sup>*<sup>1</sup> (Ω).

Now for every *<sup>λ</sup>* <sup>&</sup>gt; 0 we define the nonlinear operator *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> by

$$\begin{split} \langle A\_{\lambda}(u), v \rangle &= \int\_{\Omega} A(\mathbf{x}, \nabla u) \cdot \nabla v \, d\mathbf{x} + \int\_{\Omega} a(\mathbf{x}) |u|^{p-2} u v \, d\mathbf{x} + \int\_{\partial \Omega} \beta(\mathbf{x}) |u|^{p-2} u v \, d\sigma \\ &+ \quad \lambda \int\_{\Omega} \pi(\mathbf{x}, u) v \, d\mathbf{x} - \int\_{\Omega} f(\mathbf{x}, \mathbf{T}u, \nabla \mathbf{u}) v \, d\mathbf{x} \quad \text{for all } u, v \in \mathcal{W}^{1, p}(\Omega). \end{split} \tag{11}$$

Hypothesis (*H*) guarantees that the operator *A<sup>λ</sup>* in (11) is well defined.

Due to (10), we may consider the Nemytskij operator Π : *Lp*<sup>∗</sup> (Ω) → *L <sup>p</sup>*∗(*p*−*r*1) *<sup>r</sup>*<sup>1</sup> (Ω), associated to the function *<sup>π</sup>* in (10), namely <sup>Π</sup>(*u*) = *<sup>π</sup>*(·, *<sup>u</sup>*(·)) for all *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>p*<sup>∗</sup> (Ω). It is well defined, continuous and bounded. The condition in (*H*) that *<sup>r</sup>*<sup>1</sup> < *<sup>p</sup>* (*p*∗) is equivalent to *<sup>p</sup>*∗(*p*−*r*1) *<sup>r</sup>*<sup>1</sup> > (*p*∗) . Hence, by the Rellich–Kondrachov compact embedding theorem, the Nemytskij operator <sup>Π</sup> : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is completely continuous.

Thanks to hypothesis (*H*) we also have the Nemytskij operator *Nf* : [*u*, *<sup>u</sup>*] <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> on the ordered interval [*u*, *<sup>u</sup>*] which is associated to the function *<sup>f</sup>* : <sup>Ω</sup> <sup>×</sup> <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>, that is

$$\langle N\_f(\mu), v \rangle = \int\_{\Omega} f(\mathfrak{x}, \mathfrak{u}(\mathfrak{x}), \nabla \mathfrak{u}(\mathfrak{x})) v(\mathfrak{x}) \, d\mathfrak{x}$$

for all *<sup>u</sup>* <sup>∈</sup> [*u*, *<sup>u</sup>*] and *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω). Using (*H*) we see that *<sup>f</sup>*(·, *<sup>u</sup>*(·), <sup>∇</sup>*u*(·)) <sup>∈</sup> *<sup>L</sup> p <sup>r</sup>*<sup>1</sup> (Ω). As *v* ∈ *Lp*<sup>∗</sup> (Ω) and *<sup>p</sup> <sup>r</sup>*<sup>1</sup> > (*p*∗) , the above integral exists. By virtue of the strict inequality *<sup>p</sup> <sup>r</sup>*<sup>1</sup> > (*p*∗) , the Rellich–Kondrachov compact embedding theorem implies that the Nemytskij operator *Nf* : [*u*, *u*] → (*W*1,*p*(Ω))<sup>∗</sup> is completely continuous.

Again through the Rellich–Kondrachov compact embedding theorem we can show that the operator *<sup>B</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> given by

$$\langle B(\mu), v \rangle = \int\_{\Omega} \mathfrak{a}(\mathfrak{x}) |\mathfrak{u}(\mathfrak{x})|^{p-2} \mathfrak{u}(\mathfrak{x}) v(\mathfrak{x}) \, d\mathfrak{x}$$

for all *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) is completely continuous.

Consider also the operator <sup>Γ</sup> : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> given by

$$\langle \Gamma(u), v \rangle = \int\_{\partial \Omega} \beta(\sigma) |u(\sigma)|^{p-2} u(\sigma) v(\sigma) d\sigma \tag{12}$$

for all *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω), where the integration is done with respect to the (*<sup>N</sup>* <sup>−</sup> 1)-dimensional Hausdorff (surface) measure on *∂*Ω.

Let us check that the map <sup>Γ</sup> : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is completely continuous. To this end, let *un <sup>u</sup>* in *<sup>W</sup>*1,*p*(Ω). Then the compactness of the trace map *<sup>γ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> *<sup>L</sup>p*(*∂*Ω) ensures the strong convergence *un* <sup>≡</sup> *<sup>γ</sup>*(*un*) <sup>→</sup> *<sup>u</sup>* <sup>≡</sup> *<sup>γ</sup>*(*u*) in *<sup>L</sup>p*(*∂*Ω), thus the strong convergence <sup>|</sup>*un*<sup>|</sup> *<sup>p</sup>*−2*un* → |*u*<sup>|</sup> *<sup>p</sup>*−2*u* in *L<sup>p</sup>* (*∂*Ω). Taking into account (12) we deduce that <sup>Γ</sup>(*un*) <sup>→</sup> <sup>Γ</sup>(*u*) in (*W*1,*p*(Ω))∗, so <sup>Γ</sup> : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is completely continuous.

For every *<sup>λ</sup>* <sup>&</sup>gt; 0, the operator *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> in (11) has the expression

$$A\_{\lambda} = \bar{A} + B + \Gamma + \lambda \Pi - N\_f \circ T. \tag{13}$$

The composition *Nf* ◦ *T* makes sense because *T* takes values in the ordered interval [*u*, *u*] as seen from (7). The following theorem asserts the solvability of the equation

$$A\_{\lambda}(u) = 0.\tag{14}$$

**Theorem 2.** *Assume that the conditions* (*A*1)*,* (*A*2) *and* (*H*) *are satisfied. Then Equation* (14) *possesses at least a solution u* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) *provided <sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup> *is sufficiently large.*

**Proof.** In order to prove the solvability of operator Equation (14) we apply Theorem 1. We have to prove that the operator *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> in (13) is bounded, pseudomonotone and coercive.

By (3) and hypothesis (*A*1), in conjunction with Hölder's inequality and the Sobolev embedding theorem, we find that

$$\begin{aligned} \|\bar{A}(u)\|\_{\left(W^{1,p}(\Omega)\right)^{\*}}^{p} &= \sup\_{\|v\|\_{\beta;1,p} \le 1} |\langle \bar{A}(u), v\rangle| \\ &= \sup\_{\|v\|\_{\beta;1,p} \le 1} \left| \int\_{\Omega} A(x, \nabla u) \cdot \nabla v dx \right| \\ &\le c\_2 \sup\_{\|v\|\_{\beta;1,p} \le 1} \int\_{\Omega} (|\nabla u|^{p-1} + 1) |\nabla v| dx \\ &\le \mathcal{C} (\|u\|\_{\beta;1,p}^{p-1} + 1) \end{aligned}$$

for all *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω), with a constant *<sup>C</sup>* <sup>&</sup>gt; 0. This shows that the operator *<sup>A</sup>*˜ : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is bounded.

The composed operator *Nf* ◦ *T* is bounded because *T* is bounded and *Nf* is completely continuous. Since *<sup>B</sup>*, <sup>Π</sup> and <sup>Γ</sup> are completely continuous, it follows from (13) that *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is bounded.

We claim that *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is a pseudomonotone operator. Let a sequence {*un*} ⊂ *<sup>W</sup>*1,*p*(Ω) satisfy *un u* in *W*1,*p*(Ω) and

$$\limsup\_{n \to \infty} \langle A\_{\lambda}(u\_n), u\_n - u \rangle \le 0. \tag{15}$$

The complete continuity of the operators *B*, Π and Γ yields the strong convergent sequences *<sup>B</sup>*(*un*) <sup>→</sup> *<sup>B</sup>*(*u*), <sup>Π</sup>(*un*) <sup>→</sup> <sup>Π</sup>(*u*) and <sup>Γ</sup>(*un*) <sup>→</sup> <sup>Γ</sup>(*u*) in (*W*1,*p*(Ω))∗. This results in

$$\begin{aligned} \lim\_{n \to \infty} \langle B(u\_n), u\_n - v \rangle &= \langle B(u), u - v \rangle, \lim\_{n \to \infty} \langle \Pi(u\_n), u\_n - v \rangle = \langle \Pi(u), u - v \rangle, \\ \lim\_{n \to \infty} \langle \Gamma(u\_n), u\_n - u \rangle &= \langle \Gamma(u), u - v \rangle \end{aligned} \tag{16}$$

for all *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω). We infer that

$$\lim\_{n \to \infty} \langle B(u\_{\hbar}), u\_{\hbar} - u \rangle = \lim\_{n \to \infty} \langle \Pi(u\_{\hbar}), u\_{\hbar} - u \rangle = \lim\_{n \to \infty} \langle \Gamma(u\_{\hbar}), u\_{\hbar} - u \rangle = 0,$$

so (15) reduces to

$$\limsup\_{n \to \infty} \langle \tilde{A}(u\_n), u\_n - u \rangle \le 0. \tag{17}$$

Inequality (17) enables us to apply Proposition 1 ensuring that the strong convergence *un* → *u* in *W*1,*p*(Ω) holds.

At this point, we know that the strong convergence <sup>∇</sup>(*un*) → ∇(*u*) holds in (*Lp*(Ω))*N*, so the second inequality in (*A*1) entails *<sup>A</sup>*(·, <sup>∇</sup>*un*(·)) <sup>→</sup> *<sup>A</sup>*(·, <sup>∇</sup>*u*(·)) strongly in (*L<sup>p</sup>* (Ω))*N*. Then for each *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) one has

$$\begin{split} \lim\_{n \to \infty} \langle \bar{A}(u\_n), u\_n - v \rangle &= \lim\_{n \to \infty} \int\_{\Omega} A(\mathbf{x}, \nabla u\_n) \cdot \nabla (u\_n - v) d\mathbf{x} \\ &= \int\_{\Omega} A(\mathbf{x}, \nabla u) \cdot \nabla (u - v) d\mathbf{x} \\ &= \langle \bar{A}(u), u - v \rangle. \end{split} \tag{18}$$

Taking into account of (13), (16) and (18), we arrive at

$$\lim\_{n \to \infty} \langle A\_{\lambda}(\mu\_n), \mu\_n - \upsilon \rangle = \langle A\_{\lambda}(\mu), \mu - \upsilon \rangle$$

for all *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) and *<sup>λ</sup>* <sup>&</sup>gt; 0. Therefore the operator *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is pseudomonotone.

Next we show that the operator *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is coercive whenever *<sup>λ</sup>* <sup>&</sup>gt; 0 is sufficiently large.

Since *<sup>α</sup>* <sup>∈</sup> *<sup>L</sup>*∞(Ω), *<sup>α</sup>* <sup>≥</sup> 0, from (11) we note that

$$\langle A\_{\lambda}(u), u \rangle \ge \langle \vec{A}(u), u \rangle + \int\_{\partial \Omega} \beta(\sigma) |u(\sigma)|^{p} \, d\sigma + \lambda \int\_{\Omega} \pi(\mathbf{x}, u) u \, d\mathbf{x} - \int\_{\Omega} f(\mathbf{x}, \mathbf{T}u, \nabla(\mathbf{T}u)) u \, d\mathbf{x} \tag{19}$$

for all *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω). We estimate from below the terms in the right-hand side of (19). Assumption (*A*1) and (3) yield

$$\langle \bar{A}u, u \rangle \ge c\_1 \| |\nabla u| \|\_{L^p(\Omega)}^p \text{ for all } u \in \mathcal{W}^{1, p}(\Omega). \tag{20}$$

From (9) we derive that

$$\int\_{\Omega} \pi(\mathbf{x}, \boldsymbol{\mu}(\mathbf{x})) \boldsymbol{\mu}(\mathbf{x}) \, d\mathbf{x} \ge b\_1 \|\boldsymbol{\mu}\|\_{L^{\frac{p}{p-r\_1}}(\Omega)}^{\frac{p}{p-r\_1}} - b\_2 \text{ for all } \boldsymbol{\mu} \in \mathcal{W}^{1,p}(\Omega), \tag{21}$$

with positive constants *b*<sup>1</sup> and *b*<sup>2</sup> (see [3]).

In view of (7), we have that *<sup>u</sup>* <sup>≤</sup> *Tu* <sup>≤</sup> *<sup>u</sup>* a.e. in <sup>Ω</sup> whenever *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω). Consequently, we may set *s* = (*Tu*)(*x*) in the statement of hypothesis (*H*). Then, for each *ε* > 0, we obtain through Hölder's and Young's inequalities and the Sobolev embedding theorem the estimate

$$\begin{split} \left| \int\_{\Omega} f(\mathbf{x}, \mathbf{T}u, \nabla(\mathbf{T}u)) u \, d\mathbf{x} \right| &\leq \int\_{\Omega} \left( \sigma |u| + a |\nabla(\mathbf{T}u)|^{r\_1} |u| \right) \, d\mathbf{x} \\ &\leq \varepsilon \| |\nabla u| \|\_{L^{p}(\Omega)}^{p} + c(\varepsilon) \| |u| \|\_{L^{\frac{p}{p-r\_1}}(\Omega)}^{\frac{p}{p}} + d \| u \|\_{\beta; 1, p\prime} \end{split} \tag{22}$$

with positive constants *c*(*ε*) (depending on *ε*) and *d*.

Gathering (19)–(22) leads to

$$\langle A\_{\lambda}(u), u \rangle \ge (c\_1 - \varepsilon) \|\nabla u\|\_{L^p(\Omega)}^p + \int\_{\partial \Omega} \beta(\sigma) |u(\sigma)|^p \, d\sigma + (\lambda b\_1 - \varepsilon(\varepsilon)) \|u\|\_{L^{\frac{p}{p-p\_1}}(\Omega)}^{\frac{p}{p-r\_1}} - d \|u\|\_{\mathcal{S};1,p} - \lambda b\_2 \tag{23}$$

for all *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) and *<sup>λ</sup>* <sup>&</sup>gt; 0. Now we fix *<sup>ε</sup>* and *<sup>λ</sup>* to verify *<sup>ε</sup>* <sup>∈</sup> (0, *<sup>c</sup>*1) and *<sup>λ</sup>* <sup>&</sup>gt; *<sup>c</sup>*(*ε*) *<sup>b</sup>*<sup>1</sup> . From (2) and (23) it is clear that

$$\langle \mathcal{A}\_{\lambda}(u), u \rangle \ge c\_0 ||u||\_{\beta, 1, p}^p - d||u||\_{\beta, 1, p} - \lambda b\_2$$

for all *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω), with a constant *<sup>c</sup>*<sup>0</sup> <sup>&</sup>gt; 0. Due to the fact that *<sup>p</sup>* <sup>&</sup>gt; 1, it turns out

$$\lim\_{||\mathfrak{u}||\_{\beta,1,p}\to+\infty} \frac{\langle \mathcal{A}\_{\lambda}(\mathfrak{u}), \mathfrak{u} \rangle}{||\mathfrak{u}||\_{\beta,1,p}} = +\infty.$$

thereby the operator *A<sup>λ</sup>* is coercive.

Summarizing, we have proved that the operator *<sup>A</sup><sup>λ</sup>* : *<sup>W</sup>*1,*p*(Ω) <sup>→</sup> (*W*1,*p*(Ω))<sup>∗</sup> is bounded, pseudomonotone and coercive. This allows us to apply Theorem 1 with A = *A<sup>λ</sup>* for *λ* > 0 sufficiently large. The surjectivity of *<sup>A</sup><sup>λ</sup>* implies the existence of a solution *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) of Equation (14), thus completing the proof.

**Remark 1.** *As a consequence of* (23)*, we can precisely determine the threshold of λ* > 0 *in the statement of Theorem 2.*

#### **4. Main Abstract Result for Problem** (1)

Our result regarding the method of sub-supersolution for problem (1) is stated as follows.

**Theorem 3.** *Assume that the conditions* (*A*1)*,* (*A*2) *and* (*H*) *are satisfied. Then problem* (*P*) *possesses a solution <sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) *satisfying <sup>u</sup>* <sup>≤</sup> *<sup>u</sup>* <sup>≤</sup> *<sup>u</sup> a.e. in* <sup>Ω</sup>*, where <sup>u</sup> and <sup>u</sup> are the subsolution and the supersolution that are postulated in assumption* (*H*)*.*

**Proof.** According to Theorem 2 we can fix *λ* > 0 sufficiently large such that equation (14) admits a solution *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω). Explicitly, this reads as

$$\begin{split} & \langle \tilde{A}(\mathbf{u}), \mathbf{v} \rangle + \int\_{\Omega} \mathfrak{a}(\mathbf{x}) |\mathbf{u}|^{p-2} \mathbf{u} \mathbf{v} d\mathbf{x} + \lambda \int\_{\Omega} \mathfrak{a}(\mathbf{x}, \mathbf{u}) \mathbf{v} d\mathbf{x} + \int\_{\partial \Omega} \mathfrak{f}(\mathbf{x}) |\mathbf{u}|^{p-2} \mathbf{u} \mathbf{v} d\sigma \\ & = \int\_{\Omega} f(\mathbf{x}, \mathbf{T} \mathbf{u}, \nabla(\mathbf{T} \mathbf{u})) \mathbf{v} d\mathbf{x} \text{ for all } \mathbf{v} \in \mathcal{W}^{1, p}(\Omega). \end{split} \tag{24}$$

Let us prove that *<sup>u</sup>* <sup>≤</sup> *<sup>u</sup>* a.e. in <sup>Ω</sup>. Inserting *<sup>v</sup>* = (*<sup>u</sup>* <sup>−</sup> *<sup>u</sup>*)<sup>+</sup> <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) in (6) and (24) renders

$$\begin{split} \left\langle \bar{A}(\overline{u}), (u - \overline{u})^{+} \right\rangle &+ \int\_{\Omega} a(\mathbf{x}) |\overline{u}|^{p-2} \overline{u} (u - \overline{u})^{+} dx + \int\_{\partial \Omega} \beta(\mathbf{x}) |\overline{u}|^{p-2} \overline{u} (u - \overline{u})^{+} d\sigma \\ &\geq \int\_{\Omega} f(\mathbf{x}, \overline{u}, \nabla \overline{u}) (u - \overline{u})^{+} dx \end{split} \tag{25}$$

and

$$\begin{split} & \langle \tilde{A}(u), (u - \overline{u})^{+} \rangle + \int\_{\Omega} \mathfrak{a}(\mathbf{x}) |u|^{p-2} u (u - \overline{u})^{+} dx + \lambda \int\_{\Omega} \pi(\mathbf{x}, u) (u - \overline{u})^{+} dx \\ & + \int\_{\partial \Omega} \beta(\mathbf{x}) |u|^{p-2} u (u - \overline{u})^{+} d\sigma \\ & = \int\_{\Omega} f(\mathbf{x}, Tu, \nabla (Tu)) (u - \overline{u})^{+} d\mathbf{x}. \end{split} \tag{26}$$

Subtract (25) from (26) and use (3) and (7) to deduce that

$$\begin{split} &\int\_{\Omega} \left( A(\mathbf{x}, \nabla u) - A(\mathbf{x}, \nabla \overline{u}) \right) \nabla (u - \overline{u})^{+} dx + \int\_{\partial \Omega} \beta(\mathbf{x}) (|u|^{p-2} u - |\overline{u}|^{p-2} \overline{u}) (u - \overline{u})^{+} dx \\ &+ \int\_{\Omega} a(\mathbf{x}) ((|u|^{p-2} u - |\overline{u}|^{p-2} \overline{u}) (u - \overline{u})^{+} dx + \lambda \int\_{\Omega} \pi(\mathbf{x}, u) (u - \overline{u})^{+} dx \\ &\leq \int\_{\Omega} \left( f(\mathbf{x}, \mathbf{T} u, \nabla (\overline{u} u)) - f(\mathbf{x}, \overline{u}, \nabla \overline{u}) \right) (u - \overline{u})^{+} dx \\ &= \int\_{\{u > \overline{u}\}} \left( f(\mathbf{x}, \mathbf{T} u, \nabla (\overline{u} u)) - f(\mathbf{x}, \overline{u}, \nabla \overline{u}) \right) (u - \overline{u}) dx = 0. \end{split} \tag{27}$$

The monotonicity of *A*(*x*, ·), guaranteed by assumption (*A*2), and the monotonicity of the map *ξ* → |*ξ*| *<sup>p</sup>*−2*ξ* on R*<sup>N</sup>* give

$$\begin{split} &\quad \int\_{\Omega} (A(\mathbf{x}, \nabla \underline{u}) - A(\mathbf{x}, \nabla \overline{\underline{u}})) \nabla (\underline{u} - \overline{\underline{u}})^{+} d\mathbf{x} \\ &= \quad \int\_{\{\underline{u} > \overline{\underline{u}}\}} (A(\mathbf{x}, \nabla \underline{u}) - A(\mathbf{x}, \nabla \overline{\underline{u}})) (\nabla \underline{u} - \nabla \overline{\underline{u}}) d\mathbf{x} \geq 0, \\ &\quad \quad \int\_{\Omega} \mathbf{a}(\mathbf{x}) (|\boldsymbol{u}|^{p-2} \underline{u} - |\overline{\boldsymbol{u}}|^{p-2} \overline{\underline{u}}) (\boldsymbol{u} - \overline{\underline{u}})^{+} d\mathbf{x} \\ &= \quad \int\_{\{\underline{u} > \overline{\underline{u}}\}} \mathbf{a}(\mathbf{x}) (|\boldsymbol{u}|^{p-2} \underline{u} - |\overline{\boldsymbol{u}}|^{p-2} \overline{\underline{u}}) (\boldsymbol{u} - \overline{\underline{u}}) d\mathbf{x} \geq 0, \\ &\quad \quad \int\_{\partial \Omega} \beta(\sigma) (|\boldsymbol{u}|^{p-2} \underline{u} - |\overline{\boldsymbol{u}}|^{p-2} \overline{\underline{u}}) (\boldsymbol{u} - \overline{\underline{u}})^{+} d\sigma \\ &= \quad \int\_{\{\sigma \in \partial \Omega \, : \, : \, \underline{u} \geq \overline{\underline{u}}\}} \beta(\sigma) (|\boldsymbol{u}|^{p-2} \underline{u} - |\overline{\boldsymbol{u}}|^{p-2} \overline{\underline{u}}) (\boldsymbol{u} - \overline{\underline{u}}) d\sigma \geq 0. \end{split}$$

From (27) and (9) we obtain

$$\int\_{\{\mathfrak{u} > \overline{\mathfrak{u}}\}} (\mathfrak{u} - \overline{\mathfrak{u}})^{\frac{p}{p-r\_1}} d\mathfrak{x} = \int\_{\Omega} \pi(\mathfrak{x}, \mathfrak{u}) (\mathfrak{u} - \overline{\mathfrak{u}})^+ d\mathfrak{x} \le 0,$$

where *u* ≤ *u* a.e in Ω.

Next we show that *<sup>u</sup>* <sup>≤</sup> *<sup>u</sup>* a.e in <sup>Ω</sup>. Setting *<sup>v</sup>* = (*<sup>u</sup>* <sup>−</sup> *<sup>u</sup>*)<sup>+</sup> <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) in (5) and (24) produces

$$\begin{split} \left| \langle \bar{A}(\underline{u}), (\underline{u} - u)^{+} \rangle + \int\_{\Omega} a(\mathbf{x}) |\underline{u}|^{p-2} \underline{u}(\underline{u} - u)^{+} d\mathbf{x} + \int\_{\partial\Omega} \beta(\mathbf{x}) |\underline{u}|^{p-2} \underline{u}(\underline{u} - u)^{+} d\sigma \right. \\ \left. \right|\_{\Omega} \leq \int\_{\Omega} f(\mathbf{x}, \underline{u}, \nabla \underline{u}) (\underline{u} - u)^{+} d\mathbf{x} \end{split} \tag{28}$$

and

$$\begin{split} & \langle \tilde{A}(\boldsymbol{u}), (\underline{\boldsymbol{u}} - \boldsymbol{u})^{+} \rangle + \int\_{\Omega} \boldsymbol{u}(\mathbf{x}) |\boldsymbol{u}|^{p-2} \boldsymbol{u}(\underline{\boldsymbol{u}} - \boldsymbol{u})^{+} d\mathbf{x} + \lambda \int\_{\Omega} \boldsymbol{\pi}(\mathbf{x}, \boldsymbol{u}) (\underline{\boldsymbol{u}} - \boldsymbol{u})^{+} d\mathbf{x} \\ & \quad + \int\_{\partial \Omega} \beta(\mathbf{x}) |\boldsymbol{u}|^{p-2} \boldsymbol{u}(\underline{\boldsymbol{u}} - \boldsymbol{u})^{+} d\boldsymbol{\sigma} \\ & \quad = \int\_{\Omega} f(\mathbf{x}, \boldsymbol{T} \boldsymbol{u}, \boldsymbol{\nabla}(\boldsymbol{T} \boldsymbol{u})) (\underline{\boldsymbol{u}} - \boldsymbol{u})^{+} d\mathbf{x} . \end{split} \tag{29}$$

By subtracting (29) from (28) and taking into account (3) we arrive at

$$\begin{split} &\int\_{\Omega} \left( A(\mathbf{x}, \nabla \underline{u}) - A(\mathbf{x}, \nabla u) \right) \nabla (\underline{u} - u)^{+} d\mathbf{x} + \int\_{\partial \Omega} \beta(\mathbf{x}) (|\underline{u}|^{p-2} \underline{u} - |u|^{p-2} u) (\underline{u} - u)^{+} d\mathbf{x} \\ &+ \int\_{\Omega} a(\mathbf{x}) (|\underline{u}|^{p-2} \underline{u} - |u|^{p-2} u) (\underline{u} - u)^{+} d\mathbf{x} - \lambda \int\_{\Omega} \pi(\mathbf{x}, u) (\underline{u} - u)^{+} d\mathbf{x} \\ &\leq \int\_{\Omega} \left( f(\mathbf{x}, \underline{u}, \nabla \underline{u}) - f(\mathbf{x}, T u, \nabla (\mathrm{T} u)) (\underline{u} - u)^{+} d\mathbf{x} \right. \\ &= \int\_{\{\underline{u} \geq u\}} \left( f(\mathbf{x}, \underline{u}, \nabla \underline{u}) - f(\mathbf{x}, T u, \nabla (\mathrm{T} u)) \right) (\underline{u} - u)^{+} d\mathbf{x} = 0. \end{split} \tag{30}$$

Along (9) and proceeding as above, (30) results in

$$-\int\_{\{\underline{\mu}\geq u\}} -(\underline{u}-u)^{\frac{p}{p-r\_1}}d\mathbf{x} = -\int\_{\Omega} \pi(\mathbf{x},\boldsymbol{\mu})(\underline{\mu}-\boldsymbol{\mu})^+d\mathbf{x} \leq 0,$$

which entails that *u* ≤ *u* a.e in Ω, thus proving the claim.

Therefore the solution *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) of the operator equation (14) verifies the enclosure property *u* ≤ *u* ≤ *u* a.e. in Ω. Then we obtain from (7) and (9) that *Tu* = *u* and Π(*u*) = 0. Hence for our function *<sup>u</sup>* the equalities (24) and (4) coincide. We see that *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) is a solution of the original problem (1) fulfilling in addition *u* ≤ *u* ≤ *u* a.e. in Ω. This completes the proof.

#### **5. An Application**

The aim of this section is to apply Theorem 3 to establish the existence of positive solutions of Robin problem (1). The main point is to find appropriate ordered sub-supersolutions. The approach can be used to get other types of solutions.

In order to simplify the presentation, we focus on problem (1) driven by the Robin *p*-Laplacian, 1 < *p* < +∞, and when *α*(*x*) ≡ 0 and the *x*-dependence in the convection term *f*(*x*,*s*, *ξ*) is dropped. We emphasize that *α* ≡ 0 marks a sharp distinction in regard to the Neumann problem. Specifically, we consider the (purely) Robin problem

$$\begin{cases} -\Delta\_{\mathbb{P}}\mu = f(\boldsymbol{u}, \nabla \boldsymbol{u}) & \text{in } \Omega\\ |\nabla \boldsymbol{u}|^{p-2} \nabla \boldsymbol{u} \cdot \nu(\boldsymbol{x}) + \boldsymbol{\beta}(\boldsymbol{x}) |\boldsymbol{u}|^{p-2} \boldsymbol{u} = \boldsymbol{0} & \text{on } \partial\Omega \end{cases} \tag{31}$$

with *β*(*x*) ≥ 0 for a.e. *x* ∈ *∂*Ω, *β* ≡ 0.

We suppose that *<sup>f</sup>* : <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup> is a continuous function verifying the following assumption: (*H* ) There exist constants *<sup>a</sup>*<sup>0</sup> <sup>&</sup>gt; 0, *<sup>a</sup>*<sup>1</sup> <sup>&</sup>gt; 0, *<sup>b</sup>* <sup>&</sup>gt; 0 and *<sup>r</sup>*<sup>1</sup> <sup>∈</sup> (0, *<sup>p</sup>* (*p*∗) ) such that

$$|f(\mathbf{s}, \boldsymbol{\xi})| \le a\_1 (1 + |\boldsymbol{\xi}|^{r\_1}) \text{ for all } \mathbf{s} \in (0, b], \boldsymbol{\xi} \in \mathbb{R}^N,\tag{32}$$

$$\lambda\_1 \mathbf{s}^{p-1} \le f(\mathbf{s}, \boldsymbol{\xi}) \text{ for all } \mathbf{s} \in (0, a\_0), \, |\boldsymbol{\xi}| \prec a\_0 \tag{33}$$

and

$$f(b,0) = 0.\tag{34}$$

The condition (33) involves the first eigenvalue *λ*<sup>1</sup> of the (negative) Dirichlet *p*-Laplacian as given in (8). Let us note that *u* = *b* is not a solution to problem (31) because the boundary condition is not verified. We formulate the following result concerning problem (31).

**Theorem 4.** *Assume that the conditions* (*A*1)*,* (*A*2) *and* (*H* ) *are satisfied. Then the Robin problem* (31) *possesses a (positive) solution u* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) *satisfying* <sup>0</sup> <sup>&</sup>lt; *<sup>u</sup>* <sup>≤</sup> *b a.e. in* <sup>Ω</sup>*.*

**Proof.** Fix an eigenfunction *<sup>ϕ</sup>*<sup>1</sup> of <sup>−</sup>Δ*<sup>p</sup>* on *<sup>W</sup>*1,*<sup>p</sup>* <sup>0</sup> (Ω), with *ϕ*<sup>1</sup> > 0 in Ω, corresponding to the first eigenvalue *<sup>λ</sup>*<sup>1</sup> (see (8) and the related comments). Since *<sup>ϕ</sup>*<sup>1</sup> <sup>∈</sup> *<sup>C</sup>*1(Ω), we can choose an *<sup>ε</sup>* <sup>&</sup>gt; 0 such that

$$\varepsilon \varrho\_1(\mathbf{x}) < a\_0 \text{ and } \varepsilon |\nabla \varrho\_1(\mathbf{x})| < a\_0 \text{ for all } \mathbf{x} \in \Omega,\tag{35}$$

where *a*<sup>0</sup> is the positive constant prescribed in hypothesis (*H* ).

We note that *u* = *εϕ*<sup>1</sup> is a subsolution in the sense of (5) for the Robin problem (31). Indeed, by (8), (33) and (35) and since the trace of *u* on *∂*Ω vanishes, we infer that

$$\begin{split} \int\_{\Omega} |\nabla \underline{u}|^{p-2} \nabla \underline{u} \cdot \nabla v \mathrm{d}x + \int\_{\partial \Omega} \beta(\mathbf{x}) |\underline{u}|^{p-2} \underline{\mu} v d\sigma &= \varepsilon^{p-1} \int\_{\Omega} |\nabla \varphi\_1(\mathbf{x})|^{p-2} \nabla \varphi\_1(\mathbf{x}) \cdot \nabla v(\mathbf{x}) d\mathbf{x} \\ &= \lambda\_1 \int\_{\Omega} (\varepsilon \varphi\_1(\mathbf{x}))^{p-1} v(\mathbf{x}) d\mathbf{x} \\ &\leq \int\_{\Omega} f(\varepsilon \varphi\_1(\mathbf{x}), \varepsilon \nabla \varphi\_1(\mathbf{x})) v(\mathbf{x}) d\mathbf{x} \\ &= \int\_{\Omega} f(\underline{u}(\mathbf{x}), \nabla \underline{u}(\mathbf{x})) v(\mathbf{x}) d\mathbf{x} \text{ for all } v \in \mathcal{W}^{1,p}(\Omega), \ v \geq 0. \end{split}$$

This proves that *u* = *εϕ*<sup>1</sup> is a subsolution of problem (31).

Now we observe that the constant function *u* = *b* is a supersolution of problem (31). Indeed, let us notice from assumption (34) that

$$\begin{aligned} \int\_{\Omega} |\nabla \overline{u}|^{p-2} \nabla \overline{u} \cdot \nabla v dx + \int\_{\partial \Omega} \beta(\mathbf{x}) |\overline{u}|^{p-2} \overline{u} v d\sigma &= \int\_{\partial \Omega} \beta(\mathbf{x}) b^{p-1} v(\mathbf{x}) d\sigma \\ &\ge 0 = \int\_{\Omega} f(b, 0) v(\mathbf{x}) d\mathbf{x} \\ &= \int\_{\Omega} f(\overline{u}(\mathbf{x}), \nabla \overline{u}(\mathbf{x})) v(\mathbf{x}) d\mathbf{x} \end{aligned}$$

for all *<sup>v</sup>* <sup>∈</sup> *<sup>W</sup>*1,*p*(Ω) with *<sup>v</sup>* <sup>≥</sup> 0, which confirms that *<sup>u</sup>* <sup>=</sup> *<sup>b</sup>* is a supersolution of problem (31) in the sense of (6).

For a possibly smaller *ε* > 0 to be fulfilled *εϕ*1(*x*) ≤ *b* whenever *x* ∈ Ω, the inequality *u* ≤ *u* holds true. The growth condition in (*H*) is satisfied due to (32) because the pointwise intervals [*u*(*x*), *u*(*x*)] are all included in the bounded interval (0, *b*]. Altogether we are in a position to apply Theorem 3, which yields the desired conclusion.

We provide a simple example illustrating the applicability of Theorem 4.

**Example 1.** *Let f* : <sup>R</sup> <sup>×</sup> <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup> *be defined by*

$$f(\mathbf{s}, \mathfrak{z}) = \mathfrak{g}(\mathbf{s}) + h(\mathfrak{z}) \text{ for all } (\mathbf{s}, \mathfrak{z}) \in \mathbb{R} \times \mathbb{R}^N,$$

*with g* : <sup>R</sup> <sup>→</sup> <sup>R</sup> *defined by*

$$g(s) = \begin{cases} 0 & \text{if } s < 0 \text{ or } s > 2\\ \lambda\_1 s^{p-1} & \text{if } 0 \le s \le 1\\ \lambda\_1 (2-s)^{p-1} & \text{if } 1 < s \le 2 \end{cases}$$

*and any continuous function h* : <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup> *satisfying h*(*ξ*) <sup>≥</sup> <sup>0</sup>*, h*(0) = <sup>0</sup> *and*

$$0 \le h(\emptyset) \le a\_2 (1 + |\emptyset|^{r\_1}) \text{ for all } \emptyset \in \mathbb{R}^N,$$

*with constants a*<sup>2</sup> <sup>&</sup>gt; <sup>0</sup> *and r*<sup>1</sup> <sup>∈</sup> (0, *<sup>p</sup>* (*p*∗) )*. We note that f*(2, 0) = 0 *and*

$$f(\mathbf{s}, \boldsymbol{\mathfrak{s}}) = \mathbf{g}(\mathbf{s}) + h(\boldsymbol{\mathfrak{s}}) \ge \lambda\_1 \mathbf{s}^{p-1} \text{ for all } 0 \le \mathbf{s} \le \mathbf{1}, \boldsymbol{\mathfrak{s}} \in \mathbb{R}^N.$$

*Hypothesis* (*H* ) *is verified taking a*<sup>0</sup> = 1 *and b* = 2*. Theorem 4 can be applied to problem* (31) *with f*(*s*, *ξ*) *given above.*

**Author Contributions:** Conceptualization, D.M., A.S. and E.T. All authors contributed equally to this paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The last two authors are members of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM). The paper is partially supported by PRIN 2017—Progetti di Ricerca di rilevante Interesse Nazionale, Nonlinear Differential Problems via Variational, Topological and Set-valued Methods. The authors thank the referees for careful reading and useful comments that helped to improve the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **The Topological Transversality Theorem for Multivalued Maps with Continuous Selections**

#### **Donal O'Regan**

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway H91 TK33, Ireland; donal.oregan@nuigalway.ie

Received: 22 October 2019; Accepted: 14 November 2019; Published: 15 November 2019

**Abstract:** This paper considers a topological transversality theorem for multivalued maps with continuous, compact selections. Basically, this says, if we have two maps *F* and *G* with continuous compact selections and *F* ∼= *G*, then one map being essential guarantees the essentiality of the other map.

**Keywords:** essential maps; homotopy; selections

**MSC:** 47H10; 54H25

#### **1. Introduction**

In this paper, we consider multivalued maps *F* and *G* with continuous, compact selections and *F* ∼= *G* in this setting. The topological transversality theorem will state that *F* is essential if and only if *G* is essential (essential maps were introduced by Granas [1] and extended by Precup [2], Gabor, Gorniewicz, and Slosarsk [3], and O'Regan [4,5]). For an approach to other classes of maps, we refer the reader to O'Regan [6], where one sees that ∼= in the appropriate class can be challenging. However, the topological transversality theorem for multivalued maps with continuous compact selections has not been considered in detail. In this paper, we present a simple result that immediately yields a topological transversality theorem in this setting. In particular, we show that, for two maps *F* and *G* with continuous compact selections and *F* ∼= *G*, then one map being essential (or *d*–essential) guarantees that the other is essential (or *d*–essential). We also discuss these maps in the weak topology setting.

#### **2. Topological Transversality Theorem**

We will consider a class **A** of maps. Let *E* be a completely regular space (i.e., a Tychonoff space) and *U* an open subset of *E*.

**Definition 1.** *We say f* ∈ *D*(*U*, *E*) *if f* : *U* → *E is a continuous, compact map; here, U denotes the closure of U in E.*

**Definition 2.** *We say f* ∈ *D∂U*(*U*, *E*) *if f* ∈ *D*(*U*, *E*) *and x* = *f*(*x*) *for x* ∈ *∂U; here, ∂U denotes the boundary of U in E.*

**Definition 3.** *We say <sup>F</sup>* <sup>∈</sup> *<sup>A</sup>*(*U*, *<sup>E</sup>*) *if <sup>F</sup>* : *<sup>U</sup>* <sup>→</sup> <sup>2</sup>*<sup>E</sup> with <sup>F</sup>* <sup>∈</sup> **<sup>A</sup>**(*U*, *<sup>E</sup>*) *and there exists a selection <sup>f</sup>* <sup>∈</sup> *<sup>D</sup>*(*U*, *<sup>E</sup>*) *of F; here,* 2*<sup>E</sup> denotes the family of nonempty subsets of E.*

**Remark 1.** *Let Z and W be subsets of Hausdorff topological vector spaces Y*<sup>1</sup> *and Y*<sup>2</sup> *and F a multifunction. We say <sup>F</sup>* <sup>∈</sup> *PK*(*Z*, *<sup>W</sup>*) *if <sup>W</sup> is convex and there exists a map <sup>S</sup>* : *<sup>Z</sup>* <sup>→</sup> *<sup>W</sup> with <sup>Z</sup>* <sup>=</sup> ∪ {*int S*−1(*w*) : *<sup>w</sup>* <sup>∈</sup> *<sup>W</sup>*}*, co* (*S*(*x*)) <sup>⊆</sup> *<sup>F</sup>*(*x*) *for <sup>x</sup>* <sup>∈</sup> *<sup>Z</sup> and <sup>S</sup>*(*x*) <sup>=</sup> <sup>∅</sup> *for each <sup>x</sup>* <sup>∈</sup> *Z; here, <sup>S</sup>*−1(*w*) = {*<sup>z</sup>* : *<sup>w</sup>* <sup>∈</sup> *<sup>S</sup>*(*z*)}*. Let <sup>E</sup> be a Hausdorff topological vector space (note topological vector spaces are completely regular), U an open subset*

*of E and U paracompact. In this case, we say F* ∈ **A**(*U*, *E*) *if F* ∈ *PK*(*U*, *E*) *is a compact map. Now, [7] guarantees that there exists a continuous, compact selection f* : *U* → *E of F.*

**Definition 4.** *We say F* ∈ *A∂U*(*U*, *E*) *if F* ∈ *A*(*U*, *E*) *and x* ∈/ *F*(*x*) *for x* ∈ *∂U.*

**Definition 5.** *We say F* ∈ *A∂U*(*U*, *E*) *is essential in A∂U*(*U*, *E*) *if for any selection f* ∈ *D*(*U*, *E*) *of F and any map g* ∈ *D∂U*(*U*, *E*) *with f* |*∂<sup>U</sup>* = *g*|*∂<sup>U</sup> there exists a x* ∈ *U with x* = *g*(*x*)*.*

**Remark 2.** *If F* ∈ *A∂U*(*U*, *E*) *is essential in A∂U*(*U*, *E*) *and if f* ∈ *D*(*U*, *E*) *is any selection of F then there exists a x* ∈ *U with x* = *f*(*x*) *(take g* = *f in Definition 5), so in particular there exists a x* ∈ *U with x* ∈ *F*(*x*)*.*

**Definition 6.** *Let f* , *g* ∈ *D∂U*(*U*, *E*)*. We say f* ∼= *g in D∂U*(*U*, *E*) *if there exists a continuous, compact map h* : *U* × [0, 1] → *E with x* = *ht*(*x*) *for any x* ∈ *∂ U and t* ∈ (0, 1) *(here ht*(*x*) = *h*(*x*, *t*)*), h*<sup>0</sup> = *f and h*<sup>1</sup> = *g.*

**Remark 3.** *A standard argument guarantees that* ∼= *in D∂U*(*U*, *E*) *is an equivalence relation.*

**Definition 7.** *Let F*, *G* ∈ *A∂U*(*U*, *E*)*. We say F* ∼= *G in A∂U*(*U*, *E*) *if for any selection f* ∈ *D∂U*(*U*, *E*) *(respectively, g* ∈ *D∂U*(*U*, *E*)*) of F (respectively, of G) we have f* ∼= *g in D∂U*(*U*, *E*)*.*

**Theorem 1.** *Let E be a completely regular topological space, U an open subset of E, F* ∈ *A∂U*(*U*, *E*) *and G* ∈ *A∂U*(*U*, *E*) *is essential in A∂U*(*U*, *E*)*. In addition, suppose*

$$\begin{cases} \text{ for any selection } f \in D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E}) \text{ (respectively, } \mathcal{g} \in D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E}))\\ \text{ of } F \text{ (respectively, of } \mathcal{G} \text{) and any map } \theta \in D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E})\\ \text{ with } \theta \mid\_{\partial \mathcal{U}} = f \mid\_{\partial \mathcal{U}} \text{ we have } \mathcal{g} \cong \theta \text{ in } D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E}). \end{cases} \tag{1}$$

*Then, F is essential in A∂U*(*U*, *E*)*.*

**Proof.** Let *f* ∈ *D∂U*(*U*, *E*) be any selection of *F* and consider any map *θ* ∈ *D∂U*(*U*, *E*) with *θ*|*∂<sup>U</sup>* = *f* |*∂U*. We must show that there exists a *x* ∈ *U* with *x* = *θ*(*x*). Let *g* ∈ *D∂U*(*U*, *E*) be any selection of *G*. Now, (1) guarantees that there exists a continuous, compact map *h* : *U* × [0, 1] → *E* with *x* = *ht*(*x*) for any *x* ∈ *∂ U* and *t* ∈ (0, 1) (here, *ht*(*x*) = *h*(*x*, *t*)), *h*<sup>0</sup> = *g* and *h*<sup>1</sup> = *θ*. Let

$$\Omega = \{ \mathbf{x} \in \overline{\mathcal{U}} \colon \mathbf{x} = h(\mathbf{x}, t) \text{ for some } t \in [0, 1] \}\dots$$

Now, Ω = ∅ (note *G* is essential in *A∂U*(*U*, *E*)) and Ω is closed (note *h* is continuous) and so Ω is compact (note *h* is a compact map). In addition, note Ω ∩ *∂U* = ∅ since *x* = *ht*(*x*) for any *x* ∈ *∂ U* and *t* ∈ [0, 1]. Then, since *E* is Tychonoff, there exists a continuous map *μ* : *U* → [0, 1] with *μ*(*∂U*) = 0 and *μ*(Ω) = 1. Define the map *r* by *r*(*x*) = *h*(*x*, *μ*(*x*)) = *h* ◦ *g*(*x*), where *g* : *U* → *U* × [0, 1] is given by *g*(*x*)=(*x*, *μ*(*x*)). Note that *r* ∈ *D∂U*(*U*, *E*) (i.e., *r* is a continuous compact map) with *r*|*∂<sup>U</sup>* = *g*|*∂<sup>U</sup>* (note if *x* ∈ *∂U* then *r*(*x*) = *h*(*x*, 0) = *g*(*x*)) so since *G* is essential in *A∂U*(*U*, *E*) there exists a *x* ∈ *U* with *x* = *r*(*x*) (i.e., *x* = *hμ*(*x*)(*x*)). Thus, *x* ∈ Ω so *μ*(*x*) = 1 and thus *x* = *h*1(*x*) = *θ*(*x*).

Let *E* be a topological vector space. Before we prove the topological transversality theorem, we note the following:

(a) If *f* , *g* ∈ *D∂U*(*U*, *E*) with *f* |*∂<sup>U</sup>* = *g*|*∂U*, then *f* ∼= *g* in *D∂U*(*U*, *E*). To see this, let *h*(*x*, *t*) = (1 − *t*) *f*(*x*) + *t g*(*x*) and note *h* : *U* × [0, 1] → *E* is a continuous, compact map with *x* = *ht*(*x*) for any *x* ∈ *∂ U* and *t* ∈ (0, 1) (note *f* |*∂<sup>U</sup>* = *g*|*∂U*).

**Theorem 2.** *Let E be a topological vector space and U an open subset of E. Suppose that F and G are two maps in A∂U*(*U*, *E*) *with F* ∼= *G in A∂U*(*U*, *E*)*. Now, F is essential in A∂U*(*U*, *E*) *if and only if G is essential in A∂U*(*U*, *E*)*.*

**Proof.** Assume *G* is essential in *A∂U*(*U*, *E*). We will use Theorem 1 to show *F* is essential in *A∂U*(*U*, *E*). Let *f* ∈ *D∂U*(*U*, *E*) be any selection of *F*, *g* ∈ *D∂U*(*U*, *E*) be any selection of *G* and consider any map *θ* ∈ *D∂U*(*U*, *E*) with *θ*|*∂<sup>U</sup>* = *f* |*∂U*. Now, (a) above guarantees that *f* ∼= *θ* in *D∂U*(*U*, *E*) and this together with *F* ∼= *G* in *A∂U*(*U*, *E*) (so *f* ∼= *g* in *D∂U*(*U*, *E*)) and Remark 3 guarantees that *g* ∼= *θ* in *D∂U*(*U*, *E*). Thus, (1) holds so Theorem 1 guarantees that *F* is essential in *A∂U*(*U*, *E*). A similar argument shows that, if *F* is essential in *A∂U*(*U*, *E*), then *G* is essential in *A∂U*(*U*, *E*).

**Theorem 3.** *Let E be a Hausdorff locally convex topological vector space, U an open subset of E and* 0 ∈ *U. Assume the zero map is in* **A**(*U*, *E*)*. Then, the zero map is essential in A∂U*(*U*, *E*)*.*

**Proof.** Note *F*(*x*) = {0} for *x* ∈ *U* (i.e., *F* is the zero map) and let *f* ∈ *D∂U*(*U*, *E*) be any selection of *F*. Note *f*(*x*) = 0 for *x* ∈ *U*. Consider any map *g* ∈ *D∂U*(*U*, *E*) with *g*|*∂<sup>U</sup>* = *f* |*∂<sup>U</sup>* = {0}. We must show there exists a *x* ∈ *U* with *x* = *g*(*x*). Let

$$r(\mathbf{x}) = \begin{cases} \ \mathbf{g}(\mathbf{x}), & \mathbf{x} \in \overline{\mathcal{U}}, \\ \ 0, & \mathbf{x} \in E \backslash \overline{\mathcal{U}}. \end{cases}$$

Note *r* : *E* → *E* is a continuous, compact map so [8] guarantees that there exists a *x* ∈ *E* with *x* = *r*(*x*). If *x* ∈ *E* \*U*, then *r*(*x*) = 0, a contradiction since 0 ∈ *U*. Thus, *x* ∈ *U* and so *x* = *g*(*x*).

Now, we consider the above in the weak topology setting. Let *X* be a Hausdorff locally convex topological vector space and *U* a weakly open subset of *C* where *C* is a closed convex subset of *X*. Again, we consider a class **A** of maps.

**Definition 8.** *We say f* ∈ *WD*(*Uw*, *C*) *if f* : *U<sup>w</sup>* → *C is a weakly continuous, weakly compact map; here, U<sup>w</sup> denotes the weak closure of U in C.*

**Definition 9.** *We say f* ∈ *WD∂U*(*Uw*, *C*) *if f* ∈ *WD*(*Uw*, *C*) *and x* = *f*(*x*) *for x* ∈ *∂U; here, ∂U denotes the weak boundary of U in C.*

**Definition 10.** *We say <sup>F</sup>* <sup>∈</sup> *WA*(*Uw*, *<sup>C</sup>*) *if <sup>F</sup>* : *<sup>U</sup><sup>w</sup>* <sup>→</sup> <sup>2</sup>*<sup>C</sup> with <sup>F</sup>* <sup>∈</sup> **<sup>A</sup>**(*Uw*, *<sup>C</sup>*) *and there exists a selection f* ∈ *WD*(*Uw*, *C*) *of F.*

**Definition 11.** *We say F* ∈ *WA∂U*(*Uw*, *C*) *if F* ∈ *WA*(*Uw*, *C*) *and x* ∈/ *F*(*x*) *for x* ∈ *∂U.*

**Definition 12.** *We say F* ∈ *WA∂U*(*Uw*, *C*) *is essential in WA∂U*(*Uw*, *C*) *if for any selection f* ∈ *WD*(*Uw*, *C*) *of F and any map g* ∈ *WD∂U*(*Uw*, *C*) *with f* |*∂<sup>U</sup>* = *g*|*∂<sup>U</sup> there exists a x* ∈ *U with x* = *g*(*x*)*.*

**Definition 13.** *Let f* , *g* ∈ *WD∂U*(*Uw*, *C*)*. We say f* ∼= *g in WD∂U*(*Uw*, *C*) *if there exists a weakly continuous, weakly compact map h* : *U<sup>w</sup>* × [0, 1] → *C with x* = *ht*(*x*) *for any x* ∈ *∂ U and t* ∈ (0, 1) *(here ht*(*x*) = *h*(*x*, *t*)*), h*<sup>0</sup> = *f and h*<sup>1</sup> = *g.*

**Definition 14.** *Let F*, *G* ∈ *WA∂U*(*Uw*, *C*)*. We say F* ∼= *G in WA∂U*(*Uw*, *C*) *if for any selection f* ∈ *WD∂U*(*Uw*, *C*) *(respectively, g* ∈ *WD∂U*(*Uw*, *C*)*) of F (respectively, of G) we have f* ∼= *g in WD∂U*(*Uw*, *C*)*.*

**Theorem 4.** *Let X be a Hausdorff locally convex topological vector space and U a weakly open subset of C*, *where C is a closed convex subset of X. Suppose F* ∈ *WA∂U*(*Uw*, *C*) *and G* ∈ *WA∂U*(*Uw*, *C*) *is essential in WA∂U*(*Uw*, *C*) *and*

$$\begin{cases} \text{ for any selection } f \in \mathcal{WD}\_{\partial \mathcal{U}}(\overline{\Pi^w}, \mathbb{C}) \text{ (respectively, } \underline{\mathcal{g}} \in \mathcal{WD}\_{\partial \mathcal{U}}(\overline{\Pi^w}, \mathbb{C}))\\ \text{ of } F \text{ (respectively, of } \mathcal{G} \text{) and any map } \theta \in \mathcal{WD}\_{\partial \mathcal{U}}(\overline{\Pi^w}, \mathbb{C})\\ \text{ with } \theta|\_{\partial \mathcal{U}} = f|\_{\partial \mathcal{U}} \text{ we have } \underline{\mathcal{g}} \cong \theta \text{ in } \mathcal{WD}\_{\partial \mathcal{U}}(\overline{\Pi^w}, \mathbb{C}). \end{cases} \tag{2}$$

*Then, F is essential in WA∂U*(*Uw*, *C*)*.*

**Proof.** Let *f* ∈ *WD∂U*(*Uw*, *C*) be any selection of *F* and consider any map *θ* ∈ *WD∂U*(*Uw*, *C*) with *θ*|*∂<sup>U</sup>* = *f* |*∂U*. Let *g* ∈ *WD∂U*(*Uw*, *C*) be any selection of *G*. Now, (2) guarantees that there exists a weakly continuous, weakly compact map *h* : *U<sup>w</sup>* × [0, 1] → *C* with *x* = *ht*(*x*) for any *x* ∈ *∂ U* and *t* ∈ (0, 1) (here *ht*(*x*) = *h*(*x*, *t*)), *h*<sup>0</sup> = *g* and *h*<sup>1</sup> = *θ*. Let

$$\Omega = \left\{ \mathbf{x} \in \overline{\mathcal{U}^w} \, : \, \mathbf{x} = h(\mathbf{x}, t) \quad \text{for some} \quad t \in [0, 1] \right\} \dots$$

Recall that *X* = (*X*, *w*), the space *X* endowed with the weak topology, is completely regular. Now, Ω = ∅ is weakly closed and is in fact weakly compact with Ω ∩ *∂U* = ∅. Thus, there exists a weakly continuous map *μ* : *U<sup>w</sup>* → [0, 1] with *μ*(*∂U*) = 0 and *μ*(Ω) = 1. Define the map *r* by *r*(*x*) = *h*(*x*, *μ*(*x*)) and note *r* ∈ *WD∂U*(*Uw*, *C*) with *r*|*∂<sup>U</sup>* = *g*|*∂U*. Since *G* is essential in *WA∂U*(*Uw*, *C*), there exists a *x* ∈ *U* with *x* = *r*(*x*). Thus, *x* ∈ Ω so *x* = *h*1(*x*) = *θ*(*x*).

An obvious modification of the argument in Theorem 2 immediately yields the following result.

**Theorem 5.** *Let X be a Hausdorff locally convex topological vector space and U a weakly open subset of C*, *where C is a closed convex subset of X. Suppose F and G are two maps in WA∂U*(*U*, *C*) *with F* ∼= *G in WA∂U*(*U*, *C*)*. Now, F is essential in WA∂U*(*U*, *C*) *if and only if G is essential in WA∂U*(*U*, *C*)*.*

Now, we consider a generalization of essential maps, namely the *d*–essential maps [2]. Let *E* be a completely regular topological space and *U* an open subset of *E*. For any map *f* ∈ *D*(*U*, *E*), let *<sup>f</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>f</sup>* : *<sup>U</sup>* <sup>→</sup> *<sup>U</sup>* <sup>×</sup> *<sup>E</sup>*, with *<sup>I</sup>* : *<sup>U</sup>* <sup>→</sup> *<sup>U</sup>* given by *<sup>I</sup>*(*x*) = *<sup>x</sup>*, and let

$$d: \left\{ \left( f^{\star} \right)^{-1} \left( \mathcal{B} \right) \right\} \cup \left\{ \mathcal{D} \right\} \to K \tag{3}$$

be any map with values in the nonempty set *K*; here, *B* = (*x*, *<sup>x</sup>*) : *<sup>x</sup>* <sup>∈</sup> *<sup>U</sup>* .

**Definition 15.** *Let <sup>F</sup>* <sup>∈</sup> *<sup>A</sup>∂U*(*U*, *<sup>E</sup>*) *with <sup>F</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *F. We say <sup>F</sup>* : *<sup>U</sup>* <sup>→</sup> <sup>2</sup>*U*×*<sup>E</sup> is d–essential if, for any selection f* ∈ *D*(*U*, *E*) *of F and any map g* ∈ *D∂U*(*U*, *E*) *with f* |*∂<sup>U</sup>* = *g*|*∂<sup>U</sup> , we have that d* (*f* ) <sup>−</sup><sup>1</sup> (*B*) = *d* (*g*) <sup>−</sup><sup>1</sup> (*B*) <sup>=</sup> *<sup>d</sup>*(∅)*; here, f* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *f and g* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *g.*

**Remark 4.** *If F is d–essential, then, for any selection f* <sup>∈</sup> *<sup>D</sup>*(*U*, *<sup>E</sup>*) *of F (with f* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *f ), we have*

$$(\bigotimes \neq (f^\star)^{-1} \ (B) = \{ \mathbf{x} \in \overline{\mathcal{U}} \colon (\mathbf{x}, f(\mathbf{x})) \in B \},$$

*so there exists a x* ∈ *U with x* = *f*(*x*) *(so, in particular, x* ∈ *F*(*x*)*).*

**Theorem 6.** *Let E be a completely regular topological space, U an open subset of E, B* = (*x*, *<sup>x</sup>*) : *<sup>x</sup>* <sup>∈</sup> *<sup>U</sup> , <sup>d</sup> is defined in(3), <sup>F</sup>* <sup>∈</sup> *<sup>A</sup>∂U*(*U*, *<sup>E</sup>*)*, <sup>G</sup>* <sup>∈</sup> *<sup>A</sup>∂U*(*U*, *<sup>E</sup>*) *with <sup>F</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>F</sup> and <sup>G</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *G. Suppose <sup>G</sup> is d–essential and*

$$\begin{cases} \text{ for any selection } f \in D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E}) \text{ (respectively, } \mathcal{g} \in D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E}))\\ \text{ of } F \text{ (respectively, of } \mathcal{G} \text{) and any map } \theta \in D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E})\\ \text{with } \theta|\_{\partial \mathcal{U}} = f|\_{\partial \mathcal{U}} \text{ we have } \mathcal{g} \cong \theta \text{ in } D\_{\partial \mathcal{U}}(\overline{\mathcal{U}}, \mathcal{E}) \text{ and}\\ d\left( (f^{\star})^{-1} \left( B \right) \right) = d\left( (\mathcal{g}^{\star})^{-1} \left( B \right) \right); \text{ here } f^{\star} = I \times f \text{ and } \mathcal{g}^{\star} = I \times \mathcal{g}. \end{cases} \tag{4}$$

*Then, F is d–essential.*

**Proof.** Let *f* ∈ *D∂U*(*U*, *E*) be any selection of *F* and consider any map *θ* ∈ *D∂U*(*U*, *E*) with *θ*|*∂<sup>U</sup>* = *f* |*∂U*. We must show *d* (*f* ) <sup>−</sup><sup>1</sup> (*B*) = *d* (*θ*) <sup>−</sup><sup>1</sup> (*B*) <sup>=</sup> *<sup>d</sup>*(∅); here, *<sup>f</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>f</sup>* and *<sup>θ</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>θ</sup>*. Let *g* ∈ *D∂U*(*U*, *E*) be any selection of *G*. Now, (4) guarantees that there exists a continuous, compact map *h* : *U* × [0, 1] → *E* with *x* = *ht*(*x*) for any *x* ∈ *∂ U* and *t* ∈ (0, 1) (here *ht*(*x*) = *h*(*x*, *t*)), *h*<sup>0</sup> = *g*, *h*<sup>1</sup> = *θ* and *d* (*f* ) <sup>−</sup><sup>1</sup> (*B*) = *d* (*g*) <sup>−</sup><sup>1</sup> (*B*) ; here, *<sup>g</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>g</sup>*. Let *<sup>h</sup>* : *<sup>U</sup>* <sup>×</sup> [0, 1] <sup>→</sup> *<sup>U</sup>* <sup>×</sup> *<sup>E</sup>* be given by *h*(*x*, *t*)=(*x*, *h*(*x*, *t*)) and let

$$\Omega = \left\{ \mathbf{x} \in \overline{\mathcal{U}} \, : \, h^\*(\mathbf{x}, t) \in B \text{ for some } t \in [0, 1] \right\} \dots$$

Now, Ω = ∅ is closed, compact and Ω ∩ *∂U* = ∅ so there exists a continuous map *μ* : *U* → [0, 1] with *<sup>μ</sup>*(*∂U*) = 0 and *<sup>μ</sup>*(Ω) = 1. Define the map *<sup>r</sup>* by *<sup>r</sup>*(*x*) = *<sup>h</sup>*(*x*, *<sup>μ</sup>*(*x*)) and *<sup>r</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>r</sup>*. Now, *<sup>r</sup>* <sup>∈</sup> *<sup>D</sup>∂U*(*U*, *<sup>E</sup>*) with *<sup>r</sup>*|*∂<sup>U</sup>* <sup>=</sup> *<sup>g</sup>*|*∂U*. Since *<sup>G</sup>* is *<sup>d</sup>*–essential, then

$$d\left(\left(g^{\star}\right)^{-1}\left(B\right)\right) = d\left(\left(r^{\star}\right)^{-1}\left(B\right)\right) \neq d\left(\mathcal{O}\right).\tag{5}$$

Now, since *μ*(Ω) = 1, we have

$$\begin{aligned} \left( \left( r^\star \right)^{-1} \left( B \right) \right) &= \left\{ \mathbf{x} \in \overline{\mathcal{U}} : \left( \mathbf{x}, h(\mathbf{x}, \mu(\mathbf{x})) \right) \in B \right\} = \left\{ \mathbf{x} \in \overline{\mathcal{U}} : \left( \mathbf{x}, h(\mathbf{x}, 1) \right) \in B \right\} \\ &= \left( \left( \theta^\star \right)^{-1} \left( B \right) \right) \end{aligned}$$

so, from the above and Equation (5), we have *d* (*f* ) <sup>−</sup><sup>1</sup> (*B*) = *d* (*θ*) <sup>−</sup><sup>1</sup> (*B*) = *d*(∅).

**Theorem 7.** *Let E be a completely regular topological space, U an open subset of E, B* = (*x*, *<sup>x</sup>*) : *<sup>x</sup>* <sup>∈</sup> *<sup>U</sup> and <sup>d</sup> is defined in (3). Suppose <sup>F</sup> and <sup>G</sup> are two maps in <sup>A</sup>∂U*(*U*, *<sup>E</sup>*) *with <sup>F</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *F, <sup>G</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>G</sup> and F* ∼= *G in A∂U*(*U*, *E*)*. Then, F is d–essential if and only if G is d–essential.*

**Proof.** Assume *<sup>G</sup>* is *<sup>d</sup>*–essential. Let *<sup>f</sup>* <sup>∈</sup> *<sup>D</sup>∂U*(*U*, *<sup>E</sup>*) be any selection of *<sup>F</sup>*, *<sup>g</sup>* <sup>∈</sup> *<sup>D</sup>∂U*(*U*, *<sup>E</sup>*) be any selection of *<sup>G</sup>* and consider any map *<sup>θ</sup>* <sup>∈</sup> *<sup>D</sup>∂U*(*U*, *<sup>E</sup>*) with *<sup>θ</sup>*|*∂<sup>U</sup>* <sup>=</sup> *<sup>f</sup>* <sup>|</sup>*∂U*. If we show (4), then *<sup>F</sup>* is *d*–essential from Theorem 6. Now, *f* ∼= *θ* in *D∂U*(*U*, *E*) together with *F* ∼= *G* in *A∂U*(*U*, *E*) (so *f* ∼= *g* in *D∂U*(*U*, *E*)) guarantees that *g* ∼= *θ* in *D∂U*(*U*, *E*). To complete (4), we need to show *d* (*f* ) <sup>−</sup><sup>1</sup> (*B*) = *d* (*g*) <sup>−</sup><sup>1</sup> (*B*) ; here, *<sup>f</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>f</sup>* and *<sup>g</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>g</sup>*. We will show this by following the argument in Theorem 6. Note *G* ∼= *F* in *A∂U*(*U*, *E*) and let *h* : *U* × [0, 1] → *E* be a continuous, compact map with *x* = *ht*(*x*) for any *x* ∈ *∂ U* and *t* ∈ (0, 1) (here *ht*(*x*) = *h*(*x*, *t*)), *h*<sup>0</sup> = *g* and *h*<sup>1</sup> = *f* . Let *<sup>h</sup>* : *<sup>U</sup>* <sup>×</sup> [0, 1] <sup>→</sup> *<sup>U</sup>* <sup>×</sup> *<sup>E</sup>* be given by *<sup>h</sup>*(*x*, *<sup>t</sup>*)=(*x*, *<sup>h</sup>*(*x*, *<sup>t</sup>*)) and let

$$\Omega = \left\{ \mathbf{x} \in \overline{\mathcal{U}} \, : \, h^\*(\mathbf{x}, t) \in B \quad \text{for some} \quad t \in [0, 1] \right\} \dots$$

Now, Ω = ∅ and there exists a continuous map *μ* : *U* → [0, 1] with *μ*(*∂U*) = 0 and *μ*(Ω) = 1. Define the map *<sup>r</sup>* by *<sup>r</sup>*(*x*) = *<sup>h</sup>*(*x*, *<sup>μ</sup>*(*x*)) and *<sup>r</sup>* <sup>=</sup> *<sup>I</sup>* <sup>×</sup> *<sup>r</sup>*. Now, *<sup>r</sup>* <sup>∈</sup> *<sup>D</sup>∂U*(*U*, *<sup>E</sup>*) with *<sup>r</sup>*|*∂<sup>U</sup>* <sup>=</sup> *<sup>g</sup>*|*∂<sup>U</sup>* so, since *G* is *d*–essential, then *d* (*g*) <sup>−</sup><sup>1</sup> (*B*) = *d* (*r*) <sup>−</sup><sup>1</sup> (*B*) = *d*(∅). Now, since *μ*(Ω) = 1, we have (see the argument in Theorem 6) (*r*) <sup>−</sup><sup>1</sup> (*B*) = (*f* ) <sup>−</sup><sup>1</sup> (*B*) and, as a result, we have *d* (*f* ) <sup>−</sup><sup>1</sup> (*B*) = *d* (*g*) <sup>−</sup><sup>1</sup> (*B*) .

**Remark 5.** *It is also easy to extend the above ideas to other natural situations. Let E be a (Hausdorff) topological vector space (so automatically completely regular), Y a topological vector space, and U an open subset of E. In addition, let L* : *dom L* ⊆ *E* → *Y be a linear (not necessarily continuous) single valued map; here, dom L is a vector subspace of E. Finally, T* : *E* → *Y will be a linear, continuous single valued map with L* + *T* : *dom L* → *Y an isomorphism (i.e., a linear homeomorphism); for convenience we say T* ∈ *HL*(*E*,*Y*)*.* *We say <sup>F</sup>* <sup>∈</sup> *<sup>A</sup>*(*U*,*Y*; *<sup>L</sup>*, *<sup>T</sup>*) *if* (*<sup>L</sup>* <sup>+</sup> *<sup>T</sup>*)−<sup>1</sup> (*<sup>F</sup>* <sup>+</sup> *<sup>T</sup>*) <sup>∈</sup> *<sup>A</sup>*(*U*, *<sup>E</sup>*) *and we could discuss essential and d–essential in this situation.*

Now, we present an example to illustrate our theory.

**Example 1.** *Let E be a Hausdorff locally convex topological vector space, U an open subset of E ,* 0 ∈ *U and U paracompact. In this case, we say that F* ∈ **A**(*U*, *E*) *if F* ∈ *PK*(*U*, *E*) *(see Remark 1) is a compact map. Let F* ∈ *A∂U*(*U*, *E*) *and assume x* ∈/ *λF* (*x*) *for x* ∈ *∂U and λ* ∈ (0, 1)*. Then, F* ∼= 0 *in A∂U*(*U*, *E*)*. To see this, let f* ∈ *D∂U*(*U*, *E*) *be any selection of F and let h* : *U* × [0, 1] *be given by h*(*x*, *t*) = *t f*(*x*)*. Note that h*<sup>0</sup> = 0, *h*<sup>1</sup> = *f and x* ∈/ *ht*(*x*) *for x* ∈ *∂U and λ* ∈ (0, 1) *so f* ∼= 0 *in D∂U*(*U*, *E*)*. Now, Theorems 2 and 3 guarantee that F is essential in A∂U*(*U*, *E*)*.*

#### **3. Conclusions**

In this paper, we prove that, for two set-valued maps *F* and *G* with continuous compact selections and *F* ∼= *G*, then one being essential (or *d*–essential) guarantees that the other is essential (or *d*–essential).

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **A Class of Equations with Three Solutions**

#### **Biagio Ricceri**

Department of Mathematics and Informatics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy; ricceri@dmi.unict.it

Received: 5 March 2020; Accepted: 29 March 2020; Published: 1 April 2020

**Abstract:** Here is one of the results obtained in this paper: Let <sup>Ω</sup> <sup>⊂</sup> **<sup>R</sup>***<sup>n</sup>* be a smooth bounded domain, let *<sup>q</sup>* <sup>&</sup>gt; 1, with *<sup>q</sup>* <sup>&</sup>lt; *<sup>n</sup>*+<sup>2</sup> *<sup>n</sup>*−<sup>2</sup> if *<sup>n</sup>* <sup>≥</sup> 3 and let *<sup>λ</sup>*<sup>1</sup> be the first eigenvalue of the problem <sup>−</sup>Δ*<sup>u</sup>* <sup>=</sup> *<sup>λ</sup><sup>u</sup>* in <sup>Ω</sup>, *<sup>u</sup>* <sup>=</sup> 0 on *<sup>∂</sup>*Ω. Then, for every *<sup>λ</sup>* <sup>&</sup>gt; *<sup>λ</sup>*<sup>1</sup> and for every convex set *<sup>S</sup>* <sup>⊆</sup> *<sup>L</sup>*∞(Ω) dense in *<sup>L</sup>*2(Ω), there exists *<sup>α</sup>* <sup>∈</sup> *<sup>S</sup>* such that the problem <sup>−</sup>Δ*<sup>u</sup>* <sup>=</sup> *<sup>λ</sup>*(*u*<sup>+</sup> <sup>−</sup> (*u*+)*<sup>q</sup>*) + *<sup>α</sup>*(*x*) in <sup>Ω</sup>, *<sup>u</sup>* <sup>=</sup> 0 on *<sup>∂</sup>*Ω, has at least three weak solutions, two of which are global minima in *H*<sup>1</sup> <sup>0</sup> (Ω) of the functional *u* → 1 2 - <sup>Ω</sup> |∇*u*(*x*)| <sup>2</sup>*dx* <sup>−</sup> *<sup>λ</sup>* - Ω 1 <sup>2</sup> <sup>|</sup>*u*+(*x*)<sup>|</sup> <sup>2</sup> <sup>−</sup> <sup>1</sup> *<sup>q</sup>*+<sup>1</sup> <sup>|</sup>*u*+(*x*)<sup>|</sup> *q*+1 *dx* <sup>−</sup> - <sup>Ω</sup> *<sup>α</sup>*(*x*)*u*(*x*)*dx* where *<sup>u</sup>*<sup>+</sup> <sup>=</sup> max{*u*, 0}.

**Keywords:** minimax; multiplicity; global minima

#### **1. Introduction**

There is no doubt that the study of nonlinear PDEs lies in the core of Nonlinear Analysis. In turn, one of the most studied topics concerning nonlinear PDEs is the multiplicity of solutions. On the other hand, the study of the global minima of integral functionals is essentially the central subject of the Calculus of Variations. In the light of these facts, it is hardly understable why the number of the known results on multiple global minima of integral functionals is extremely low. Certainly, this is not due to a lack of intrinsic mathematical interest. Probably, the reason could reside in the fact that there is not an abstract tool which has the same popularity as the one that, for instance, the Lyusternik–Schnirelmann theory and the Morse theory have in dealing with multiple solutions for nonlinear PDEs.

Abstract results on the multiplicity of global minima, however, are already present in the literature. We allude to the result first obtained in [1] and then extended in [2,3] which ensures the existence of at least two global minima provided that a strict minimax inequality holds. We already have obtained a variety of applications upon different ways of checking the required strict inequality ([4–6]).

The aim of the present paper is to establish an application of Theorem 1 of [7] which is itself an application of the main result in [3]. Precisely, we first establish a general result which ensures the existence of three solutions for a certain equation provided that another related one has no non-zero solutions (Theorem 1). Then, we present an application to nonlinear elliptic equations (Theorem 2).

#### **2. Results**

In the sequel, (*X*, ·*X*) is a reflexive real Banach space, (*Y*,·, ·*Y*) is a real Hilbert space, *I*, *ψ* : *<sup>X</sup>* <sup>→</sup> **<sup>R</sup>** are two *<sup>C</sup>*<sup>1</sup> functionals, with *<sup>I</sup>*(0) = *<sup>ψ</sup>*(0) = 0 and sup**<sup>R</sup>** *<sup>ψ</sup>* <sup>&</sup>gt; 0, *<sup>ϕ</sup>* : *<sup>X</sup>* <sup>→</sup> *<sup>Y</sup>* is a *<sup>C</sup>*<sup>1</sup> operator, with *ϕ*(0) = 0. For each fixed *y* ∈ *Y*, we denote by *∂xϕ*(·), *y* the derivative of the functional *x* → *ϕ*(*x*), *y*. Clearly, one has

$$
\partial\_{\mathfrak{x}} \langle \mathfrak{p}(\mathfrak{x}), \mathfrak{y} \rangle (\mathfrak{u}) = \langle \mathfrak{p}'(\mathfrak{x})(\mathfrak{u}), \mathfrak{y} \rangle
$$

for all *x*, *u* ∈ *X*.

We say that *<sup>I</sup>* is coercive if lim*xX*→+<sup>∞</sup> *<sup>I</sup>*(*x*)=+∞. We also say that *<sup>I</sup>* admits a continuous inverse on *X*<sup>∗</sup> if there exists a continuous operator *T* : *X*<sup>∗</sup> → *X* such that *T*(*I* (*x*)) = *x* for all *x* ∈ *X*.

Here is our abstract result:

**Theorem 1.** *Let I be weakly lower semicontinuous and coercive, and let I admit a continuous inverse on X*∗*. Moreover, assume that the operators ϕ and ψ are compact and that*

$$\lim\_{\|\mathbf{x}\|\chi\to+\infty} \frac{\langle\varrho(\mathbf{x}),\mathbf{y}\rangle\_Y}{I(\mathbf{x})} = 0 \tag{1}$$

*for all y in a convex and dense set V* ⊆ *Y.*

*Set*

$$\theta^\* := \inf\_{\mathfrak{x} \in \psi^{-1}(]0, +\infty[)} \frac{I(\mathfrak{x})}{\Psi(\mathfrak{x})} \text{ .} $$

˜ *θ* := lim inf*x*∈*ψ*−1(]0,+∞[),*xX*→+<sup>∞</sup> *I*(*x*) *<sup>ψ</sup>*(*x*) *if <sup>ψ</sup>*−1(]0, <sup>+</sup>∞[) *is unbounded* +∞ *otherwise*

*and assume that*

$$
\theta^\* < \delta\_\* 
$$

*Then, for each <sup>λ</sup>* <sup>∈</sup>]*θ*∗, ˜ *θ*[*, with λ* ≥ 0*, either the equation*

$$I'(x) = -\partial\_x \langle \phi(x), \phi(x) \rangle + \lambda \psi'(x)$$

*has a non-zero solution, or, for each convex set S* ⊆ *V dense in Y, there exists y*˜ ∈ *S such that the equation*

$$I'(\mathfrak{x}) = \partial\_{\mathfrak{x}} \langle \mathfrak{q}(\mathfrak{x})\_{\prime} \tilde{\mathfrak{y}} \rangle\_{Y} + \lambda \mathfrak{y}^{\prime}(\mathfrak{x})\_{\prime}$$

*has at least three solutions, two of which are global minima in X of the functional*

$$\mathbf{x} \rightarrow I(\mathbf{x}) - \langle \boldsymbol{\varrho}(\mathbf{x}), \boldsymbol{\mathfrak{y}} \rangle\_Y - \lambda \boldsymbol{\psi}(\mathbf{x})\,\,\,\,\,$$

As it was said in the Introduction, the main tool to prove Theorem 1 is a result recently obtained in [7]. For reader's convenience, we now recall its statement:

**Theorem 2.** *([7], Theorem 1). - Let <sup>X</sup>*, *<sup>E</sup> be two real reflexive Banach spaces and let* <sup>Φ</sup> : *<sup>X</sup>* <sup>×</sup> *<sup>E</sup>* <sup>→</sup> **<sup>R</sup>** *be a <sup>C</sup>*<sup>1</sup> *functional satisfying the following conditions:*


*Then, either the system*

$$\begin{cases} \Phi\_x'(x,y) = 0\\ \Phi\_y'(x,y) = 0 \end{cases}$$

*has a solution* (*x*∗, *y*∗) *such that*

$$\Phi(x^\*, y^\*) = \inf\_{x \in X} \Phi(x, y^\*) = \sup\_{y \in E} \Phi(x^\*, y) \text{ .}$$

*or, for every convex set T* ⊆ *S dense in E, there exists y*˜ ∈ *T such that equation*

$$
\Phi\_x'(x,\bar{y}) = 0
$$

*has at least three solutions, two of which are global minima in X of the functional* Φ(·, *y*˜)*.*

**Proof of Theorem 1.** Fix *<sup>λ</sup>* <sup>∈</sup>]*θ*∗, ˜ *θ*[, with *λ* ≥ 0. Assume that the equation

$$I'(x) = -\partial\_x \langle \boldsymbol{\varrho}(\boldsymbol{x}), \boldsymbol{\varrho}(\boldsymbol{x}) \rangle + \lambda \boldsymbol{\psi}'(\boldsymbol{x})$$

has no non-zero solution. Fix a convex set *S* ⊆ *Y* dense in *Y*. We have to show that there exists *y*˜ ∈ *S* such that the equation

$$I'(\mathfrak{x}) = \partial\_{\mathfrak{x}} \langle \mathfrak{p}(\mathfrak{x}), \mathfrak{y} \rangle\_{Y} + \lambda \mathfrak{y}'(\mathfrak{x})$$

has at least three solutions, two of which are global minima in *X* of the functional *x* → *I*(*x*) − *ϕ*(*x*), *y*˜*<sup>Y</sup>* − *λψ*(*x*). To this end, let us apply Theorem 2. Consider the functional Φ : *X* × *Y* → **R** defined by

$$\Phi(\mathbf{x}, \mathbf{y}) = I(\mathbf{x}) - \frac{1}{2} \|\mathbf{y}\|\_{Y}^{2} - \langle \boldsymbol{\varrho}(\mathbf{x}), \mathbf{y} \rangle - \lambda \boldsymbol{\psi}(\mathbf{x})$$

for all (*x*, *<sup>y</sup>*) <sup>∈</sup> *<sup>X</sup>* <sup>×</sup> *<sup>Y</sup>*. Of course, <sup>Φ</sup> is *<sup>C</sup>*<sup>1</sup> and, for each *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*, <sup>Φ</sup>(*x*, ·) is concave and <sup>−</sup>Φ(*x*, ·) is coercive. Fix *y* ∈ *Y*. Let us show that the operator *∂xϕ*(·), *y* is compact. To this end, let {*xn*} be a bounded sequence in *X*. Since *ϕ* is compact, up to a subsequence, {*ϕ* (*xn*)} converges in L(*X*,*Y*) to some *η*. That is

$$\lim\_{n \to \infty} \sup\_{||u||\_X = 1} ||q'(\mathfrak{x}\_n)(u) - \eta(u)||\_Y = 0$$

On the other hand, we have

$$\sup\_{\|u\|\_{X}=1} |\partial\_{\mathbf{x}} \langle \rho(\mathbf{x}\_{n}), \mathbf{y} \rangle (u) - \langle \eta(u), \mathbf{y} \rangle| = \sup\_{\|u\|\_{X}=1} |\langle \rho'(\mathbf{x}\_{n})(u), \mathbf{y} \rangle - \langle \eta(u), \mathbf{y} \rangle| $$

$$\leq \sup\_{\|u\|\_{X}=1} \|\rho'(\mathbf{x}\_{n})(u) - \eta(u)\|\_{Y} \|\mathbf{y}\|\_{Y}$$

and so the sequence {*∂xϕ*(*xn*), *y*(·)} converges in *X*<sup>∗</sup> to *η*(·)(*y*). Then, since *ψ* is compact, the operator *∂xϕ*(·), *y* + *λψ* (·) is compact too. From this, it follows that *ϕ*(·), *y* + *λψ*(·) is sequentially weakly continuous ([8], Corollary 41.9). If *x<sup>X</sup>* is large enough, we have *I*(*x*) > 0 and so we can write

$$\Phi(\mathbf{x}, \mathbf{y}) = I(\mathbf{x}) \left( 1 - \frac{\frac{1}{2} \|\mathbf{y}\|\|\_{Y}^{2} + \langle \varrho(\mathbf{x}), \mathbf{y} \rangle + \lambda \psi(\mathbf{x})}{I(\mathbf{x})} \right) \,. \tag{2}$$

In view of (1), we also have

$$\liminf\_{\|\mathbf{x}\|\|\mathbf{x}\|\to+\infty} \left(1 - \frac{\frac{1}{2} \|\|y\|\|\_{Y}^{2} + \langle \boldsymbol{\varphi}(\mathbf{x}), y \rangle + \lambda \boldsymbol{\psi}(\mathbf{x})}{I(\mathbf{x})}\right) = 1 - \limsup\_{\|\mathbf{x}\|\|\mathbf{x}\|\to+\infty} \frac{\lambda \boldsymbol{\psi}(\mathbf{x})}{I(\mathbf{x})}.\tag{3}$$

We claim that

$$\limsup\_{||\mathfrak{x}||\to+\infty} \frac{\lambda \psi(\mathfrak{x})}{I(\mathfrak{x})} < 1\,. \tag{4}$$

This is clear if either *<sup>λ</sup>* <sup>=</sup> 0 or lim sup*xX*→+<sup>∞</sup> *ψ*(*x*) *<sup>I</sup>*(*x*) <sup>≤</sup> 0. If *<sup>λ</sup>* <sup>&</sup>gt; 0 and lim sup*xX*→+<sup>∞</sup> *ψ*(*x*) *<sup>I</sup>*(*x*) > 0, then (4) is equivalent to

$$\limsup\_{||x||\chi\to+\infty} \frac{\psi(x)}{l(x)} < +\infty$$

and

$$
\lambda < \frac{1}{\limsup\_{\|\mathbf{x}\|\_{X} \to +\infty} \frac{\Psi(\mathbf{x})}{I(\mathbf{x})}} \,. \tag{5}
$$

But

$$\frac{1}{\limsup\_{\|x\|\_X \to +\infty} \frac{\Psi(x)}{I(x)}} = \liminf\_{\substack{x \in \psi^{-1}([0,+\infty[), \|x\|\_X \to +\infty} \frac{I(x)}{\Psi(x)}} \frac{I(x)}{\sqrt{\!}} \,\, ^t$$

and so (5) is satisfied just since *λ* < ˜ *θ*. Since *I* is coercive and weakly lower semicontinuous, the functional Φ(·, *y*) turns out to be coercive, in view of (2), (3), (4), and weakly lower semicontinuous, in view of the Eberlein-Smulyan theorem. Finally, since *I* admits a continuous inverse on *X*∗, Φ(·, *y*) satisfies the Palais-Smale condition in view of Example 38.25 of [8]. Hence, Φ satisfies the assumptions of Theorem 2. Now, we claim that there is no solution (*x*∗, *y*∗) of the system

$$\begin{cases} \Phi'\_x(x,y) = 0\\ \Phi'\_y(x,y) = 0 \end{cases}$$

such that

$$\Phi(x^\*, y^\*) = \inf\_{x \in X} \Phi(x, y^\*) \, .$$

Arguing by contradiction, assume that such a (*x*∗, *y*∗) does exist. This amounts to say that

$$\begin{cases} \begin{aligned} I'(x^\*) &= \partial\_{\mathbf{x}} \langle \boldsymbol{\varrho}(\mathbf{x}^\*) \boldsymbol{\varrho}^\* \rangle + \lambda \boldsymbol{\psi}'(\mathbf{x}^\*), \\ \boldsymbol{\mathfrak{y}}^\* &= -\boldsymbol{\varrho}(\mathbf{x}^\*) \end{aligned} \end{cases}$$

and

$$I(\mathbf{x}^\*) - \langle \varrho(\mathbf{x}^\*), \mathbf{y}^\* \rangle - \lambda \psi(\mathbf{x}^\*) = \inf\_{\mathbf{x} \in X} (I(\mathbf{x}) - \langle \varrho(\mathbf{x}), \mathbf{y}^\* \rangle - \lambda \psi(\mathbf{x})) \,. \tag{6}$$

Therefore

$$I'(\mathfrak{x}^\*) = -\partial\_{\mathfrak{x}} \langle \not\!\! \! \!/ (\mathfrak{x}^\*), \not\!\! \! \! (\mathfrak{x}^\*) \rangle + \lambda \psi'(\mathfrak{x}^\*) \dots$$

So, by the initial assumption, we have *x*∗ = 0 and hence *y*∗ = 0 (recall that *ϕ*(0) = 0). As a consequence, since *I*(0) = *ψ*(0) = 0, (6) becomes

$$\inf\_{x \in X} \left( I(x) - \lambda \psi(x) \right) = 0 \,. \tag{7}$$

Now, notice that (7) contradicts the fact that *λ* > *θ*∗. Hence, *a fortiori*, the system

$$\begin{cases} \Phi\_x'(x,y) = 0\\ \Phi\_y'(x,y) = 0 \end{cases}$$

has no solution (*x*∗, *y*∗) such that

$$\Phi(\mathbf{x}^\*, \mathbf{y}^\*) = \inf\_{\mathbf{x} \in X} \Phi(\mathbf{x}, \mathbf{y}^\*) = \sup\_{\mathbf{y} \in Y} \Phi(\mathbf{x}^\*, \mathbf{y})$$

and then the existence of *y*˜ ∈ *S* is directly ensured by Theorem 2.

We now present an application of Theorem <sup>1</sup> to a class of nonlinear elliptic equations. Let <sup>Ω</sup> <sup>⊂</sup> **<sup>R</sup>***<sup>n</sup>* be a smooth bounded domain. We denote by A the class of all Carathéodory's functions *f* : Ω × **R** → **R** such that, for each *<sup>u</sup>*, *<sup>v</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω), the function *<sup>x</sup>* <sup>→</sup> *<sup>f</sup>*(*x*, *<sup>u</sup>*(*x*))*v*(*x*) lies in *<sup>L</sup>*1(Ω). For *<sup>f</sup>* ∈ A, we consider the Dirichlet problem

$$\begin{cases} -\Delta u = f(\mathfrak{x}, u) & \text{in } \Omega \\\ u = 0 & \text{on } \partial\Omega \end{cases} $$

As usual, a weak solution of the problem is any *<sup>u</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω) such that

$$\int\_{\Omega} \nabla u(\mathbf{x}) \nabla v(\mathbf{x}) d\mathbf{x} = \int\_{\Omega} f(\mathbf{x}, u(\mathbf{x})) v(\mathbf{x}) d\mathbf{x}$$

for all *<sup>v</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω). Also, we denote by *λ*<sup>1</sup> the first eigenvalue of the Dirichlet problem

$$\begin{cases} -\Delta u = \lambda u & \text{in } \Omega \\\ u = 0 & \text{on } \partial\Omega \end{cases} $$

For any continuous function *<sup>f</sup>* : **<sup>R</sup>** <sup>→</sup> **<sup>R</sup>**, we set *<sup>F</sup>*(*ξ*) = *ξ* <sup>0</sup> *f*(*t*)*dt* for all *ξ* ∈ **R**.

**Theorem 3.** *Let f* , *g* : **R** → **R** *be two continuous functions satisfying the following growth conditions:*

(*a*) *if n* ≤ 3*, one has*

$$\lim\_{|\xi| \to +\infty} \frac{|F(\xi)|}{\xi^2} = 0 \; ; \; $$

(*b*) *if n* <sup>≥</sup> <sup>2</sup>*, there exist p*, *<sup>q</sup>* <sup>&</sup>gt; <sup>0</sup>*, with p* <sup>&</sup>lt; <sup>2</sup> *<sup>n</sup>*−<sup>2</sup> *, q* <sup>&</sup>lt; *<sup>n</sup>*+<sup>2</sup> *<sup>n</sup>*−<sup>2</sup> *if n* <sup>≥</sup> <sup>3</sup>*, such that*

$$\begin{aligned} \sup\_{\substack{\mathfrak{z} \in \mathbb{R} \\ \mathfrak{z} \in \mathbb{R}}} \frac{|f(\mathfrak{z})|}{1 + |\mathfrak{z}|^p} &< +\infty \,\,\mu \\\\ \sup\_{\substack{\mathfrak{z} \in \mathbb{R} \\ \mathfrak{z} \in \mathbb{R}}} \frac{|\mathfrak{g}(\mathfrak{z})|}{1 + |\mathfrak{z}|^q} &< +\infty \,\,\,\mu \end{aligned}$$

*Set*

$$\rho := \limsup\_{|\xi| \to +\infty} \frac{G(\xi)}{\xi^2},$$

*ξ*∈**R**

$$\sigma := \max \left\{ \liminf\_{\xi \to 0^+} \frac{G(\xi)}{\xi^2}, \liminf\_{\xi \to 0^-} \frac{G(\xi)}{\xi^2} \right\},$$

*and assume that*

max{*ρ*, 0} < *σ* .

$$\text{Then, for every } \lambda \in \left] \frac{\lambda\_1}{2\sigma}, \frac{\lambda\_1}{2\max\{\rho, 0\}} \right[ \text{ (with the conventions } \frac{\lambda\_1}{+\infty} = 0, \frac{\lambda\_1}{0} = +\infty \text{), either the problem } \frac{\lambda\_1}{2\max\{\rho, 0\}} \text{ are equal to the same.}$$

$$\begin{cases} -\Delta u = -F(u)f(u) + \lambda g(u) & \text{in } \Omega\\ u = 0 & \text{on } \partial\Omega \end{cases} \tag{8}$$

*has a non-zero weak solution, or, for every convex set <sup>S</sup>* <sup>⊆</sup> *<sup>L</sup>*∞(Ω) *dense in <sup>L</sup>*2(Ω)*, there exists <sup>α</sup>* <sup>∈</sup> *<sup>S</sup> such that the problem*

$$\begin{cases} -\Delta u = a(\mathbf{x})f(u) + \lambda \mathbf{g}(u) & \text{in } \Omega\\ u = 0 & \text{on } \partial\Omega \end{cases} \tag{9}$$

*has at least three weak solutions, two of which are global minima in H*<sup>1</sup> <sup>0</sup> (Ω) *of the functional*

$$\mu \to \frac{1}{2} \int\_{\Omega} |\nabla u(\mathbf{x})|^2 d\mathbf{x} - \int\_{\Omega} a(\mathbf{x}) F(u(\mathbf{x})) d\mathbf{x} - \lambda \int\_{\Omega} G(u(\mathbf{x})) d\mathbf{x} \,\,\, \mu$$

**Proof.** We are going to apply Theorem 1 taking *X* = *H*<sup>1</sup> <sup>0</sup> (Ω), *<sup>Y</sup>* = *<sup>L</sup>*2(Ω), with their usual scalar products (that is, *u*, *<sup>v</sup><sup>X</sup>* <sup>=</sup> - <sup>Ω</sup> <sup>∇</sup>*u*(*x*)∇*v*(*x*)*dx* and *u*, *<sup>v</sup><sup>Y</sup>* <sup>=</sup> - <sup>Ω</sup> *<sup>u</sup>*(*x*)*v*(*x*)*dx*), *<sup>V</sup>* <sup>=</sup> *<sup>L</sup>*∞(Ω) and

$$I(\boldsymbol{\mu}) = \frac{1}{2} \|\boldsymbol{\mu}\|\_{X'}^2$$

$$\boldsymbol{\varphi}(\boldsymbol{\mu}) = \boldsymbol{F} \circ \boldsymbol{\mu}$$

$$\boldsymbol{\psi}(\boldsymbol{\mu}) = \int\_{\Omega} G(\boldsymbol{\mu}(\boldsymbol{x})) d\boldsymbol{x}$$

for all *u* ∈ *X*. In view of (*b*), thanks to the Sobolev embedding theorem, the operator *ϕ* and the functional *ψ* are *C*1, with compact derivative. Moreover, the solutions of the equation

$$I'(\mu) = -\partial\_{\mathfrak{u}} \langle \boldsymbol{\varrho}(\mathfrak{u}), \boldsymbol{\varrho}(\mathfrak{u}) \rangle\_Y + \lambda \boldsymbol{\psi}'(\mathfrak{u})$$

are weak solutions of (8) and, for each *α* ∈ *Y*, the solutions of the equation

$$I'(\mu) = \partial\_{\mathfrak{u}} \langle \mathfrak{p}(\mathfrak{u}), \mathfrak{a} \rangle\_Y + \lambda \mathfrak{p}'(\mathfrak{u})$$

are weak solutions of (9). Moreover, condition (1) follows readily from (*a*) which is automatically satisfied when *<sup>n</sup>* <sup>≥</sup> 4 since *<sup>p</sup>* <sup>&</sup>lt; <sup>2</sup> *<sup>n</sup>*−<sup>2</sup> . We claim that

$$\limsup\_{||u||\_X \to +\infty} \frac{\psi(u)}{||u||\_X^2} \le \frac{\rho}{\lambda\_1} \,. \tag{10}$$

Indeed, fix *ν* > *ρ*. Then, there exists *δ* > 0 such that

$$G(\xi) \le \nu \xi^2 \tag{11}$$

for all *x* ∈ **R** \ [−*δ*, *δ*]. Fix *u* ∈ *X* \ {0}. From (11) we clearly obtain

$$\psi(u) \le \nu \|u\|\_Y^2 + \text{meas}(\Omega) \sup\_{[-\delta,\delta]} G \le \nu \frac{\|u\|\_X^2}{\lambda\_1} + \text{meas}(\Omega) \sup\_{[-\delta,\delta]} G$$

and so

$$\limsup\_{||u||\_X \to +\infty} \frac{\psi(u)}{||u||\_X^2} \le \frac{\nu}{\lambda\_1} \,. \tag{12}$$

Now, we get (10) passing in (12) to the limit for *ν* tending to *ρ*. We also claim that

$$\frac{\sigma}{\lambda\_1} \le \sup\_{u \in X \backslash \{0\}} \frac{\psi(u)}{||u||\_X^2} \,. \tag{13}$$

Indeed, fix *<sup>η</sup>* <sup>&</sup>lt; *<sup>σ</sup>*. For instance, let *<sup>σ</sup>* <sup>=</sup> lim inf*ξ*→0<sup>+</sup> *<sup>G</sup>*(*ξ*) *<sup>ξ</sup>*<sup>2</sup> . Then, there exists *<sup>η</sup>* > 0 such that

$$G(\xi) \ge \eta \xi^2 \tag{14}$$

for all *<sup>ξ</sup>* <sup>∈</sup> [0, *<sup>η</sup>*]. Fix any *<sup>v</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω) such that *v*<sup>2</sup> *<sup>X</sup>* <sup>=</sup> *<sup>λ</sup>*1*v*<sup>2</sup> *<sup>Y</sup>* and *v*(Ω) ⊆ [0, *η*]. From (14) we obtain

$$
\psi(v) \ge \eta \|v\|\_Y^2
$$

and so

$$\sup\_{\mu \in X \backslash \{0\}} \frac{\psi(\mu)}{\|\mu\|\_X^2} \ge \frac{\psi(v)}{\|v\|\_X^2} \ge \frac{\eta}{\lambda\_1} \,. \tag{15}$$

Now, (13) is obtained from (15) passing to the limit for *η* tending to *σ*. Now, fix *λ* ∈ *λ*1 <sup>2</sup>*<sup>σ</sup>* , *<sup>λ</sup>*<sup>1</sup> 2 max{*ρ*,0} . Then, from (10) and (13), we obtain

$$\limsup\_{||u||\_X \to +\infty} \frac{\psi(u)}{I(u)} < \frac{1}{\lambda} < \sup\_{u \in X \backslash \{0\}} \frac{\psi(u)}{I(u)} \cdot \lambda$$

This readily implies that *θ*<sup>∗</sup> < *λ* < ˜ *θ* and the conclusion is directly provided by Theorem 1. **Corollary 1.** *Let the assumptions of Theorem 3 be satisfied and let λ* ∈ *λ*1 <sup>2</sup>*<sup>σ</sup>* , *<sup>λ</sup>*<sup>1</sup> 2 max{*ρ*,0} *satisfy*

$$\sup\_{\xi \in \mathbf{R}} (\lambda g(\xi) - F(\xi)f(\xi))\xi \le 0 \,. \tag{16}$$

*Then, for every convex set S* <sup>⊆</sup> *<sup>L</sup>*∞(Ω) *dense in L*2(Ω)*, there exists <sup>α</sup>* <sup>∈</sup> *S such that the problem*

$$\begin{cases} -\Delta u = \alpha(\mathfrak{x})f(u) + \lambda \mathfrak{g}(u) & \text{in } \Omega\\ u = 0 & \text{on } \partial\Omega \end{cases}$$

*has at least three weak solutions, two of which are global minima in H*<sup>1</sup> <sup>0</sup> (Ω) *of the functional*

$$\mu \to \frac{1}{2} \int\_{\Omega} |\nabla u(\mathbf{x})|^2 d\mathbf{x} - \int\_{\Omega} a(\mathbf{x}) F(u(\mathbf{x})) d\mathbf{x} - \lambda \int\_{\Omega} G(u(\mathbf{x})) d\mathbf{x} \dots$$

**Proof.** It suffices to observe that, in view of (16), 0 is the only solution of (8) and then to apply Theorem 3.

Finally, notice the following remarkable corollary of Corollary 1:

**Corollary 2.** *Let <sup>q</sup>* <sup>&</sup>gt; <sup>1</sup>*, with <sup>q</sup>* <sup>&</sup>lt; *<sup>n</sup>*+<sup>2</sup> *<sup>n</sup>*−<sup>2</sup> *if <sup>n</sup>* <sup>≥</sup> <sup>3</sup>*. Let <sup>h</sup>* : **<sup>R</sup>** <sup>→</sup> **<sup>R</sup>** *be a non-negative continuous function, with* inf[0,1] *h* > 0*, satisfying conditions* (*a*) *and* (*b*) *of Theorem 3 for f* = *h.*

*Then, for every <sup>λ</sup>* <sup>&</sup>gt; *<sup>λ</sup>*<sup>1</sup> *and for every convex set <sup>S</sup>* <sup>⊆</sup> *<sup>L</sup>*∞(Ω) *dense in <sup>L</sup>*2(Ω)*, there exists <sup>α</sup>* <sup>∈</sup> *<sup>S</sup> such that the problem*

$$\begin{cases} -\Delta u = \mathfrak{a}(\mathfrak{x})h(\mathfrak{u}) + \lambda(\mathfrak{u}^+ - (\mathfrak{u}^+)^q) & \text{in } \Omega\\\ u = 0 & \text{on } \partial\Omega \end{cases}$$

*has at least three weak solutions, two of which are global minima in H*<sup>1</sup> <sup>0</sup> (Ω) *of the functional*

$$\mathcal{U}u \to \frac{1}{2} \int\_{\Omega} |\nabla u(\mathbf{x})|^2 d\mathbf{x} - \int\_{\Omega} u(\mathbf{x}) H(u(\mathbf{x})) d\mathbf{x} - \lambda \int\_{\Omega} \left(\frac{1}{2} |u^+(\mathbf{x})|^2 - \frac{1}{q+1} |u^+(\mathbf{x})|^{q+1}\right) d\mathbf{x} \dots$$

**Proof.** Fix *λ* > *λ*1. Notice that, since inf[0,1] *h* > 0, the number

$$\gamma := \inf\_{\xi \in [0,1]} \frac{H(\xi)h(\xi)}{\xi}.$$

is positive. Now, we are going to apply Corollary 1 taking

$$f(\vec{\xi}) = \sqrt{\frac{\lambda}{\gamma}} h(\vec{\xi})$$

and

$$\lg(\xi) = \xi^+ - (\xi^+)^q \cdot$$

Of course (with the notations of Theorem 3), *ρ* = 0 and *σ* = <sup>1</sup> <sup>2</sup> . Since *f* in non-negative, *F f* is so in [0, +∞[ and non-positive in ] − ∞, 0]. Therefore, (16) is satisfied for all *ξ* ∈ **R** \ [0, 1] since *g* has the opposite sign of *F f* in that set. Now, let *ξ* ∈]0, 1]. We have

$$\frac{F(\xi^{\chi})f(\xi^{\chi})}{\xi} = \frac{\lambda}{\gamma} \frac{H(\xi^{\chi})h(\xi)}{\xi} \ge \lambda \ge \lambda (1 - \xi^{q-1})$$

which gives (16). Now, let *<sup>S</sup>* <sup>⊆</sup> *<sup>L</sup>*∞(Ω) be any convex set dense in *<sup>L</sup>*2(Ω). Then, the set \$*<sup>γ</sup> <sup>λ</sup> S* is convex and dense in *L*2(Ω) and the conclusion follows applying Corollary 1 with this set.

**Remark 1.** *We are not aware of known results close enough to Theorems 1 and 3 in order to do a proper comparison. We refer to the monographs [9,10] for an account on multiplicity results for nonlinear PDEs.*

**Funding:** This research received no external funding.

**Acknowledgments:** The author has been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by the Università degli Studi di Catania, "Piano della Ricerca 2016/2018 Linea di intervento 2".

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Correction* **Correction: Ricceri, B. A Class of Equations with Three Solutions.** *Mathematics* **2020,** *8***, 478**

#### **Biagio Ricceri**

Department of Mathematics and Informatics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy; ricceri@dmi.unict.it

Received: 15 October 2020; Accepted: 22 December 2020; Published: 5 January 2021

The author wishes to make the following correction to this paper [1]:

Everywhere it occurs, the phrase "for every convex set *<sup>S</sup>* <sup>⊆</sup> *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω) dense in *<sup>H</sup>*<sup>1</sup> <sup>0</sup> (Ω)" should be replaced with "for every convex set *<sup>S</sup>* <sup>⊆</sup> *<sup>L</sup>*∞(Ω) dense in *<sup>L</sup>*2(Ω)".

Actually, thanks to (*b*) of Theorem 2, condition (1) can be weakened to

$$\lim\_{\|\mathbf{x}\|\_{X}\to+\infty} \frac{\langle \boldsymbol{\varrho}(\mathbf{x}), \boldsymbol{y} \rangle\_{Y}}{I(\mathbf{x})} = 0 \tag{1}$$

for all *y* in a convex and dense set *V* ⊆ *Y*. Then, in the conclusion of Theorem 1, we can replace "*S* ⊆ *Y*" with "*<sup>S</sup>* <sup>⊆</sup> *<sup>V</sup>*". Finally, in the proof of Theorem 3, we take *<sup>V</sup>* <sup>=</sup> *<sup>L</sup>*∞(Ω), so that condition (*a*) is actually enough to prove equality (1).

The author would like to apologize for any inconvenience caused to the readers by these changes. The changes do not affect the scientific results. The original article has been updated.

**Conflicts of Interest:** The author declare no conflict of interest.

#### **Reference**

1. Ricceri, B. A Class of Equations with Three Solutions. *Mathematics* **2020**, *8*, 478. [CrossRef]

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