*Article* **Positive Solutions for Discontinuous Systems via a Multivalued Vector Version of Krasnosel'ski˘ı's Fixed Point Theorem in Cones**

#### **Rodrigo López Pouso 1, Radu Precup 2,\* and Jorge Rodríguez-López <sup>1</sup>**


Received: 12 February 2019; Accepted: 14 May 2019; Published: 20 May 2019

**Abstract:** We establish the existence of positive solutions for systems of second–order differential equations with discontinuous nonlinear terms. To this aim, we give a multivalued vector version of Krasnosel'ski˘ı's fixed point theorem in cones which we apply to a regularization of the discontinuous integral operator associated to the differential system. We include several examples to illustrate our theory.

**Keywords:** Krasnosel'ski˘ı's fixed point theorem; positive solutions; discontinuous differential equations; differential system

#### **1. Introduction**

We study the existence and localization of positive solutions for the system

$$\begin{cases} \boldsymbol{u}\_1^{\prime\prime}(t) + \mathbf{g}\_1(t)f\_1(t,\boldsymbol{u}\_1(t),\boldsymbol{u}\_2(t)) = 0\_\prime\\ \boldsymbol{u}\_2^{\prime\prime}(t) + \mathbf{g}\_2(t)f\_2(t,\boldsymbol{u}\_1(t),\boldsymbol{u}\_2(t)) = 0\_\prime \end{cases}$$

subject to the Sturm–Liouville boundary conditions (7).

The novelties in this paper are in two directions. On the one hand, we allow the functions *fi* (*i* = 1, 2) to be discontinuous with respect to the unknown over some time-dependent sets, see Definitions 1 and 2. On the other hand, in order to localize the solutions of the system, we shall establish a multivalued vector version of Krasnosel'ski˘ı's fixed point theorem which allows different asymptotic behaviors in the nonlinearities *f*<sup>1</sup> and *f*2, see Remark 3.

The existence of discontinuities in the functions *f*<sup>1</sup> or *f*<sup>2</sup> makes impossible to apply directly the standard fixed point theorems in cones for compact operators since the integral operator corresponding to the differential problem is not necessarily continuous. In order to avoid this difficulty, we regularize the possibly discontinuous operator obtaining an upper semicontinuous multivalued one. Then we look for fixed points of this multivalued mapping that are proved to be Carathéodory solutions for the differential system. In the case of scalar problems, similar ideas appear in the papers [1–3].

This approach of using set-valued analysis in the study of discontinuous problems is a classical one, see [4]. Nevertheless, the regularization is usually made in the nonlinearities transforming the problem into a differential inclusion and the solutions are often given in the sense of the set-valued analysis (Krasovskij and Filippov solutions [5,6]), see e.g., [7,8]. Similar ideas are also used in the papers [5,9] where there are provided some sufficient conditions for the Krasovskij solutions to be Carathéodory solutions. Recently, second-order scalar discontinuous problems have been investigated by using variational methods [10–12]. However, in these papers there are not considered time-dependent discontinuity sets. Observe also that a lot of existence results for discontinuous differential problems are based on monotonicity hypotheses on their nonlinear parts, see [13], but such assumptions are not necessary in our approach.

Going from scalar discontinuous problems to systems of discontinuous equations is not trivial and it makes possible to consider two different notions for the discontinuity sets. The first approach (see Definition 1 and Theorem 3) allows to study the discontinuities in each variable independently. For instance, it guarantees the existence of a positive solution for the following particular system

$$\begin{cases} -\mathbf{x}^{\prime\prime}(t) = \mathbf{x}^2 + \mathbf{x}^2 y^2 H(1-\mathbf{x}) H(1-y) \iota \\ -y^{\prime\prime}(t) = \sqrt{\mathbf{x}} + \sqrt{\mathbf{y}} + H(\mathbf{x}-1) H(y-1) \iota \end{cases}$$

subject to the Sturm–Liouville boundary conditions, where *<sup>H</sup>* : <sup>R</sup> <sup>→</sup> <sup>R</sup> is the Heaviside step function given by

$$H(\mathbf{x}) = \begin{cases} \ 0, & \text{if } \mathbf{x} \le 0, \\\ 1, & \text{if } \mathbf{x} > 0, \end{cases}$$

see Example 1. Notice that the nonlinearities in this example are discontinuous at *x* = 1 for each *<sup>y</sup>* <sup>∈</sup> <sup>R</sup><sup>+</sup> and at *<sup>y</sup>* <sup>=</sup> 1 for every *<sup>x</sup>* <sup>∈</sup> <sup>R</sup>+. Moreover, the first nonlinearity has a superlinear behavior and the second one has a sublinear one. Our second approach allows to study functions which are discontinuous over time-dependent curves in R<sup>2</sup> <sup>+</sup> and the conditions imposed to these curves are local, see Definition 2 and Theorem 4. In particular, we establish the existence of a positive solution for the system

$$\begin{cases} -\mathfrak{x}^{\prime\prime}(t) = (\mathfrak{x}y)^{1/3}\prime \\ -\mathfrak{y}^{\prime\prime}(t) = \left(1 + (\mathfrak{x}y)^{1/3}\right)H(\mathfrak{x}^2 + \mathfrak{y}^2)\prime \end{cases}$$

subject to the Sturm–Liouville boundary conditions.

As mentioned above, our results rely on fixed point theory for multivalued operators in cones. We finish this introductory part by recalling the version of Krasnosel'ski˘ı's fixed point theorem for set-valued maps given by Fitzpatrick–Petryshyn [14].

**Theorem 1.** *Let X be a Fréchet space with a cone K* ⊂ *X. Let d be a metric on X and let r*1,*r*<sup>2</sup> ∈ (0, ∞)*, <sup>r</sup>* <sup>=</sup> min {*r*1,*r*2}*, <sup>R</sup>* <sup>=</sup> max {*r*1,*r*2} *and <sup>F</sup>* : *BR*(0) <sup>∩</sup> *<sup>K</sup>* −→ <sup>2</sup>*<sup>K</sup> usc and condensing. Suppose there exists a continuous seminorm p such that* (*I* − *F*) *Br*<sup>1</sup> (0) ∩ *K is p-bounded. Moreover, suppose that F satisfies:*


*Then F has a fixed point x*<sup>0</sup> *with r* ≤ *d*(*x*0, 0) ≤ *R.*

In the case of a Banach space (*X*, ·*X*) and of an operator *<sup>F</sup>* = (*F*1, *<sup>F</sup>*2) : *<sup>K</sup>* <sup>⊂</sup> *<sup>X</sup>*<sup>2</sup> <sup>→</sup> <sup>2</sup>*<sup>K</sup>* under the hypotheses of the previous theorem, we obtain the existence of a fixed point *x* = (*x*1, *x*2) for *F* such that *<sup>r</sup>* <sup>≤</sup> *x* <sup>≤</sup> *<sup>R</sup>*, where · denotes a norm in *<sup>X</sup>*2, for example, (*x*1, *<sup>x</sup>*2) <sup>=</sup> *x*1*<sup>X</sup>* <sup>+</sup> *x*2*X*. Then 0 ≤ *x*1*<sup>X</sup>* ≤ *R* and 0 ≤ *x*2*<sup>X</sup>* ≤ *R*, but it is not possible to obtain a lower bound for the norm of every component. This fact motivates the use of a vector version of Krasnosel'ski˘ı's fixed point theorem. Such a version was introduced in [15] for single-valued operators. Another advantage of the vector approach is that it allows different behaviors in each component of the system.

#### **2. Multivalued Vector Version of Krasnosel'ski˘ı's Fixed Point Theorem**

In the sequel, let (*X*, ·) be a Banach space, *K*1, *K*<sup>2</sup> ⊂ *X* two cones and *K* := *K*<sup>1</sup> × *K*<sup>2</sup> the corresponding cone of *<sup>X</sup>*<sup>2</sup> <sup>=</sup> *<sup>X</sup>* <sup>×</sup> *<sup>X</sup>*. For *<sup>r</sup>*, *<sup>R</sup>* <sup>∈</sup> <sup>R</sup><sup>2</sup> <sup>+</sup>, *r* = (*r*1,*r*2), *R* = (*R*1, *R*2), we denote

$$\begin{aligned} (\mathbb{K}\_i)\_{r\_i, \mathbb{R}\_i} &:= \{ u \in \mathbb{K}\_i \,:\, r\_i \le \|u\| \le \mathbb{R}\_i \} \quad (i = 1, 2), \\ \mathbb{K}\_{r, \mathbb{R}} &:= \{ u \in \mathbb{K} \,:\, r\_i \le \|u\_i\| \le \mathbb{R}\_i \text{ for } i = 1, 2 \} . \end{aligned}$$

The following fixed point theorem is an extension of the vector version of Krasnosel'ski˘ı's fixed point theorem given in [15,16] to the class of upper semicontinuous (usc, for short) multivalued mappings.

**Theorem 2.** *Let αi*, *β<sup>i</sup>* > 0 *with α<sup>i</sup>* = *βi, ri* = min{*αi*, *βi*} *and Ri* = max {*αi*, *βi*} *for i* = 1, 2*. Assume that <sup>N</sup>* : *Kr*,*<sup>R</sup>* <sup>→</sup> <sup>2</sup>*K, <sup>N</sup>* = (*N*1, *<sup>N</sup>*2)*, is an usc map with nonempty closed and convex values such that <sup>N</sup>*(*Kr*,*R*) *is compact, and there exist hi* ∈ *Ki* \ {0}*, i* = 1, 2*, such that for each i* ∈ {1, 2} *the following conditions are satisfied:*

$$
\lambda \mu\_i \notin \mathcal{N}\_i \mu \quad \text{for any } \mathfrak{u} \in \mathcal{K}\_{r, \mathbb{R}} \text{ with } ||\mathfrak{u}\_i|| = a\_i \text{ and } \arg \lambda > 1; \tag{1}
$$

$$
\mu\_i \notin \mathcal{N}\_i \mathfrak{u} + \mu h\_i \quad \text{for any } \mathfrak{u} \in \mathcal{K}\_{r, \mathbb{R}} \text{ with } ||\mathfrak{u}\_i|| = \beta\_i \text{ and any } \mu > 0. \tag{2}
$$

*Then N has a fixed point u* = (*u*1, *u*2) *in K, that is, u* ∈ *Nu, with ri* ≤ *ui* ≤ *Ri for i* = 1, 2*.*

**Proof.** We shall consider the four possible combinations of compression-expansion conditions for *N*<sup>1</sup> and *N*2.

1. Assume first that *β<sup>i</sup>* < *α<sup>i</sup>* for both *i* = 1, 2 (compression for *N*<sup>1</sup> and *N*2). Then *ri* = *β<sup>i</sup>* and *Ri* = *α<sup>i</sup>* for *<sup>i</sup>* <sup>=</sup> 1, 2. Denote *<sup>h</sup>* = (*h*1, *<sup>h</sup>*2) and define the map *<sup>N</sup>*˜ : *<sup>K</sup>* <sup>→</sup> *<sup>K</sup>* given, for *<sup>u</sup>* <sup>∈</sup> *<sup>K</sup>*, by

$$N\hat{u} = \min\left\{\frac{\|u\_1\|}{r\_1}, \frac{\|u\_2\|}{r\_2}, 1\right\} N\left(\delta\_1(u\_1) \frac{u\_1}{\|u\_1\|}, \delta\_2(u\_2) \frac{u\_2}{\|u\_2\|}\right) + \left(1 - \min\left\{\frac{\|u\_1\|}{r\_1}, \frac{\|u\_2\|}{r\_2}, 1\right\}\right) h\_r$$

where *δi*(*ui*) = max{min{*ui*, *Ri*},*ri*} for *i* = 1, 2.

The map *N*˜ is usc (the composition of usc maps is usc, see [17], Theorem 17.23) and *N*˜ (*K*) is relatively compact since its values belong to the compact set co (*N*(*Kr*,*R*) ∪ {*h*}). Then Kakutani's fixed point theorem implies that there exists *<sup>u</sup>* <sup>∈</sup> *<sup>K</sup>* such that *<sup>u</sup>* <sup>∈</sup> *Nu*˜ .

It remains to prove that *u* ∈ *Kr*,*R*. It is clear that *ui* > 0 since *hi* = 0 for *i* = 1, 2. Assume 0 < *u*1 <sup>&</sup>lt; *<sup>r</sup>*<sup>1</sup> and 0 <sup>&</sup>lt; *u*2 <sup>&</sup>lt; *<sup>r</sup>*2. If min !*u*1 *<sup>r</sup>*<sup>1</sup> , *u*2 *r*2 " = *u*1 *<sup>r</sup>*<sup>1</sup> , then

$$\mu \in \frac{||\boldsymbol{u}\_1||}{r\_1} \mathcal{N}\left(\frac{r\_1}{||\boldsymbol{u}\_1||} \boldsymbol{u}\_1, \frac{r\_2}{||\boldsymbol{u}\_2||} \boldsymbol{u}\_2\right) + \left(1 - \frac{||\boldsymbol{u}\_1||}{r\_1}\right) \boldsymbol{h}\_r$$

so 
$$\frac{r\_1}{||u\_1||}u\_1 \in N\_1 \left(\frac{r\_1}{||u\_1||}u\_{1'}\frac{r\_2}{||u\_2||}u\_2\right) + \frac{r\_1}{||u\_1||}\left(1 - \frac{||u\_1||}{r\_1}\right)h\_{1'}$$

what contradicts (2) for *i* = 1. Analogously, we can obtain contradictions for any other point *u* ∈ *Kr*,*R*, as done in [15,16] for single-valued maps.

2. Assume that *β*<sup>1</sup> < *α*<sup>1</sup> (compression for *N*1) and *β*<sup>2</sup> > *α*<sup>2</sup> (expansion for *N*2). Let *N*<sup>∗</sup> *<sup>i</sup>* : *Kr*,*<sup>R</sup>* → *Ki* (*i* = 1, 2) be given by

$$\begin{aligned} N\_1^\* u &= N\_1 \left( u\_{1\prime} \left( \frac{R\_2}{||u\_2||} + \frac{r\_2}{||u\_2||} - 1 \right) u\_2 \right), \\ N\_2^\* u &= \left( \frac{R\_2}{||u\_2||} + \frac{r\_2}{||u\_2||} - 1 \right)^{-1} N\_2 \left( u\_{1\prime} \left( \frac{R\_2}{||u\_2||} + \frac{r\_2}{||u\_2||} - 1 \right) u\_2 \right). \end{aligned} \tag{3}$$

Notice that the map *N*∗ = (*N*∗ <sup>1</sup> , *N*<sup>∗</sup> <sup>2</sup> ) is in case 1, and thus *N*<sup>∗</sup> has a fixed point *v* ∈ *Kr*,*R*. Further, the point *u* defined as *u*<sup>1</sup> = *v*<sup>1</sup> and *u*<sup>2</sup> = *<sup>R</sup>*<sup>2</sup> *v*2 <sup>+</sup> *<sup>r</sup>*<sup>2</sup> *v*2 <sup>−</sup> <sup>1</sup> *v*<sup>2</sup> is a fixed point of the operator *N*.

3. The case *β*<sup>1</sup> > *α*<sup>1</sup> (expansion for *N*1) and *β*<sup>2</sup> < *α*<sup>2</sup> (compression for *N*2) is similar to the previous one by taking the map *N*∗ = (*N*∗ <sup>1</sup> , *N*<sup>∗</sup> <sup>2</sup> ) defined as

$$\begin{split} N\_1^\* u &= \left( \frac{R\_1}{||u\_1||} + \frac{r\_1}{||u\_1||} - 1 \right)^{-1} N\_1 \left( \left( \frac{R\_1}{||u\_1||} + \frac{r\_1}{||u\_1||} - 1 \right) u\_1, u\_2 \right), \\ N\_2^\* u &= N\_2 \left( \left( \frac{R\_1}{||u\_1||} + \frac{r\_1}{||u\_1||} - 1 \right) u\_1, u\_2 \right). \end{split} \tag{4}$$

4. The case *β<sup>i</sup>* > *α<sup>i</sup>* for *i* = 1, 2 (expansion for *N*<sup>1</sup> and *N*2) reduces to case 1, if we consider the map *N*∗ = (*N*∗ <sup>1</sup> , *N*<sup>∗</sup> <sup>2</sup> ) where *N*<sup>∗</sup> <sup>1</sup> is defined by (4) and *N*<sup>∗</sup> <sup>2</sup> , by (3).

Therefore, the proof is over.

**Remark 1** (Multiplicity)**.** *Although we are interested in fixed points for the operator N satisfying that both components are nonzero, if we replace conditions* (1) *and* (2) *in Theorem 2 by the following ones:*

$$\begin{aligned} \lambda u\_i \notin \mathcal{N}\_i u &\quad \text{for } ||u\_i|| = \mathfrak{a}\_{i\prime} \; ||u\_j|| \le \mathcal{R}\_j \; (j \ne i) \; \text{and} \; \lambda \ge 1;\\ u\_i \notin \mathcal{N}\_i u + \mu h\_i &\quad \text{for } ||u\_i|| = \mathcal{R}\_i \; ||u\_j|| \le \mathcal{R}\_j \; (j \ne i) \; \text{and} \; \mu \ge 0. \end{aligned}$$

*then we can achieve multiplicity results.*

*Indeed, if β<sup>i</sup>* > *α<sup>i</sup> for i* = 1 *or i* = 2*, then the operator N has one additional fixed point v* = (*v*1, *v*2) *such that vi* <sup>&</sup>lt; *ri and rj* <sup>&</sup>lt; % %*vj* % % < *Rj with j* = *i. Furthermore, if β<sup>i</sup>* > *α<sup>i</sup> for i* = 1, 2*, then N has three nontrivial fixed points. Such cases are considered in the paper [18] in connection with* (*p*, *q*)*-Laplacian systems.*

Our purpose is to apply Theorem 2 to a multivalued regularization of a discontinuous system of single-valued operators associated to a system of differential equations with discontinuous nonlinearities. Our aim is to obtain new existence and localization results for such kind of problems.

In order to do that, we need the following definitions and results.

Let *U* be a relatively open subset of the cone *K* := *K*<sup>1</sup> × *K*<sup>2</sup> and *T* : *U* → *K*, *T* = (*T*1, *T*2), an operator not necessarily continuous. We associate to the operator *T* the following multivalued map <sup>T</sup> : *<sup>U</sup>* <sup>→</sup> <sup>2</sup>*<sup>K</sup>* given by

$$\mathbb{T} = (\mathbb{T}\_1, \mathbb{T}\_2), \quad \mathbb{T}\_i u = \bigcap\_{\iota > 0} \overline{\infty} \, T\_i \left( \overline{\mathbb{B}}\_{\iota}(u) \cap \overline{\mathbb{U}} \right) \quad \text{for every} \; u \in \overline{\mathbb{U}} \quad (i = 1, 2), \tag{5}$$

where *Bε*(*u*) := *<sup>v</sup>* <sup>∈</sup> *<sup>X</sup>*<sup>2</sup> : *ui* <sup>−</sup> *vi* <sup>≤</sup> *<sup>ε</sup>* for *<sup>i</sup>* <sup>=</sup> 1, 2 , *U* denotes the closure of the set *U* with the relative topology of *<sup>K</sup>* and co means closed convex hull. The map T*<sup>i</sup>* is called the closed-convex envelope of *Ti* and it satisfies the following properties, see [2].

**Proposition 1.** *Let* <sup>T</sup> *be the closed-convex envelope of an operator <sup>T</sup>* : *<sup>U</sup>* −→ *K. The following properties are satisfied:*

*1. If T maps bounded sets into relatively compact sets, then* T *assumes compact values and it is usc;*

*2. If T U is relatively compact, then* T *U is relatively compact too.*

**Remark 2.** *The following two statements are equivalent:*


$$\left\| y - \sum\_{j=1}^{m} \lambda\_j |T\_i x\_j| \right\| < \rho.$$

#### **3. Positive Solutions of Discontinuous Systems**

We study the existence and localization of positive solutions for the following second-order coupled differential system

$$\begin{cases} \boldsymbol{u}\_1^{\prime\prime}(t) + \mathcal{g}\_1(t) f\_1(t, \boldsymbol{u}\_1(t), \boldsymbol{u}\_2(t)) = 0, \\\boldsymbol{u}\_2^{\prime\prime}(t) + \mathcal{g}\_2(t) f\_2(t, \boldsymbol{u}\_1(t), \boldsymbol{u}\_2(t)) = 0, \end{cases} \tag{6}$$

for *t* ∈ *I* = [0, 1], with the following boundary conditions

$$a\_i u\_i(0) - b\_i u\_i'(0) = 0, \quad c\_i u\_i(1) + d\_i u\_i'(1) = 0,\tag{7}$$

for *<sup>i</sup>* <sup>=</sup> 1, 2, where *ai*, *bi*, *ci*, *di* <sup>∈</sup> <sup>R</sup><sup>+</sup> <sup>≡</sup> [0, <sup>∞</sup>) and *<sup>ρ</sup><sup>i</sup>* :<sup>=</sup> *bici* <sup>+</sup> *aici* <sup>+</sup> *aidi* <sup>&</sup>gt; 0 for *<sup>i</sup>* <sup>=</sup> 1, 2. Assume that, for *i* = 1, 2,

(*H*1) *gi* <sup>∈</sup> *<sup>L</sup>*1(*I*), *gi*(*t*) <sup>≥</sup> 0 for a.e. *<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>* and - 3/4 1/4 *g*(*s*) *ds* > 0; (*H*2) *fi* : *<sup>I</sup>* <sup>×</sup> <sup>R</sup><sup>2</sup> <sup>+</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> satisfies that


$$f\_i(t, \mu\_1, \mu\_2) \le R\_{i,r} \quad \text{for } \mu\_1, \mu\_2 \in [0, \rho] \text{ and a.e. } t \in I.$$

Notice that condition (*H*2) (*i*) is satisfied if *fi*(·, *u*1, *u*2) is measurable for all constants *u*1, *u*2, and if *fi*(*t*, ·, ·) is continuous for a.a. *t*, which is not necessarily the case in this paper.

Let *X* = C(*I*) be the space of continuous functions defined on *I* endowed with the usual norm *v* := *v*<sup>∞</sup> = max*t*∈*<sup>I</sup>* |*v*(*t*)| and let *P* be the cone of all nonnegative functions of *X*. A positive solution to (6)–(7) is a function *<sup>u</sup>* = (*u*1, *<sup>u</sup>*2) with *ui* <sup>∈</sup> *<sup>P</sup>* <sup>∩</sup> *<sup>W</sup>*2,1(*I*), *ui* <sup>≡</sup> 0 (*<sup>i</sup>* <sup>=</sup> 1, 2) such that *<sup>u</sup>* satisfies (6) for a.a. *t* ∈ *I* and the boundary conditions (7). The existence of positive solutions to problems (6)–(7) is equivalent to the existence of fixed points of the integral operator *<sup>T</sup>* : *<sup>P</sup>*<sup>2</sup> <sup>→</sup> *<sup>P</sup>*2, *<sup>T</sup>* = (*T*1, *<sup>T</sup>*2), given by

$$(T\_i u)(t) = \int\_0^1 G\_i(t, s) g\_i(s) f\_i(s, \mu\_1(s), \mu\_2(s)) \, ds, \qquad i = 1, 2,\tag{8}$$

where *Gi*(*t*,*s*) are the corresponding Green's functions which are explicitly given by

$$G\_{l}(t,s) = \frac{1}{\rho\_{i}} \begin{cases} (c\_{i} + d\_{i} - c\_{i}t)(b\_{i} + a\_{i}s), & \text{if } 0 \le s \le t \le 1, \\\ (b\_{i} + a\_{i}t)(c\_{i} + d\_{i} - c\_{i}s), & \text{if } 0 \le t \le s \le 1. \end{cases}$$

Denote

$$M\_i := \min \left\{ \frac{c\_i + 4d\_i}{4(c\_i + d\_i)}, \frac{a\_i + 4b\_i}{4(a\_i + b\_i)} \right\},$$

then it is possible to check the following inequalities:

$$\begin{array}{rcl} \mathcal{G}\_{l}(t,s) & \leq \mathcal{G}\_{l}(s,s) \quad \text{for } t,s \in I, \\ M\_{l}\mathcal{G}\_{l}(s,s) & \leq \mathcal{G}\_{l}(t,s) \quad \text{for } t \in [1/4, 3/4], \ s \in I. \end{array}$$

Consider in *X* the cones *K*<sup>1</sup> and *K*<sup>2</sup> defined as

$$K\_i = \{ v \in P : v(t) \ge M\_i \, \| \, \|v\|\_{\infty} \text{ for all } t \in [1/4, 3/4] \} \text{ \textquotedblleft}$$

and the corresponding cone *<sup>K</sup>* :<sup>=</sup> *<sup>K</sup>*<sup>1</sup> <sup>×</sup> *<sup>K</sup>*<sup>2</sup> in *<sup>X</sup>*2. Then, *<sup>T</sup>*(*K*) <sup>⊂</sup> *<sup>K</sup>*. Indeed, for *<sup>u</sup>* <sup>∈</sup> *<sup>K</sup>* and *<sup>i</sup>* <sup>=</sup> 1, 2,

$$\begin{aligned} \|M\_i \| \| T\_i u \| &= M\_i \max\_{t \in [0, 1]} \int\_0^1 G\_i(t, s) g\_i(s) f\_i(s, u\_1(s), u\_2(s)) \, ds \\ &\le M\_i \int\_0^1 G\_i(s, s) g\_i(s) f\_i(s, u\_1(s), u\_2(s)) \, ds \le \min\_{t \in [1/4, 3/4]} T\_i u(t) \dots \end{aligned}$$

Hence, *Tiu* ∈ *Ki* for every *u* ∈ *K* and *i* = 1, 2.

Therefore, it must be clear that we intend to apply Theorem 2 in a subset of *K* to the multivalued operator T associated to the discontinuous operator *T*. Later, we shall provide conditions about the functions *fi* (*<sup>i</sup>* <sup>=</sup> 1, 2) which guarantee that Fix(T) <sup>⊂</sup> Fix(*T*), where Fix(*S*) stands for the set of fixed points of the mapping *S*. As a consequence, we obtain some results concerning the existence of positive solutions for system (6)–(7).

Let us introduce some notations. For *αi*, *β<sup>i</sup>* > 0 with *α<sup>i</sup>* = *β<sup>i</sup>* and *ε* > 0, we let *ri* = min{*αi*, *βi*}, *Ri* = max{*αi*, *βi*} (*i* = 1, 2) and

$$\begin{split} f\_1^{\mathbb{R},x} &:= \inf \{ f\_1(t,u\_1,u\_2) \; : \; t \in [1/4, 3/4], \; M\_1(\beta\_1 - \varepsilon) \le u\_1 \le \beta\_1 + \varepsilon, \; M\_2r\_2 \le u\_2 \le R\_2 \}, \\ f\_2^{\mathbb{R},x} &:= \inf \{ f\_2(t,u\_1,u\_2) \; : \; t \in [1/4, 3/4], \; M\_1r\_1 \le u\_1 \le R\_1, \; M\_2(\beta\_2 - \varepsilon) \le u\_2 \le \beta\_2 + \varepsilon \}, \\ f\_1^{u,\varepsilon} &:= \sup \{ f\_1(t,u\_1,u\_2) \; : \; t \in [0,1], \; 0 \le u\_1 \le u\_1 + \varepsilon, \; 0 \le u\_2 \le R\_2 \}, \\ f\_2^{u,\varepsilon} &:= \sup \{ f\_2(t,u\_1,u\_2) \; : \; t \in [0,1], \; 0 \le u\_1 \le R\_1, \; 0 \le u\_2 \le a\_2 + \varepsilon \}. \end{split}$$

Also, denote

$$A\_i := \inf\_{t \in [1/4, 3/4]} \int\_{1/4}^{3/4} \mathcal{G}\_i(t, s) \mathcal{g}\_i(s) \, ds, \qquad B\_i := \sup\_{t \in [0, 1]} \int\_0^1 \mathcal{G}\_i(t, s) \mathcal{g}\_i(s) \, ds$$

for *i* = 1, 2.

**Lemma 1.** *Assume that there exist αi*, *β<sup>i</sup>* > 0 *with α<sup>i</sup>* = *βi, i* = 1, 2*, and ε* > 0 *such that*

$$B\_i f\_i^{\alpha, x} < u\_i, \quad A\_i f\_i^{\beta, x} > \beta\_i \quad \text{for } i = 1, 2. \tag{9}$$

*Then, for each i* ∈ {1, 2}*, the following conditions are satisfied:*

*<sup>λ</sup>ui* <sup>∈</sup> <sup>T</sup>*iu for any u* <sup>∈</sup> *Kr*,*<sup>R</sup> with ui*<sup>∞</sup> <sup>=</sup> *<sup>α</sup><sup>i</sup> and any <sup>λ</sup>* <sup>&</sup>gt; 1; (10)

$$\|u\_i \notin \mathbb{T}\_i u + \mu\|\_i \quad \text{for any } u \in K\_{r, \mathbb{R}} \text{ with } \|u\_i\|\_{\infty} = \beta\_i \text{ and any } \mu > 0,\tag{11}$$

*where h*<sup>1</sup> *and h*<sup>2</sup> *are constant functions equal to* 1*.*

*Moreover, the map* T *defined as in (5) has at least one fixed point in Kr*,*R.*

**Proof.** First, observe that if *v* ∈ *Kr*,*R*, then

$$M\_i r\_i \le v\_i(t) \le R\_i \quad \text{for all } t \in \left[\frac{1}{4}, \frac{3}{4}\right] \quad (i = 1, 2),$$

and if *v* ∈ *Bε*(*u*) ∩ *Kr*,*<sup>R</sup>* for some *u* ∈ *Kr*,*R*, and *u*1<sup>∞</sup> = *α*1, then *v*1(*t*) ≤ *α*<sup>1</sup> + *ε* for all *t* ∈ [0, 1] and

$$M\_1(\alpha\_1 - \varepsilon) \le v\_1(t) \le \alpha\_1 + \varepsilon \quad \text{for all } t \in \left[\frac{1}{4}, \frac{3}{4}\right]$$

.

Now we prove (10) for *<sup>i</sup>* <sup>=</sup> 1. Assume that *u*1<sup>∞</sup> <sup>=</sup> *<sup>α</sup>*<sup>1</sup> and let us see that *<sup>λ</sup>u*<sup>1</sup> <sup>∈</sup> <sup>T</sup>1*<sup>u</sup>* for *<sup>λ</sup>* <sup>&</sup>gt; 1. First, we shall show that given a family of vectors *vk* ∈ *Bε*(*u*) ∩ *Kr*,*<sup>R</sup>* and numbers *λ<sup>k</sup>* ∈ [0, 1] such that ∑ *λ<sup>k</sup>* = 1 (*k* = 1, . . . , *m*), then

$$
\lambda u\_1 \neq \sum\_{k=1}^{m} \lambda\_k \, T\_1 v\_{k\prime}
$$

what implies that *<sup>λ</sup>u*<sup>1</sup> <sup>∈</sup> co *T*1 *Bε*(*u*) ∩ *Kr*,*<sup>R</sup>* . Indeed, if not, taking the supremum for *<sup>t</sup>* <sup>∈</sup> [0, 1],

$$\begin{aligned} \lambda \alpha\_1 &\le \sup\_{t \in [0,1]} \sum\_{k=1}^m \lambda\_k \int\_0^1 G\_1(t,s) \varrho\_1(s) f\_1(s, \upsilon\_{k,1}(s), \upsilon\_{k,2}(s)) \, ds \\ &\le \sum\_{k=1}^m \lambda\_k \sup\_{t \in [0,1]} \int\_0^1 G\_1(t,s) \varrho\_1(s) f\_1(s, \upsilon\_{k,1}(s), \upsilon\_{k,2}(s)) \, ds \\ &\le \sum\_{k=1}^m \lambda\_k f\_1^{\alpha, \varepsilon} B\_1 = f\_1^{\alpha, \varepsilon} B\_1 < \alpha\_{1\prime} \end{aligned}$$

a contradiction. Notice that if *<sup>λ</sup>u*<sup>1</sup> <sup>∈</sup> co *T*1 *Bε*(*u*) ∩ *Kr*,*<sup>R</sup>* , then it is the limit of a sequence of functions satisfying the previous inequality and thus, as a limit, it satisfies *λ α*<sup>1</sup> ≤ *α*<sup>1</sup> which is also a contradiction since *<sup>λ</sup>* <sup>&</sup>gt; 1. Therefore, *<sup>λ</sup>u*<sup>1</sup> <sup>∈</sup> <sup>T</sup>1*<sup>u</sup>* for *<sup>λ</sup>* <sup>&</sup>gt; 1.

In order to prove (11) for *<sup>i</sup>* <sup>=</sup> 1, assume that *u*1<sup>∞</sup> <sup>=</sup> *<sup>β</sup>*<sup>1</sup> and *<sup>u</sup>*<sup>1</sup> <sup>=</sup> <sup>∑</sup>*<sup>m</sup> <sup>k</sup>*=<sup>1</sup> *λ<sup>k</sup> T*1*vk* + *μ* for some family of vectors *vk* ∈ *Bε*(*u*) ∩ *Kr*,*<sup>R</sup>* and numbers *λ<sup>k</sup>* ∈ [0, 1] such that ∑ *λ<sup>k</sup>* = 1 (*k* = 1, ... , *m*) and some *μ* > 0. Then for *t* ∈ [1/4, 3/4], we have

$$\begin{aligned} \mu\_1(t) &= \sum\_{k=1}^m \lambda\_k \int\_0^1 G\_1(t,s) \mathcal{g}\_1(s) f\_1(s, v\_{k,1}(s), v\_{k,2}(s)) \, ds + \mu \\ &\ge \sum\_{k=1}^m \lambda\_k \int\_{1/4}^{3/4} G\_1(t,s) \mathcal{g}\_1(s) f\_1(s, v\_{k,1}(s), v\_{k,2}(s)) \, ds + \mu \\ &\ge \sum\_{k=1}^m \lambda\_k f\_1^{\otimes \varepsilon} \int\_{1/4}^{3/4} G\_1(t,s) \mathcal{g}\_1(s) \, ds + \mu \\ &\ge f\_1^{\otimes \varepsilon} A\_1 + \mu > \beta\_1 + \mu, \end{aligned}$$

so *<sup>β</sup>*<sup>1</sup> <sup>&</sup>gt; *<sup>β</sup>*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>*, a contradiction. Hence, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> co *T*1 *Bε*(*u*) ∩ *Kr*,*<sup>R</sup>* + *μh*1. As before,

$$
\mu\_1 \notin \overline{\text{co}}\left(T\_1\left(\overline{\mathcal{B}}\_\varepsilon(\mu) \cap \mathcal{K}\_{r,\mathbb{R}}\right)\right) + \mu h\_1,
$$

because in that case we arrive to the inequality *<sup>β</sup>*<sup>1</sup> <sup>≥</sup> *<sup>β</sup>*<sup>1</sup> <sup>+</sup> *<sup>μ</sup>* for *<sup>μ</sup>* <sup>&</sup>gt; 0. Therefore, *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> <sup>T</sup>1(*u*) + *<sup>μ</sup>h*1.

Similarly, it is possible to prove conditions (10) and (11) for *i* = 2.

To finish, the conclusion is obtained by applying Theorem 2 to the operator T.

**Remark 3** (Asymptotic conditions)**.** *The existence of αi*, *β<sup>i</sup>* > 0 *with α<sup>i</sup>* = *βi, i* = 1, 2*, and ε* > 0 *satisfying* (9) *is guaranteed, in the autonomous case, by the following sufficient conditions:*

(*a*) *f*<sup>1</sup> *has a superlinear behavior and f*2*, a sublinear one, that is,*

$$\begin{aligned} \lim\_{x \to \infty} \frac{f\_1(\mathbf{x}, y)}{\mathbf{x}} &= +\infty & \text{for all } y > 0, & \lim\_{x \to 0} \frac{f\_1(\mathbf{x}, y)}{\mathbf{x}} &= 0 & \text{for all } y \ge 0;\\ \lim\_{y \to \infty} \frac{f\_2(\mathbf{x}, y)}{y} &= 0 & \text{for all } \mathbf{x} \ge 0, & \lim\_{y \to 0} \frac{f\_2(\mathbf{x}, y)}{y} &= +\infty & \text{for all } \mathbf{x} > 0. \end{aligned}$$

(*b*) *Both f*<sup>1</sup> *and f*<sup>2</sup> *have a superlinear behavior, that is,*

$$\begin{aligned} \lim\_{x \to \infty} \frac{f\_1(\mathbf{x}, y)}{\mathbf{x}} &= +\infty & \text{for all } y > 0, & \lim\_{x \to 0} \frac{f\_1(\mathbf{x}, y)}{\mathbf{x}} &= 0 & \text{for all } y \ge 0;\\ \lim\_{y \to \infty} \frac{f\_2(\mathbf{x}, y)}{y} &= +\infty & \text{for all } \mathbf{x} > 0, & \lim\_{y \to 0} \frac{f\_2(\mathbf{x}, y)}{y} &= 0 & \text{for all } \mathbf{x} \ge 0. \end{aligned}$$

(*c*) *Both f*<sup>1</sup> *and f*<sup>2</sup> *have a sublinear behavior, that is,*

$$\begin{aligned} \lim\_{x \to \infty} \frac{f\_1(x, y)}{x} &= 0 & \text{for all } y \ge 0, \quad \lim\_{x \to 0} \frac{f\_1(x, y)}{x} &= +\infty & \text{for all } y > 0;\\ \lim\_{y \to \infty} \frac{f\_2(x, y)}{y} &= 0 & \text{for all } x \ge 0, \quad \lim\_{y \to 0} \frac{f\_2(x, y)}{y} &= +\infty & \text{for all } x > 0. \end{aligned}$$

**Remark 4.** *If f*<sup>1</sup> *and f*<sup>2</sup> *are monotone in both variables, it is possible to specify the numbers f α*,*ε <sup>i</sup> and f β*,*ε i (i* = 1, 2*), so in this case, conditions* (9) *only depend on the behavior of the functions at four points in* R<sup>2</sup> +*, see [15,16].*

Note that Lemma 1 gives us sufficient conditions for the existence of a fixed point in *Kr*,*<sup>R</sup>* of the multivalued operator T. Hence, it remains to provide hypothesis on the functions *fi* (*<sup>i</sup>* = 1, 2) which imply Fix(T) <sup>⊂</sup> Fix(*T*) in order to obtain a solution for the system (6)–(7). Observe also that no continuity hypotheses were required to the functions *fi* until now.

The following definition introduces some curves where we allow the functions *fi* to be discontinuous in each variable. The idea of using such curves can be found in some recent papers for second-order discontinuous scalar problems [1–3] and, in some sense, it recalls the notion of time-depending discontinuity sets from [9].

**Definition 1.** *We say that* <sup>Γ</sup><sup>1</sup> : [*a*1, *<sup>b</sup>*1] <sup>⊂</sup> *<sup>I</sup>* = [0, 1] <sup>→</sup> <sup>R</sup>+*,* <sup>Γ</sup><sup>1</sup> <sup>∈</sup> *<sup>W</sup>*2,1(*a*1, *<sup>b</sup>*1)*, is an inviable discontinuity curve with respect to the first variable <sup>u</sup>*<sup>1</sup> *if there exist <sup>ε</sup>* <sup>&</sup>gt; <sup>0</sup> *and <sup>ψ</sup>*<sup>1</sup> <sup>∈</sup> *<sup>L</sup>*1(*a*1, *<sup>b</sup>*1), *<sup>ψ</sup>*1(*t*) <sup>&</sup>gt; <sup>0</sup> *for a.e. <sup>t</sup>* <sup>∈</sup> [*a*1, *<sup>b</sup>*1] *such that either*

$$
\Gamma\_1^{\prime\prime}(t) + \psi\_1(t) < -g\_1(t)f\_1(t,y,z) \text{ for a.e. } t \in [a\_1, b\_1], \text{ all } y \in [\Gamma\_1(t) - \varepsilon, \Gamma\_1(t) + \varepsilon] \text{ and all } z \in \mathbb{R}\_+, \tag{12}
$$

*or*

$$\Gamma\_1^{\prime\prime}(t) - \psi\_1(t) > -g\_1(t)f\_1(t,y,z) \text{ for a.e.} \ t \in [a\_1, b\_1], \text{ all } y \in [\Gamma\_1(t) - \varepsilon, \Gamma\_1(t) + \varepsilon] \text{ and all } z \in \mathbb{R}\_+. \tag{13}$$

*Similarly, we say that* <sup>Γ</sup><sup>2</sup> : [*a*2, *<sup>b</sup>*2] <sup>⊂</sup> *<sup>I</sup>* = [0, 1] <sup>→</sup> <sup>R</sup>+*,* <sup>Γ</sup><sup>2</sup> <sup>∈</sup> *<sup>W</sup>*2,1(*a*2, *<sup>b</sup>*2)*, is an inviable discontinuity curve with respect to the second variable <sup>u</sup>*<sup>2</sup> *if there exist <sup>ε</sup>* <sup>&</sup>gt; <sup>0</sup> *and <sup>ψ</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>*1(*a*2, *<sup>b</sup>*2), *<sup>ψ</sup>*2(*t*) <sup>&</sup>gt; <sup>0</sup> *for a.e. t* ∈ [*a*2, *b*2] *such that either*

$$
\Gamma\_2''(t) + \psi\_2(t) < -g\_2(t) f\_2(t, y, z) \text{ for a.e. } t \in [a\_2, b\_2], \text{ all } y \in \mathbb{R}\_+ \text{ and all } z \in \left[\Gamma\_2(t) - \varepsilon, \Gamma\_2(t) + \varepsilon\right],
$$

*or*

$$
\Gamma\_2''(t) - \psi\_2(t) > -g\_2(t) f\_2(t, y, z) \text{ for a.e.} \ t \in [a\_2, b\_2], \text{ all } y \in \mathbb{R}\_+ \text{ and all } z \in \left[\Gamma\_2(t) - \varepsilon, \Gamma\_2(t) + \varepsilon\right].
$$

*Mathematics* **2019**, *7*, 451

Now we state some technical results that we need in the proof of the condition Fix(T) <sup>⊂</sup> Fix(*T*). Their proofs can be found in [3]. In the sequel, *m* denotes the Lebesgue measure in R.

**Lemma 2** ([3], Lemma 4.1)**.** *Let <sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> <sup>R</sup>*, <sup>a</sup>* <sup>&</sup>lt; *b, and let <sup>g</sup>*, *<sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*1(*a*, *<sup>b</sup>*)*, <sup>g</sup>* <sup>≥</sup> <sup>0</sup> *a.e., and <sup>h</sup>* <sup>&</sup>gt; <sup>0</sup> *a.e. in* (*a*, *<sup>b</sup>*)*. For every measurable set J* ⊂ (*a*, *b*) *with m*(*J*) > 0 *there is a measurable set J*<sup>0</sup> ⊂ *J with m*(*J* \ *J*0) = 0 *such that for every τ*<sup>0</sup> ∈ *J*<sup>0</sup> *we have*

$$\lim\_{t \to \tau\_0^+} \frac{\int\_{\lceil \tau\_0 t \rceil \vee I} \operatorname{g}(s) \, ds}{\int\_{\tau\_0}^t h(s) \, ds} = 0 = \lim\_{t \to \tau\_0^-} \frac{\int\_{\lceil t, \tau\_0 \rceil \vee I} \operatorname{g}(s) \, ds}{\int\_t^{\tau\_0} h(s) \, ds}.$$

**Corollary 1** ([3], Corollary 4.2)**.** *Let <sup>a</sup>*, *<sup>b</sup>* <sup>∈</sup> <sup>R</sup>*, <sup>a</sup>* <sup>&</sup>lt; *b, and let <sup>h</sup>* <sup>∈</sup> *<sup>L</sup>*1(*a*, *<sup>b</sup>*) *be such that <sup>h</sup>* <sup>&</sup>gt; <sup>0</sup> *a.e. in* (*a*, *<sup>b</sup>*)*. For every measurable set J* ⊂ (*a*, *b*) *with m*(*J*) > 0 *there is a measurable set J*<sup>0</sup> ⊂ *J with m*(*J* \ *J*0) = 0 *such that for all τ*<sup>0</sup> ∈ *J*<sup>0</sup> *we have*

$$\lim\_{t \to \tau\_0^+} \frac{\int\_{[\tau\_0, t] \cap f} h(s) \, ds}{\int\_{\tau\_0}^t h(s) \, ds} = 1 = \lim\_{t \to \tau\_0^-} \frac{\int\_{[t, \tau\_0] \cap f} h(s) \, ds}{\int\_t^{\tau\_0} h(s) \, ds}.$$

We shall also need the following lemma, see [2], Lemma 3.11.

**Lemma 3.** *If M* <sup>∈</sup> *<sup>L</sup>*1(0, 1)*, M* <sup>≥</sup> <sup>0</sup> *almost everywhere, then the set*

$$Q = \left\{ u \in \mathcal{C}^1([0,1]) : \left| u'(t) - u'(s) \right| \le \int\_s^t M(r) \, dr \quad \text{whenever } 0 \le s \le t \le 1 \right\}$$

*is closed in* C([0, 1]) *endowed with the maximum norm topology.*

*Moreover, if un* <sup>∈</sup> *<sup>Q</sup> for all <sup>n</sup>* <sup>∈</sup> <sup>N</sup> *and un* <sup>→</sup> *<sup>u</sup> uniformly in* [0, 1]*, then there exists a subsequence* {*unk*} *which tends to u in the* <sup>C</sup><sup>1</sup> *norm.*

Now we are ready to present the following existence and localization result for the differential system (6)–(7).

**Theorem 3.** *Suppose that the functions fi and gi (i* = 1, 2*) satisfy conditions* (*H*1)*,* (*H*2) *and*

(*H*3) *There exist inviable discontinuity curves* <sup>Γ</sup>1,*<sup>n</sup>* : *<sup>I</sup>*1,*<sup>n</sup>* := [*a*1,*n*, *<sup>b</sup>*1,*n*] <sup>⊂</sup> *<sup>I</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> *with respect to the first variable, <sup>n</sup>* <sup>∈</sup> <sup>N</sup>*, and inviable discontinuity curves* <sup>Γ</sup>2,*<sup>n</sup>* : *<sup>I</sup>*2,*<sup>n</sup>* := [*a*2,*n*, *<sup>b</sup>*2,*n*] <sup>⊂</sup> *<sup>I</sup>* <sup>→</sup> <sup>R</sup><sup>+</sup> *with respect to the second variable, <sup>n</sup>* <sup>∈</sup> <sup>N</sup>*, such that for each <sup>i</sup>* ∈ {1, 2} *and for a.e. <sup>t</sup>* <sup>∈</sup> *<sup>I</sup> the function* (*u*1, *u*2) → *fi*(*t*, *u*1, *u*2) *is continuous on*

$$\left( \mathbb{R}\_+ \mid \bigcup\_{\{n:t \in I\_{1,n}\}} \{\Gamma\_{1,n}(t)\} \right) \times \left( \mathbb{R}\_+ \mid \bigcup\_{\{n:t \in I\_{2,n}\}} \{\Gamma\_{2,n}(t)\} \right).$$

*Moreover, assume that there exist αi*, *β<sup>i</sup>* > 0 *with α<sup>i</sup>* = *βi, i* = 1, 2*, and ε* > 0 *such that*

*Bi f α*,*ε <sup>i</sup>* < *αi*, *Ai f β*,*ε <sup>i</sup>* > *β<sup>i</sup> for i* = 1, 2.

*Then system* (6)*–*(7) *has at least one solution in Kr*,*R.*

**Proof.** The operator *T* : *Kr*,*<sup>R</sup>* → *K*, *T* = (*T*1, *T*2), given by (8) is well-defined and the hypotheses (*H*1) and (*H*2) imply that *T*(*Kr*,*R*) is relatively compact as an immediate consequence of the Ascoli–Arzelá theorem. Moreover, by (*H*1) and (*H*2), there exist functions *<sup>η</sup><sup>i</sup>* <sup>∈</sup> *<sup>L</sup>*1(*I*) (*<sup>i</sup>* <sup>=</sup> 1, 2) such that

$$g\_i(t)f\_i(t, \mu\_1, \mu\_2) \le \eta\_i(t) \quad \text{for a.e.} \ t \in I \text{ and all } \mu\_1 \in [0, R\_1], \ \mu\_2 \in [0, R\_2]. \tag{14}$$

Therefore, *T*(*Kr*,*R*) ⊂ *Q*<sup>1</sup> × *Q*2, where

$$Q\_i = \left\{ u \in \mathcal{C}^1([0,1]) : \left| u'(t) - u'(s) \right| \le \int\_s^t \eta\_i(r) \, dr \quad \text{whenever } 0 \le s \le t \le 1 \right\},$$

for *i* = 1, 2, which by virtue of Lemma 3 is a closed and convex subset of *X* = C(*I*). Then, by 'convexification', <sup>T</sup>(*Kr*,*R*) <sup>⊂</sup> *<sup>Q</sup>*<sup>1</sup> <sup>×</sup> *<sup>Q</sup>*2, where <sup>T</sup> is the multivalued map associated to *<sup>T</sup>* defined as in (5).

By Lemma 1, the multivalued map T has a fixed point in *Kr*,*R*. Hence, if we show that all the fixed points of the operator T are fixed points of *T*, the conclusion is obtained. To do so, we fix an arbitrary function *u* ∈ *Kr*,*<sup>R</sup>* ∩ (*Q*<sup>1</sup> × *Q*2) and we consider three different cases.

Case 1: *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*1,*<sup>n</sup>* : *<sup>u</sup>*1(*t*) = <sup>Γ</sup>1,*n*(*t*)}∪{*<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*2,*<sup>n</sup>* : *<sup>u</sup>*2(*t*) = <sup>Γ</sup>2,*n*(*t*)}) = 0 for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Let us prove that *<sup>T</sup>* is continuous at *<sup>u</sup>*, which implies that <sup>T</sup>*<sup>u</sup>* <sup>=</sup> {*Tu*}, and therefore the relation *<sup>u</sup>* <sup>∈</sup> <sup>T</sup>*<sup>u</sup>* gives that *u* = *Tu*.

The assumption implies that for a.a. *t* ∈ *I* the mappings *f*1(*t*, ·) and *f*2(*t*, ·) are continuous at *u*(*t*)=(*u*1(*t*), *u*2(*t*)). Hence if *uk* → *u* in *Kr*,*<sup>R</sup>* then

$$f\_i(t, \mu\_k(t)) \to f\_i(t, \mu(t)) \quad \text{for a.a. } t \in I \text{ and for } i = 1, 2, \dots$$

which, along with (14), yield *Tuk* <sup>→</sup> *Tu* in <sup>C</sup>(*I*)2, so *<sup>T</sup>* is continuous at *<sup>u</sup>*.

Case 2: *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*1,*<sup>n</sup>* : *<sup>u</sup>*1(*t*) = <sup>Γ</sup>1,*n*(*t*)}) <sup>&</sup>gt; 0 for some *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. In this case we can prove that *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> <sup>T</sup>1*u*, and thus *<sup>u</sup>* <sup>∈</sup> <sup>T</sup>*u*.

To this aim, first, we fix some notation. Let us assume that for some *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> we have *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*1,*<sup>n</sup>* : *<sup>u</sup>*1(*t*) = <sup>Γ</sup>1,*n*(*t*)}) <sup>&</sup>gt; 0 and there exist *<sup>ε</sup>* <sup>&</sup>gt; 0 and *<sup>ψ</sup>* <sup>∈</sup> *<sup>L</sup>*1(*I*1,*n*), *<sup>ψ</sup>*(*t*) <sup>&</sup>gt; 0 for a.a. *<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*1,*n*, such that (13) holds with Γ<sup>1</sup> replaced by Γ1,*n*. (The proof is similar if we assume (12) instead of (13), so we omit it.)

We denote *J* = {*t* ∈ *I*1,*<sup>n</sup>* : *u*1(*t*) = Γ1,*n*(*t*)}, and we deduce from Lemma 2 that there is a measurable set *J*<sup>0</sup> ⊂ *J* with *m*(*J*0) = *m*(*J*) > 0 such that for all *τ*<sup>0</sup> ∈ *J*<sup>0</sup> we have

$$\lim\_{t \to \tau\_0^+} \frac{2 \int\_{\left[\tau\_0 t\right] \backslash J} \eta\_1(s) \, ds}{(1/4) \int\_{\tau\_0}^t \psi(s) \, ds} = 0 = \lim\_{t \to \tau\_0^-} \frac{2 \int\_{\left[t, \tau\_0\right] \backslash J} \eta\_1(s) \, ds}{(1/4) \int\_t^{\tau\_0} \psi(s) \, ds}. \tag{15}$$

By Corollary 1 there exists *J*<sup>1</sup> ⊂ *J*<sup>0</sup> with *m*(*J*<sup>0</sup> \ *J*1) = 0 such that for all *τ*<sup>0</sup> ∈ *J*<sup>1</sup> we have

$$\lim\_{t \to \tau\_0^+} \frac{\int\_{[\tau\_0, t] \cap l\_0} \psi(s) \, ds}{\int\_{\tau\_0}^t \psi(s) \, ds} = 1 = \lim\_{t \to \tau\_0^-} \frac{\int\_{[t, \tau\_0] \cap l\_0} \psi(s) \, ds}{\int\_t^{\tau\_0} \psi(s) \, ds}. \tag{16}$$

Let us now fix a point *<sup>τ</sup>*<sup>0</sup> <sup>∈</sup> *<sup>J</sup>*1. From (15) and (16) we deduce that there exist *<sup>t</sup>*<sup>−</sup> <sup>&</sup>lt; ˜*t*<sup>−</sup> <sup>&</sup>lt; *<sup>τ</sup>*<sup>0</sup> and *<sup>t</sup>*<sup>+</sup> <sup>&</sup>gt; ˜*t*<sup>+</sup> <sup>&</sup>gt; *<sup>τ</sup>*0, *<sup>t</sup>*<sup>±</sup> sufficiently close to *<sup>τ</sup>*<sup>0</sup> so that the following inequalities are satisfied for all *<sup>t</sup>* ∈ [˜*t*+, *<sup>t</sup>*+]:

$$2\int\_{\left[\tau\_{0},t\right]\backslash J} \eta\_{1}(s) \, ds < \frac{1}{4} \int\_{\tau\_{0}}^{t} \psi(s) \, ds,\tag{17}$$

$$\int\_{\{\mathbf{r}\_0, t\} \cap I} \Psi(s) \, ds \ge \int\_{\{\mathbf{r}\_0, t\} \cap I\_0} \Psi(s) \, ds > \frac{1}{2} \int\_{\mathbf{r}\_0}^t \Psi(s) \, ds,\tag{18}$$

and for all *<sup>t</sup>* <sup>∈</sup> [*t*−, ˜*t*−]:

$$2\int\_{[t,\tau\_0]\backslash\int} \eta\_1(s) \, ds < \frac{1}{4} \int\_t^{\tau\_0} \psi(s) \, ds,\tag{19}$$

$$\int\_{\{t,\tau\_0\}\cap J} \psi(s) \, ds > \frac{1}{2} \int\_t^{\tau\_0} \psi(s) \, ds. \tag{20}$$

Finally, we define a positive number

$$\bar{\rho} = \min \left\{ \frac{1}{4} \int\_{\bar{l}\_{-}}^{\tau\_{0}} \psi(s) \, ds, \frac{1}{4} \int\_{\tau\_{0}}^{\bar{l}\_{+}} \psi(s) \, ds \right\},\tag{21}$$

and we are ready to prove that *<sup>u</sup>*<sup>1</sup> <sup>∈</sup> <sup>T</sup>1*u*. It suffices to prove the following claim:

Claim: let *ε* > 0 be given by our assumptions over Γ1,*<sup>n</sup>* as Definition 1 shows, and let *ρ* = *ρ*˜ <sup>2</sup> min {˜*t*<sup>−</sup> <sup>−</sup> *<sup>t</sup>*−, *<sup>t</sup>*<sup>+</sup> <sup>−</sup> ˜*t*+}, where *<sup>ρ</sup>*˜ is as in (21). For every finite family *xj* <sup>∈</sup> *<sup>B</sup>ε*(*u*) <sup>∩</sup> *Kr*,*<sup>R</sup>* and *<sup>λ</sup><sup>j</sup>* <sup>∈</sup> [0, 1] (*j* = 1, 2, . . . , *m*), with ∑ *λ<sup>j</sup>* = 1, we have *u*<sup>1</sup> − ∑ *λjT*1*xj*<sup>∞</sup> ≥ *ρ*.

Let *xj* and *λ<sup>j</sup>* be as in the Claim and, for simplicity, denote *y* = ∑ *λjT*1*xj*. For a.a. *t* ∈ *J* = {*t* ∈ *I*1,*<sup>n</sup>* : *u*1(*t*) = Γ1,*n*(*t*)} we have

$$y^{\prime\prime}(t) = \sum\_{j=1}^{m} \lambda\_j (T\_1 \mathbf{x}\_j)^{\prime\prime}(t) = -\sum\_{j=1}^{m} \lambda\_j \lg\_1(t) f\_1(t, \mathbf{x}\_{j,1}(t), \mathbf{x}\_{j,2}(t)).\tag{22}$$

On the other hand, for every *j* ∈ {1, 2, . . . , *m*} and every *t* ∈ *J* we have

$$|x\_{\dot{\jmath},1}(t) - \Gamma\_{1,n}(t)| = |x\_{\dot{\jmath},1}(t) - u\_1(t)| < \varepsilon\_{\prime}$$

and then the assumptions on Γ1,*<sup>n</sup>* ensure that for a.a. *t* ∈ *J* we have

$$y''(t) = -\sum\_{j=1}^{m} \lambda\_j \lg\_1(t) f\_1(t, \mathbf{x}\_{j,1}(t), \mathbf{x}\_{j,2}(t)) < \sum\_{j=1}^{m} \lambda\_j \left(\Gamma\_{1,n}^{\prime\prime}(t) - \psi(t)\right) = u\_1^{\prime\prime}(t) - \psi(t). \tag{23}$$

Now for *<sup>t</sup>* <sup>∈</sup> [*t*−, ˜*t*−] we compute

$$\begin{split} y'(\tau\_0) - y'(t) &= \int\_t^{\eta\_0} y''(s) \, ds = \int\_{[t,\eta\_0] \cap \int} y''(s) \, ds + \int\_{[t,\eta\_0] \backslash \int} y''(s) \, ds \\ &< \int\_{[t,\eta\_0] \cap \int} u\_1''(s) \, ds - \int\_{[t,\eta\_0] \cap \int} \psi(s) \, ds \\ &\quad + \int\_{[t,\tau\_0] \backslash \int} \eta\_1(s) \, ds \qquad \text{(by (23), (22) and (14))} \\ &= u\_1'(\tau\_0) - u\_1'(t) - \int\_{[t,\eta\_0] \backslash \int} u\_1''(s) \, ds - \int\_{[t,\eta\_0] \cap \int} \psi(s) \, ds + \int\_{[t,\tau\_0] \backslash \int} \eta\_1(s) \, ds \\ &\le u\_1'(\tau\_0) - u\_1'(t) - \int\_{[t,\eta\_0] \cap \int} \psi(s) \, ds + 2 \int\_{[t,\eta\_0] \backslash \int} \eta\_1(s) \, ds \\ &< u\_1'(\tau\_0) - u\_1'(t) - \frac{1}{4} \int\_t^{\eta\_0} \psi(s) \, ds \qquad \text{(by (19) and (20))}, \end{split}$$

hence *y* (*t*) − *u* <sup>1</sup>(*t*) ≥ *ρ*˜ provided that *y* (*τ*0) ≥ *u* <sup>1</sup>(*τ*0). Therefore, by integration we obtain

$$y(\vec{t}\_{-}) - u\_1(\vec{t}\_{-}) = y(t\_{-}) - u\_1(t\_{-}) + \int\_{t\_{-}}^{\vec{t}\_{-}} (y'(t) - u\_1'(t)) \, dt \ge y(t\_{-}) - u\_1(t\_{-}) + \tilde{\rho}(\vec{t}\_{-} - t\_{-}).$$

So, if *y*(*t*−) − *u*1(*t*−) ≤ −*ρ*, then *y* − *u*1<sup>∞</sup> ≥ *ρ*. Otherwise, if *y*(*t*−) − *u*1(*t*−) > −*ρ*, then we have *<sup>y</sup>*(˜*t*−) <sup>−</sup> *<sup>u</sup>*1(˜*t*−) <sup>&</sup>gt; *<sup>ρ</sup>* and thus *<sup>y</sup>* <sup>−</sup> *<sup>u</sup>*1<sup>∞</sup> <sup>≥</sup> *<sup>ρ</sup>*, too.

Similar computations in the interval [˜*t*+, *<sup>t</sup>*+] instead of [*t*−, ˜*t*−] show that if *<sup>y</sup>* (*τ*0) ≤ *u* <sup>1</sup>(*τ*0) then we have *u* <sup>1</sup>(*t*) − *y* (*t*) <sup>≥</sup> *<sup>ρ</sup>*˜ for all *<sup>t</sup>* <sup>∈</sup> [˜*t*+, *<sup>t</sup>*+] and this also implies *<sup>y</sup>* <sup>−</sup> *<sup>u</sup>*1 <sup>≥</sup> *<sup>ρ</sup>*. The claim is proven.

Case 3: *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> *<sup>I</sup>*2,*<sup>n</sup>* : *<sup>u</sup>*2(*t*) = <sup>Γ</sup>2,*n*(*t*)}) <sup>&</sup>gt; 0 for some *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. In this case it is possible to prove that *<sup>u</sup>*<sup>2</sup> <sup>∈</sup> <sup>T</sup>2*u*. The details are similar to those in Case 2, with obvious changes, so we omit them.

**Remark 5.** *Observe that Definition 1 allows to study the discontinuities of the functions fi independently in each variable u*<sup>1</sup> *and u*2*, as shown in condition* (*H*3)*.*

*In addition, a continuum set of discontinuity points is possible: for instance, the function f*<sup>1</sup> *may be discontinuous at the point <sup>u</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> *for all <sup>u</sup>*<sup>2</sup> <sup>∈</sup> <sup>R</sup><sup>+</sup> *provided that the constant function* <sup>Γ</sup><sup>1</sup> <sup>≡</sup> <sup>1</sup> *is an inviable discontinuity curve with respect to the first variable. This fact improves the ideas given in [5] for first-order autonomous systems where "only" a countable set of discontinuity points are allowed.*

**Remark 6.** *Notice that conditions* (12) *and* (13) *are not local in the last variable. However, the condition*

$$\inf\_{t \in I, \mathbf{x}, \mathbf{y} \in \mathbb{R}\_+} f\_1(t, \mathbf{x}, \mathbf{y}) > 0$$

*implies that any constant function stands for an inviable discontinuity curve with respect to the first variable (since condition* (13) *holds). Moreover, any function with strictly positive second derivative is always an inviable discontinuity curve with respect to the variable u*<sup>1</sup> *without any additional condition on f*1*.*

Now we illustrate our existence result by some examples.

**Example 1.** *Consider the coupled system*

$$\begin{cases} -\mathbf{x}^{\prime\prime}(t) = \mathbf{x}^2 + \mathbf{x}^2 y^2 H(a - \mathbf{x}) H(b - y), \\\ -y^{\prime\prime}(t) = \sqrt{\mathbf{x}} + \sqrt{\mathbf{y}} + H(\mathbf{x} - c) H(y - d), \end{cases} \tag{24}$$

*subject to the boundary conditions* (7) *(replacing u*<sup>1</sup> *and u*<sup>2</sup> *by x and y, respectively) where a*, *b*, *c*, *d* > 0 *and H denotes the Heaviside function.*

*The existence of numbers α<sup>i</sup> and β<sup>i</sup> in the conditions of* (9) *is guaranteed by Remark 3* (*a*) *since f*1(*x*, *y*) = *<sup>x</sup>*<sup>2</sup> <sup>+</sup> *<sup>x</sup>*2*y*2*H*(*<sup>a</sup>* <sup>−</sup> *<sup>x</sup>*)*H*(*<sup>b</sup>* <sup>−</sup> *<sup>y</sup>*) *is a superlinear function and <sup>f</sup>*2(*x*, *<sup>y</sup>*) = <sup>√</sup>*<sup>x</sup>* <sup>+</sup> <sup>√</sup>*<sup>y</sup>* <sup>+</sup> *<sup>H</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>c</sup>*)*H*(*<sup>y</sup>* <sup>−</sup> *<sup>d</sup>*) *is a sublinear function.*

*On the other hand, the function* (*x*, *<sup>y</sup>*) → *<sup>f</sup>*1(*x*, *<sup>y</sup>*) *is continuous on* (R<sup>+</sup> \ {*a*}) <sup>×</sup> (R<sup>+</sup> \ {*b*}) *and the constant function* Γ<sup>1</sup> ≡ *a stands for an inviable curve with respect to the first variable. Indeed,*

$$-\Gamma\_1^{\prime\prime}(t) + \frac{a^2}{8} = \frac{a^2}{8} < f\_1(y, z) \quad \text{for } a.a. \ t \in [0, 1] \text{ and for all } y \in \left[\frac{a}{2}, \frac{3a}{2}\right] \text{ and } z \in \mathbb{R}\_+. $$

*hence* (13) *holds with <sup>ψ</sup>*<sup>1</sup> <sup>≡</sup> *<sup>a</sup>*2/8*.*

*Moreover, the constant function* Γ<sup>2</sup> ≡ *b is an inviable curve with respect to the second variable, according to Remark 6 since*

$$\inf\_{\mathbf{x}, y \in \mathbb{R}\_+} f\_2(\mathbf{x}, y) > 0.$$

*Similarly, the function <sup>f</sup>*2(*x*, *<sup>y</sup>*) = <sup>√</sup>*<sup>x</sup>* <sup>+</sup> <sup>√</sup>*<sup>y</sup>* <sup>+</sup> *<sup>H</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>c</sup>*)*H*(*<sup>y</sup>* <sup>−</sup> *<sup>d</sup>*) *satisfies the hypothesis* (*H*3) *in Theorem 3, so the system* (7)*–*(24) *has at least one positive solution.*

**Example 2.** *Consider the system*

$$\begin{cases} -x''(t) = x^2 + x^2 y^2 H(a + t^2 - x) H(b + mt - y), \\ -y''(t) = \sqrt{x} + \sqrt{y} + H(x - c) H(y - d), \end{cases} \tag{25}$$

*subject to the boundary conditions* (7)*, where a*, *<sup>b</sup>*, *<sup>c</sup>*, *<sup>d</sup>* <sup>&</sup>gt; <sup>0</sup> *and m* <sup>∈</sup> <sup>R</sup>*.*

*Now, for a.a. t* ∈ *I, the function* (*x*, *y*) → *f*1(*t*, *x*, *y*)*, where*

$$f\_1(t, x, y) = x^2 + x^2 y^2 H(a + t^2 - x) H(b + mt - y),$$

*is continuous on* <sup>R</sup><sup>+</sup> \ {*<sup>a</sup>* <sup>+</sup> *<sup>t</sup>* 2} <sup>×</sup> (R<sup>+</sup> \ {*<sup>b</sup>* <sup>+</sup> *mt*}) *and the curve* <sup>Γ</sup>1(*t*) = *<sup>a</sup>* <sup>+</sup> *<sup>t</sup>* <sup>2</sup> *is inviable with respect to the first variable. Indeed,* (13) *is satisfied with ψ*<sup>1</sup> ≡ 1*, since*

$$-\Gamma\_1^{\prime\prime}(t) + 1 = -1 < f\_1(t, y, z) \quad \text{for a.e.} \ t \in [0, 1] \text{ and for all } y, z \in \mathbb{R}\_+. $$

*On the other hand, the curve* Γ2(*t*) = *b* + *mt is inviable with respect to the variable y, according to Remark 6, since* Γ <sup>2</sup> (*t*) ≡ <sup>0</sup> *and* inf*x*,*y*∈R<sup>+</sup> *<sup>f</sup>*2(*x*, *<sup>y</sup>*) > <sup>0</sup>*.*

*Therefore, Theorem 3 ensures the existence of one positive solution for problem* (7)*–*(25)*.*

Nevertheless, the conditions of Definition 1 are too strong for functions *f*<sup>1</sup> which are discontinuous at a single isolated point (*x*0, *<sup>y</sup>*0) or, more generally, over a curve (*γ*1(*t*), *<sup>γ</sup>*2(*t*)) for *<sup>t</sup>* <sup>∈</sup> ¯*<sup>I</sup>* <sup>⊂</sup> *<sup>I</sup>*. This is the motivation for another definition of the notion of discontinuity curves. This notion will be a generalization of the admissible curves presented in [2] for one equation.

**Definition 2.** *We say that <sup>γ</sup>* = (*γ*1, *<sup>γ</sup>*2) : [*a*, *<sup>b</sup>*] <sup>⊂</sup> *<sup>I</sup>* = [0, 1] <sup>→</sup> <sup>R</sup><sup>2</sup> <sup>+</sup>*, <sup>γ</sup><sup>i</sup>* <sup>∈</sup> *<sup>W</sup>*2,1(*a*, *<sup>b</sup>*) *(i* <sup>=</sup> 1, 2*), is an admissible discontinuity curve for the differential equation u* <sup>1</sup> = −*g*1(*t*)*f*1(*t*, *u*1(*t*), *u*2(*t*)) *if one of the following conditions holds:*


$$\begin{aligned} \gamma\_1''(t) + \psi(t) &< -g\_1(t)f\_1(t,y,z) \quad \text{for a.e.} \ t \in [a,b] \text{ all } y \in \left[\gamma\_1(t) - \varepsilon, \gamma\_1(t) + \varepsilon\right],\\ &\text{and all } z \in \left[\gamma\_2(t) - \varepsilon, \gamma\_2(t) + \varepsilon\right], \end{aligned}$$

*or*

$$\begin{aligned} \gamma\_1''(t) - \psi(t) &> -g\_1(t)f\_1(t,y,z) \quad \text{for a.e.} \ t \in [a,b] \text{ all } y \in \left[\gamma\_1(t) - \varepsilon, \gamma\_1(t) + \varepsilon\right] \\ &\quad \text{and all } z \in \left[\gamma\_2(t) - \varepsilon, \gamma\_2(t) + \varepsilon\right]. \end{aligned}$$

*In this case we say that γ is inviable.*

*Similarly, we can define admissible discontinuity curves for u* <sup>2</sup> = −*g*2(*t*)*f*2(*t*, *u*1(*t*), *u*2(*t*))*.*

**Theorem 4.** *Suppose that the functions fi and gi (i* = 1, 2*) satisfy conditions* (*H*1)*,* (*H*2) *and*


*Moreover, assume that there exist αi*, *β<sup>i</sup>* > 0 *with α<sup>i</sup>* = *βi, i* = 1, 2*, and ε* > 0 *such that*

$$B\_i f\_i^{\alpha, x} < \alpha\_i \quad A\_i f\_i^{\beta, x} > \beta\_i \quad \text{for } i = 1, 2.$$

*Then the differential system* (6)*–*(7) *has at least one solution in Kr*,*R.*

**Proof.** Notice that in virtue of Lemma <sup>1</sup> it is sufficient to show that Fix(T) <sup>⊂</sup> Fix(*T*). Reasoning as in the proof of Theorem 3, if we fix a function *u* ∈ *Kr*,*<sup>R</sup>* ∩ (*Q*<sup>1</sup> × *Q*2), we have to consider three different cases.

Case 1: *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> *In* : *<sup>u</sup>*(*t*) = *<sup>γ</sup>n*(*t*)}∪{*<sup>t</sup>* <sup>∈</sup> ˜*In* : *<sup>u</sup>*(*t*) = *<sup>γ</sup>*˜*n*(*t*)}) = 0 for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Then *<sup>T</sup>* is continuous at *u*.

Case 2: *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> *In* : *<sup>u</sup>*(*t*) = *<sup>γ</sup>n*(*t*)}) <sup>&</sup>gt; 0 or *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> ˜*In* : *<sup>u</sup>*(*t*) = *<sup>γ</sup>*˜*n*(*t*)}) <sup>&</sup>gt; 0 for some *<sup>γ</sup><sup>n</sup>* or *<sup>γ</sup>*˜*<sup>n</sup>* inviable. Then *<sup>u</sup>* <sup>∈</sup> <sup>T</sup>*u*. The proof follows the ideas from Case 2 in Theorem 3.

Case 3: *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> *In* : *<sup>u</sup>*(*t*) = *<sup>γ</sup>n*(*t*)}) <sup>&</sup>gt; 0 or *<sup>m</sup>*({*<sup>t</sup>* <sup>∈</sup> ˜*In* : *<sup>u</sup>*(*t*) = *<sup>γ</sup>*˜*n*(*t*)}) <sup>&</sup>gt; 0 only for viable curves. Then the relation *<sup>u</sup>* <sup>∈</sup> <sup>T</sup>*<sup>u</sup>* implies *<sup>u</sup>* <sup>=</sup> *Tu*. In this case the idea is to show that *<sup>u</sup>* is a solution of the differential system. The proof is analogus to that of the equivalent case in [2], Theorem 3.12 or [3], Theorem 4.4, so we omit it here.

**Remark 7.** *Notice that, in the case of a function* (*u*1, *u*2) → *f*1(*t*, *u*1, *u*2) *which is discontinuous at a single point* (*x*0, *y*0)*, Definition 2 requires that one of the following two conditions holds:*

(*i*) *f*1(*t*, *x*0, *y*0) = 0 *for a.e. t* ∈ [0, 1]*;*

(*ii*) *there exist <sup>ε</sup>* <sup>&</sup>gt; <sup>0</sup> *and <sup>ψ</sup>* <sup>∈</sup> *<sup>L</sup>*1(0, 1)*, <sup>ψ</sup>*(*t*) <sup>&</sup>gt; <sup>0</sup> *for a.e. t* <sup>∈</sup> *I such that*

$$0 < \psi(t) < \lg\_1(t) f\_1(t, \mathbf{x}, y) \text{ for a.e. } t \in I, \text{ all } \mathbf{x} \in [\mathbf{x}\_0 - \varepsilon, \mathbf{x}\_0 + \varepsilon] \text{ and all } y \in [\underline{y}\_0 - \varepsilon, \underline{y}\_0 + \varepsilon].$$

*In particular, for (ii), it suffices that there exist ε*, *δ* > 0 *such that*

$$0 < \delta < f\_1(t, \mathbf{x}, y) \text{ for a.e. } t \in I, \text{ all } \mathbf{x} \in [\mathbf{x}\_0 - \varepsilon, \mathbf{x}\_0 + \varepsilon] \text{ and all } y \in [y\_0 - \varepsilon, y\_0 + \varepsilon].$$

To finish, we present two simple examples which fall outside of the applicability of Theorem 3, but which can be studied by means of Theorem 4.

**Example 3.** *Consider the problem*

$$\begin{cases} -\mathbf{x}''(t) = f\_1(\mathbf{x}, \mathbf{y}) = (\mathbf{x}y)^{1/3} \left( 2 - \cos\left( 1/((\mathbf{x} - 1)^2 + (\mathbf{y} - 1)^2) \right) H\left( (\mathbf{x} - 1)^2 + (\mathbf{y} - 1)^2 \right) \right), \\\ -\mathbf{y}''(t) = f\_2(\mathbf{x}, \mathbf{y}) = (\mathbf{x}y)^{1/3}, \end{cases} \tag{26}$$

*subject to the boundary conditions* (7)*.*

*It is clear that f*<sup>1</sup> *and f*<sup>2</sup> *have a sublinear behavior, see Remark 3.*

*The function* (*x*, *<sup>y</sup>*) → *<sup>f</sup>*1(*x*, *<sup>y</sup>*) *is continuous on* <sup>R</sup><sup>2</sup> <sup>+</sup> \ {(1, 1)} *and the constant function γ*(*t*) = (*γ*1(*t*), *γ*2(*t*)) ≡ (1, 1) *is an inviable admissible discontinuity curve for the differential equation* −*x* (*t*) = *<sup>f</sup>*1(*x*, *<sup>y</sup>*) *since* <sup>0</sup> <sup>&</sup>lt; 1/√<sup>3</sup> <sup>4</sup> <sup>≤</sup> *<sup>f</sup>*1(*x*, *<sup>y</sup>*) *for all x* <sup>∈</sup> [1/2, 3/2] *and all y* <sup>∈</sup> [1/2, 3/2]*; and <sup>γ</sup>* <sup>1</sup> (*t*) = 0*.*

*Therefore, Theorem 4 guarantees the existence of a positive solution for problem* (7)*–*(26)*.*

**Example 4.** *Consider the following system*

$$\begin{cases} -\mathbf{x}^{\prime\prime}(t) = f\_1(\mathbf{x}, \mathbf{y}) = (\mathbf{x}\mathbf{y})^{1/3}, \\\ -\mathbf{y}^{\prime\prime}(t) = f\_2(\mathbf{x}, \mathbf{y}) = \left(1 + (\mathbf{x}\mathbf{y})^{1/3}\right) H(\mathbf{x}^2 + \mathbf{y}^2), \end{cases} \tag{27}$$

*subject to the boundary conditions* (7)*.*

*The nonlinearities of the system have again a sublinear behavior. Now, the function* (*x*, *y*) → *f*2(*x*, *y*) *is continuous on* R<sup>2</sup> <sup>+</sup> \ {(0, 0)} *and the constant function γ*(*t*) = (*γ*1(*t*), *γ*2(*t*)) ≡ (0, 0) *is a viable admissible discontinuity curve for the differential equation.*

*Hence, by application of Theorem 4, one obtains that the system* (7)*–*(27) *has at least one positive solution.*

**Author Contributions:** The authors contributed equally to this work.

**Funding:** R. López Pouso was partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, Project MTM2016-75140-P, and Xunta de Galicia ED341D R2016/022 and GRC2015/004. Jorge Rodríguez-López was partially supported by Xunta de Galicia Scholarship ED481A-2017/178.

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

#### **References**


© 2019 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* **Existence of Solutions for Nonhomogeneous Choquard Equations Involving p-Laplacian**

**Xiaoyan Shi 1, Yulin Zhao 1,\* and Haibo Chen <sup>2</sup>**


Received: 26 August 2019; Accepted: 15 September 2019; Published: 19 September 2019

**Abstract:** This paper is devoted to investigating a class of nonhomogeneous Choquard equations with perturbation involving p-Laplacian. Under suitable hypotheses about the perturbation term, the existence of at least two nontrivial solutions for the given problems is obtained using Nehari manifold and minimax methods.

**Keywords:** p-Laplacian; choquard equation; nonhomogeneous; nehari method; minimax methods

#### **1. Introduction and Main Results**

In this paper we are interested in the following generalized nonlinear Choquard equation with perturbation involving p-Laplacian

$$-\Delta\_{\mathbb{P}}\mu + V(\mathbf{x})|\boldsymbol{\mu}|^{p-2}\boldsymbol{\mu} = \left(\int\_{\mathbb{R}^N} \frac{\left|\boldsymbol{\mu}(\boldsymbol{y})\right|^q}{\left|\mathbf{x} - \boldsymbol{y}\right|^\mu} d\boldsymbol{y}\right) |\boldsymbol{\mu}|^{q-2}\boldsymbol{\mu} + \boldsymbol{g}(\mathbf{x}), \qquad \mathbf{x} \in \mathfrak{R}^N \tag{1}$$

where *<sup>N</sup>* <sup>≥</sup> 3, 2 <sup>≤</sup> *<sup>p</sup>* <sup>&</sup>lt; *<sup>N</sup>*, 0 <μ< *<sup>N</sup>*, *<sup>p</sup>* <sup>2</sup> (<sup>2</sup> <sup>−</sup> <sup>μ</sup> *<sup>N</sup>* ) <sup>&</sup>lt; *<sup>q</sup>* <sup>&</sup>lt; *<sup>p</sup>*<sup>∗</sup> <sup>2</sup> (<sup>2</sup> <sup>−</sup> <sup>μ</sup> *<sup>N</sup>* ), 0 <sup>&</sup>lt; *<sup>V</sup>* <sup>∈</sup> *<sup>C</sup>*1(-*<sup>N</sup>*, -), Δ*<sup>p</sup>* = *div*( ∇*u p*−2∇*u*) is the p-Laplacian operator, and *<sup>g</sup>* : -*<sup>N</sup>* → is perturbation. Here *p*∗ = *Np*/(*N* − *p*) denotes the Sobolev conjugate of *p*.

The homogeneous, a.e. *g*(*x*) ≡ 0, which means zero is a solution of problem (1). It was investigated in [1]. A special case of problem (1) is the well-known Choquard-Pekar equation

$$-\Delta u + u = \left(\frac{1}{|\mathbf{x}|^{\mu}} \* |u|^{2}\right)u, \qquad \mathbf{x} \in \mathfrak{R}^{N} \tag{2}$$

which was investigated by Pekar [2] in relationship with the quantum field theory of a polaron. In particular, when *u* is a solution to (2), we know that φ(*x*, *t*) = *u*(*x*)*e*−*it* is a solitary wave of the following Hartree equation

$$i\frac{\partial\phi}{\partial t} = -\Delta\phi - \left(\frac{1}{|\mathbf{x}|^{\mu}} \star |\phi|^2\right)\phi\_\prime \text{ in } \mathfrak{R}^3 \times \mathfrak{R}\_+.$$

which was introduced by Choquard in 1976 to describe an electron trapped in its own hole as approximation to Hartree-Fock theory of a one-component plasma; see [3,4]. This equation was also proposed by Penrose in [5] as a model of self-gravitating matter and is usually known in that context as the nonlinear Schrödinger-Newton equation. For more details, discussion about the physical aspects of the problem we refer the readers to [6–11] and the references therein.

From a mathematical point of view, the Choquard-Pekar Equation (2) and its generalizations have been widely studied. Take for instance, Lieb [4] investigated the existence and uniqueness, up to translations, of the ground state to problem (2) by using symmetric decreasing rearrangement inequalities. Later, Lions [6] proved the existence of infinitely many radially symmetric solutions to problem (2) via critical point theory. Ackermann [12] established some existence and multiplicity results for a type of periodic Choquard-Pekar equation with nonlocal superlinear part. Further interesting results on Choquard equations may be found in [13–26], the survey [27], and the references therein.

In [15], Ma and Zhao investigated the generalized stationary nonlinear Choquard equation

$$-\Delta u + u = \left( \int\_{\mathfrak{R}^N} \frac{\left| u(y) \right|^q}{\left| x - y \right|^\mu} dy \right) |u|^{q-2} u, \qquad x \in \mathfrak{R}^N \tag{3}$$

where *N* ≥ 3, 0 <μ< *N*, *q* ≥ 2 Under the suitable conditions on μ, *N*, and *q*, which include the classical case, they showed that every positive solution to problem (3) is radially symmetric and monotone decreasing on some point. Using the same condition, Cingolani et al. [9] treated (3) with the case where both the vector and the scalar potential have some symmetries, and they established the regularity and some decay asymptotically at infinity of the ground states to problem (3). In [28], Moroz and Van Schaftingen eliminated this restriction and in the optimal range of parameters they derived the regularity, positivity, and radial symmetry of the ground states, and also gave decay asymptotically at infinity for them.

When the potential *V*(*x*) is continuous and bounded below in -*<sup>N</sup>*, Alves and Yang [13] studied the multiplicity and concentration behavior of positive solutions for quasilinear Choquard equation involving p-Laplacian:

$$-\varepsilon^{p}\Delta\_{\mathbb{P}}u + V(\mathbf{x})|u|^{p-2}u = \varepsilon^{\mu-N} \left(\int\_{\mathbb{R}^N} \frac{Q(y)F(u(y))}{\left|x-y\right|^\mu} dy\right) \mathbb{Q}(\mathbf{x})f(u), \; \mathbf{x} \in \mathfrak{R}^N \tag{4}$$

where *N* ≥ 3, 0 <μ< *N*, *V*, and *Q* are two continuous real functions in -*<sup>N</sup>*, ε is a positive parameter and *F*(*t*) be the primate function of *f*(*t*), and Δ*<sup>p</sup>* = *div*( ∇*u p*−2∇*u*) is p-Laplacian operator, 1 <sup>&</sup>lt; *<sup>p</sup>* <sup>&</sup>lt; *<sup>N</sup>* In [1], suppose that the potential *V* and the nonlinearity *f* satisfy suitable assumption, Sun considered the case ε = 1 and *Q* = 1, and proved the existence of solutions in the level of mountain pass for problem (4). Further, Alves et al. [29] considered a class of generalized Choquard equation with the nonlinearities involving *N*-functions, and they obtained the existence of solutions for the given Choquard equation involving the ΔΦ-Laplacian operator, where ΔΦ = *div*(φ( ∇*u* )∇*u*) and <sup>Φ</sup> : -→ is a *N*-function. Other related results about Choquard equation involving p-Laplacian can be found in [25,30–36] and the references therein.

In 2003, Küpper et al. [37] studied the existence of positive solutions and the bifurcation point for the following Choquard equation

$$-\Delta u + u = \left(\int\_{\mathfrak{R}^3} \frac{\left|\mu(y)\right|^2}{\left|x - y\right|} dy \right) u + \lambda g(x), \qquad x \in \mathfrak{R}^3 \tag{5}$$

where *<sup>g</sup>*(*x*) <sup>∈</sup> *<sup>H</sup>*−1(-<sup>3</sup>), *<sup>g</sup>*(*x*) <sup>≥</sup> 0, *<sup>g</sup>*(*x*) <sup>≡</sup> 0. They proved that there exist positive constants <sup>λ</sup><sup>∗</sup> and <sup>λ</sup>∗∗ such that problem (5) has at least two positive solutions for λ ∈ (0, λ∗), and no positive solution for λ>λ∗∗ Furthermore, they showed that λ<sup>∗</sup> = λ∗∗ is a bifurcation point of problem (5).

Very recently, Xie et al. [23] showed the following nonhomogeneous Choquard equation

$$-\Delta u + V(\mathbf{x})u = \left(\int\_{\mathfrak{R}^N} \frac{\left|u(y)\right|^q}{\left|\mathbf{x} - y\right|^\mu} dy\right) |u|^{q-2}u + \mathbf{g}(\mathbf{x}), \qquad \mathbf{x} \in \mathfrak{R}^N$$

had two nontrivial solutions if 2 − μ/*N* < *q* < (2*N* − μ)/(*N* − 2) satisfies the following compactness condition:

(*A*1) *V* ∈ *C*(-*<sup>N</sup>*, -+) is coercive, i.e., lim|*x*|→+∞*V*(*x*)=+∞.

In [24], Zhang, Xu and Zhang also investigated the bound and ground states for nonhomogeneous Choquard equation under the following assumption.

(*A*2) *V* ∈ *C*(-*<sup>N</sup>*, -+), inf-*<sup>N</sup> V* > 0, and there exists a positive constant *r* such that, for any *M* > 0, *meas x* ∈ -*<sup>N</sup>* : *x* − *y* <sup>≤</sup> *<sup>r</sup>*, *<sup>V</sup>*(*x*) <sup>≤</sup> *<sup>M</sup>* <sup>→</sup> 0 as *y* <sup>→</sup> +∞, where *meas* stands for the Lebesgue measure.

In [38], Shen, Gao and Yang considered a class of critical nonhomogeneous Choquard equation

$$-\Delta u = \left(\int\_{\Re^N} \frac{\left|u(y)\right|^{q\_\*}}{\left|x-y\right|^\mu} dy\right) |u|^{q\_\*-2} u + \lambda u + g(x), \qquad x \in \Omega$$

where Ω is a smooth bounded domain of -*<sup>N</sup>*, 0 in interior of Ω, λ ∈ -, 0 <μ< *N*, *N* ≥ 7, *q*<sup>∗</sup> = (2*N* − μ)/(*N* − 2) is the upper critical exponent. By applying variational methods, they obtain the existence of multiple solutions for the above problem when λ ∈ (0, λ1), where λ<sup>1</sup> is the first eigenvalue of −Δ. Other related results about non-homogeneous Choquard equation can be found in [1,29,33,39–43] and the references therein.

Our work is motivated by the above work [23,37,41,44] where authors used the structure of associated Nehari manifold to obtain the multiplicity of solutions for the studied problems. Concerning the nonhomogeneous problem, Wang [41] dealt with the problem (1) in the case *p* = 2, *V* ≡ 1 and obtained the multiple solutions of problem (1). In this paper, we investigate the nonhomogeneous problem (1) in case of 2 ≤ *p* < *N* and extend the results in the literatures [23,24,41,44]. The used approach of our paper comes from the literatures [23,24,41]. However, owe to dealing with p-Laplacian and nonlocal terms the calculation of our problem will be more complicated.

Before giving our main results, we need the following function spaces. *<sup>W</sup>*1,*p*(-*<sup>N</sup>*) is the usual Sobolev space with norm

$$\left\|\boldsymbol{\mu}\right\|\_{1}^{p} = \int\_{\mathbb{R}^{N}} (\left|\nabla \boldsymbol{\mu}\right|\_{p} + \left|\boldsymbol{\mu}\right|^{p}) d\boldsymbol{x}$$

and *L<sup>r</sup>* (-*<sup>N</sup>*), for 1 ≤ *r* ≤ ∞ denotes the Lebesgue space with the norm

$$\|\boldsymbol{u}\|\_{\mathcal{V}} = \left(\int\_{\mathbb{R}^N} |\boldsymbol{u}|^r d\boldsymbol{x}\right)^{1/r} , \text{if } 1 \le r < \infty$$

In what follows, we consider the following Banach space

$$E\_V = \left\{ \mu \in \mathcal{W}^{1,p}(\mathfrak{R}^N) : \int\_{\mathfrak{R}^N} V(x) |\mu|^p dx < +\infty \right\}.$$

endowed with the inner product and norm

$$\langle u, v \rangle = \int\_{\mathfrak{R}^N} |\nabla u|^{p-2} \nabla u \nabla v dx + \int\_{\mathfrak{R}^N} |u|^{p-2} u v dx,\\ \|u\|^p = \int\_{\mathfrak{R}^N} (|\nabla u|^p + V(\mathbf{x}) |u|^p dx)$$

Throughout this paper, we assume the following condition on the function *V*.

(*A*0) *V* ∈ *C*(-*<sup>N</sup>*, -), inf*x*∈-*<sup>N</sup> V*(*x*) > 0 and there exists a constant *M* > 0 such that *meas x* ∈ -*<sup>N</sup>* : *V*(*x*) ≤ *M* < ∞, where *meas* is the Lebesgue measure.

Now we recall the well-known embedding results in [45] (Lemma 2.1).

#### **Lemma 1.** *The following statements hold.*

*(i) There exists a continuous embedding from W*1,*p*(-*<sup>N</sup>*) *into Lr* (-*<sup>N</sup>*) *for any r* ∈ [*p*, *p*<sup>∗</sup> ). *(ii) Under the condition* (*A*0) *on V, the embedding from EV into Lr* (-*<sup>N</sup>*) *is compact for any r* ∈ [*p*, *p*<sup>∗</sup> )*.*

Denote *Sr* be the best constant of the embedding from *EV* into *Lr* (-*<sup>N</sup>*) as

$$|u|\_{r} \leq S\_{T} \|u\|\_{\nu} \qquad \forall u \in E\_{V}$$

To obtain our result, we make the following assumption on perturbation term *g*:

(G). The perturbation function g ∈ *L* 2*Nq* <sup>2</sup>*N*(*q*−1)+<sup>μ</sup> (-*<sup>N</sup>*), *g* is nonzero, and there is a positive constant 

$$a = \alpha(N, p, q, \mu, S\_{\frac{2Nq}{2N - \mu}}), \text{ such that } \left| g \right|\_{\frac{2Nq}{2N(q - 1) + \mu}} < \alpha.$$
 
$$\text{Obviously if } a = 0, \text{ then we always need a solution.}$$

Obviously, if *g* = 0, then we always get a solution for problem (1) that is the trivial solution. Now, the main result of this article reads as follows.

**Theorem 1.** *Suppose* (*A*0), *g* ≡ 0 *, and* (G) *hold. Then problem (1) admits two weak solutions. One of which is a local minimum solution with the ground state energy, and another is bound state solution. In additional, if g* ≥ 0 *then the two weak solutions are nonnegative.*

This paper is organized as follows. In Section 2, we introduce the variational setting for problem (1) and give some related preliminaries. In Section 3, we study the Palais-Smale sequences and the minimization problems. Finally, we give the proof of Theorem 1 in Section 4.

#### **2. Variational Setting and Fibering Map Analysis**

This section is devoted to stating the variational setting and giving some lemmas which will be used as tools to prove our main results. The key inequality is the following classical Hardy-Littlewood-Sobolev inequality [3].

**Lemma 2.** *(Hardy-Littlewood-Sobolev inequality [3]). Let t*,*s* > 1 *, and* 0 <μ< *N with* μ/*N* + 1/*s* + 1/*t* = 2 *, f* ∈ *L<sup>t</sup>* (*RN*) *and g* ∈ *L<sup>s</sup>* (*RN*)*. Then there exists a constant C*(*N*, *t*, μ,*s*) *independent of f*, *g such that*

$$\int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{f(\boldsymbol{x})\,\boldsymbol{g}(\boldsymbol{y})}{\left|\boldsymbol{x}-\boldsymbol{y}\right|^\mu} d\boldsymbol{x} d\boldsymbol{y} \leq C(N, t, \mu, \boldsymbol{s}) \left|f\right|\_{L^1} \cdot \left|\boldsymbol{g}\right|\_{L^s}.$$

By the Hardy-Littlewood-Sobolev inequality we have that

$$\int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|u(x)|\_q |u(y)|^q}{|x - y|^\mu} dx dy$$

is well defined if *u <sup>q</sup>* <sup>∈</sup> *Lt* (*RN*) for some *t* > 1 satisfying

$$\frac{\mu}{N} + \frac{2}{t} = 2$$

For we will be working in the space *<sup>W</sup>*1,*p*(-*<sup>N</sup>*), by Sobolev embedding theorem we obtain that *qt* ∈ [*p*, *p*<sup>∗</sup> ], where *p*<sup>∗</sup> = *Np*/(*N* − *p*); that is

$$\frac{p}{2}(2-\frac{\mu}{N}) \le q \le \frac{p^\*}{2}(2-\frac{\mu}{N}) = \frac{p}{2}\Big(\frac{2N-\mu}{N-p}\Big)$$

Define

$$q\_l := \frac{p}{2}(2 - \frac{\mu}{N}), \text{ and } q^\mu := \frac{p}{2}(\frac{2N - \mu}{N - p})$$

Therefore, *ql* and *q<sup>u</sup>* are called as lower and upper critical exponents in the sense of the Hardy-Littlewood-Sobolev inequality. We constrain our discussion only when *q* ∈ (*ql*, *qu*) We define the energy functional corresponding to problem (1) as

$$I(u) = \frac{1}{p} \int\_{\mathfrak{R}^N} (|\nabla u|^p + V(\mathbf{x}) |u|^{\
\mathsf{T}}) d\mathbf{x} - \frac{1}{2q} \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|u(\mathbf{x})|^q |u(y)|^{\mathsf{T}}}{|\mathbf{x} - y|^{\mathsf{T}}} d\mathbf{x} dy - \int\_{\mathfrak{R}^N} g(\mathbf{x}) u d\mathbf{x}, u \in E\_V$$

By the condition (*G*), Hardy-Littlewood-Sobolev inequality and Sobolev inequality, we have

$$\int\_{\Re^N} \int\_{\Re^N} \frac{|u(x)|^q |u(y)|^q}{|x - y|^\mu} dx dy \le \mathbb{C}(N, \mu) \|u^q\|\_{\frac{2Nq}{2N - \mu}}^2 \le \mathbb{C}(N, \mu) S\_{\frac{2Nq}{2N - \mu}}^{2q} \|u\|^{2q} \tag{6}$$

and

$$\int\_{\mathfrak{R}^N} \operatorname{g}(\mathbf{x}) \mu d\mathbf{x} \le \left| \operatorname{g} \right|\_{\frac{2Nq}{2N(q-1)+\mu}} \|\mu\|\_{\frac{2Nq}{2N-\mu}} \le \left| \mathcal{S} \right|\_{\frac{2Nq}{2N(q-1)+\mu}} \mathcal{S}\_{\frac{2Nq}{2N-\mu}} \|\mu\|\tag{7}$$

for any *uq* ∈ *Lr* (-*<sup>N</sup>*),*r* > 1, μ ∈ (0, *N*) and *ql* ≤ *q* ≤ *qu*, *g* ∈ *L* 2*Nq* <sup>2</sup>*N*(*q*−1)+<sup>μ</sup> (-*<sup>N</sup>*). Therefore, one knows that *I* is well defined and *<sup>I</sup>*(*u*) <sup>∈</sup> *<sup>C</sup>*2(*EV*, -) and its critical points are weak solutions of problem (1). Moreover,

$$\langle l'(\boldsymbol{u}), \boldsymbol{v} \rangle = \int\_{\mathfrak{R}^N} (|\nabla \boldsymbol{u}|^{p-2} \nabla \boldsymbol{u} \nabla \boldsymbol{v} + V(\boldsymbol{x}) |\boldsymbol{u}|^{p-2} \boldsymbol{u} \boldsymbol{v}) d\boldsymbol{x}$$

$$- \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|\boldsymbol{u}(\boldsymbol{y})|^{q} |\boldsymbol{u}(\boldsymbol{x})|^{q-2} \boldsymbol{u}(\boldsymbol{x}) \boldsymbol{v}(\boldsymbol{x})}{\left| \boldsymbol{x} - \boldsymbol{y} \right|^{\mu}} d\boldsymbol{x} d\boldsymbol{y} - \int\_{\mathfrak{R}^N} \boldsymbol{g}(\boldsymbol{x}) \boldsymbol{v} d\boldsymbol{x} \cdot \boldsymbol{u}$$

for all *v* ∈ *EV*. Thus, we will constrain our functional *I* on the Nehari manifold

Λ = {*u* ∈ *EV* : *I* (*u*), *u* = 0 

Clearly, every nontrivial weak solution of problem (1) belongs to Λ. Denote Ψ(*u*) = *I* (*u*), *u* , so we can see that

$$\langle I'(\mathfrak{u}), \mathfrak{u} \rangle = \|\mathfrak{u}\|^p - \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|\mathfrak{u}(\mathfrak{x})|^q |\mathfrak{u}(\mathfrak{y})|^q}{\left|\mathfrak{x} - \mathfrak{y}\right|^\mu} dx d\mathfrak{y} - \int\_{\mathfrak{R}^N} \mathfrak{g}(\mathfrak{x}) \mathfrak{u}(\mathfrak{x}) d\mathfrak{x} \,\mathfrak{y}$$

and

$$\langle \Psi'(u), u \rangle = p \| u \|^{p} - 2q \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|u(x)|^q |u(y)|^q}{|x - y|^{\mu}} dx dy - \int\_{\mathfrak{R}^N} g(x) u(x) dx$$

Notice that, if *u*<sup>0</sup> is a local minimum solution of the functional *I*, one has

$$\langle I'(\mu\_0), \mu\_0 \rangle = 0, \qquad \langle \Psi'(\mu\_0) \ , \mu\_0 \rangle \ge 0$$

Thus, we can subdivide the Nehari manifold Λ into three parts as follows:

$$\begin{aligned} \Lambda^+ &= \{ \mathfrak{u} \in \Lambda : \langle \Psi'(\mathfrak{u}), \mathfrak{u} \rangle > 0 \} \\\\ \Lambda^- &= \{ \mathfrak{u} \in \Lambda : \langle \Psi'(\mathfrak{u}), \mathfrak{u} \rangle < 0 \} \\\\ \Lambda^0 &= \{ \mathfrak{u} \in \Lambda : \langle \Psi'(\mathfrak{u}), \mathfrak{u} \rangle = 0 \} \end{aligned}$$

Clearly, only <sup>Λ</sup><sup>0</sup> contains the element 0. It is easy to see that <sup>Λ</sup><sup>0</sup> <sup>∪</sup> <sup>Λ</sup><sup>+</sup> and <sup>Λ</sup><sup>0</sup> <sup>∪</sup> <sup>Λ</sup><sup>−</sup> are closed subsets of *EV*. In the due course of this paper, we will subsequently give reason to divide the set Λ into above three subsets.

For the convenience of calculations, for *u* ∈ *EV*, we denote

$$A := A(\boldsymbol{\mu}) = \int\_{\mathfrak{R}^N} (|\nabla \boldsymbol{\mu}|^{p-1} \nabla \boldsymbol{\mu} + V(\boldsymbol{x}) |\boldsymbol{\mu}|^{p-1} \boldsymbol{\mu}) d\boldsymbol{x} = \|\boldsymbol{\mu}\|^p$$

$$B := B(\boldsymbol{\mu}) = \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|\boldsymbol{\mu}(\boldsymbol{x})|^q |\boldsymbol{\mu}(\boldsymbol{y})|^q}{\left|\boldsymbol{x} - \boldsymbol{y}\right|^\mu} d\boldsymbol{x} d\boldsymbol{y}$$

$$\mathcal{C} := \mathcal{C}(\mathfrak{u}) = \int\_{\mathfrak{R}^N} \mathfrak{g}(\mathfrak{x}) \mathfrak{u} d\mathfrak{x}$$

For *u* ∈ *EV*, we define the fibering map ϕ : (0, +∞) → as

$$\varphi(t) := I(tu) = \frac{A}{p}t^p - \frac{B}{2q}t^{2q} - \text{C}t, \qquad t > 0 \tag{8}$$

From (8) we have

$$\varphi'(t) = \frac{1}{t} \langle l'(tu), tu \rangle = \frac{\Psi(tu)}{t} = At^{p-1} - Bt^{2q-1} - \mathcal{C} \tag{9}$$

which implies that *u* ∈ Λ if and only if ϕ (1) = 0. It is easy to see that *tu* ∈ Λ with *t* > 0 if and only if ϕ (*t*) = 0, i.e., Λ = *u* ∈ *EV* : ϕ (*t*) = 0 . Moreover,

$$\varphi''(t) = \frac{\langle \Psi'(tu), tu \rangle - \Psi(tu)}{t^2} = (p-1)At^{p-2} - (2q-1)Bt^{2q-2} \tag{10}$$

which implies that for *u* ∈ Λ, Ψ (*tu*), *tu* > 0 or < 0 if and only if ϕ(*t*) > 0 or < 0, respectively. That is to say, from the sign of ϕ(*t*) the stationary points of ϕ(*t*) can be divided into three types, namely local minimum, local maximum, and turning point. Thus, Λ<sup>±</sup> and Λ<sup>0</sup> can also be written as

$$\Lambda^{\pm} = \{ \mathfrak{t}u \in \Lambda : \varphi''(t) > 0 \text{ or } < 0 \}, \text{and } \Lambda^0 = \{ \mathfrak{u} \in \Lambda : \varphi''(t) = 0 \}$$

**Lemma 3.** *Assume that g* ≡ 0 *and satisfies* (*G*)*. Then for any u* ∈ *EV*\{0}*, there exists a unique t*<sup>1</sup> = *t*1(*u*) > 0 *such that t*1*u* ∈ Λ−*. In particular,*

$$t\_1 > \left[\frac{(p-1)A}{(2q-1)B}\right]^{1/(2q-p)} := t\_0$$

*and I*(*t*1*u*) = max*t*≥0*I*(*tu*) *for* -*<sup>N</sup> gudx* ≤ 0.

Moreover, if -*<sup>N</sup> gudx* <sup>&</sup>gt; 0, then there exist unique 0 <sup>&</sup>lt; *<sup>t</sup>*<sup>2</sup> <sup>=</sup> *<sup>t</sup>*2(*u*) <sup>&</sup>lt; *<sup>t</sup>*<sup>3</sup> <sup>=</sup> *<sup>t</sup>*3(*u*) such that *<sup>t</sup>*2*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>+. In particular, *I*(*t*3*u*) = max*t*≥*t*<sup>2</sup> *I*(*tu*), *I*(*t*2*u*) = min 0≤*t*≤*t*<sup>3</sup> *I*(*tu*).

**Proof.** Set *<sup>k</sup>*(*t*) = *Atp*−<sup>1</sup> <sup>−</sup> *Bt*2*q*−1, then <sup>ϕ</sup> (*t*) = *k*(*t*) − *C* and *k* (*t*) = ϕ(*t*) Obviously, lim*t*→0<sup>+</sup> *k*(*t*) = 0, lim*t*→+∞*k*(*t*) = −∞ and *k*(*t*) > 0 for *t* > 0 sufficiently small. Due to 2*q* > *p*, if *k* (*t*0) = 0, then *<sup>t</sup>*<sup>0</sup> = ( (*p*−1)*<sup>A</sup>* (2*q*−1)*<sup>B</sup>* ) 1/(2*q*−*p*) . Thus, we have *k* (*t*) > 0 for *t* ∈ (0, *t*0), and *k* (*t*) < 0 for *t* ∈ (*t*0, +∞).

In the case *<sup>C</sup>* = -*<sup>N</sup> <sup>g</sup>*(*x*)*udx* <sup>≤</sup> 0, there exists a unique *<sup>t</sup>*<sup>1</sup> with *<sup>t</sup>*<sup>1</sup> <sup>&</sup>gt; *<sup>t</sup>*<sup>0</sup> such that *<sup>k</sup>*(*t*1) = -*<sup>N</sup> gudx* and *k* (*t*1) < 0. Therefore,

$$\langle l'(t\_1 u), t\_1 u \rangle = At\_1^p - Bt\_1^{2q} - Ct\_1 = t\_1(At\_1^{p-1} - Bt\_1^{2q-1} - C) = t\_1(k(t\_1) - C) = 0$$

This implies *t*1*u* ∈ Λ. Moreover,

$$<\langle \Psi'(t\_1 u), t\_1 u \rangle = A p t\_1^p - 2B q t\_1^{2q} - \text{C} t\_1 = (p - 1) A t\_1^p - (2q - 1) B t\_1^{2q} = t\_1^2 k'(t\_1) < 0$$

which implies that *t*1*u* ∈ Λ−, and *I*(*t*1*u*) = max*t*≥0*I*(*tu*).

In the case *<sup>C</sup>* = -*<sup>N</sup> g*(*x*)*udx* > 0, for any *u* ∈ *E*1, where *E*<sup>1</sup> = {*u* ∈ *EV* : *u* = 1}. By the assumption (*G*) and*t*<sup>0</sup> <sup>=</sup>*t*0(*u*)=( *<sup>p</sup>*−<sup>1</sup> (2*q*−1)*B*(*u*) ) 1/(2*q*−*p*) , we have

$$\begin{split} \max\_{t\geq 0} \varphi'(t) &\qquad \geq \varphi'(\overline{t}o) = \overline{t}\_0^{p-1} - \overline{Bt}\_0^{2p-1} - \overline{C} \\ &= \left[ \frac{p-1}{(2q-1)\overline{B}} \right]^{\frac{p-1}{2q-p}} - \overline{B} \left[ \frac{p-1}{(2q-1)\overline{B}}^{\frac{2q-1}{2q-p}} - \overline{C} \right] \\ &= \left[ \frac{p-1}{(2q-1)\overline{B}} \right]^{\frac{p}{2q-p}} - \overline{B} \left[ \frac{p-1}{(2q-1)\overline{B}}^{\frac{p-1}{2q-p}} \frac{p-1}{(2q-1)\overline{B}}^{\frac{p-1}{2q-p}} - \overline{C} \right] \\ &= \left[ \frac{p-1}{(2q-1)\overline{B}} \right]^{\frac{p-1}{2q-p}} - \frac{p-1}{2q-1} \left[ \frac{p-1}{(2q-1)\overline{B}}^{\frac{p-1}{2q-p}} - \overline{C} \right] \\ &\geq \frac{(p-1)^{\frac{p-1}{2q-p}} (2q-p)}{(2q-1)\frac{2q-p}{2q-p} \frac{p-1}{2q-p}} - \left[ \ $g\right]\_{\frac{2Nq}{2N(q-1)+\mu}}^{\frac{p-1}{2q-p}} \frac{S\_{\frac{2Nq}{2N-p}}}{2N-p} \\ &\geq (a-\left[ \$ g\right]\_{\frac{2Nq}{2N(q-1)+\mu}}^{\frac{2Nq}{2N-p}}) S\_{\frac{2Nq}{2N-p}} > 0 \end{split} \tag{11}$$

where

$$B\_0 = \sup\_{\|\mathbf{u}\|=1} \int\_{\mathbb{R}^N} \int\_{\mathbb{R}^N} \frac{|u(x)|^q |u(y)|^q}{|\mathbf{x} - y|^\mu} dx dy,\text{ and}\\a = a(N, p, q, \mu, S\_{\frac{2Nq}{2N-p}}) := \frac{(p-1)^{\frac{p-1}{2q-p}} (2q-p)!}{(2q-1)^{\frac{2q-1}{2q-p}} B\_0^{\frac{p-1}{2q-p}} S\_{\frac{2Nq}{2N-p}}}$$

From (27), we have for *u* ∈ *E*1,

$$\lim\_{t \to 0^+} k(t) = 0 < \int\_{\mathfrak{R}^N} \operatorname{g}(\mathbf{x}) \mathsf{u} d\mathbf{x} \le \left| \operatorname{g} \right|\_{\frac{2Nq}{2N(q-1)+\mu}} S\_{\frac{2Nq}{2N-\mu}} \left\| \mathbf{u} \right\| = \left| \operatorname{g} \right|\_{\frac{2Nq}{2N(q-1)+\mu}} S\_{\frac{2Nq}{2N-\mu}} \le k(\overline{t\_0}) < \infty$$

Hence, there exist unique 0 < *t*<sup>2</sup> = *t*2(*u*) < *t*<sup>0</sup> < *t*<sup>3</sup> = *t*3(*u*) such that

$$k(t\_2) = \int\_{\mathfrak{R}^N} g(\mathbf{x}) \, d\mathbf{x} = k(t\_3) \text{ and } k'(t\_3) < 0 < k'(t\_2)$$

Consequently, *<sup>t</sup>*2*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup><sup>+</sup> and *<sup>t</sup>*3*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup><sup>−</sup> It is easy to see that *<sup>d</sup> dtI*(*tu*) = ϕ (*t*) = 0 for *t* = *t*<sup>2</sup> or *t* = *t*3, and ϕ(*t*) > 0 for*t* ∈ (0,*t*0) and ϕ(*t*) < 0 for*t* ∈ (*t*0, +∞). Then *I*(*t*3*u*) = max *t*≥*t*<sup>2</sup> *I*(*tu*), *I*(*t*2*u*) = min 0≤*t*≤*t*<sup>3</sup> *I*(*tu*). This proof is completed. -

**Lemma 4.** *For g* <sup>≡</sup> <sup>0</sup>*, the condition* (*G*) *is satisfied, then* <sup>Λ</sup><sup>0</sup> = {0}*.*

**Proof.** To prove <sup>Λ</sup><sup>0</sup> = {0}, we need to show that for any *<sup>u</sup>* <sup>∈</sup> *EV*\{0}, the fibering map <sup>ϕ</sup>(*t*) has no critical point that is a turning point. For any *<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>−, set *<sup>u</sup>* <sup>=</sup> *<sup>u</sup>*( *<sup>u</sup>* ) −1 , then *<sup>u</sup>* <sup>∈</sup> *<sup>E</sup>*1. By the proof of Lemma 3, *<sup>k</sup>*(*t*) has a unique global maximum point *<sup>t</sup>*<sup>0</sup> = ( *<sup>p</sup>*−<sup>1</sup> (2*q*−1)*B*(*u*)) 1/(2*q*−*p*) , and

$$k(\tilde{t}\_0) = \frac{(2q-p)}{2q-1} \left[ \frac{p-1}{(2q-1)B(\overline{u})} \right]^{\frac{p-1}{2q-p}} := k\_0.$$

According to (8)–(10), we deduce that if 0 <sup>&</sup>lt; *<sup>C</sup>*(*u*) <sup>&</sup>lt; *<sup>k</sup>*0, the equation <sup>ϕ</sup> (*t*) = 0 has exactly two roots *<sup>t</sup>*1,*t*<sup>2</sup> satisfying 0 <sup>&</sup>lt; *<sup>t</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>t</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>t</sup>*<sup>2</sup> and if *<sup>C</sup>*(*u*) <sup>≤</sup> 0, <sup>ϕ</sup> (*t*) = 0 has only one point *t*<sup>3</sup> such that *t*<sup>3</sup> > *t*0. Since ϕ(*t*) = *k* (*t*), we have <sup>ϕ</sup>(*t*1) <sup>&</sup>gt; 0, <sup>ϕ</sup>(*t*2) <sup>&</sup>lt; 0 and <sup>ϕ</sup>(*t*3) <sup>&</sup>lt; 0. Hence, if 0 <sup>&</sup>lt; *<sup>C</sup>*(*u*) <sup>&</sup>lt; *<sup>k</sup>*0, then *<sup>t</sup>*1*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>+, *<sup>t</sup>*2*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup><sup>−</sup> and if *<sup>C</sup>*(*u*) <sup>≤</sup> 0, then *<sup>t</sup>*3*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>−. This implies <sup>Λ</sup><sup>±</sup> <sup>∩</sup> *<sup>u</sup>* <sup>∈</sup> *EV* : *<sup>u</sup>* <sup>∈</sup> *<sup>E</sup>*1, 0 <sup>&</sup>lt; *<sup>C</sup>*(*u*) <sup>&</sup>lt; *<sup>k</sup>*<sup>0</sup> - <sup>∅</sup> and <sup>Λ</sup>−∩ {*<sup>u</sup>* <sup>∈</sup> *EV* : *<sup>u</sup>* <sup>∈</sup> *<sup>E</sup>*1,*C*(*u*) <sup>≤</sup> <sup>0</sup> - ∅. As a consequence, we infer that Λ<sup>±</sup> are nonempty. It is easy to see that for any sign of *<sup>C</sup>*(*u*), critical point of the fibering map <sup>ϕ</sup>(*t*) is either a point of local

maximum or local minimum which implies <sup>Λ</sup><sup>0</sup> <sup>=</sup> {0}. Therefore, it remains to show that *<sup>k</sup>*<sup>0</sup> <sup>&</sup>gt; *<sup>C</sup>*(*u*). By the condition (*G*) and Lemma 3 we have

$$k\_0 - \mathcal{C}(\overleftarrow{\boldsymbol{\mu}}) = k(\overleftarrow{t}\_0) - \mathcal{C}(\overleftarrow{\boldsymbol{\mu}}) = \overleftarrow{\boldsymbol{\tilde{r}}}\_0^{p-1} - B(\overleftarrow{\boldsymbol{\mu}}) \boldsymbol{\tilde{t}}\_0^{2q-1} - \mathcal{C}(\overleftarrow{\boldsymbol{\mu}}) > 0$$

This completes the proof. -

**Lemma 5.** *Assume the condition* (*G*) *holds, then* Λ− *is closed.*

**Proof.** Let *cl*(Λ−) denote the closure of Λ−. Due to *cl*(Λ−) ⊂ Λ<sup>−</sup> ∪ {0}, it is sufficient to prove that <sup>0</sup> *cl*(Λ−) or equivalently the distance *dist*(*u*, <sup>Λ</sup>−) <sup>&</sup>gt; 0. Set *<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup><sup>−</sup> and denote *<sup>u</sup>* <sup>=</sup> *<sup>u</sup>*( *<sup>u</sup>* ) −1 , then *<sup>u</sup>* <sup>∈</sup> *<sup>E</sup>*1. Under the assumption (*G*) and the proof of Lemma 4, one has

$$C(\overline{u}) < \overline{I}\_0^{p-1} - B(\overline{u}) \overline{I}\_0^{2q-1} = \left[ \frac{p-1}{(2q-1)B(\overline{u})} \right]^{\frac{p-1}{2q-p}} - B(\overline{u}) \left[ \frac{p-1}{(2q-1)B(\overline{u})} \right]^{\frac{2q-1}{2q-p}} = \frac{2q-p}{2q-1} \cdot \left[ \frac{p-1}{(2q-1)B(\overline{u})} \right]^{\frac{p-1}{2q-p}} = k\_0 \tag{12}$$

Moreover, we have that if *<sup>C</sup>*(*u*) <sup>≤</sup> 0 then <sup>ϕ</sup> (*t*) = 0 has only one point *<sup>t</sup>*<sup>3</sup> <sup>&</sup>gt; *<sup>t</sup>*<sup>0</sup> such that *<sup>t</sup>*3*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>−. Then we have *<sup>t</sup>*3*<sup>u</sup>* <sup>=</sup> *<sup>u</sup>* with *u* <sup>=</sup> *<sup>t</sup>*<sup>3</sup> <sup>&</sup>gt; *<sup>t</sup>*<sup>0</sup> . Also, if 0 <sup>&</sup>lt; *<sup>C</sup>*(*u*) <sup>&</sup>lt; *<sup>k</sup>*0, the equation <sup>ϕ</sup> (*t*) = 0 has exactly two roots *<sup>t</sup>*1,*t*<sup>2</sup> with 0 <sup>&</sup>lt; *<sup>t</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>t</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>t</sup>*<sup>2</sup> such that *<sup>t</sup>*1*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup><sup>+</sup> and *<sup>t</sup>*2*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>−. Hence, we have *<sup>t</sup>*2*<sup>u</sup>* <sup>=</sup> *<sup>u</sup>* and *u* <sup>=</sup> *<sup>t</sup>*<sup>2</sup> <sup>&</sup>gt; *<sup>t</sup>*<sup>0</sup> . In a word, for any *<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>−, we get *u* <sup>&</sup>gt; *<sup>t</sup>*<sup>0</sup> . By (7) we know that *<sup>B</sup>*(*u*) is bounded from above. It follows from definition of *t*<sup>0</sup> that

$$\overline{t}\_0 = \left[\frac{p-1}{(2q-1)B(\overline{u})}\right]^{1/(2q-p)} \ge \left[\frac{p-1}{(2q-1)\overline{B}\_0}\right]^{1/(2q-p)} := \tau \tag{13}$$

where

$$\widetilde{B}\_0 = \sup\_{||\widetilde{u}|| = 1} \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|\widetilde{u}(\boldsymbol{x})|^q |\widetilde{u}(\boldsymbol{y})|^q}{\left| \boldsymbol{x} - \boldsymbol{y} \right|^\mu} d\boldsymbol{x} d\boldsymbol{y}$$

which implies that *dist*(*u*, <sup>Λ</sup>−) = inf*u*∈Λ<sup>−</sup> *u* <sup>≥</sup> τ > 0. Hence *cl*(Λ−) = <sup>Λ</sup><sup>−</sup> and this proves the Lemma. -

**Lemma 6.** *Assume* (*A*0) *and* (*G*) *hold. Then the functional I*(*u*) *is coercive and bounded below on* Λ *Thus I*(*u*) *is bounded below on* Λ<sup>+</sup> *and* Λ−.

**Proof.** Let *<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>, from *I* (*u*), *u* = 0 and (7) we derive that

$$\begin{array}{ll} I(u) &= \frac{1}{p} \int\_{\mathfrak{R}^N} \left( |\nabla u|^p + V(\mathbf{x}) |u|^p \right) d\mathbf{x} - \frac{1}{2q} \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|u(x)|^p |u(y)|^q}{|x - y|^p} dx dy - \int\_{\mathfrak{R}^N} \mathbf{g}(\mathbf{x}) u d\mathbf{x} \\ &= \frac{1}{p} A(u) - \frac{1}{2q} B(u) - C(u) \\ &= \left( \frac{1}{p} - \frac{1}{2q} A(u) - (1 - \frac{1}{2q}) \int\_{\mathfrak{R}^N} \mathbf{g}(\mathbf{x}) u d\mathbf{x} \\ &\geq \frac{2q - p}{2pq} \|u\|^p - \frac{2q - 1}{2q} \left| \mathbf{g} \right|\_{\frac{2Nq}{2N(q-1) + \mu}} \mathop{\mathcal{S}}\_{\frac{2Nq}{2N - \mu}} \|u\| \| \end{array} \tag{14}$$

where *S* <sup>2</sup>*Nq* 2*N*−μ denotes the best constant of the embedding from *EV* into *L* 2*Nq* <sup>2</sup>*N*−<sup>μ</sup> (-*<sup>N</sup>*). It is to see that *I* is coercive and bounded below in the manifold Λ. This completes the proof. -

#### **3. Minimization Problems and Palais-Smale Analysis**

According to Lemma 6, we can define the following two minimization problems:

$$\bar{u}^- := \inf\_{\mathfrak{u} \in \Lambda^-} I(\mathfrak{u}) \tag{15}$$

$$\dot{u}^+ := \inf\_{u \in \Lambda^+} I(u) \tag{16}$$

Clearly, if the infimum of (15) and (16) are achieved, then we can show that they produce a weak solution of our problem (1).

**Lemma 7.** *If u is a local minimizers of I on* Λ<sup>+</sup> *and* Λ<sup>−</sup> *respectively, then I* (*u*) = 0*.*

**Proof.** If *u* is a local minimizers of *I* on Λ±, then ∇(*I <sup>N</sup>*<sup>±</sup> )(*u*) = 0. Using Theorem 4.1.1 of [46] we infer that there exists Lagrangian multiplier λ ∈ such that

$$\langle I'(\mathfrak{u}), \mathfrak{u} \rangle = \lambda \langle \Psi(\mathfrak{u}), \mathfrak{u} \rangle$$

Since *u* ∈ Λ±, *I* (*u*), *u* = 0 and Ψ(*u*), *u* - 0. This implies λ = 0. Thus *u* is a nontrivial weak solution of our problem (1). -

By Lemma 6 we know that the problem of investigating solutions of problem (1) can be translated into that of studying minimizers of (15) and (16).

**Lemma 8.** *Assume* (*A*0) *and* (*G*) *are satisfied. Then the functional I*(*u*) *satisfies (PS)c condition with c* ∈ -. *That is, if* {*un*} *is a sequence in EV satisfying*

$$I(u\_n) \to c \text{ and } I'(u\_n) \to 0, \text{ as } n \to +\infty \tag{17}$$

*for some c* ∈ -*, then* {*un*} *possesses a convergent subsequence.*

**Proof.** If {*un*} be a sequence in *EV* satisfies (17), then similar to Lemma 6 we get that *un* is bounded in *EV*. Since *EV* is reflexive Banach space, up to a subsequence, we may assume that *un* weakly converges to *u* in *EV*. By using compact embedding of *EV i*n *L<sup>r</sup>* (-*<sup>N</sup>*) for *r* ∈ [*p*, *p*<sup>∗</sup> ), *un* strongly converges to *u* in *Lr* (-*<sup>N</sup>*). Since *<sup>q</sup>* <sup>∈</sup> (*ql*, *qu*) and *<sup>p</sup>* <sup>&</sup>lt; <sup>2</sup>*Nq* <sup>2</sup>*N*−<sup>μ</sup> <sup>&</sup>lt; *<sup>p</sup>*<sup>∗</sup> , it follows from Hardy-Littlewood-Sobolev inequality that

$$\int\_{\Re^N} \int\_{\Re^N} \frac{|u\_n(y)|^q |u\_n(\mathbf{x})|^{q-2} u\_n(\mathbf{x}) [u\_n(\mathbf{x}) - u(\mathbf{x})]}{|\mathbf{x} - \mathbf{y}|^\mu} d\mathbf{x} d\mathbf{y} \le \mathbb{C}(N, \mu) |u\_n|^{\frac{2q-1}{2N-\mu}}\_{\frac{2Nq}{2N-\mu}} |u\_n - \mathbf{u}|\_{\frac{2Nq}{2N-\mu}} \to 0$$

as *n* → ∞. Then, we also get

$$\int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|u\_n(y)|^q \left| u\_n(\mathbf{x}) \right|^{q-2} u\_n(\mathbf{x}) \left[ u\_n(\mathbf{x}) - u(\mathbf{x}) \right]}{\left| \mathbf{x} - y \right|^\mu} d\mathbf{x} dy \to 0, \qquad n \to \infty$$

Thus

$$\begin{split} o(1) &= \langle I'(\boldsymbol{\mu}\_{\mathrm{il}}) - I'(\boldsymbol{\mu}\_{\mathrm{i}}), \boldsymbol{\mu}\_{\mathrm{il}} - \boldsymbol{\mu} \rangle \\ &= ||\boldsymbol{\mu}\_{\mathrm{il}} - \boldsymbol{\mu}||^{p} - \int\_{\mathfrak{R}^{N}} \int\_{\mathfrak{R}^{N}} \frac{|\boldsymbol{\mu}\_{\mathrm{u}}(\boldsymbol{y})|^{q} |\boldsymbol{\mu}\_{\mathrm{u}}(\boldsymbol{x})|^{q-2} \boldsymbol{\mu}\_{\mathrm{u}}(\boldsymbol{x}) [\boldsymbol{\mu}\_{\mathrm{n}}(\boldsymbol{x}) - \boldsymbol{\mu}(\boldsymbol{x})]}{\left\| \boldsymbol{x} - \boldsymbol{y} \right\|^{\mu}} d\boldsymbol{x} d\boldsymbol{y} \\ &+ \int\_{\mathfrak{R}^{N}} \int\_{\mathfrak{R}^{N}} \frac{|\boldsymbol{\mu}(\boldsymbol{y})|^{q} |\boldsymbol{\mu}(\boldsymbol{x})|^{q-2} \boldsymbol{\mu}(\boldsymbol{x}) [\boldsymbol{\mu}\_{\mathrm{n}}(\boldsymbol{x}) - \boldsymbol{u}(\boldsymbol{x})]}{\left\| \boldsymbol{x} - \boldsymbol{y} \right\|^{\mu}} d\boldsymbol{x} d\boldsymbol{y} \\ &= \left\| \boldsymbol{\mu}\_{\mathrm{n}} - \boldsymbol{u} \right\|\_{\mathcal{P}} + o(1) \end{split}$$

which implies that *un* → *u* in *EV* and consequently ends the proof. -

The following result is an observation regarding the minimizers of Λ<sup>+</sup> and Λ−.

**Lemma 9.** *Assume* (*A*0) *and* (*G*) *are satisfied. Then i*<sup>+</sup> < 0 *and i*<sup>−</sup> > 0*.*

**Proof.** Let *<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>, by the proof of Lemma 4 we have that if 0 <sup>&</sup>lt; *<sup>C</sup>*(*u*) <sup>&</sup>lt; *<sup>k</sup>*<sup>0</sup> corresponding to *<sup>u</sup>* <sup>=</sup> *<sup>u</sup>*( *u* ) <sup>−</sup><sup>1</sup> <sup>∈</sup> *<sup>E</sup>*1, then <sup>ϕ</sup> (*t*) = 0 has exactly two roots *<sup>t</sup>*1,*t*<sup>2</sup> such that 0 <sup>&</sup>lt; *<sup>t</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>t</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>t</sup>*<sup>2</sup> and *<sup>t</sup>*1*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup><sup>+</sup> and *<sup>t</sup>*2*<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>−. Since <sup>ϕ</sup> (*t*) = *t <sup>p</sup>*−<sup>1</sup> <sup>−</sup> *<sup>B</sup>*(*u*)*<sup>t</sup>* <sup>2</sup>*q*−<sup>1</sup> <sup>−</sup> *<sup>C</sup>*(*u*), we get that lim*t*→0<sup>+</sup> <sup>ϕ</sup> (*t*) = <sup>−</sup>*C*(*u*) <sup>&</sup>lt; 0 and ϕ(*t*) > 0 for any *t* ∈ (0,*t*0). Due to *t*<sup>1</sup> is point of local minimum of ϕ(t) and *t*<sup>1</sup> > 0, we have that ϕ(*t*1) < lim*t*→0<sup>+</sup> ϕ(*t*) = 0 and then *i* <sup>+</sup> <sup>≤</sup> *<sup>I</sup>*(*t*1*u*) = <sup>ϕ</sup>(*t*1) <sup>&</sup>lt; 0. Moreover *<sup>i</sup>* :<sup>=</sup> inf*u*∈Λ*I*(*u*) <sup>≤</sup> inf*u*∈Λ<sup>+</sup> *<sup>I</sup>*(*u*) = *i* <sup>+</sup> < 0.

Now we claim that *i* <sup>−</sup> <sup>&</sup>gt; 0. In fact, from (7) we know that *<sup>B</sup>* <sup>≤</sup> *<sup>M</sup>*0||*u*||2*<sup>q</sup>* , where *M*<sup>0</sup> = *C*(*N*, μ)*S* 2*q* 2*Nq* 2*N*−μ . This implies that there is a positive constant *M*<sup>1</sup> which is independent of *u* such that

$$\frac{\left(\left\|\mu\right\|\left|\mu\right|\right)^{2q/(2q-p)}}{B^{p/(2q-p)}} = \frac{A^{2q/(2q-p)}}{B^{p/(2q-p)}} \ge M\_1 \tag{18}$$

By the given assumption and (18) we discuss ϕ(*t*0) corresponding to *u* as

$$\begin{split} q(t\_0) &= \frac{1}{p} \Big| \mu \Big|\_{p} \Big|\_{p} t\_0^p - \frac{1}{2q} B t\_0^{2q} - \mathsf{C} t\_0 \\ \leq \frac{A}{p} \Big[ \frac{(p-1)A}{(2q-1)B} \Big]^{p/(2q-p)} - \frac{B}{2q} \Big[ \frac{(p-1)A}{(2q-1)B} \Big]^{2q/(2q-p)} - \mathsf{C} \Big[ \frac{(p-1)A}{(2q-1)B} \Big]^{1/(2q-p)} \\ &= \frac{(2q-p)(2q+p-1)}{2pq(2q-1)} \cdot \frac{A^{2q/(2q-p)}}{B^{p/(2q-p)}} - \mathsf{C} \Big( \frac{p-1}{2q-1} \Big)^{1/(2q-p)} \cdot \frac{A^{1/(2q-p)}}{B^{1/(2q-p)}} \\ &\geq \frac{(2q-p)(2q+p-1)}{2pq(2q-1)} \cdot \frac{A^{2q/(2q-p)}}{B^{p/(2q-p)}} \\ &\geq \frac{(2q-p)(2q+p-1)}{2pq(2q-1)} \cdot M\_1 := M\_\* \end{split}$$

where the positive constant *M*<sup>∗</sup> is independent of *u*. Hence,

$$\mu^- = \inf\_{\mu \in \Lambda \backslash \{0\}} \max \{ I(\mu) \} \ge \inf\_{\mu \in \Lambda \backslash \{0\}} \wp(t\_0) \ge M\_\* > 0.$$

This completes the proof. -

Now we study the nature of minimizing sequences for the functional *I*(*u*). Using the idea of [44] to obtain a (*PS*)*<sup>i</sup>* <sup>+</sup> sequence from the minimization sequence of our problem (1). The following lemma is a consequence of Lemma 4.

**Lemma 10.** *Assume* (*A*0) *and* (*G*) *hold. Then for <sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>+*, there exists a constant* ρ > <sup>0</sup> *and a di*ff*erentiable function* <sup>η</sup><sup>+</sup> : *<sup>B</sup>*(0, <sup>ρ</sup>) → -<sup>+</sup> := (0, <sup>+</sup>∞) *such that* <sup>η</sup>+(0) = <sup>1</sup>*,* <sup>η</sup>+(*w*)(*<sup>u</sup>* <sup>−</sup> *<sup>w</sup>*) <sup>∈</sup> <sup>Λ</sup>+*, and*

$$\begin{split} \langle (\eta^{+})'(0), w \rangle &= M^\*[p \int\_{\mathbb{R}^N} (|\nabla u|^{p-2} \nabla u \nabla w + V(\mathbf{x}) |u|^{p-2} u w) dx \\ &- 2q \int\_{\mathbb{R}^N} \int\_{\mathbb{R}^N} \frac{|u(y)|^q |u(x)|^{q-2} u(x) w(x)}{|\mathbf{x} - y|^\mu} dx dy - \int\_{\mathbb{R}^N} g w dx] \end{split} \tag{19}$$

*for any w* ∈ *B*(0, ρ)*, where B*(0, ρ) *denotes the ball centered at 0 with radius* ρ*, and M*<sup>∗</sup> = [(*p* − 1) *u <sup>p</sup>* <sup>−</sup> (2*<sup>q</sup>* <sup>−</sup> <sup>1</sup>)*B*(*u*)]−<sup>1</sup> *.*

**Proof.** Fixing a function *<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>+, we define a *<sup>C</sup>*<sup>1</sup> mapping <sup>Φ</sup> : - × *EV* → as follows

$$\Phi(t, w) = t^{p-1} \|u - w\|^p - t^{2q-1} \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{\left| (u - w) \left( y \right) \right|^q \left| (u - w)(x) \right|^q}{\left| x - y \right|^\mu} dx dy - \int\_{\mathfrak{R}^N} g(x) (u - w) dx$$

Notice that Φ(1, 0) = *I* (*u*), *u* = 0. Moreover

$$\Phi(t,0) = At^{p-1} - t^{2q-1}B - \mathbb{C} = q^\prime(t)$$

where <sup>ϕ</sup> is the fibering map defined in (8). Since *<sup>u</sup>* <sup>∈</sup> <sup>Λ</sup>+, we have <sup>ϕ</sup>(1) <sup>&</sup>gt; 0, and then so Φ*t*(1, 0) = ϕ(1) - 0. By Applying the implicit function theorem at point (1,0), we get that there is <sup>ρ</sup> = <sup>ρ</sup>(*u*) <sup>&</sup>gt; 0 and a differentiable function <sup>η</sup><sup>+</sup> : *<sup>B</sup>*(0, <sup>ρ</sup>) → -<sup>+</sup> such that <sup>η</sup>+(0) = 1, <sup>η</sup>+(*w*)(*<sup>u</sup>* <sup>−</sup> *<sup>w</sup>*) <sup>∈</sup> <sup>Λ</sup> for any *w* ∈ *B*(0, ρ), and

$$\langle (\eta^+)'(0), w \rangle = -\frac{\langle \Phi\_{\text{w}}(1,0), w \rangle}{\Phi\_t(1,0)}$$

Now we only show that <sup>η</sup>+(*w*)(*<sup>u</sup>* <sup>−</sup> *<sup>w</sup>*) <sup>∈</sup> <sup>Λ</sup><sup>+</sup> for any *<sup>w</sup>* <sup>∈</sup> *<sup>B</sup>*(0, <sup>ρ</sup>). In fact, from Lemma 5 it follows that <sup>Λ</sup><sup>−</sup> <sup>∪</sup> <sup>Λ</sup><sup>0</sup> is closed, then the distance *dist*(*u*, <sup>Λ</sup><sup>−</sup> <sup>∪</sup> <sup>Λ</sup>0) <sup>&</sup>gt; 0. Since the function <sup>η</sup>+(*w*)(*<sup>u</sup>* <sup>−</sup> *<sup>w</sup>*) is continuous with respect to *w*, taking ρ = ρ(*u*) > 0 sufficiently small, such that

$$\|\|\eta^+(w)(u-w)-u\|\| < \frac{1}{4}dist(u, \Lambda^- \cup \Lambda^0), \forall w \in B(0,\rho).$$

Then <sup>η</sup>+(*w*)(*<sup>u</sup>* <sup>−</sup> *<sup>w</sup>*) does not belong to <sup>Λ</sup><sup>−</sup> <sup>∪</sup> <sup>Λ</sup>0. Thus <sup>η</sup>+(*w*)(*<sup>u</sup>* <sup>−</sup> *<sup>w</sup>*) <sup>∈</sup> <sup>Λ</sup>+. Finally, (19) can be obtained by direct differentiating Φ(*w*, η+(*w*)) = 0 with respect to *w*.

This completes the proof. -

To derive a sequence (*PS*)*<sup>i</sup>* <sup>−</sup> from the minimizing sequence of our problem (1), similar to Lemma 10 we can obtain the following proposition.

**Proposition 1.** *If* (*A*0) *and* (*G*) *are satisfied. Then for u* ∈ Λ−*, there exists a constant* ρ > 0 *and a di*ff*erentiable function* η<sup>−</sup> : *B*(0, ρ) → -<sup>+</sup> *such that* η−(0) = 1*,* η−(*w*)(*u* − *w*) ∈ Λ−*, and*

$$\begin{cases} (\eta^{-})'(0), w \rangle = \mathcal{M}^\*[p \int\_{\mathfrak{R}^N} (|\nabla u|^{p-2} \nabla u \nabla w + V(\mathbf{x}) |u|^{p-2} u w) dx \\ \qquad - 2q \int\_{\mathfrak{R}^N} \int\_{\mathfrak{R}^N} \frac{|u(y)|^q |u(x)|^{q-2} u(x) w(x)}{|\mathbf{x} - \mathbf{y}|^{\mathfrak{l}}} d\mathbf{x} dy - \int\_{\mathfrak{R}^N} g w d\mathbf{x}] \end{cases}$$

*for any w* ∈ *B*(0, ρ)*, and M*<sup>∗</sup> = [(*p* − 1) *u <sup>p</sup>* <sup>−</sup> (2*<sup>q</sup>* <sup>−</sup> <sup>1</sup>)*B*(*u*)]−<sup>1</sup> *.*

**Lemma 11.** *If* (*A*0) *and* (*G*) *are satisfied. There exists a positive constant M such that*

$$-\frac{(2q-p)(p-1)}{2pq}\theta^{\frac{p}{p-1}} \le i = \inf\_{u \in \Lambda} I(u) \le -\frac{(2pq-2q-p)(2q-p)}{4pq^2} \cdot M \tag{20}$$

*where* θ = <sup>2</sup>*q*−<sup>1</sup> 2*q*−*p g* <sup>2</sup>*Nq* 2*N*(*q*−1) *S* <sup>2</sup>*Nq* 2*N*−μ

**Proof.** For any *u* ∈ Λ, According to (13) one has

$$I(u) \geq \frac{2q - p}{2pq} \|u\|^p - \frac{2q - 1}{2q} \|g\|\_{\frac{2\mathcal{N}q}{2\mathcal{N}(q-1) + \mu}} S\_{\frac{2\mathcal{N}q}{2\mathcal{N} - \mu}} \|u\| \geq -\frac{(2q - p)(p - 1)}{2pq} \theta^{\frac{p}{p - 1}}$$

Thus,

$$i \ge -\frac{(2q-p)(p-1)}{2pq} \theta^{\frac{p}{p-1}}$$

On the other hand, set *u*<sup>0</sup> ∈ Λ be the unique solution of the following equation

$$-\Delta\_p u + V(\mathfrak{x}) \Big| u \big|\_{p-1} u = \mathfrak{g}(\mathfrak{x}), \qquad \forall \mathfrak{x} \in \mathfrak{R}^N$$

Due to *g* - 0, -*<sup>N</sup> g*(*x*)*u*0*dx* = ||*u*0||*<sup>p</sup>* > 0. Then by Lemma 4, there exists *t*<sup>1</sup> > 0 such that *<sup>t</sup>*1*u*<sup>0</sup> <sup>∈</sup> <sup>Λ</sup>+. Therefore,

$$\begin{array}{l} I(t\_1 \mu\_0) = \frac{1-p}{p} \|\mu\_0\|\|^p t\_1^p + \frac{2q-1}{2q} t\_1^{2q} B(\mu\_0) \\ < \frac{1-p}{p} \|\mu\_0\|\|^p t\_1^p + \frac{p(2q-1)}{4q^2} t\_1^p \|\mu\_0\|\|^p \\ = -\frac{(2pq-2q-p)(2q-p)}{4pq^2} t\_1^p \|\|\mu\_0\|\|^p < 0 \end{array}$$

Choose *M* = *t p* 1 *u*0 *<sup>p</sup>* we obtain the result. -

**Lemma 12.** *If* (*A*0) *and* (*G*) *are satisfied, then there exists a sequence* {*un*} <sup>⊂</sup> <sup>Λ</sup><sup>+</sup> *such that <sup>I</sup>*(*un*) <sup>→</sup> *<sup>i</sup>* <sup>+</sup> *and I* (*un*) → 0 *as n* → ∞.

**Proof.** From Lemma 6, we already show that *<sup>I</sup>* is bounded from below on <sup>Λ</sup>, and <sup>Λ</sup><sup>+</sup> <sup>∪</sup> {0} is closed in Λ. Obviously Ekeland's variational principle (see [44]) applies to the minimization problem (16). It admits a minimizing sequence {*un*} <sup>⊂</sup> <sup>Λ</sup><sup>+</sup> such that

(i) *<sup>I</sup>*(*un*) <sup>&</sup>lt; inf*u*∈Λ+∪{0} *I*(*u*) + <sup>1</sup> *<sup>n</sup>* , and (ii) *<sup>I</sup>*(*w*) <sup>≥</sup> *<sup>I</sup>*(*un*) <sup>−</sup> <sup>1</sup> *n w* − *un* , <sup>∀</sup>*<sup>w</sup>* <sup>∈</sup> <sup>Λ</sup><sup>+</sup> <sup>∪</sup> {0} Then by (i) we have

$$I(u) = \frac{2q - p}{p} \|u\_n\|^p - \frac{2q - 1}{2q} \int\_{\mathcal{R}^N} g(x) u\_n dx < i + \frac{1}{n} \tag{21}$$

for *n* large enough. This together with Lemma 11 shows

$$\int\_{\mathfrak{R}^N} g(x)u\_{\mathfrak{R}} dx \ge \frac{(2pq - 2q - p)(2q - p)}{2pq(2q - 1)}M > 0\tag{22}$$

which implies *un* - 0 for any *<sup>n</sup>*. By Lemma 4, we know *<sup>i</sup>* <sup>≤</sup> inf*u*∈Λ<sup>+</sup> *<sup>I</sup>*(*u*) = *<sup>i</sup>* <sup>+</sup> < 0. Notice that *I*(0) = 0, then inf*u*∈Λ+∪{0} *I*(*u*) = *i* <sup>+</sup>. Hence *<sup>I</sup>*(*un*) <sup>→</sup> *<sup>i</sup>* <sup>+</sup> as *<sup>n</sup>* → ∞, and we can assume that *un* <sup>∈</sup> <sup>Λ</sup>+. Then ||*un*||*<sup>p</sup>* <sup>=</sup> *<sup>B</sup>*(*un*) + *<sup>C</sup>*(*un*). Furthermore, we deduce from (13) and (i) that

$$\ln i^{+} + \frac{1}{n} \ge I(u\_n) \ge \frac{2q - p}{2pq} \|u\|^p - \frac{2q - 1}{2q} \|g\|\_{\frac{2Nq}{2N(q-1) + \mu}} S\_{\frac{2Nq}{2N - \mu}} \|u\| \tag{23}$$

which implies that {*un*} is bounded. Now we claim that inf*n*||*un*||≥ ξ > 0 for some constant ξ. In fact, if not, by (23), *I*(*un*) → 0, as *n* → ∞. Using (23) which is a contradiction to first assertion. Therefore, there exist positive constants ξ<sup>2</sup> > ξ<sup>1</sup> such that

$$
\mathbb{Z}\_1 \le \|u\_n\| \le \mathbb{Z}\_2 \tag{24}
$$

Now to finish the proof, we only need to show that *I* (*un*) → 0, as *n* → ∞. By Lemma 10, for each *n*, we get a differentiable function η<sup>+</sup> *<sup>n</sup>* : *B*(0, ε) → -+ for ε > 0 as follows

$$\eta\_n^+ (\delta) := \eta\_n^+ (\delta h\_n)\_\prime - \varepsilon < \delta < \varepsilon\_n$$

where *hn* = *<sup>I</sup>* (*un*) ||*I*(*un*)||. According to Lemma 10, we get <sup>η</sup><sup>+</sup> *<sup>n</sup>* (0) = 1, and

$$w\_{\delta} := \eta\_n^+ (\delta) \left[ u\_n - \delta h\_n \right] \in \Lambda^+.$$

By Taylor's expansion and (ii), since *<sup>w</sup>*<sup>δ</sup> <sup>∈</sup> <sup>Λ</sup><sup>+</sup> we have

$$\begin{aligned} \frac{1}{n} \left\| w\_{\delta} - u\_{n} \right\| &\geq I(u\_{\mathfrak{n}}) - I(w\_{\delta}) \\ = (1 - \eta\_{n}^{+}(\delta)) \left\langle l'(w\_{\delta}), u\_{\mathfrak{n}} \right\rangle + \delta \eta\_{n}^{+}(\delta) \langle l'(w\_{\delta}), h\_{\mathfrak{n}} \rangle + o(\left\| w\_{\delta} - u\_{\mathfrak{n}} \right\|) \end{aligned}$$

*Mathematics* **2019**, *7*, 871

which implies

$$(\frac{1}{n} + o(1)) \|w\_{\delta} - u\_n\| \ge (1 - \eta\_n^+(\delta)) \langle l'(w\_{\delta}), u\_n \rangle + \delta \eta\_n^+(\delta) \langle l'(w\_{\delta}), h\_n \rangle \tag{25}$$

Dividing (25) by δ for δ -0 and passing to the limit as δ → 0, we obtain

$$(\frac{1}{n} + o(1))(1 + \left| (\eta\_n^+)'(0) \right| \| u\_n \|) \ge -(\eta\_n^+)'(0) \langle l'(u\_n), u\_n \rangle + \| l'(u\_n) \|\tag{26}$$

Since *un* <sup>∈</sup> <sup>Λ</sup>+, it follows from (26) that

$$\|\|{l'(\mu\_n)}\|\le \left(\frac{1}{n} + o(1)\right) (1 + \left| (\eta\_n^+)'(0) \right| \cdot \|\|{\mu\_n}\|\|\right) \tag{27}$$

From (24) we know that *un* is bounded. Then it remains to prove that (η<sup>+</sup> *<sup>n</sup>* ) (0) is uniformly bounded with respect to *n*. In fact, according to the definition of η<sup>+</sup> *<sup>n</sup>* and Lemma 5, we have

$$\begin{split} \langle (\eta\_{n}^{+})'(0), h\_{\hbar} \rangle &= \frac{1}{(p-1) \| u\_{\hbar} \|\_{p} - (2q-1) \mathcal{B}(u\_{\hbar})} \Big[ p \int\_{\mathbb{R}^{N}} \left( |\nabla u\_{\hbar}|^{p-2} \nabla u\_{\hbar} \nabla h\_{\hbar} + V(\mathbf{x}) |u\_{\hbar}|^{p-2} u\_{\hbar} h\_{\hbar} \right) d\mathbf{x} \\ &- 2q \int\_{\mathbb{R}^{N}} \int\_{\mathbb{R}^{N}} \frac{|u\_{\hbar}(\mathbf{y})|^{q} |u\_{\hbar}(\mathbf{x})|^{p-2} u\_{\hbar}(\mathbf{x}) h\_{\hbar}(\mathbf{x})}{|\mathbf{x} - \mathbf{y}|^{\boldsymbol{\mu}}} d\mathbf{x} d\mathbf{y} - \int\_{\mathbb{R}^{N}} g h\_{\hbar} d\mathbf{x} \Big] \end{split} \tag{28}$$

By the boundedness of *un* and (28), we say that there exists a constant λ such that

$$\left| (\eta\_n^+)'(0) \right| = \left| \left\langle (\eta\_n^+)'(0), h\_n \right\rangle \right| \le \frac{\lambda}{(p-1) \left| \left\| \mu\_n \right\| \right|\_p - (2q-1)B(\mu\_n) \right| }$$

Therefore, it remains to show that χ(*un*) := (*p* − 1) *un <sup>p</sup>* <sup>−</sup> (2*<sup>q</sup>* <sup>−</sup> <sup>1</sup>)*B*(*un*) possesses a positive lower bound.

To prove the existence of positive lower bound of χ(*un*), passing to a subsequence, we assume

$$\chi(\mu\_n) = (p-1)\|\mu\_n\|^p - (2q-1)B(\mu\_n) = o(1), n \to \infty \tag{29}$$

Since *un* <sup>∈</sup> <sup>Λ</sup>+, we obtain

$$\|\mu\_{\mathfrak{N}}\|^p - B(\mathfrak{u}\_{\mathfrak{N}}) = \mathbb{C}(\mathfrak{u}\_{\mathfrak{N}})$$

This along with (29) gives

$$C(u\_n) = \frac{2q - p}{2q - 1} \|u\_n\|^p + o(1) \tag{30}$$

It follows from the condition (*G*) that there is a sufficiently small μ > 0 such that *g* <sup>2</sup>*Nq* 2*N*(*q*−1)+μ ≤ (1 − μ)α. Similarly to the proof of (12), we have

$$\mathcal{C}(u) < \frac{2q - p}{2q - 1} (1 - \mu) \left( \frac{p - 1}{(2q - 1)B(u)} \right)^{\frac{p - 1}{2q - p}} \tag{31}$$

for any *u* ∈ *E*1. Therefore, by the principle of homogeneity,

$$\frac{2q-p}{2q-1} + \frac{o(1)}{\|\mu\_n\|^p} = \frac{\mathbb{C}(u\_n)}{\|u\_n\|^p} < \frac{2q-p}{2q-1} (1-\tau) \left(\frac{(p-1)\|\|u\_n\|\|^p}{(2q-1)B(u\_n)}\right)^{\frac{p-1}{2q-p}}\tag{32}$$

If *un* → 0, then similar to (7) one has *C*(*un*) → 0. Therefore

$$\langle \mathbf{u}^+ + o\_n(1) = I(u\_n) - \frac{1}{2q} \langle I'(u\_n), u\_n \rangle = \frac{2q - p}{2pq} \| u\_n \|^p - \frac{2q - 1}{2q} \mathcal{C}(u\_n) \to 0$$

which is a contradiction with *i* <sup>+</sup> <sup>&</sup>lt; 0. Thus *un*<sup>→</sup> 0, as *<sup>n</sup>* → ∞. Consequently, from (30)–(32) we can deduce that

$$\frac{2q-p}{2q-1} \le \frac{2q-p}{2q-1}(1-\mu), n \to \infty$$

which is a contradiction. Therefore, we conclude that *I* (*un*) → 0, as *n* → ∞. The proof is completed. -

**Proposition 2.** *Under assumptions* (*A*0) *and* (*G*)*, there exists a sequence* {*u*ˆ*n*} ⊂ Λ<sup>−</sup> *such that I*(*u*ˆ*n*) → *i* − *and I* (*u*ˆ*n*) → 0 *as n* → ∞*.*

**Proof.** By Lemma 5 we know that Λ− is closed. Thus, by Ekeland's variational principle on Λ− we get a sequence {*u*ˆ*n*} ⊂ Λ<sup>−</sup> such that

(iii) *<sup>I</sup>*(*u*ˆ*n*) <sup>&</sup>lt; inf*u*∈Λ<sup>−</sup> *I*(*u*) + <sup>1</sup> *<sup>n</sup>* , and (iv) *<sup>I</sup>*(*w*) <sup>≥</sup> *<sup>I</sup>*(*u*ˆ*n*) <sup>−</sup> <sup>1</sup> *n <sup>w</sup>* <sup>−</sup> *<sup>u</sup>*ˆ*<sup>n</sup>* , <sup>∀</sup>*<sup>w</sup>* <sup>∈</sup> <sup>Λ</sup><sup>−</sup> .

From (24) we know that *u*ˆ*<sup>n</sup>* is bounded. By coercivity of *I*, {*u*ˆ*n*} forms a bounded sequence in Λ. Moreover, from Lemma 5 we know that inf*u*∈Λ<sup>−</sup> *u* <sup>≥</sup> τ > 0, which implies that <sup>Λ</sup><sup>−</sup> stays away from the origin. Then using Proposition 1 and following the proof of Lemma 12 we conclude the result. -

#### **4. The Proof of Theorem 1**

In this section, we show that the minimums are achieved for *i* <sup>+</sup> and *i* −, and also give the proof of Theorem 1.

**Proposition 3.** *Assume g* - 0,(*A*0) *and* (*G*) *are satisfied. Then i can be achieved at point u*<sup>∗</sup> ∈ Λ*, which is a weak solution of problem (1). Moreover, u*<sup>∗</sup> <sup>∈</sup> <sup>Λ</sup><sup>+</sup> *and u*<sup>∗</sup> *is a local minimum for I on EV*.

**Proof.** By Lemma 8, there exists a sequence {*un*} ⊂ Λ such that *I*(*un*) → *i* and *I* (*un*) → 0 as *n* → ∞. Set *u*<sup>∗</sup> be the weak limit of the sequence {*un*} on *EV*, then *un* ∈ Λ satisfies (22) we get

$$\int\_{\mathfrak{R}^N} g(\mathbf{x}) \mu\_\*(\mathbf{x}) d\mathbf{x} > 0 \tag{33}$$

On the other hand, *I* (*un*) → 0 as *n* → ∞ implies that

$$\langle I'(\mu\_\*), v \rangle = 0, \text{ for every } v \in \Lambda.$$

i.e., *u*<sup>∗</sup> is a weak solution of problem (1). In particular, *u*<sup>∗</sup> ∈ Λ, and

$$i \le I(\mu\_\*) \le \lim\_{n \to +\infty} \inf \{ I(\mu\_n) \} = i$$

This implies that *u*<sup>∗</sup> is the minimum of *I* over *EV*.

For *u*<sup>∗</sup> ∈ Λ be such that *i* = *I*(*u*∗), using Lemma 9 we have *I*(*u*∗) < 0. Then we get *u*<sup>∗</sup> - 0. Therefore *u*<sup>∗</sup> is a nontrivial weak solution of problem (1). Since (33) holds, applying Lemma 4 we see that there exist *<sup>t</sup>*1,*t*<sup>2</sup> <sup>&</sup>gt; 0 such that *<sup>u</sup>*<sup>1</sup> :<sup>=</sup> *<sup>t</sup>*1*u*<sup>∗</sup> <sup>∈</sup> <sup>Λ</sup><sup>+</sup> and *<sup>t</sup>*2*u*<sup>∗</sup> <sup>∈</sup> <sup>Λ</sup>−. We claim that *<sup>t</sup>*<sup>1</sup> <sup>=</sup> 1 i.e., *<sup>u</sup>*<sup>∗</sup> <sup>∈</sup> <sup>Λ</sup>+. If *<sup>t</sup>*<sup>1</sup> <sup>&</sup>lt; 1, then *<sup>t</sup>*<sup>2</sup> <sup>=</sup> 1 which means *<sup>u</sup>*<sup>∗</sup> <sup>∈</sup> <sup>Λ</sup>−. Now *<sup>I</sup>*(*t*1*u*∗) <sup>≤</sup> *<sup>I</sup>*(*u*∗) = *<sup>i</sup>* <sup>&</sup>lt; 0 which is a contradiction with *<sup>t</sup>*1*u*<sup>∗</sup> <sup>∈</sup> <sup>Λ</sup>+.

Next we will prove that *u*<sup>∗</sup> is also a local minimum of *I* on *EV*. Obviously, for any *u* ∈ Λ with *C*(*u*) > 0 we can deduce that

*I*(*t*2*u*) ≤ *I*(*tu*) for any*t* ∈ (0, *t*0)

where *<sup>t</sup>*<sup>0</sup> = ( (*p*−1)*<sup>A</sup>* (2*q*−1)*<sup>B</sup>* ) 1/(2*q*−*p*) ,*t*<sup>2</sup> is corresponding to *u*. Moreover, if *u* = *u*<sup>∗</sup> then

$$\overline{f\_2} = 1 < \widehat{t}\_0 = \left[ \frac{(p-1)A(\mu\_\*)}{(2q-1)B(\mu\_\*)} \right]^{1/(2q-p)}$$

Taking ρ > 0 small enough so that

$$1 < \hat{t}\_w = \left[\frac{(p-1)A(u\_\*-w)}{(2q-1)B(u\_\*-w)}\right]^{1/(2q-p)} \|w\|\| < p\tag{34}$$

Thus, it follows from Lemma 10 that there exists a differentiable map <sup>η</sup><sup>+</sup> : *<sup>B</sup>*(0, <sup>ρ</sup>) → -+ such that <sup>η</sup>+(*w*)(*u*<sup>∗</sup> <sup>−</sup> *<sup>w</sup>*) <sup>∈</sup> <sup>Λ</sup><sup>+</sup> for *w* < ρ small. Then for any*<sup>t</sup>* <sup>∈</sup> (0, <sup>ˆ</sup>*tw*) we have

$$I(\overline{t}(\mu\_\ast - w)) \ge I(\eta^+(w)(\mu\_\ast - w)) \ge I(\mu\_\ast) \tag{35}$$

Since (34) holds, taking*<sup>t</sup>* <sup>=</sup> 1 in (35) we get *<sup>I</sup>*(*u*∗) <sup>≤</sup> *<sup>I</sup>*(*u*<sup>∗</sup> <sup>−</sup> *<sup>w</sup>*) for *w* < ρ, which implies that *<sup>u</sup>*<sup>∗</sup> is a local minimum of *I* on *EV*. The proof is completed. -

**Proof of Theorem 1.** Firstly, we deal with the minimization problem (16). According to Proposition 3, we only need to show that there exist a nonnegative solution on <sup>Λ</sup><sup>+</sup> if *<sup>g</sup>* <sup>≥</sup> 0. Assume *<sup>g</sup>* <sup>≥</sup> 0, from the proof of Lemma 3, it is easy to see that *B*(*u*∗) = *B*( *u*∗ ) and *<sup>C</sup>*(*u*∗) <sup>≤</sup> *<sup>C</sup>*( *u*∗ ). Moreover, it follows from the proof of Lemma 4 that there exists *t*<sup>1</sup> > 0 such that *t*<sup>1</sup> *u*∗ <sup>∈</sup> <sup>Λ</sup><sup>+</sup> and *<sup>t</sup>*1|*u*∗|<sup>&</sup>gt; 0. If <sup>ϕ</sup>*u*(*t*) denotes the fibering map corresponding to *u* ∈ *EV* as introduced in Section 2, we have ϕ |*u*∗| (1) ≤ ϕ *u*∗ (1) = 0. Since *t*<sup>1</sup> is the point of local minimum of ϕ|*u*∗|(*t*) for *t* ∈ (0, *t*0( *u*∗ )), where

$$t\_0(|\mu\_\*|) = \left[\frac{(p-1)A(\left|\mu\_\*\right|)}{(2q-1)B(\left|\mu\_\*\right|)}\right]^{1/(2q-p)}$$

and *t*<sup>1</sup> ≥ 1. Consequently, we have that *I*(*t*<sup>1</sup> *u*∗ ) <sup>≤</sup> *<sup>I</sup>*( *u*∗ ). Then

$$\mu^+ \le I(t\_1 \left| \mu\_\* \right|) \le I(\left| \mu\_\* \right|) \le I(\mu\_\*) = i^+$$

This means that *t*1|*u*∗| solves the minimization problem (16). Therefore, we find a nonnegative solution for problem (1) using the maximum principle.

Now we show that the infimum *i* − is achieved and the minimizer is second weak solution of problem (1). Consider the minimization problem (15). From Proposition 2, we know that there exists a sequence {*u*ˆ*n*} ⊂ Λ<sup>−</sup> such that *I*(*u*ˆ*n*) → *i* − and *I* (*u*ˆ*n*) → 0 as *n* → ∞. By Lemma 4, we get that there exists *u*<sup>∗</sup> <sup>∈</sup> *cl*(Λ−) = <sup>Λ</sup><sup>−</sup> such that *<sup>I</sup>*(*u*∗) = *<sup>i</sup>* −, *I* (*u*∗) = 0. Therefore, Lemma 7 implies that *u*<sup>∗</sup> is a weak solution of problem (1). In addition, if *g* ≥ 0, it follows from the proof of Lemma 4 and Proposition 1 that there exists *t*<sup>2</sup> > 0 such that *t*<sup>2</sup> *u*∗ <sup>∈</sup> <sup>Λ</sup><sup>−</sup> . Let

$$t\_0(\left|\overline{\mu}\_\*\right|) = \left[\frac{(p-1)A(\left|\overline{\mu}\_\*\right|)}{(2q-1)B(\left|\overline{\mu}\_\*\right|)}\right]^{1/(2q-p)}$$

then since *u*<sup>∗</sup> <sup>∈</sup> <sup>Λ</sup>−, taking account of the graph of the fibering map corresponding to *u*<sup>∗</sup> we can deduce that

$$\overline{u}^- \le I(t\_2 | \overleftarrow{u}\_\* |) \le I(t\_2 \overleftarrow{u}\_\*) \le \max\_{t \ge t\_0(\overleftarrow{[u}\_\* |)} \{I(t\_2 \overleftarrow{u}\_\*)\} = I(\overleftarrow{u}\_\*) = \overline{u}^-$$

This means that *t*<sup>2</sup> *u*∗ solves the minimization problem (15) and then we know that it is a nonnegative weak solution of problem (1) using the maximum principle again. Due to <sup>Λ</sup><sup>+</sup> <sup>∩</sup> <sup>Λ</sup><sup>−</sup> <sup>=</sup> <sup>∅</sup> and Lemma 9 shows that *i* <sup>+</sup> < *i* <sup>−</sup>, then *u*<sup>∗</sup> *u*∗. This ends the proof. -

#### **5. Conclusions**

In this work, we study a class of nonhomogeneous Choquard equations with perturbation involving p-Laplacian. We give sufficient conditions of the existence of at least two nontrivial solutions for problems (1). Next it is worth investigating infinitely many solutions for nonhomogeneous Choquard equations involving p-Laplacian.

**Author Contributions:** Supervision, Y.Z. and H.C.; Writing-original draft, X.S.; Writing-review & editing, X.S. and Y.Z.

**Funding:** The research was supported by Hunan Provincial Natural Science Foundation of China (No.2019JJ40068).

**Acknowledgments:** The authors thank the anonymous referees for their careful reading and helpful suggestions, which help to improve the quality of this paper.

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

#### **References**


© 2019 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* **New Generalized Mizoguchi-Takahashi's Fixed Point Theorems for Essential Distances and** *e***0-Metrics**

#### **Binghua Jiang 1, Huaping Huang <sup>2</sup> and Wei-Shih Du 3,\***


Received: 9 November 2019; Accepted: 9 December 2019; Published: 11 December 2019

**Abstract:** In this paper, we present some new generalizations of Mizoguchi-Takahashi's fixed point theorem which also improve and extend Du-Hung's fixed point theorem. Some new examples illustrating our results are also given. By applying our new results, some new fixed point theorems for essential distances and *e*0-metrics were established.

**Keywords:** MT -function; MT (*λ*)-function; *<sup>τ</sup>*-function; essential distance (*e*-distance); *<sup>e</sup>*0-metric; Du-Hung's fixed point theorem; Mizoguchi-Takahashi's fixed point theorem; Nadler's fixed point theorem; Banach contraction principle

**MSC:** 47H10; 54H25

#### **1. Introduction**

Let (*W*, *ρ*) be a metric space. For each *a* ∈ *W* and any nonempty subset *M* of *W*, let

$$\rho(a,M) = \inf\_{b \in M} \rho(a,b).$$

Denote by N (*W*), the family of all nonempty subsets of *W*, and by CB(*W*), the class of all nonempty closed and bounded subsets of *W*. A function H : CB(*W*) × CB(*W*) → [0, +∞) defined by

$$\mathcal{H}(\mathbb{C}, D) = \max \left\{ \sup\_{a \in D} \rho(a, \mathbb{C}), \sup\_{a \in \mathbb{C}} \rho(a, D) \right\}$$

is said to be the *Hausdorff metric* on CB(*W*) induced by the metric *ρ* on *W*. A point *z* in *W* is a fixed point of a mapping *T* if *z* = *Tz* (when *T* : *W* → *W* is a single-valued mapping) or *z* ∈ *Tz* (when *T* : *W* → N (*W*) is a multivalued mapping). The set of fixed points of *T* is denoted by F(*T*).

Fixed point theory is a fascinating mathematical theory that has a wide range of applications in many areas of mathematics, including nonlinear analysis, optimization, variational inequality problems, integral and differential equations and inclusions, critical point theory, nonsmooth analysis, dynamic system theory, control theory, economics, game theory, finance mathematics and so on. The famous Banach contraction principle [1] is undoubtedly one of the most important and applicable fixed point theorems which has played a significant role in nonlinear analysis and applied mathematical analysis. Many authors have devoted their attentions to study generalizations in various different directions of the Banach contraction principle; see, e.g., [2–23] and references therein.

**Theorem 1.** *(Banach [1]) Let (W, ρ ) be a complete metric space and T:W*→ *W be a selfmapping. Assume that there exists a nonnegative number λ* < 1 *such that*

$$
\rho(Ta, Tb) \le \lambda \rho(a, b) \text{ for all } a, b \in \mathcal{W}.
$$

*Then T has a unique fixed point in W.*

Nadler's fixed point theorem [21] was established in 1969 to extend the Banach contraction principle for multivalued mappings.

**Theorem 2.** *(Nadler [21]) Let (W, ρ ) be a complete metric space and T* : *W* → CB(*W*) *be a multivalued mapping. Suppose that there exists a nonnegative number λ* < 1 *such that*

$$\mathcal{H}(Ta, Tb) \le \lambda \rho(a, b) \quad \text{for all } a, b \in \mathcal{W}.$$

*Then T has a fixed point in W.*

Later, in 1989, Mizoguchi and Takahashi [20] presented a celebrated generalization of Nadler's fixed point theorem. In 2008, Suzuki gave an example [22] (Example 1) to show that Mizoguchi-Takahashi's fixed point theorem is a real generalization of Nadler's fixed point theorem.

**Theorem 3.** *(MizoguchiandTakahashi [20]) Let (W, ρ) be a complete metric space and T* : *W* → CB(*W*) *be a multivalued mapping. Assume that*

$$\mathcal{H}(Ta, Tb) \le \mu(\rho(a, b))\rho(a, b) \quad \text{for all } a, b \in \mathcal{W}\_\*$$

*where μ:* [0, +∞) → [0, 1) *is an* MT *-function; that is, μ satisfies* lim sup *<sup>x</sup>*→*s*<sup>+</sup> *μ*(*x*) < 1 *for all s* ∈ [0, +∞)*. Then T has a fixed point in W.*

A number of generalizations of Mizoguchi-Takahashi's fixed point theorem were studied; see [2,4,8–13,15,16] and references therein. In 2016, Du and Hung [10] established the following generalized Mizoguchi-Takahashi's fixed point theorem.

**Theorem 4.** *(Du and Hung [10]) Let* (*W*, *ρ*) *be a complete metric space, T* : *W* → *CB*(*W*) *be a multivalued mapping and μ* : [0, +∞) → [0, 1) *be an* MT *-function. Suppose that*

min{H(*Ta*, *Tb*), *ρ*(*a*, *Ta*)} ≤ *μ*(*ρ*(*a*, *b*))*ρ*(*a*, *b*) *for all a*, *b* ∈ *W with a* = *b.*

*Then T admits a fixed point in W.*

Theorem 4 is different from known generalizations in the existing literature and was illustrated by [7] (Example A) in which Mizoguchi-Takahashi's fixed point theorem is not applicable.

In this paper, we establish some new generalizations of Mizoguchi-Takahashi's fixed point theorem which also improve and extend Du-Hung's fixed point theorem. Some new examples illustrating our results are also given. By applying our new results, we obtained some new fixed point theorems for essential distances and *e*0-metrics.

#### **2. Preliminaries**

Let (*W*, *<sup>ρ</sup>*) be a metric space. A real valued function *<sup>f</sup>* : *<sup>W</sup>* <sup>→</sup> <sup>R</sup> is called *lower semicontinuous* if {*<sup>x</sup>* <sup>∈</sup> *<sup>W</sup>* : *<sup>f</sup>*(*x*) <sup>≤</sup> *<sup>r</sup>*} is *closed* for any *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>. Recall that a function *<sup>κ</sup>* : *<sup>W</sup>* <sup>×</sup> *<sup>W</sup>* <sup>→</sup> [0, <sup>+</sup>∞) is called a *w*-*distance* [14,18], if the following are satisfied:


A function *κ* : *W* × *W* → [0, +∞) is said to be a *τ-function* [2,3,6,8,9,17,19], if the following conditions hold:


It is obvious that the metric *ρ* is a *w*-distance and any *w*-distance is a *τ*-function, but the converse is not true; see [2,17] for more details.

The following result is useful in our proofs.

**Lemma 1.** *(See [6] (Lemma 1.1).) If condition* (*τ*4) *is weakened to the following condition* (*τ*4) :

(*τ*4) *for any a* ∈ *W with κ*(*a*, *a*) = 0*, if κ*(*a*, *b*) = 0 *and κ*(*a*, *c*) = 0*, then b* = *c,*

*then* (*τ*3) *implies* (*τ*4) *.*

In 2016, Du [6] introduced the concept of essential distance; see also [8].

**Definition 1.** *(See [6] (Definition 1.2).) Let* (*W*, *d*) *be a metric space. A function κ* : *W* × *W* → [0, +∞) *is called an essential distance (abbreviated as "e-distance") if conditions* (*τ*1)*,* (*τ*2) *and* (*τ*3) *hold.*

#### **Remark 1.**


The following known result is crucial in this paper.

**Lemma 2.** *(See [3] (Lemma 2.1).) Let* (*W*, *ρ*) *be a metric space and κ* : *W* × *W* → [0, +∞) *be a function. Assume that <sup>κ</sup> satisfies the condition* (*τ*3)*. If a sequence* {*an*} *in <sup>W</sup> with* lim*n*→<sup>∞</sup> sup{*κ*(*an*, *am*) : *<sup>m</sup>* <sup>&</sup>gt; *<sup>n</sup>*} <sup>=</sup> <sup>0</sup>*, then* {*an*} *is a Cauchy sequence in W.*

In 2016, Du introduced the concept of MT (*λ*)-function [5] as follows (see also [7]).

**Definition 2.** *Let λ* > 0*. A function τ* : [0, +∞) → [0, *λ*) *is said to be an* MT (*λ*)*-function [5] if* lim sup *<sup>x</sup>*→*γ*<sup>+</sup> *τ*(*x*) < *λ for all γ* ∈ [0, +∞)*. As usual, we simply write "*MT *-function" instead of "*MT (1)*-function".*

A useful characterization theorem for MT (*λ*)-functions was established by Du [5] in 2016 as follows.

**Theorem 5.** *(See [5] (Theorem 2.4).) Let λ* > 0 *and let τ* : [0, +∞) → [0, *λ*) *be a function. Then the following statements are equivalent.*

*(1) τ is an* MT (*λ*)*-function.*


Let *κ* be an *e*-distance on a metric space (*W*, *ρ*). For each *a* ∈ *W* and any nonempty subset *G* of *W*, we define *κ*(*a*, *G*) by

$$\kappa(a, G) = \inf\_{b \in G} \kappa(a, b).$$

The following Lemma is essentially proved in [2].

**Lemma 3.** *(See [2] (Lemma 1.2).) Let G be a closed subset of a metric space* (*W*, *ρ*) *and κ be a function satisfying the condition* (*τ*3)*. Suppose that there exists c* ∈ *W such that κ*(*c*, *c*) = 0*. Then κ*(*c*, *G*) = 0 *if and only if c* ∈ *G.*

Very recently, Du introduced and studied the concept of *e*0-distance [9].

**Definition 3.** *(See [9] (Definition 1.3).) Let* (*W*, *ρ*) *be a metric space. A function κ* : *W* × *W* → [0, +∞) *is called an e*0*-distance if it is an e-distance on W with <sup>κ</sup>*(*a*, *<sup>a</sup>*) = <sup>0</sup> *for all a* <sup>∈</sup> *W.*

**Remark 2.** *By applying Lemma 1, if <sup>κ</sup> is an e*0*-distance on W, then for a*, *<sup>b</sup>* <sup>∈</sup> *W, <sup>κ</sup>*(*a*, *<sup>b</sup>*) = <sup>0</sup> ⇐⇒ *<sup>a</sup>* <sup>=</sup> *b.*

**Example 1.** *Let <sup>W</sup>* <sup>=</sup> <sup>R</sup> *with the metric <sup>ρ</sup>*(*a*, *<sup>b</sup>*) = <sup>|</sup>*<sup>a</sup>* <sup>−</sup> *<sup>b</sup>*|*. Then* (*W*, *<sup>ρ</sup>*) *is a metric space. Define the function κ* : *W* × *W* → [0, +∞) *by*

$$\kappa(x, y) = \max\{9(x - y), 5(y - x)\}.$$

*Therefore κ is not a metric due to its asymmetry. It is easy to see that κ is an e*0*-distance on W.*

The following concept of *e*0-metric was studied by Du in [9] which generalizes the concept of Hausdorff metric.

**Definition 4.** *(See [9] (Definition 1.4).) Let* (*W*, *ρ*) *be a metric space and κ be an e*0*-distance. For any E, F* ∈ CB(*W*)*, define a function* D*<sup>κ</sup>* : CB(*W*) × CB(*W*) → [0, +∞) *by*

$$\mathcal{D}\_{\kappa}(E,F) = \max \{ \mathcal{J}\_{\kappa}(E,F), \mathcal{J}\_{\kappa}(F,E) \}\_{\kappa}$$

*where ξκ*(*E*, *<sup>F</sup>*) = sup*x*∈*<sup>E</sup> <sup>κ</sup>*(*x*, *<sup>F</sup>*)*, and then* <sup>D</sup>*<sup>κ</sup> is said to be the e*0*-metric on* CB(*W*) *induced by <sup>κ</sup>.*

The following result presented in [9] (Theorem 1.3) is quite important in our proofs. Although its proof is similar to the proof of [2] (Theorem 1.2), we give it here for the sake of completeness and the readers convenience.

**Theorem 6.** *(See [9] (Theorem 1.3).) Let* (*W*, *<sup>ρ</sup>*) *be a metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric defined as in Definition <sup>4</sup> on* CB(*W*) *induced by an e*0*-distance <sup>κ</sup>. Then, for E*, *<sup>F</sup>*, *<sup>G</sup>* ∈ CB(*W*)*, the following hold:*


**Proof.** To see (i), if *ξκ*(*E*, *F*) = 0, then *κ*(*a*, *F*) = 0 for all *a* ∈ *E*. By Lemma 3, we get *E* ⊆ *F*. Conversely, if *E* ⊆ *F*, by Lemma 3 again, we obtain *ξκ*(*E*, *F*) = 0 and (i) is proven. Fix *a* ∈ *E* and *c* ∈ *G*. Then we have

$$
\kappa(a, F) \le \kappa(a, b) \le \kappa(a, c) + \kappa(c, b) \quad \text{for all } b \in F\_{\kappa}
$$

which deduces

$$
\kappa(a, F) \le \kappa(a, c) + \kappa(c, F).
$$

So, for any *a* ∈ *E*, we obtain

$$
\kappa(a, F) \le \inf \{ \kappa(a, c) + \kappa(c, F) : c \in G \} \le \kappa(a, G) + \zeta\_\kappa(G, F).
$$

Taking the supremum on both sides of the last inequality over all *a* ∈ *E*, we can obtain (ii). Finally, we verify (iii). Obviously, D*κ*(*E*, *F*) ≥ 0 and D*κ*(*E*, *F*) = D*κ*(*F*, *E*). By using (i), we have D*κ*(*E*, *F*) = 0 ⇐⇒ *E* = *F*. Applying (ii), we have

$$\begin{aligned} \mathcal{D}\_{\mathbb{X}}(E,F) &= \max \{ \mathfrak{f}\_{\mathbb{X}}(E,F), \mathfrak{f}\_{\mathbb{X}}(F,E) \} \\ &\leq \max \{ \mathfrak{f}\_{\mathbb{X}}(E,G) + \mathfrak{f}\_{\mathbb{X}}(G,F), \mathfrak{f}\_{\mathbb{X}}(F,G) + \mathfrak{f}\_{\mathbb{X}}(G,E) \} \\ &\leq \mathcal{D}\_{\mathbb{X}}(E,G) + \mathcal{D}\_{\mathbb{X}}(G,F). \end{aligned}$$

These arguments show that D*<sup>κ</sup>* is a metric on CB(*W*).

**Definition 5.** *Let U be a nonempty subset of a metric space* (*W*, *ρ*) *and κ be an e-distance on W. A multivalued mapping T:U* → N (*W*) *is said to have the κ-approximate fixed point property in U provided* inf *<sup>a</sup>*∈*<sup>U</sup> <sup>κ</sup>*(*a*, *Ta*) = <sup>0</sup>*. In particular, if κ* ≡ *ρ, then T is said to have the approximate fixed point property in U.*

**Remark 3.** *Let U be a nonempty subset of a metric space* (*W*, *ρ*) *and T* : *U* → N (*W*) *be a multivalued mapping. Clearly,* F(*T*) ∩ *U* = ∅ *implies that T has the approximate fixed point property in U.*

#### **3. Main Results**

In this section, we first prove a new generalized Mizoguchi-Takahashi's fixed point theorem with a new nonlinear condition.

**Theorem 7.** *Let* (*W*, *<sup>ρ</sup>*) *be a metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance <sup>κ</sup>. Let T* : *W* → CB(*W*) *be a multivalued mapping and ϕ* : [0, +∞) → [0, 1) *be an* MT *-function. Assume that*

$$\kappa(a, \mathbf{x}) \le \kappa(\mathbf{x}, a) \qquad \text{for all } a \in T\mathbf{x} \tag{1}$$

*and*

$$\min \{ \mathcal{D}\_{\mathbb{R}}(Tu, Tv), \kappa(u, Tu) \} \le \varrho(\kappa(u, v))\kappa(u, v) \quad \text{for all } u, v \in \mathbb{W} \text{ with } u \ne v. \tag{2}$$

*Then, the following statements hold:*

*(a) For any <sup>z</sup>*<sup>0</sup> <sup>∈</sup> *W, there exists a Cauchy sequence* {*zn*}<sup>∞</sup> *<sup>n</sup>*=<sup>0</sup> *in W started at z*<sup>0</sup> *satisfying zn* ∈ *Tzn*−<sup>1</sup> *for each n* <sup>∈</sup> <sup>N</sup> *and*

$$\lim\_{n \to \infty} \kappa(z\_n, z\_{n-1}) = \lim\_{n \to \infty} \kappa(z\_{n-1}, z\_n) = \inf\_{n \in \mathbb{N}} \kappa(z\_n, z\_{n-1}) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0;$$

*(b) T has the κ-approximate fixed point property in W.*

*Moreover, if W is complete and T further satisfies one of the following conditions:*


*then T admits a fixed point in W.*

**Proof.** Let *τ* : [0, +∞) → [0, 1) be defined by

$$
\pi(\mathbf{x}) = \frac{1}{2}(\boldsymbol{\varrho}(\mathbf{x}) + 1) \quad \text{for all } \mathbf{x} \in [0, +\infty).
$$

Hence 0 ≤ *ϕ*(*x*) < *τ*(*x*) < 1 for all *x* ∈ [0, ∞). Given *b* ∈ *W*. Take *z*<sup>0</sup> = *b* ∈ *W* and choose *z*<sup>1</sup> ∈ *Tz*0. If *z*<sup>1</sup> = *z*0, then *z*<sup>0</sup> ∈ F(*T*) and we are done. Otherwise, if *z*<sup>1</sup> = *z*0, then *κ*(*z*1, *z*0) > 0 and we obtain from (2) that

$$\min\{\mathcal{D}\_{\mathbf{x}}(Tz\_1, Tz\_0), \mathbf{x}(z\_1, Tz\_1)\} \le \varphi(\mathbf{x}(z\_1, z\_0))\mathbf{x}(z\_1, z\_0) < \tau(\mathbf{x}(z\_1, z\_0))\kappa(z\_1, z\_0). \tag{3}$$

Since

$$\kappa(z\_1, Tz\_1) \le \sup\_{w \in Tx\_0} \kappa(w, Tz\_1) \le \mathcal{D}\_{\kappa}(Tz\_0, Tz\_1) = \mathcal{D}\_{\kappa}(Tz\_1, Tz\_0),$$

we get

$$\min \{ \mathcal{D}\_{\kappa}(Tz\_1, Tz\_0), \kappa(z\_1, Tz\_1) \} = \kappa(z\_1, Tz\_1). \tag{4}$$

Hence, by (3) and (4), we obtain

$$
\kappa(z\_1, Tz\_1) < \tau(\kappa(z\_1, z\_0))\kappa(z\_1, z\_0).
$$

which deduces that there exists *z*<sup>2</sup> ∈ *Tz*<sup>1</sup> such that

$$
\kappa(z\_1, z\_2) < \pi(\kappa(z\_1, z\_0))\kappa(z\_1, z\_0).
$$

Since *z*<sup>2</sup> ∈ *Tz*1, by (1), we have

$$
\kappa(z\_2, z\_1) < \pi(\kappa(z\_1, z\_0))\kappa(z\_1, z\_0).
$$

Next, if *z*<sup>2</sup> = *z*1, then *z*<sup>1</sup> ∈ F(*T*) and we finish the proof. Otherwise, since

$$\kappa(z\_2, Tz\_2) = \min\{\mathcal{D}\_\kappa(Tz\_2, Tz\_1), \kappa(z\_2, Tz\_2)\} < \pi(\kappa(z\_2, z\_1))\kappa(z\_2, z\_1),$$

there exists *z*<sup>3</sup> ∈ *Tz*<sup>2</sup> such that

*κ*(*z*2, *z*3) < *τ*(*κ*(*z*2, *z*1))*κ*(*z*2, *z*1).

By (1), we have

$$
\kappa(z\_{3\prime}z\_2) < \pi(\kappa(z\_{2\prime}z\_1))\kappa(z\_{2\prime}z\_1).
$$

So, by induction, we can obtain a sequence {*zn*}*n*∈N∪{0} in *<sup>W</sup>* satisfying the following: for each *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>,


By (iii), the sequence {*κ*(*zn*, *zn*−1)}*n*∈<sup>N</sup> is strictly decreasing in [0, +∞). Hence

$$\lim\_{n \to \infty} \kappa(z\_n, z\_{n-1}) = \inf\_{n \in \mathbb{N}} \kappa(z\_n, z\_{n-1}) \quad \text{exists.} \tag{5}$$

Since *ϕ* is an MT -function, by applying (8) of Theorem 5 with *λ* = 1, we obtain

$$0 \le \sup\_{\mathfrak{n} \in \mathbb{N}} \mathfrak{p}(\mathfrak{x}(z\_{\mathfrak{n}\prime} z\_{n-1})) < 1.$$

So we get

$$0 < \sup\_{n \in \mathbb{N}} \tau(\kappa(z\_{n\prime}, z\_{n-1})) = \frac{1}{2} \left[ 1 + \sup\_{n \in \mathbb{N}} \varrho(\kappa(z\_{n\prime}, z\_{n-1})) \right] < 1.$$

Put *γ* := sup *<sup>n</sup>*∈<sup>N</sup> *<sup>τ</sup>*(*κ*(*zn*, *zn*−1)). Thus *<sup>γ</sup>* <sup>∈</sup> (0, 1). For any *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, by (iii) again, we have

$$\kappa(z\_{n+1}, z\_n) \preccurlyeq \tau(\kappa(z\_{n}, z\_{n-1})) \\
\kappa(z\_n, z\_{n-1}) \le \gamma \kappa(z\_n, z\_{n-1}).\tag{6}$$

By (6), we get

$$\kappa(z\_{n+1}, z\_n) < \gamma \kappa(z\_n, z\_{n-1}) < \dots < \gamma^n \kappa(z\_1, z\_0) \quad \text{for each } n \in \mathbb{N}.\tag{7}$$

Since 0 < *γ* < 1, by taking the limit as *n* → ∞ in (7), we obtain

$$\lim\_{n \to \infty} \kappa(z\_n, z\_{n-1}) = 0. \tag{8}$$

Taking into account (5) and (8), we obtain

$$\lim\_{n \to \infty} \kappa(z\_{n\prime} z\_{n-1}) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n\prime} z\_{n-1}) = 0.$$

On the other hand, from (ii) and using (1), we have

$$
\kappa \kappa (z\_n, z\_{n+1}) \preccurlyeq \chi \kappa (z\_n, z\_{n-1}) \le \gamma \kappa (z\_{n-1}, z\_n) \quad \text{for each } n \in \mathbb{N}.
$$

which shows that the sequence {*κ*(*zn*−1, *zn*)}*n*∈<sup>N</sup> is also strictly decreasing in [0, +∞), and hence, we can deduce

$$
\kappa(z\_{n}, z\_{n+1}) < \gamma^{n} \kappa(z\_{0}, z\_{1}) \quad \text{for each } n \in \mathbb{N}.\tag{9}
$$

So, by (9), we get

$$\lim\_{n \to \infty} \kappa(z\_{n-1}, z\_n) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0. \tag{10}$$

Since *zn* <sup>∈</sup> *Tzn*−<sup>1</sup> for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, by (10), we prove

$$\inf\_{a \in \mathcal{W}} \kappa(a, \operatorname{Ta}) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0;$$

that is, *<sup>T</sup>* has the *<sup>κ</sup>*-approximate fixed point property in *<sup>W</sup>*. Next, we claim that {*zn*}*n*∈N∪{0} is a Cauchy sequence in *<sup>W</sup>*. For *<sup>m</sup>*, *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> with *<sup>m</sup>* <sup>&</sup>gt; *<sup>n</sup>*, we have from (9) that

$$\kappa(z\_n, z\_m) \le \sum\_{j=n}^{m-1} \kappa(z\_j, z\_{j+1}) < \frac{\gamma^n}{1-\gamma} \kappa(z\_0, z\_1). \tag{11}$$

Since 0 < *γ* < 1, the last inequality implies

$$\lim\_{m \to \infty} \sup \{ \kappa(z\_n, z\_m) : m > n \} = 0. \tag{12}$$

Applying Lemma 2, we prove that {*zn*}*n*∈N∪{0} is a Cauchy sequence in *W*.

Now, we assume that *W* is complete. We want to show F(*T*) = ∅ if *T* further satisfies one of conditions (D1)–(D5). Since {*zn*}*n*∈N∪{0} is Cauchy in *W* and *W* is complete, there exists *w* ∈ *W* such that *zm* → *w* as *m* → ∞. From (*τ*2) and (11), we have

$$\kappa(z\_{\nu}, w) \le \frac{\gamma^{\text{II}}}{1 - \gamma} \kappa(z\_0, z\_1) \quad \text{for all } n \in \mathbb{N}.\tag{13}$$

In order to finish the proof, it is sufficient to show *w* ∈ F(*T*). If (D1) holds, since *T* is closed and *zn* ∈ *Tzn*−<sup>1</sup> and *zn* → *w* as *n* → ∞, we get *w* ∈ *Tw*. If (D2) holds, by the lower semicontinuity of *f* , *zn* → *w* as *n* → ∞ and (10), we obtain

$$\begin{aligned} \kappa(w\_\prime Tw) &= f(w) \\ &\le \liminf\_{n \to \infty} \kappa(z\_{n\prime} T z\_n) \\ &\le \lim\_{n \to \infty} k(z\_{n\prime} z\_{n+1}) = 0. \end{aligned}$$

By Lemma 3, *w* ∈ F(*T*). Suppose that (D3) is satisfied. Since {*zn*} is Cauchy, we have lim*n*→<sup>∞</sup> *ρ*(*zn*, *zn*+1) = 0. So, by the lower semicontinuity of *g* and *zn* → *w* as *n* → ∞, we get

$$\rho(w, Tw) = \mathfrak{g}(w) \le \lim\_{n \to \infty} \rho(z\_n, z\_{n+1}) = 0.$$

By the closedness of *Tw*, we show *w* ∈ F(*T*). Assume that (D4) holds. By (12), there exists {*un*}⊂{*zn*} with lim sup*n*→∞{*κ*(*un*, *um*) : *<sup>m</sup>* <sup>&</sup>gt; *<sup>n</sup>*} <sup>=</sup> 0 and {*vn*} ⊂ *Tw* such that lim*n*→<sup>∞</sup> *<sup>κ</sup>*(*un*, *vn*) = 0. By (*τ*3), lim*n*→<sup>∞</sup> *ρ*(*un*, *vn*) = 0. Since *ρ*(*vn*, *w*) ≤ *ρ*(*vn*, *un*) + *ρ*(*un*, *w*), we have *vn* → *w* as *n* → ∞. By the closedness of *Tw*, we obtain *w* ∈ *Tw*. Finally, suppose that (D5) holds. If *w* ∈/ *Tw*, then, by (11) and (13), we obtain

$$\begin{aligned} 0 &< \inf\_{a \in W} \{ k(a, w) + k(a, Ta) \} \\ &\le \inf\_{n \in \mathbb{N}} \{ k(z\_{n\prime} w) + k(z\_{n\prime} Tz\_n) \} \\ &\le \inf\_{n \in \mathbb{N}} \{ k(z\_{n\prime} w) + k(z\_{n\prime} z\_{n+1}) \} \\ &\le \lim\_{n \to \infty} \frac{2\gamma^n}{1 - \gamma} \kappa(z\_{0\prime} z\_1) \\ &= 0, \end{aligned}$$

which leads to a contradiction. Therefore, it must be *w* ∈ F(*T*). The proof is completed.

Here, we give a simple example illustrating Theorem 7.

**Example 2.** *Let W* = [0, +∞) *with the metric ρ*(*x*, *y*) = |*x* − *y*| *for x*, *y* ∈ *W. Let Tx* = [0, *x*] *for x* ∈ *W. It is obvious that each x* ∈ *W is a fixed point of T. Let ϕ be any* MT *-function. Let κ* : *W* × *W* → [0, +∞) *be defined by*

$$\kappa(u,v) = \max\{\theta(u-v), \mathfrak{z}(v-u)\} \quad \text{for } u,v \in \mathcal{W}.$$

*Then, <sup>κ</sup> is an e*0*-metric on W. Given x* <sup>∈</sup> *W. For any a* <sup>∈</sup> *Tx* = [0, *<sup>x</sup>*]*, we have*

$$\kappa(a,\mathbf{x}) = \mathfrak{F}(\mathbf{x} - a) \le \mathfrak{P}(\mathbf{x} - a) = \kappa(\mathbf{x}, a)\_{\prime}$$

*which shows that* (1) *holds. Clearly, the function x* → *ρ*(*x*, *Tx*) *is a zero function on W, so it is lower semicontinuous. Hence (D3) holds. We now claim*

$$\min \{ \mathcal{D}\_{\mathbf{x}}(T\boldsymbol{u}, T\boldsymbol{v}), \kappa(\boldsymbol{u}, T\boldsymbol{u}) \} \leq \boldsymbol{\varrho}(\kappa(\boldsymbol{u}, \boldsymbol{\upsilon})) \kappa(\boldsymbol{u}, \boldsymbol{\upsilon}) \quad \text{for all } \boldsymbol{u}, \boldsymbol{\upsilon} \in \mathcal{W} \text{ with } \boldsymbol{u} \neq \boldsymbol{\upsilon}.$$

*We consider the following two possible cases:*

*Case 1. If* 0 ≤ *u* < *v, we have*

$$\kappa(u, Tu) = 0\_{\prime}$$

$$\mathcal{D}\_{\kappa}(Tu, Tv) = \max\left\{ \sup\_{z \in Tu} \kappa(z, Tv), \sup\_{z \in Tv} \kappa(z, Tu) \right\} = 9(v - u)^{\kappa}$$

*and*

$$
\kappa(u, v) = 5(v - u).
$$

*So,* min{D*κ*(*Tu*, *Tv*), *κ*(*u*, *Tu*)} = 0 ≤ *ϕ*(*κ*(*u*, *v*))*κ*(*u*, *v*)*. Case 2. If* 0 ≤ *v* < *u, we obtain*

$$\kappa(\mathfrak{u}, T\mathfrak{u}) = 0,$$

$$\mathcal{D}\_{\mathfrak{x}}(T\mathfrak{u}, T\mathfrak{v}) = 9(\mathfrak{u} - \mathfrak{v})$$

*and*

$$
\kappa(\mu, v) = 9(\mu - v).
$$

*Hence,* min{D*κ*(*Tu*, *Tv*), *κ*(*u*, *Tu*)} = 0 ≤ *ϕ*(*κ*(*u*, *v*))*κ*(*u*, *v*)*.*

*By Cases 1 and 2, our claim is verified, and hence,* (2) *holds. Therefore, all the assumptions of Theorem 7 are satisfied and we also show that T has a fixed point in W from Theorem 7. Notice that*

$$\mathcal{H}(T(5), T(9)) = 4 > \wp(\wp(5,9))\wp(5,9).$$

*so Mizoguchi-Takahashi's fixed point theorem is not applicable here. This example shows that Theorem 7 is a real generalization of Mizoguchi-Takahashi's fixed point theorem.*

**Remark 4.** *Du-Hung's fixed point theorem (i.e., Theorem 4) can be proven immediately from Theorem 7. Indeed, let κ* ≡ *ρ. Then,* (1) *and* (2)*, as in Theorem 7, are satisfied. We claim that (D4) as in Theorem 7 holds. Let* {*zn*} *in X with zn*+<sup>1</sup> <sup>∈</sup> *Tzn, n* <sup>∈</sup> <sup>N</sup> *and* lim*n*→<sup>∞</sup> *zn* <sup>=</sup> *w. We obtain*

$$\begin{aligned} \lim\_{n \to \infty} \rho(z\_{n+1}, Tw) &\le \lim\_{n \to \infty} \mathcal{H}(Tz\_n, Tw) \\ &\le \lim\_{n \to \infty} \{\rho(\rho(z\_{n\prime} w))\rho(z\_{n\prime} w)\} = 0, \end{aligned}$$

*which shows that (D4) holds. Therefore, all the assumptions of Theorem 7 are satisfied. By applying Theorem 7, we prove* F(*T*) = ∅*.*

In Theorem 7, if *T* : *W* → *W* is a self-mapping, then we obtain the following new fixed point theorem which generalizes Banach contraction principle.

**Corollary 1.** *Let* (*W*, *ρ*) *be a metric space, T* : *W* → *W be a self-mapping and ϕ* : [0, +∞) → [0, 1) *be an* MT *-function. Assume that*

$$
\kappa(a, \mathbf{x}) \le \kappa(\mathbf{x}, a) \qquad \text{for all } a \in T\mathbf{x}.
$$

*and*

$$\min\{\kappa(Tu, Tv), \kappa(u, Tu)\} \le \varrho(\kappa(u, v))\kappa(u, v) \quad \text{for all } u, v \in W \text{ with } u \ne v.$$

*Then the following statements hold:*

*(a) For any <sup>z</sup>*<sup>0</sup> <sup>∈</sup> *W, there exists a Cauchy sequence* {*zn*}<sup>∞</sup> *<sup>n</sup>*=<sup>0</sup> *in W started at z*<sup>0</sup> *satisfying zn* = *Tzn*−<sup>1</sup> *for each n* <sup>∈</sup> <sup>N</sup> *and*

$$\lim\_{n \to \infty} \kappa(z\_n, z\_{n-1}) = \lim\_{n \to \infty} \kappa(z\_{n-1}, z\_n) = \inf\_{n \in \mathbb{N}} \kappa(z\_n, z\_{n-1}) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0;$$

*(b) T has the κ-approximate fixed point property in W.*

*Moreover, if W is complete and T further satisfies one of conditions (D1)-(D5) as in Theorem 7, then T admits a fixed point in W.*

By applying Theorem 7, we establish some new fixed point theorems for *e*0-metrics and *e*0-distances.

**Corollary 2.** *Let* (*W*, *<sup>ρ</sup>*) *be a complete metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance κ. Let ϕ* : [0, +∞) → [0, 1) *be an* MT *-function and T* : *W* → CB(*W*) *be a multivalued mapping satisfying one of conditions (D1)-(D5) as in Theorem 7. Assume that*

$$
\kappa(a, x) \le \kappa(x, a) \qquad \text{for all } a \in T\mathfrak{x}
$$

*and*

$$\mathcal{D}\_{\mathbf{x}}(\mathrm{T}\boldsymbol{u}, \mathrm{T}\boldsymbol{u}) + \kappa(\boldsymbol{u}, \mathrm{T}\boldsymbol{u}) \le 2\varrho(\mathbf{x}(\boldsymbol{u}, \boldsymbol{\upsilon})) \mathbf{x}(\boldsymbol{u}, \boldsymbol{\upsilon}) \quad \text{for all } \boldsymbol{u}, \boldsymbol{\upsilon} \in \mathbb{W} \text{ with } \boldsymbol{u} \ne \boldsymbol{\upsilon}. \tag{14}$$

*Then T admits a fixed point in W.*

**Proof.** For any *u*, *v* ∈ *W* with *u* = *v*, by (14), we have

$$\min\{\mathcal{D}\_{\mathbf{x}}(Tu, Tv), \kappa(u, Tu)\} \le \frac{1}{2} \left(\mathcal{D}\_{\mathbf{x}}(Tu, Tu) + \kappa(u, Tu)\right) \le q(\kappa(u, v))\kappa(u, v).$$

Hence the condition (2) in Theorem 7 holds. Therefore, the conclusion is immediate from Theorem 7.

**Corollary 3.** *Let* (*W*, *<sup>ρ</sup>*) *be a complete metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance κ. Let ϕ* : [0, +∞) → [0, 1) *be an* MT *-function and T* : *W* → CB(*W*) *be a multivalued mapping satisfying one of conditions (D1)-(D5) as in Theorem 7. Assume that*

$$
\kappa(a, \mathbf{x}) \le \kappa(\mathbf{x}, a) \qquad \text{for all } a \in T\mathbf{x}^\*
$$

*and*

$$\sqrt{\mathcal{D}\_{\mathbf{x}}(Tu, Tv)\kappa(u, Tu)} \le \varrho(\kappa(u, v))\kappa(u, v) \quad \text{for all } u, v \in \mathcal{W} \text{ with } u \ne v. \tag{15}$$

*Then T admits a fixed point in W.*

**Proof.** For any *u*, *v* ∈ *W* with *u* = *v*, from (15), we obtain

$$\min\{\mathcal{D}\_{\mathbf{x}}(T\mathbf{u}, T\mathbf{v}), \mathbf{x}(\boldsymbol{u}, T\mathbf{u})\} \leq \sqrt{\mathcal{D}\_{\mathbf{x}}(T\mathbf{u}, T\mathbf{v})\kappa(\boldsymbol{u}, T\mathbf{u})} \leq \varrho(\kappa(\boldsymbol{u}, \boldsymbol{v}))\kappa(\boldsymbol{u}, \boldsymbol{v}).$$

So the condition (2) in Theorem 7 holds. Hence, the conclusion is immediate from Theorem 7.

In fact, we can establish a wide generalization of Corollary 2 as follows.

**Corollary 4.** *Let* (*W*, *<sup>ρ</sup>*) *be a complete metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance κ. Let ϕ* : [0, +∞) → [0, 1) *be an* MT *-function and T* : *X* → CB(*W*) *be a multivalued mapping satisfying one of conditions (D1)-(D5) as in Theorem 7. Assume that*

$$
\kappa(a, \mathbf{x}) \le \kappa(\mathbf{x}, a) \qquad \text{for all } a \in T\mathbf{x}.
$$

$$\frac{\text{s.t.} \mathcal{D}\_{\mathbf{x}}(Tu, Tv) + t\mathbf{x}(u, Tv)}{\text{s} + t} \le \varrho(\kappa(u, v))\kappa(u, v) \quad \text{for all } u, v \in \mathcal{W} \text{ with } u \ne v,\tag{16}$$

*where s*, *t* ≥ 0 *with s* + *t* > 0*. Then T admits a fixed point in W.*

**Proof.** For any *u*, *v* ∈ *W* with *u* = *v*, by (16), we get

$$\min\{\mathcal{D}\_{\mathbf{x}}(Tu, Tv), \kappa(u, Tu)\} \le \frac{s\mathcal{D}\_{\mathbf{x}}(Tu, Tv) + t\kappa(u, Tv)}{s + t} \le \wp(\kappa(u, v))\kappa(u, v),$$

and hence the condition (2) in Theorem 7 is satisfied. So the desired conclusion follows from Theorem 7 immediately.

Now, we focus the following new fixed point theorem without the assumption (1) and satisfy another new condition

$$\min \{ \mathcal{D}\_{\mathbf{x}}(T\boldsymbol{u}, T\boldsymbol{v}), \boldsymbol{\kappa}(\boldsymbol{v}, T\boldsymbol{v}) \} \leq \boldsymbol{\varphi}(\boldsymbol{\kappa}(\boldsymbol{u}, \boldsymbol{v})) \boldsymbol{\kappa}(\boldsymbol{u}, \boldsymbol{v}) \quad \text{for all } \boldsymbol{u}, \boldsymbol{v} \in \boldsymbol{W} \text{ with } \boldsymbol{u} \neq \boldsymbol{v}\_{\boldsymbol{v}}$$

which is different from (2) as in Theorem 7. It is worth mentioning that this new fixed point theorem is meaningful because an *e*0*-distance* is asymmetric in general.

**Theorem 8.** *Let* (*W*, *<sup>ρ</sup>*) *be a metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance <sup>κ</sup>. Let T* : *W* → CB(*W*) *be a multivalued mapping and ϕ* : [0, +∞) → [0, 1) *be an* MT *-function. Assume that*

$$\min \{ \mathcal{D}\_{\mathbf{x}}(Tu, Tv), \kappa(v, Tv) \} \le \varrho(\kappa(u, v))\kappa(u, v) \quad \text{for all } u, v \in \mathbb{W} \text{ with } u \ne v. \tag{17}$$

*Then the following statements hold:*

*(a) For any <sup>z</sup>*<sup>0</sup> <sup>∈</sup> *W, there exists a Cauchy sequence* {*zn*}<sup>∞</sup> *<sup>n</sup>*=<sup>0</sup> *in W started at z*<sup>0</sup> *satisfying zn* ∈ *Tzn*−<sup>1</sup> *for each n* <sup>∈</sup> <sup>N</sup> *and*

$$\lim\_{n \to \infty} \kappa(z\_{n-1}, z\_n) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0;$$

*(b) T has the κ-approximate fixed point property in W.*

*Moreover, if W is complete and T further satisfies one of conditions (D1)-(D5) as in Theorem 7, then* F(*T*) = ∅*.*

**Proof.** Define *τ*(*x*) = <sup>1</sup> <sup>2</sup> (*ϕ*(*x*) + 1) for all *x* ∈ [0, +∞). Then 0 ≤ *ϕ*(*x*) < *τ*(*x*) < 1 for all *x* ∈ [0, +∞). Let *b* ∈ *W*. Take *z*<sup>0</sup> = *b* ∈ *W* and choose *z*<sup>1</sup> ∈ *Tz*0. If *z*<sup>1</sup> = *z*0, then *z*<sup>0</sup> ∈ F(*T*) and we are done. Otherwise, if *z*<sup>1</sup> = *z*0, then *κ*(*z*0, *z*1) > 0. By (17), we have

$$\begin{aligned} \kappa(z\_1, Tz\_1) &= \min \{ \mathcal{D}\_{\kappa}(Tz\_0, Tz\_1), \kappa(z\_1, Tz\_1) \}, \\ &\le q(\kappa(z\_0, z\_1)) \kappa(z\_0, z\_1) \\ &< \tau(\kappa(z\_0, z\_1)) \kappa(z\_0, z\_1), \end{aligned}$$

from which one can deduce that there exists *z*<sup>2</sup> ∈ *Tz*<sup>1</sup> such that

$$
\kappa(z\_1, z\_2) < \pi(\kappa(z\_0, z\_1))\kappa(z\_0, z\_1).
$$

Next, if *z*<sup>2</sup> = *z*1, then *z*<sup>1</sup> ∈ F(*T*), and we finish the proof. Otherwise, since

$$\kappa(z\_2, Tz\_2) = \min\{\mathcal{D}\_\kappa(Tz\_1, Tz\_2), \kappa(z\_2, Tz\_2)\} < \pi(\kappa(z\_1, z\_2))\kappa(z\_1, z\_2),$$

then there exists *z*<sup>3</sup> ∈ *Tz*<sup>2</sup> such that

$$
\kappa(z\_2, z\_3) < \pi(\kappa(z\_1, z\_2))\kappa(z\_1, z\_2).
$$

Hence, by induction, we can obtain a sequence {*zn*}*n*∈N∪{0} satisfying the following: for each *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>,


By (v), the sequence {*κ*(*zn*−1, *zn*)}*n*∈<sup>N</sup> is strictly decreasing in [0, +∞). So

$$\lim\_{n \to \infty} \kappa(z\_{n-1}, z\_n) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) \quad \text{exists} \tag{18}$$

Since *ϕ* is an MT -function, by applying (8) of Theorem 5 with *λ* = 1, we obtain

$$0 \le \sup\_{n \in \mathbb{N}} \wp(\kappa(z\_{n-1}, z\_n)) < 1.$$

So we get

$$0 < \sup\_{n \in \mathbb{N}} \tau(\kappa(z\_{n-1}, z\_n)) = \frac{1}{2} \left[ 1 + \sup\_{n \in \mathbb{N}} \varrho(\kappa(z\_{n-1}, z\_n)) \right] < 1.$$

Hence *c* := sup *<sup>n</sup>*∈<sup>N</sup> *<sup>τ</sup>*(*κ*(*zn*−1, *zn*)) <sup>∈</sup> (0, 1). For any *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, by (v) again, we obtain

$$
\kappa(z\_n, z\_{n+1}) < \pi(\kappa(z\_{n-1}, z\_n)) \kappa(z\_{n-1}, z\_n) \le c \kappa(z\_{n-1}, z\_n).
$$

which implies

$$
\kappa(z\_n, z\_{n+1}) < c^n \kappa(z\_0, z\_1) \quad \text{for each } n \in \mathbb{N}.\tag{19}
$$

Since 0 < *c* < 1, by taking the limit as *n* → ∞ in (19), we have

$$\lim\_{n \to \infty} \kappa(z\_n, z\_{n+1}) = 0. \tag{20}$$

Combining (18) and (20), we obtain

$$\lim\_{n \to \infty} \kappa(z\_{n-1}, z\_n) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0 \tag{21}$$

and hence (a) is proven. To see (b), since *zn* <sup>∈</sup> *Tzn*−<sup>1</sup> for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, by (21), we show that

$$\inf\_{a \in \mathcal{W}} \kappa(a, \operatorname{Ta}) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0.$$

Using a similar argument as in the proof of Theorem 7, one can verify that F(*T*) = ∅ and finish this proof.

The following example not only illustrates Theorem 8 but also shows that Theorem 8 is different from Theorem 7.

**Example 3.** *Let W* = [0, +∞) *with the metric ρ*(*x*, *y*) = |*x* − *y*| *for x*, *y* ∈ *W. Let Tx* = [0, *x*] *for x* ∈ *W. So each x* ∈ *W is a fixed point of T. Let ϕ be any* MT *-function. Let κ* : *W* × *W* → [0, +∞) *be defined by*

$$\kappa(\mathfrak{u}, \upsilon) = \max \{ 4(\mathfrak{u} - \upsilon), 7(\upsilon - \mathfrak{u}) \} \quad \text{for } \mathfrak{u}, \upsilon \in \mathcal{W}.$$

*Then <sup>κ</sup> is an <sup>e</sup>*0*-metric on W. Clearly, the function <sup>x</sup>* → *<sup>ρ</sup>*(*x*, *Tx*) *is a zero function on W, so it is lower and semicontinuous. Hence, (D3) holds. Using a similar argument as in Example 2, we can prove that*

min{D*κ*(*Tu*, *Tv*), *κ*(*v*, *Tv*)} ≤ *ϕ*(*κ*(*u*, *v*))*κ*(*u*, *v*) *for all u*, *v* ∈ *W with u* = *v.*

*Hence, all the assumptions of Theorem 8 are satisfied. Applying Theorem 8, we also prove that T has a fixed point in W. Notice that* 1 ∈ *T*(2)=[0, 2] *and*

$$
\mathfrak{x}(1,2) = 7 > 4 = \mathfrak{x}(2,1),
$$

*so* (1) *does not hold and hence Theorem 7 is not applicable here. Moreover, since*

$$\mathcal{H}(T(\mathfrak{A}), T(\mathfrak{A})) = \mathfrak{F} \succ \varrho(\rho(\mathfrak{A}, \mathfrak{k})) \rho(\mathfrak{A}, \mathfrak{k})\_{\mathfrak{q}}$$

*Mizoguchi-Takahashi's fixed point theorem is also not applicable.*

Some new fixed point theorems are established by Theorem 8 immediately.

**Corollary 5.** *Let* (*W*, *ρ*) *be a metric space, T* : *W* → *W be a selfmapping and ϕ* : [0, +∞) → [0, 1) *be an* MT *-function. Assume that*

min{*κ*(*Tu*, *Tv*), *κ*(*v*, *Tv*)} ≤ *ϕ*(*κ*(*u*, *v*))*κ*(*u*, *v*) *for all u*, *v* ∈ *W with u* = *v.*

*Then the following statements hold:*

*(a) For any <sup>z</sup>*<sup>0</sup> <sup>∈</sup> *W, there exists a Cauchy sequence* {*zn*}<sup>∞</sup> *<sup>n</sup>*=<sup>0</sup> *in W started at z*<sup>0</sup> *satisfying zn* = *Tzn*−<sup>1</sup> *for each n* <sup>∈</sup> <sup>N</sup> *and*

$$\lim\_{n \to \infty} \kappa(z\_{n-1}, z\_n) = \inf\_{n \in \mathbb{N}} \kappa(z\_{n-1}, z\_n) = 0;$$

*(b) T has the κ-approximate fixed point property in W.*

*Moreover, if W is complete and T further satisfies one of conditions (D1)-(D5) as in Theorem 7, then T admits a fixed point in W.*

**Corollary 6.** *Let* (*W*, *<sup>ρ</sup>*) *be a complete metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance κ. Let ϕ* : [0, +∞) → [0, 1) *be an* MT *-function and T* : *W* → CB(*W*) *be a multivalued mapping satisfying one of conditions (D1)–(D5) as in Theorem 7. Assume that*

$$\mathcal{D}\_{\mathbf{x}}(T\mathbf{u}, T\upsilon) + \kappa(\upsilon, T\upsilon) \le 2\varrho(\kappa(\mathbf{u}, \upsilon))\kappa(\mathbf{u}, \upsilon) \quad \text{for all } \mathbf{u}, \upsilon \in \mathbb{W} \text{ with } \mathbf{u} \ne \upsilon.$$

*Then* F(*T*) = ∅*.*

**Corollary 7.** *Let* (*W*, *<sup>ρ</sup>*) *be a complete metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance κ. Let ϕ* : [0, +∞) → [0, 1) *be an* MT *-function and T* : *W* → CB(*W*) *be a multivalued mapping satisfying one of conditions (D1)-(D5) as in Theorem 7. Assume that*

$$\sqrt{\mathcal{D}\_{\mathbf{x}}(Tu, Tv)\kappa(v, Tv)} \le \varrho(\kappa(u, v))\kappa(u, v) \quad \text{for all } u, v \in \mathcal{W} \text{ with } u \ne v.$$

*Then* F(*T*) = ∅*.*

**Corollary 8.** *Let* (*W*, *<sup>ρ</sup>*) *be a complete metric space and* <sup>D</sup>*<sup>κ</sup> be an <sup>e</sup>*0*-metric on* CB(*W*) *induced by an <sup>e</sup>*0*-distance κ. Let ϕ* : [0, +∞) → [0, 1) *be an* MT *-function and T* : *W* → CB(*W*) *be a multivalued mapping satisfying one of conditions (D1)-(D5) as in Theorem 7. Assume that*

$$\frac{s\mathcal{D}\_{\mathbf{x}}(Tu, Tv) + t\kappa(v, Tv)}{s+t} \le \varphi(\kappa(u, v))\kappa(u, v) \quad \text{for all } u, v \in \mathcal{W} \text{ with } u \ne v.$$

*where s*, *t* ≥ 0 *with s* + *t* > 0*. Then* F(*T*) = ∅*.*

#### **Remark 5.**


#### **4. Conclusions**

Our main purpose in this paper is to establish new generalizations of Mizoguchi-Takahashi's fixed point theorem for essential distances and *e*0-metrics satisfying the following new conditions:


We give new examples to illustrate our results. As applications, some new fixed point theorems for essential distances and *e*0-metrics are also established by applying these new generalized Mizoguchi-Takahashi's fixed point theorems. Our new results generalize and improve some of known results on the topic in the literature.

**Author Contributions:** All authors contributed equally to this work. All authors read and approved the final manuscript.

**Funding:** The third author is supported by grant number MOST 107-2115-M-017-004-MY2 of the Ministry of Science and Technology of the Republic of China.

**Acknowledgments:** The authors wish to express their hearty thanks to the anonymous referees for their valuable suggestions and comments.

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

#### **References**


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