**Contents**

## **Preface to "Nonlinear Analysis and Optimization with Applications"**

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century.

This book focuses on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world. It consists of eleven papers covering a number of new ideas, concepts, methods, applications and current research problems. The Guest Editors would like to sincerely thank all the authors for their valuable contributions. There are still many fundamental and important questions that remain unanswered, promising a great future for these fields. We are sure that these extremely valuable papers in this book will interest readers and will stimulate new research work, and open new perspectives over some specific problems and applications.

Finally, we would like to express our hearty thanks to the editors of the journal *Axioms*, particularly Assistant Editor Luna Shen, for their great support throughout the editing process of the Special Issue for Axioms and its present MDPI Reprint Book.

> **Wei-Shih Du, Liang-Ju Chu, Fei He, Radu Precup** *Editors*

### *Article* **New Generalized Ekeland's Variational Principle, Critical Point Theorems and Common Fuzzy Fixed Point Theorems Induced by Lin-Du's Abstract Maximal Element Principle**

**Junjian Zhao <sup>1</sup> and Wei-Shih Du 2,\***


**Abstract:** In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with critical point theorem, maximal element theorem, generalized Ekeland's variational principle and common (fuzzy) fixed point theorem for essential distances.

**Keywords:** maximal element; fixed point; sizing-up function; *μ*-bounded quasi-ordered set; critical point; fuzzy mapping; Ekeland's variational principle; Caristi's fixed point theorem; Takahashi's nonconvex minimization theorem; essential distance

**MSC:** 47H04; 47H10; 58E30

**Citation:** Zhao, J.; Du, W.-S. New Generalized Ekeland's Variational Principle, Critical Point Theorems and Common Fuzzy Fixed Point Theorems Induced by Lin-Du's Abstract Maximal Element Principle. *Axioms* **2021**, *10*, 11. https://doi.org/ 10.3390/axioms10010011

Received: 19 December 2020 Accepted: 17 January 2021 Published: 20 January 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

### **1. Introduction**

Maximal element principle (MEP, for short) is a fascinating theory that has a wide range of applications in many fields of mathematics. Various generalizations in different directions of maximal element principle have been investigated by several authors, see [1–8] and references therein. Lin and Du [3,4,7] introduced the concepts of the sizing-up function and *μ*-bounded quasi-ordered set to define sufficient conditions for a nondecreasing sequence on a quasi-ordered set to have an upper bound and used them to establish an abstract MEP.

**Definition 1** (see [3,4,7])**.** *Let <sup>E</sup> be a nonempty set. A function <sup>μ</sup>* : <sup>2</sup>*<sup>E</sup>* <sup>→</sup> [0, <sup>+</sup>∞] *defined on the power set* 2*<sup>E</sup> of E is called sizing-up if it satisfies the following properties*

*(μ*1*) μ*(∅) = 0*; (μ*2*) μ*(*C*) ≤ *μ*(*D*) *if C* ⊆ *D.*

**Definition 2** (see [3,4,7])**.** *Let <sup>E</sup> be a nonempty set and <sup>μ</sup>* : <sup>2</sup>*<sup>E</sup>* <sup>→</sup> [0, <sup>+</sup>∞] *a sizing-up function. A multivalued map <sup>T</sup>* : *<sup>E</sup>* <sup>→</sup> <sup>2</sup>*<sup>E</sup> with nonempty values is said to be of type* (*μ*) *if for each <sup>x</sup>* <sup>∈</sup> *<sup>E</sup> and -* > 0*, there exists a y* = *y*(*x*, *-*) ∈ *T*(*x*) *such that μ*(*T*(*y*)) ≤ *-.*

**Definition 3** (see [3,4,7])**.** *A quasi-ordered set* (*E*, -) *with a sizing-up function <sup>μ</sup>* : <sup>2</sup>*<sup>E</sup>* <sup>→</sup> [0, +∞]*, in short* (*E*, -, *μ*)*, is said to be μ-bounded if every* -*-nondecreasing sequence z*<sup>1</sup> *z*<sup>2</sup> - ··· *zn zn*+<sup>1</sup> -··· *in E satisfying*

$$\lim\_{n \to +\infty} \mu(\{z\_n z\_{n+1}, \dots\}) = 0$$

*has an upper bound.*

The following abstract maximal element principle of Lin and Du is established in [3,4,7].

**Theorem 1.** *Let* (*E*, -, *<sup>μ</sup>*) *be a <sup>μ</sup>-bounded quasi-ordered set with a sizing-up function <sup>μ</sup>* : <sup>2</sup>*<sup>E</sup>* <sup>→</sup> [0, <sup>+</sup>∞]*. For each <sup>x</sup>* <sup>∈</sup> *E, let <sup>S</sup>* : *<sup>E</sup>* <sup>→</sup> <sup>2</sup>*<sup>E</sup> be defined by <sup>S</sup>*(*x*) = {*<sup>y</sup>* <sup>∈</sup> *<sup>E</sup>* : *<sup>x</sup> y*}*. If S is of type* (*μ*)*, then for each z*<sup>0</sup> ∈ *E, there exists a nondecreasing sequence z*<sup>0</sup> *z*<sup>1</sup> *z*<sup>2</sup> - ··· *in E and v* ∈ *E such that*


Ekeland's variational principle [9,10] is a very important tool for the study of approximate solutions approximate solutions of nonconvex minimization problems.

**Theorem 2.** *(Ekeland's variational principle) Let* (*M*, *d*) *be a complete metric space and f* : *M* → (−∞, +∞] *be a proper lower semicontinuous and bounded below function. Let ε* > 0 *and u* ∈ *M with f*(*u*) < +∞*. Then there exists v* ∈ *M such that*


In 1976, Caristi [11] established the following famous fixed point theorem:

**Theorem 3.** *(Caristi's fixed point theorem) Let* (*M*, *d*) *be a complete metric space and f* : *M* → (−∞, +∞] *be a proper lower semicontinuous and bounded below function. Suppose that T* : *M* → *M is selfmapping, satisfying*

$$f(Tz) + d(z, Tz) \le f(z)$$

*for each z* ∈ *M. Then there exists w* ∈ *M such that Tw* = *w.*

In 1991, Takahashi [12] proved the following nonconvex minimization theorem:

**Theorem 4.** *(Takahashi's nonconvex minimization theorem) Let* (*M*, *d*) *be a complete metric space and f* : *M* → (−∞, +∞] *be a proper lower semicontinuous and bounded below function. Suppose that for any x* ∈ *M with f*(*x*) > inf*z*∈*<sup>M</sup> f*(*z*)*, there exists yx* ∈ *M with yx* = *x such that*

$$f(y\_x) + d(x, y\_x) \le f(x).$$

*Then there exists w* ∈ *M such that f*(*w*) = inf*z*∈*<sup>M</sup> f*(*z*)*.*

It is well known that Caristi's fixed point theorem, Takahashi's nonconvex minimization theorem and Ekeland's variational principle are logically equivalent; for detail, one can refer to [3,6–8,13–24]. Many authors have devoted their attention to investigating generalizations and applications in various different directions of the well-known fixed point theorems (see, e.g., [3–8,12–31] and references therein). By using Theorem 1, Du proved several versions of generalized Ekeland's variational principle and maximal element principle and established their equivalent formulations in complete metric spaces, for detail, see [3,4].

In this paper, we present some new existence theorems related with critical point theorem, generalized Ekeland's variational principle, maximal element principle, and common (fuzzy) fixed point theorem for essential distances by applying Theorem 1.

### **2. Preliminaries**

Let *E* be a nonempty set. A fuzzy set in *E* is a function of *E* into [0, 1]. Let F(*E*) be the family of all fuzzy sets in *E*. A fuzzy mapping on *E* is a mapping from *E* into F(*E*). This enables us to regard each fuzzy map as a two variable function of *E* × *E* into [0, 1]. Let *F* be a fuzzy mapping on *E*. An element *a* of *E* is said to be a fuzzy fixed point of *F* if *F*(*a*, *a*) = 1 (see, e.g., [4]). Let <sup>Γ</sup> : *<sup>E</sup>* <sup>→</sup> <sup>2</sup>*<sup>E</sup>* be a multivalued mapping. A point *<sup>x</sup>* <sup>∈</sup> *<sup>E</sup>* is called to be a *critical point* (or *stationary point* or *strict fixed point*) [4] of Γ if Γ(*v*) = {*v*}.

Let *E* be a nonempty set and "-" a quasi-order (preorder or pseudo-order; that is, a reflexive and transitive relation) on *E*. Then (*E*, -) is called a quasi-ordered set. An element *v* in *E* is called a *maximal element* of *E* if there is no element *x* of *E*, different from *v*, such that *v x*; that is, *v w* for some *w* ∈ *E* implies that *v* = *w*. Let (*E*, -) be a quasi-ordered set. A sequence {*xn*}*n*∈<sup>N</sup> is called --*nondecreasing* (resp. --*nonincreasing*) if *xn xn*+<sup>1</sup> (resp. *xn*+<sup>1</sup> *xn*) for each *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>.

Let (*X*, *<sup>d</sup>*) be a metric space. A real valued function *<sup>ϕ</sup>* : *<sup>X</sup>* <sup>→</sup> <sup>R</sup> is *lower semicontinuous* (in short *l*.*s*.*c*) (resp. *upper semicontinuous*, in short *u*.*s*.*c*) if {*x* ∈ *X* : *ϕ*(*x*) ≤ *r*} (resp. {*<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>* : *<sup>ϕ</sup>*(*x*) <sup>≥</sup> *<sup>r</sup>*}) is *closed* for each *<sup>r</sup>* <sup>∈</sup> <sup>R</sup>. A real-valued function *<sup>f</sup>* : *<sup>X</sup>* <sup>→</sup> (−∞, <sup>+</sup>∞] is said to be proper if *f* ≡ +∞. Recall that a function *p* : *X* × *X* → [0, +∞) is called a *w*-*distance* [17,23], if the following are satisfied


The concept of *τ-*function was introduced and studied by Lin and Du as follows. A function *p* : *X* × *X* → [0, ∞) is said to be a *τ-function* [4,13,15,20,22,24,25], if the following conditions hold


It is worth mentioning that a *τ-*function is nonsymmetric in general. It is known that any metric *d* is a *w*-distance and any *w*-distance is a *τ*-function, but the converse is not true, see [24] for more detail.

**Lemma 1** (see [15,16,26])**.** *If condition* (*τ*4) *is weakened to the following condition* (*τ*4) *:*

(*τ*4) *for any x* ∈ *X with p*(*x*, *x*) = 0*, if p*(*x*, *y*) = 0 *and p*(*x*, *z*) = 0*, then y* = *z,*

*then* (*τ*3) *implies* (*τ*4) *.*

The concept of essential distance was introduced by Du [15] in 2016.

**Definition 4** (see [15])**.** *Let* (*X*, *d*) *be a metric space. A function p* : *X* × *X* → [0, +∞) *is called an essential distance if conditions* (*τ*1)*,* (*τ*2)*, and* (*τ*3) *hold.*

**Remark 1.** *It is obvious that any τ-function is an essential distance. By Lemma 1, we know that if p is an essential distance, then condition* (*τ*4) *holds.*

The following known result is very crucial in our proofs.

**Lemma 2** (see [4])**.** *Let* (*X*, *d*) *be a metric space and p* : *X* × *X* → [0, +∞) *be a function. Assume that p satisfies the condition* (*τ*3)*. If a sequence* {*xn*} *in X with* lim*n*→<sup>∞</sup> sup{*p*(*xn*, *xm*) : *m* > *n*} = 0*, then* {*xn*} *is a Cauchy sequence in X.*

### **3. Main Results**

**Lemma 3.** *Let* (*M*, *d*) *be a metric space and p* : *M* × *M* → [0, +∞) *be a function satisfying p*(*x*, *x*) = 0 *for all x* ∈ *M and p*(*x*, *z*) ≤ *p*(*x*, *y*) + *p*(*y*, *z*) *for any x*, *y*, *z* ∈ *M. Suppose that the extended real-valued function L* : *M* × *M* → (−∞, +∞] *satisfies the following assumptions*

	- *Define a binary relation on M by*

*x y* ⇐⇒ *L*(*x*, *y*) + *p*(*x*, *y*) ≤ 0*.*

*Then is a quasi-order.*

**Proof.** Clearly, *x x* for all *x* ∈ *M*. If *x y* and *y z*, then

$$L(x, y) + p(x, y) \le 0$$

and

$$L(y, z) + p(y, z) \le 0.$$

By (ii), we get

$$L(\mathbf{x}, z) + p(\mathbf{x}, z) \le L(\mathbf{x}, y) + L(y, z) + p(\mathbf{x}, y) + p(y, z) \le 0\_{\star}$$

which shows that *x z*. Hence is a quasi-order.

**Lemma 4.** *Let* (*M*, *d*)*, p, L, and be the same as in Lemma 3. Assume that for each x* ∈ *M, the function y* <sup>→</sup> *<sup>p</sup>*(*x*, *<sup>y</sup>*) *is l.s.c. Define G* : *<sup>M</sup>* <sup>→</sup> <sup>2</sup>*<sup>M</sup> by*

$$G(\mathbf{x}) = \{ y \in M : \mathbf{x} \le y \} \quad \text{for } \mathbf{x} \in M.$$

*Then the following hold*


**Proof.** Obviously, the conclusion (a) holds. To see (b), let *y* ∈ *G*(*x*). Then *x y*. We claim that *G*(*y*) ⊆ *G*(*x*). Given *z* ∈ *G*(*y*). Thus *y z*. By the transitive relation, we get *x z* which means *z* ∈ *G*(*x*). Hence *G*(*y*) ⊆ *G*(*x*).

The following theorem is one of the main results of this paper.

**Theorem 5.** *Let* (*M*, *d*) *be a metric space and p be an essential distance on M with p*(*x*, ·) *is l.s.c. for each x* ∈ *M and p*(*a*, *a*) = 0 *for all a* ∈ *M. Suppose that L, and G be the same as in Lemmas 3 and 4. If,*

$$p(y, \mathbf{x}) \le p(\mathbf{x}, y) \quad \text{for all } y \in G(\mathbf{x}),$$

*then the following hold:*

*(a) G is of type* (*μp*) *where μp*(*D*) := sup{*p*(*x*, *y*) : *x*, *y* ∈ *D*} *for D* ⊆ *M;*

*(b) If M is* -*-complete, then* (*M*, -, *μp*) *is a μp-bounded quasi-ordered set.*

**Proof.** We first show that *G is of type* (*μp*). Let *x* ∈ *M* and *-* > 0 be given. Then there exists *<sup>n</sup>*<sup>0</sup> <sup>=</sup> *<sup>n</sup>*0(*ε*) <sup>∈</sup> <sup>N</sup>, such that 2−*n*<sup>0</sup> <sup>&</sup>lt; *-* <sup>2</sup> . Define a function *κ* : *M* → [−∞, +∞] by

$$\kappa(\mathbf{x}) = \inf\_{y \in G(\mathbf{x})} L(\mathbf{x}\_\prime y).$$

Let *y* ∈ *G*(*x*). If *κ*(*x*) = −∞, then 0 ≤ *p*(*x*, *y*) < −*κ*(*x*). Otherwise, if *κ*(*x*) > −∞, then

$$p(x,y) \le -L(x,y) \le -\kappa(x).$$

Hence we conclude

$$0 \le p(\mathbf{x}, y) \le -\kappa(\mathbf{x}) \quad \text{for all } y \in G(\mathbf{x}).\tag{1}$$

Set *x*<sup>1</sup> := *x* ∈ *M*. Thus one can choose *x*<sup>2</sup> ∈ *G*(*x*1) ⊆ *M*, such that

$$L(\mathbf{x}\_1, \mathbf{x}\_2) \le \kappa(\mathbf{x}\_1) + \frac{1}{2}.$$

Let *<sup>k</sup>* <sup>∈</sup> <sup>N</sup> and assume that *xk* <sup>∈</sup> *<sup>M</sup>* is already known. Then, one can choose *xk*<sup>+</sup><sup>1</sup> ∈ *G*(*xk*) such that

$$L(\mathfrak{x}\_{k\prime}\mathfrak{x}\_{k+1}) \le \mathfrak{x}(\mathfrak{x}\_k) + \frac{1}{2^k}.$$

Hence, by induction, we obtain a nondecreasing sequence *x*<sup>1</sup> *x*<sup>2</sup> - ··· in *M* such that *xn*+<sup>1</sup> ∈ *G*(*xn*) and

$$L(\mathbf{x}\_{\hbar}, \mathbf{x}\_{n+1}) \le \kappa(\mathbf{x}\_{\hbar}) + \frac{1}{2^n} \quad \text{for all } n \in \mathbb{N}.\tag{2}$$

By Lemma 4, we have *<sup>G</sup>*(*xn*+1) <sup>⊆</sup> *<sup>G</sup>*(*xn*) for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. So it follows that

$$\begin{split} \kappa(\mathbf{x}\_{n+1}) &= \inf\_{y \in G(\mathbf{x}\_{n+1})} L(\mathbf{x}\_{n+1}, y) \\ &\ge \inf\_{y \in G(\mathbf{x}\_{n})} L(\mathbf{x}\_{n+1}, y) \\ &\ge \inf\_{y \in G(\mathbf{x}\_{n})} [L(\mathbf{x}\_{n\prime} y) - L(\mathbf{x}\_{n\prime}, \mathbf{x}\_{n+1})] \\ &= \kappa(\mathbf{x}\_{n}) - L(\mathbf{x}\_{n\prime} \mathbf{x}\_{n+1}). \end{split} \tag{3}$$

Combining (2) with (3), we obtain

$$
\kappa(\mathfrak{x}\_{n+1}) + \frac{1}{2^n} \ge 0,
$$

and hence

$$0 \le -\kappa(\mathbf{x}\_{n+1}) \le \frac{1}{2^n} < \frac{\epsilon}{2} \quad \text{for all } n \ge n\_0.$$

Put *w* = *xn*0+1. Thus *w* ∈ *G*(*x*) and

$$0 \le -\kappa(w) < \frac{\epsilon}{2}.$$

If *G*(*w*) is a singleton set, then *μp*(*G*(*w*)) = 0 ≤ *-*. Assume that *G*(*w*) is not a singleton set. Let *u*, *v* ∈ *G*(*w*). By our hypothesis, we have *p*(*u*, *w*) ≤ *p*(*w*, *u*). So, by (1), we obtain

$$\begin{aligned} p(u,v) &\leq p(u,w) + p(w,v) \\ &\leq -2\kappa(w) \\ &< \epsilon \end{aligned}$$

which implies

$$\mu\_p(G(w)) = \sup \{ p(\mu, v) : \mu, v \in G(w) \} \le \epsilon.$$

Therefore *G* is of type (*μp*). Finally, we prove (b). Let *α*<sup>1</sup> *α*<sup>2</sup> - ··· be a - nondecreasing sequence in *<sup>M</sup>* satisfying lim *<sup>n</sup>*→+<sup>∞</sup> *<sup>μ</sup>p*({*αn*, *<sup>α</sup>n*+1, ···}) = 0. Since

$$\begin{aligned} 0 &= \lim\_{n \to +\infty} \mu\_p(\{a\_{n\prime} a\_{n+1\prime} \cdots \}) \\ &= \lim\_{n \to +\infty} \sup \{ p(u, v) : u, v \in \{a\_{n\prime} a\_{n+1\prime} \cdots\} \}\_{\prime \prime} \end{aligned}$$

we get

$$\lim\_{n \to +\infty} \sup \{ p(\alpha\_n, \alpha\_m) : m > n \} = 0.$$

So, by applying Lemma 2, we show that {*αn*} is a nondecreasing Cauchy sequence in *M*. By the --completeness of *M*, there exists *β* ∈ *M* such that *α<sup>n</sup>* → *β* as *n* → +∞. We claim that *<sup>β</sup>* is an upper bound of {*αn*}+<sup>∞</sup> *<sup>n</sup>*=1. For each *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, since *<sup>α</sup><sup>m</sup>* <sup>∈</sup> *<sup>G</sup>*(*αn*) for all *m* ≥ *n* and *α<sup>n</sup>* → *β*, by the closedness of *G*(*αn*), we have *β* ∈ *G*(*αn*) or *α<sup>n</sup> <sup>β</sup>* for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Therefore *β* is an upper bound of {*αn*} and hence (*M*, -, *μp*) is a *μp*-bounded quasi-ordered set. The proof is completed.

The following result is immediate from Theorem 5 and Lemmas 3 and 4.

**Corollary 1.** *Let* (*M*, *d*) *be a metric space and p be an essential distance on M with p*(*x*, ·) *a l.s.c. for each x* ∈ *M and p*(*a*, *a*) = 0 *for all a* ∈ *M. Suppose that the extended real-valued function f* : *M* → (−∞, +∞] *is proper, l.s.c. and bounded below. Let ε* > 0*. Define a binary relation* -(*ε*, *<sup>f</sup>* ,*p*) *on M by*

$$\propto \overset{\kappa}{\lesssim} (\iota, f, p) \text{ } y \iff \varepsilon p(\mathfrak{x}, y) \le f(\mathfrak{x}) - f(y).$$

*Let* <sup>Γ</sup> : *<sup>M</sup>* <sup>→</sup> <sup>2</sup>*<sup>M</sup> be defined by*

$$\Gamma(\mathfrak{x}) = \{ y \in M : \mathfrak{x} \lessdot\_{(e,f,p)} y \} \text{ for } \mathfrak{x} \in M.$$

*Then the following hold:*


**Proof.** Define *L* : *M* × *M* → (−∞, ∞] by

$$L(\mathfrak{x}, \mathfrak{y}) = \frac{1}{\varepsilon} (f(\mathfrak{y}) - f(\mathfrak{x})) .$$

Then the following hold


Therefore, applying Theorem 5 and Lemmas 3 and 4, we show the desired conclusions.

By applying Theorem 5, we obtain a new result related to common fuzzy fixed point theorem, critical point theorem, maximal element principle and generalized Ekeland's variational principle for essential distances.

**Theorem 6.** *Let* (*M*, *d*) *be a complete metric space. Suppose that p, L,* -*, and G be the same as in Theorem 5. Let I be any index set. For each i* ∈ *I, let Fi be a fuzzy mapping on M. Assume that for each* (*i*, *x*) ∈ *I* × *M, there exists y*(*i*,*x*) ∈ *G*(*x*) *such that Fi*(*x*, *y*(*i*,*x*)) = 1*. Then for every ε* > 0 *and for every u* ∈ *M, there exists v* ∈ *M such that*


**Proof.** By applying Theorem 5, *G* is of type (*μp*) and (*M*, -, *μp*) is a *μp*-bounded quasiordered set, where

$$\mu\_p(D) := \sup \{ p(\mathbf{x}, \mathbf{y}) : \mathbf{x}, \mathbf{y} \in D \} \quad \text{for } D \subseteq M. \epsilon$$

Let *u* ∈ *M* be given. Put *u*<sup>0</sup> := *u*. Since *G* is of type (*μp*), by Theorem 1, there exists a --nondecreasing sequence *u*<sup>0</sup> *u*<sup>1</sup> *u*<sup>2</sup> -··· in *M* and *v* ∈ *M* such that


From (i), we prove (c). Next, we claim that *G*(*v*) = {*v*}. Let *z* ∈ *G*(*v*). By (*μ*2) and (ii), we have

$$p(v, z) = \mu\_{\mathcal{P}}(\{v, z\}) \le \mu\_{\mathcal{P}}(G(v)) = 0,$$

which deduces *p*(*v*, *z*) = 0. Since *p*(*v*, *v*) = 0, by Lemma 1, we get *z* = *v*. Therefore *G*(*v*) = {*v*} and, equivalency, (d) holds. For each (*i*, *v*) ∈ *I* × *M*, due to *G*(*v*) = {*v*} and our hypothesis, there exists *y*(*i*,*v*) := *v* ∈ *G*(*v*) such that *Fi*(*v*, *v*) = *Fi*(*v*, *y*(*i*,*v*)) = 1. So (e) is true. Finally, we verify (a). If *v w* for some *w* ∈ *W*, then *w* ∈ *G*(*v*) = {*v*}, which implies *v* = *w*. Hence *v* is a maximal element of (*M*, -). The proof is completed.

**Corollary 2.** *Let* (*M*, *d*) *be a complete metric space and ε* > 0*. Suppose that f , p*, -(*ε*, *<sup>f</sup>* ,*p*)*, and* Γ *be the same as in Corollary 1. Let I be any index set. For each i* ∈ *I, let Fi be a fuzzy mapping on M. Assume that for each* (*i*, *x*) ∈ *I* × *M, there exists y*(*i*,*x*) ∈ Γ(*x*) *such that Fi*(*x*, *y*(*i*,*x*)) = 1*. Then for every u* ∈ *M, there exists v* ∈ *M such that*


**Proof.** Define *L* : *M* × *M* → (−∞, +∞] by

$$L(\mathfrak{x}, \mathfrak{y}) = \frac{1}{\varepsilon} (f(\mathfrak{y}) - f(\mathfrak{x})).$$

Then,

$$\mathbf{x} \overset{<}{\sim}\_{\left(\mathfrak{e},f,p\right)} y \iff L(\mathfrak{x},y) + p(\mathfrak{x},y) \le 0.1$$

So the desired conclusions follow from Theorem 6 immediately.

Let (*M*, *<sup>d</sup>*) *be a metric space and <sup>T</sup>* : *<sup>M</sup>* <sup>→</sup> <sup>2</sup>*<sup>M</sup>* be a multivalued mapping with nonempty values. Then we can define a fuzzy mapping *K* on *M* by

$$
\mathcal{K}(x, y) = \chi\_{T(x)}(y),
$$

where *χ<sup>A</sup>* is the characteristic function for an arbitrary set *A* ⊂ *M*. Note that

$$K(\mathfrak{x}, \mathfrak{y}) = 1 \iff \mathfrak{y} \in T(\mathfrak{x})\,.$$

The following new result related to critical point theorem, generalized Ekeland's variational principle, maximal element principle, and common fixed point theorem for essential distances can be established by Theorem 6 immediately.

**Theorem 7.** *Let* (*M*, *d*) *be a complete metric space. Suppose that p, L,* -*, and G are the same as in Theorem 5. Let <sup>I</sup> be any index set. For each <sup>i</sup>* <sup>∈</sup> *I, let Ti* : *<sup>M</sup>* <sup>→</sup> <sup>2</sup>*<sup>M</sup> be a multivalued mapping with nonempty values such that for each* (*i*, *x*) ∈ *I* × *M, there exists y*(*i*,*x*) ∈ *Ti*(*x*) - *G*(*x*)*. Then for every ε* > 0 *and for every u* ∈ *M, there exists v* ∈ *M such that*


*(e) v is a common fixed point for the family* {*Ti*}*i*∈*I.*

**Corollary 3.** *Let* (*M*, *d*) *be a complete metric space and ε* > 0*. Suppose that f , p*, -(*ε*, *<sup>f</sup>* ,*p*)*, and* <sup>Γ</sup> *be the same as in Corollary 1. Let <sup>I</sup> be any index set. For each <sup>i</sup>* <sup>∈</sup> *I, let Ti* : *<sup>M</sup>* <sup>→</sup> <sup>2</sup>*<sup>M</sup> be a multivalued mapping with nonempty values such that for each* (*i*, *x*) ∈ *I* × *M, there exists y*(*i*,*x*) ∈ *Ti*(*x*) -Γ(*x*)*. Then for every u* ∈ *M, there exists v* ∈ *M such that*


Finally, the following simple example is given to illustrate Corollary 3.

**Example 1.** *Let M* = [−1, 1] *with the metric d*(*x*, *y*) = |*x* − *y*| *for x*, *y* ∈ *M. Then* (*M*, *d*) *is a complete metric space. Let <sup>T</sup>*1, *<sup>T</sup>*<sup>2</sup> : *<sup>M</sup>* <sup>→</sup> <sup>2</sup>*<sup>M</sup> be defined by <sup>T</sup>*1*<sup>x</sup>* <sup>=</sup> <sup>1</sup> 2 *x and T*2*x* = <sup>1</sup> 3 *x for <sup>x</sup>* <sup>∈</sup> *M. Clearly,* <sup>0</sup> *is the unique common fixed point of <sup>T</sup>*<sup>1</sup> *and <sup>T</sup>*2*. Let <sup>f</sup>* : *<sup>M</sup>* <sup>→</sup> <sup>R</sup> *by <sup>f</sup>*(*x*) = <sup>|</sup>*x*<sup>|</sup> *for x* ∈ *M. Define a binary relation* -(1, *<sup>f</sup>* ,*d*) *on M by*

$$\propto \lesssim\_{(1,f,d)} y \iff d(\mathfrak{x},y) \le f(\mathfrak{x}) - f(y).$$

*Then* -(1, *<sup>f</sup>* ,*d*)*is a quasi-order and*

$$\Gamma(\mathbf{x}) = \{ y \in M : \mathbf{x} \nleq\_{\mathbb{Z}(1,f,d)} y \} = \{ y \in M : d(\mathbf{x}, y) \le f(\mathbf{x}) - f(y) \} \ne \mathcal{Q}.$$

*It is easy to see that for each x* ∈ *M, we have*

$$d\left(\mathbf{x}, \frac{1}{2}\mathbf{x}\right) = f(\mathbf{x}) - f\left(\frac{1}{2}\mathbf{x}\right)$$

*and*

$$d\left(\mathbf{x}, \frac{1}{3}\mathbf{x}\right) = f(\mathbf{x}) - f\left(\frac{1}{3}\mathbf{x}\right).$$

*Hence* <sup>1</sup> <sup>2</sup> *<sup>x</sup>* <sup>∈</sup> *<sup>T</sup>*1*<sup>x</sup>* <sup>∩</sup> <sup>Γ</sup>(*x*) *and* <sup>1</sup> <sup>3</sup> *x* ∈ *T*2*x* ∩ Γ(*x*) *for any x* ∈ *M. Therefore, all the assumptions of Corollary 3 are satisfied. By applying Corollary 3, for every u* ∈ *M, we can obtain v* ∈ *M (in fact, v* = 0*) such that*


### **Remark 2.**


### **4. Conclusions**

Maximal element principle is a significant theory and has already been proposed and investigated its potential applications in several areas of mathematics. In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with common (fuzzy) fixed point theorem, maximal element theorem, critical point theorem and generalized Ekeland's variational principle for essential distances.

**Author Contributions:** Writing—original draft, J.Z. and W.-S.D. Both authors have read and agreed to the published version of the manuscript.

**Funding:** The first author is partially supported by the Natural Science Foundation of Tianjin city, China (Grant No. 18JCYBJC16300). The second author is partially supported by Grant No. MOST 109-2115-M-017-002 of the Ministry of Science and Technology of the Republic of China.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

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

### **References**

