On the Existence Theorems of Equilibrium and Quasi-Equilibrium Problems in Different Spaces

Abstract

In this article, the solvability of the equilibrium problem (EP), Minty equilibrium problem (MEP), and quasi-equilibrium problem (QEP) by using the notions of cyclically monotone and cyclically antimonotone in the setting of topological vector spaces and metric spaces is investigated. Also, the concepts transfer lower continuity, transfer weakly lower continuity, lower semicontinuity, from above, and sequentially weakly lower semicontinous which are weaker notions than the lower semicontinuity for establishing the existence results for EP and QEP, and the other forms of them are applied. Moreover, by using the famous results for the minimum points of a function, some existence theorems, by using the triangle property, of solutions for EP and QEP are given when the domains of bifunctions are compact and not compact. The results of this paper can be viewed as new versions of the corresponding published results with new and mild assumptions.

1. Introduction

The equilibrium problems as we know them now were first introduced by Muu and Oettli [1], and these were later worked on by Blum and Oettli [2]. The idea of an equilibrium solution has been successfully applied to a great deal of applications; therefore, many researchers have focused their attention on this issue. Let X be a set and $F:X\times X\to \mathbb{R}$ is a bifunction. By an EP, we understand the problem of finding

$$\tilde{x}\in X\mathrm{such}\mathrm{that}F(\tilde{x},y)\ge 0,\forall y\in X.$$

(1)

Related to EP, it is natural to consider the following problem, which was called Minty equilibrium problem (MEP), which consists of finding

$$\tilde{x}\in X\mathrm{such}\mathrm{that}F(y,\tilde{x})\le 0,\forall y\in X.$$

(2)

This framework allows for the modeling of a wide range of equilibria-related problems utilizing several mathematical models, including noncooperative games, variational inequalities, and optimization. Therefore, a lot of effort has been put into the issue theoretical and numerical analysis (see, for instance, the references [3,4,5]). An EP where the feasible set fluctuates according to the point under consideration is called a QEP. Let X be a set and $T:X\to 2^X$ a set-valued map. By QEP [6], we understand the problem of finding

$$\tilde{x}\in X\mathrm{such}\mathrm{that}\tilde{x}\in T\left(\tilde{x}\right)\wedge F(\tilde{x},y)\ge 0,\forall y\in T\left(\tilde{x}\right),$$

(3)

where $F:X\times X\to \mathbb{R}$ is a bifunction. Related to QEP, the following problem, which was called the Minty quasi-equilibrium problem (MQEP), is introduced as follows

$$\tilde{x}\in X\mathrm{such}\mathrm{that}\tilde{x}\in T\left(\tilde{x}\right)\wedge F(y,\tilde{x})\le 0,\forall y\in T\left(\tilde{x}\right),$$

(4)

Many pertinent problems, including quasi-variational inequalities, Nash is EP, quasi-optimization problems, and relative optimization problems, fall within this category. For solving the existence of solutions of EP and QEP, the most often used conditions were convexity, generalized convexity, and monotonicity along with some continuity [6,7,8,9,10,11]. According to certain writers [6,9,10,11], the triangle inequality property is added to prevent any convexity assumption. Let X be a set, $F:X\times X\to \mathbb{R}$ a bifunction, and $T:X\to X$ a map. The scalar QEP (MQEP) [12] consists of finding

$$\begin{array}{cc}\hfill & \tilde{x}\in X\mathrm{such}\mathrm{that}T\left(\tilde{x}\right)=\tilde{x}\wedge F(\tilde{x},y)\ge 0,\forall y\in X,\hfill \\ \hfill (& \tilde{x}\in X\mathrm{such}\mathrm{that}T\left(\tilde{x}\right)=\tilde{x}\wedge F(y,\tilde{x})\le 0,\forall y\in X).\hfill \end{array}$$

Also, the scalar generalized quasi-equilibrium problem (GQEP) is finding $\tilde{x}\in X\mathrm{such}\mathrm{that}T\left(\tilde{x}\right)=\tilde{x},F(\tilde{x},\tilde{x})=G(\tilde{x},\tilde{x})\le 0\wedge (F+G)(y,\tilde{x})\le 0\le (F+G)(\tilde{x},y),\forall y\in X$, where $F,G:X\times X\to \mathbb{R}$ are two bifunctions.

In order to prove existence theorems for a solution of the problems EP, QEP, scalar QEP, and scalar GQEP, the idea of cyclic antimonotonicity seems to be the logical generalization of the triangle inequality property. One of the goals of this article is to solve the EP, QEP, scalar QEP, and scalar GQEP by using the notions of cyclically monotone and cyclically antimonotone in the setting of topological vector spaces and metric spaces. To do this, we will use the concepts of transfer lower continuity, transfer weakly lower continuity, and lower semicontinuity from above, which are weaker forms than the lower semicontinuity. Moreover, the main results given in [13,14,15,16] about the minimum points, in order to establish existence theorems for a solution of EP, QEP will be applied.

A well-known result by Minty (1967) states the equivalence between Minty and Stampacchia variational inequalities under continuity and monotonicity assumptions of the involved operator. This result was established for variational inequalities and then for EP by making use of a certain convexity assumption for $f(x,·)$ [17]. We restate the same result that provides a link between the (MEP) and the (EP) by replacing the convexity by introducing a new, weaker notion. By applying the well-known results and the concepts of triangle inequality and cyclic monotonicity, some existence results for EP by using suitable coercivity for bifunctions whose domains are not necessarily compact are presented.

The rest of the paper is organized as follows. Section 2 is devoted to basic concepts. In Section 3, some existence results for QEP and EP, introducing some coercivity conditions, and a lemma for making a link between the solution sets of the MEP and EP are given. The results of this article can be viewed as a generalization, extension, and improvement of the corresponding results presented in [6,9,10,11,12,18,19,20].

2. Basic Concepts

This section provides definitions and tools that will be needed throughout the article.

Definition 1.

Let E and Y be two sets and $T:E\to 2^Y$ (the power set of Y) a set-valued map. For a nonempty subset A of E, we write

$$T\left(A\right)=\bigcup _{x\in A}T\left(x\right)$$

The set $T\left(A\right)$ is called the image of A under the set-valued map T. Also if E and Y are topological spaces then T is called closed-valued (compact-valued) when $T\left(x\right)$ is closed (compact) for each x in E.

Let E and Y be two topological spaces, $F:E\times Y\to \mathbb{R}$, a bifunction and $T:E\to 2^Y$ be a set-valued map with nonempty values. The marginal function depends on F and T is defined by

$$\varphi \left(e\right)=inf\left\{F\right(e,y):y\in T(e\left)\right\},\forall e\in E.$$

Also, the map $N\left(e\right)=\{y\in T(e):F(e,y)=\varphi (e\left)\right\}$ is called the minimum (marginal) set-valued map (see, for more details, Ref. [16] and the references therein).

Example 1.

Let $E={\mathbb{R}}_{+},Y=\mathbb{R}$, $F:E\times Y\to \mathbb{R}$ be defined by $F(x,y)=xy$ and $T:E\to 2^Y$ by $T\left(x\right)=[0,x]$, then

$$\varphi \left(e\right)=inf\left\{F\right(e,y):y\in T(e\left)\right\}=inf\{ey:y\in [0,e\left]\right\}=inf\{ey:0\le y\le e\}=0$$

and

$$N\left(e\right)=\{y\in T(e):F(e,y)=\varphi (e\left)\right\}=\{y\in [0,e]:ey=0\}=\{0\le y\le e:ey=0\}=\left\{0\right\}.$$

Definition 2

([6,13,16]). Given a nonempty subset C of the topological space X, the function $g:C\to \mathbb{R}$ is said to be:

• lower semicontinuous at $x\in C$, if $g\left(x\right)>\lambda$ then there exists a neighborhood $V_x$ of x such that $g\left(y\right)>\lambda$, $\forall z\in V_x\cap C$. Equivalently, for any net $\left\{x_{\alpha }\right\}\subset C$ with $x_{\alpha }\to x$ the following inequality holds

  $$g\left(x\right)\le \underset{\alpha }{lim\; inf}g\left(x_{\alpha }\right).$$
• sequentially lower semicontinuous at $x\in C$, if, for any sequence $\left\{x_n\right\}\subset C$, with $x_n\to x$ then

  $$g\left(x\right)\le \underset{n}{lim\; inf}g\left(x_n\right).$$
• transfer lower continuous at $x\in C$, if the inequality $g\left(x\right)>g\left(y\right)$ implies there exist $w\in C$ and a neighborhood $V_x$ of x such that $g\left(z\right)>g\left(w\right)$, $\forall z\in V_x\cap C$.
• weakly transfer lower continuous at $x\in C$, if $g\left(x\right)>g\left(y\right)$ then there exists $w\in C$ and a neighborhood $V_x$ of x such that $g\left(z\right)\ge g\left(w\right)$, $\forall z\in V_x\cap C$. Equivalently for any net $\left\{x_{\alpha }\right\}\subset C$, $x_{\alpha }\to x$ there exists $s\in X$ such that

  $$g\left(s\right)\le \underset{\alpha }{lim\; inf}g\left(x_{\alpha }\right).$$
• upper semicontinuous at $x\in C$, if $-g$ is lower semicontinuous at $x\in C$.

We say that the function g is lower semicontinuous (sequentially lower semicontinuous, transfer lower continuous, weakly transfer lower continuous) on the set C if it is lower semicontinuous (sequentially lower semicontinuous, transfer lower continuous, weakly transfer lower continuous) at each point of C.

It is worth mentioning that if C is subset of a Banach space and $g:C\to \mathbb{R}$ is a function, then it is said to be sequentially weakly lower semicontinuous when it is sequentially lower semicontinuous with respect to the weak topology.

Remark 1.

Obviously the lower semicontinuity implies the transfer lower continuity, because it suffices that in the definition of the lower semicontinuity we take $y=w$ and $\lambda =g\left(y\right)$. In general, the converse may fail as demonstrated by the following example.

Example 2.

Let $X=\mathbb{R}$, $C=[-2,2]$ and $g:C\to \mathbb{R}$ be defined by

$$g\left(x\right):=\left\{\begin{array}{cc}x+1\hfill & \mathrm{if}-2\le x<0,\hfill \\ x+3\hfill & \mathrm{if}0\le x\le 2\hfill \end{array}\right.$$

The following theorem provides a necessary and sufficient condition for a function to be transfer lower continuous.

Theorem 1

([16]). Let C be a nonempty and compact subset of a topological space X and $g:C\to \mathbb{R}$ be a function. Then, the set $arg{\min }_{\mathit{C}}\mathit{g}=\{x\in C:g\left(x\right)={inf}_{\mathit{C}}g\}$ is nonempty and compact if and only if g is transfer lower continuous.

Definition 3

([21]). Given a nonempty subset C of a metric space (X,d), the function $g:C\to \mathbb{R}\cup \{-\infty ,+\infty \}$ is said to be lower semicontinuous from above if for any $x\in C$ and every sequence $\left\{x_n\right\}\subset C$, $x_n\to x$, satisfying $g\left(x_1\right)\ge g\left(x_2\right)\ge \dots \ge g\left(x_n\right)\ge \dots$ it holds

$$g\left(x\right)\le \underset{n\to \infty }{lim}g\left(x_n\right).$$

Remark 2.

It is clear that a lower semicontinuous from above is lower semicontinuous. The following example shows that the converse is not true, in general.

Example 3.

Let $X=C=\mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ is defined as

$$g\left(x\right):=\left\{\begin{array}{cc}x+{\displaystyle \frac{1}{4}}\hfill & \mathrm{if}x<0,\hfill \\ x^2+3\hfill & \mathrm{if}x\ge 0.\hfill \end{array}\right.$$

It is simple to confirm that g is lower semicontinuous from above but g is not lower semicontinuous.

Remark that we can extend Definition 3 from metric spaces to topological spaces by the similar definition.

Definition 4.

Let E and Y be two topological spaces and $T:E\to 2^Y$ a set-valued map. A bifunction $F:E\times Y\to \mathbb{R}\cup \{-\infty \}$ is said to be

• transfer lower continuous with respect to T if, for every $(e,y)\in E\times Y$ with $y\in T\left(e\right)$, $F(e,z)<F(e,y)$ for some $z\in T\left(e\right)$ then there are $w\in Y$ and a neighborhood $V_{(e,y)}$ of $(e,y)$ such that for any $(s,v)\in V_{(e,y)}$ with $v\in T\left(s\right)$, $F(s,w)<F(s,v)$ and $w\in T\left(s\right)$.

Theorem 2

([16]). Let E and Y be two topological spaces. Let $T:E\to 2^Y$ be a nonempty compact-valued and closed set-valued map and let $F:E\times Y\to \mathbb{R}\cup \{-\infty \}$ be a bifunction. Suppose F is transfer lower continuous with respect to T. Then the minimum set-valued map $N:E\to 2^Y$ defined, for each $e\in E$, as

$$N\left(e\right)=\{x\in T(e):F(e,y)\ge F(e,x),\forall y\in T(e\left)\right\}$$

is nonempty compact-valued and closed.

Definition 5

([12]). Let $l^{\infty }$ be the Banach space of bounded sequences with the supremum norm. A linear functional μ on $l^{\infty }$ is called the mean if $\mu \left(e\right)=\parallel \mu \parallel =1$, where $e=(1,1,\dots )$. For $x=(x_1,x_2,\dots )$, the value $\mu \left(x\right)$ is also denoted by ${\mu }_n\left(x_n\right)$.

Lemma 1

([12]). Let $(X,d)$ be a metric space, $\left\{x_n\right\}$ a bounded sequence in X and μ a mean on $l^{\infty }$. If $g:\mathbb{R}\to \mathbb{R}$ is defined by

$$g\left(y\right)={\mu }_n\left(d(x_n,y)\right),\forall y\in X,$$

then g is a continuous function on X.

Lemma 2

([12]). Let $(X,d)$ be a metric space, let $\left\{x_n\right\}$ be a bounded sequence in X and let μ be a mean on $l^{\infty }$. If $g:\mathbb{R}\to \mathbb{R}$ is defined by

$$g\left(z\right)={\mu }_{n}d(x_n,z),\forall z\in X,$$

then $g\left(x\right)=g\left(y\right)=0$ implies $x=y$.

Lemma 3

([12]). Let $(X,d)$ be a metric space, let $\left\{x_n\right\}$ be a bounded sequence in X and let μ be a mean on $l^{\infty }$. Suppose that $\left\{y_n\right\}$ be a sequence in X with ${lim}_{m\to \infty }{\mu }_{n}d(x_n,y_m)=0$, Then $\left\{y_n\right\}$ is a Cauchy sequence.

Definition 6

([6]). For a given nonempty set C, the bifunction $F:C\times C\to \mathbb{R}$ is said:

• to have the triangle inequality property on C if

  $$F(x,y)\le F(x,z)+F(z,y),\forall x,y,z\in C.$$

  (5)
• to be cyclically antimonotone on C if for all $n\in \mathbb{N}$ and all $x_0,x_1,\dots ,x_n\in C$, with $x_{n+1}=x_0$,

  $$\sum _{i=0}^{n}F(x_i,x_{i+1})\ge 0.$$
• to be cyclically monotone on C if $-F$ is cyclically antimonotone, in other words, for all $n\in \mathbb{N}$ and all $x_0,x_1,\dots ,x_n\in C$, with $x_{n+1}=x_0$,

  $$\sum _{i=0}^{n}F(x_i,x_{i+1})\le 0.$$

Theorem 3

([9]). Let C be a nonempty set. A bifunction $F:C\times C\to \mathbb{R}$ is cyclically antimonotone (cyclically monotone) if and only if, there exists a function $g:C\to \mathbb{R}$ such that

$$\begin{array}{c}F(x,y)\ge g\left(y\right)-g\left(x\right),\forall x,y\in C,\hfill \\ \left(F\right(x,y)\le g(y)-g(x),\forall x,y\in C).\hfill \end{array}$$

(6)

The next result is important and provides a link between the triangle inequality property and cyclical antimonotonicity.

Remark 3

([9]). If the triangle inequality property is satisfied by the bifunction F (bifunction $-F$) then the bifunction F (bifunction $-F$) is cyclically antimonotone (cyclically monotone).

3. Existence Results for Equilibrium and Quasi-Equilibrium Problems

The following theorem plays a crucial role in proving an existence result of a solution for quasi-equilibrium problem.

Theorem 4

(Kakutani–Fan–Glicksberg fixed-point theorem). Let S be a non-empty, compact and convex subset of a locally convex Hausdorff space. Let $\varphi :S\to 2^S$ be a set-valued map on S which has a closed graph and the property that $\varphi \left(x\right)$ is non-empty and convex for all $x\in S$. Then the set of fixed points of ϕ is non-empty and compact.

Theorem 5.

Let C be a compact convex subset of a locally convex Hausdorff space, $T:C\to 2^C$ a closed set-valued mapping with nonempty compact and convex values. If $F:C\times C\to \mathbb{R}$ is a bifunction satisfying the following conditions

(i) 

F is transfer lower continuous with respect to T;

(ii) 

for each x in C, the mapping $y\to F(x,y)$ is quasiconvex;

(iii) 

F satisfies the triangle inequality property.

Then the solution set of QEP is nonempty.

Proof. 

By Theorem 2, the set-valued map $G:C\to 2^C$ is defined by $e\to G\left(e\right)=\{x\in T(e):F(e,y)\ge F(e,x),\forall y\in T(x\left)\right\}$ has closed graph with nonempty compact values and moreover by (ii)the values are convex. Hence, it follows from Theorem 4 that the set-valued map G has a fixed point. Thus, there exists x in C such that $x\in T\left(x\right)$ and $F(x,y)\ge F(x,x),\forall y\in T\left(x\right)$. Therefore by (iii), we have

$$F(x,y)\ge F(x,y)-F(x,x)\ge 0,\forall y\in T\left(x\right).$$

This completes the proof. □

Example 4.

Let $C=[-2,2]$, $F:C\times C\to \mathbb{R}$ be defined by $F(x,y)=x^2-y+1$, and $T:C\to 2^C$ be defined by $T\left(x\right)=[0,|x\left|\right]$. It is obvious that $\mathrm{Fix}\left(\mathit{T}\right)=[0,2]$ and all the conditions of Theorem 5 are satisfied. An easy calculation shows that the QEP solution set is $[1,2]$.

Remark 4.

Theorem 5 is topological vector space version of Theorem 5.1 in [18] from which it differs as follows.

(a) 

We replace EP by QEP.

(b) 

We replace the lower semicontinuous from above and lower bounded $F(x,·),\forall x\in C$ by the transfer lower continuity with respect to T.

(c) 

For any fixed $y\in X$, we do not require the upper semicontinuous $F(·,y)$.

Further Theorem 5 is topological vector space version of Theorem 3.3 in [10] from which it differs as follows.

(a) 

We replace EP by QEP.

(b) 

We replace the lower semicontinuous from above $F(x,·)$, for some $x\in X$ by the transfer lower continuity with respect to T.

As a direct consequence of Theorem 3, the following theorem is equivalent to Theorem 5.

Theorem 6.

Let C be a compact convex subset of a locally convex Hausdorff space, $T:C\to 2^C$ a closed set-valued mapping with nonempty compact and convex values, and $F:C\times C\to \mathbb{R}$ be a bifunction. If $g:C\times C\to \mathbb{R}$ is lower semicontinuous and convex such that

$$F(x,y)\ge g\left(y\right)-g\left(x\right),\forall x,y\in C,$$

then the solution set of QEP is nonempty.

Proof. 

It is enough for one to apply Theorem 3 for the set-valued mapping T and bifunction $G(x,y)=g\left(y\right)-g\left(x\right)$. □

Theorem 7.

Let C be a compact convex subset of a locally convex Hausdorff space, $T:C\to 2^C$ a closed set-valued mapping with nonempty compact and convex values. If $F:C\times C\to \mathbb{R}$ is a bifunction satisfying the following conditions

(i) 

F is transfer lower continuous with respect to T;

(ii) 

for each x in C, the mapping $y\to F(x,y)$ is quasiconvex;

(iii) 

F has the triangle inequality property;

then the solution set of MQEP is nonempty.

Proof. 

The proof is similar to the proof of Theorem 2. □

We now consider the case when the values of the set-valued map $T:X\to 2^X$ are not necessarily compact. If X is a finite dimensional Euclidean space the situation is good, and interesting results are given by the authors of the reference [22], while in the infinite-dimensional case, the situation is complicated. Most authors considered Banach spaces or metrizable topological vector spaces whose topologies have the following properties:

(a)

The closed balls are compact with respect to the topology;

(b)

For every $y\in X$, the distance function $d(·,y)$ is lower semicontinuous with respect to the topology.

For instance, conditions (a) and (b) are satisfied when X is a reflexive Banach space and its topology is the weak topology. In the next theorems we are going to extend the obtained results for a solution of quasi-equilibrium problems from reflexive Banach spaces to locally convex Hausdorff topological spaces for the mappings whose domain is not necessarily compact.

Theorem 8.

Let C be a closed convex subset of a locally convex Hausdorff space, $T:C\to 2^C$ a closed set-valued mapping with nonempty convex values. If $F:C\times C\to \mathbb{R}$ is a bifunction satisfying the following conditions

(i) 

F is transfer lower continuous with respect to T;

(ii) 

for each x in C, the mapping $y\to F(x,y)$ is quasiconvex;

(iii) 

F satisfies the triangle inequality property;

(iv) 

there exists a compact and convex subset D of C such that $D\cap T\left(x\right)$ is nonempty for each x in C, and

$$\forall z\in T\left(x\right)\setminus D,\exists w\in T\left(x\right)\cap D:F(z,w)<0.$$

(7)

Then, the set solution set of QEP is nonempty and relatively compact.

Proof. 

We define $T_{{\mid }_D}:D\to 2^D$ by $T_{{\mid }_D}\left(x\right)=T\left(x\right)\cap D$. Then by Theorem 5 the solution set of QEP for $(T_{{\mid }_D},F_{{\mid }_{D\times D}})$ is nonempty, hence there exists $\overline{x}\in D\cap T\left(\overline{x}\right)$, such that

$$F(\overline{x},y)\ge 0,\forall y\in D\cap T\left(\overline{x}\right).$$

(8)

We claim that $\overline{x}$ is a solution of QEP on C. Otherwise, there exists $z\in T\left(\overline{x}\right)\setminus D$ such that $F(\overline{x},z)<0$. Thus, by (iv), there exists $w\in T\left(\overline{x}\right)\cap D$ such that $F(z,w)<0$. Consequently, it follows from (iii) that $F(\overline{x},w)\le F(\overline{x},z)+F(z,w)<0$ which is a contradiction, and the proof is completed. □

The next result provides an existence theorem for a solution of MQEP.

Theorem 9.

Let C be a closed convex subset of a locally convex Hausdorff space, $T:C\to 2^C$ a closed map with nonempty convex values. If $F:C\times C\to \mathbb{R}$ is a bifunction satisfying the following conditions:

(i) 

F is transfer lower continuous with respect to T;

(ii) 

for each x in C, the mapping $y\to F(x,y)$ is quasiconvex;

(iii) 

F satisfies the triangle inequality property;

(iv) 

there exists a compact and convex subset D of C such that $D\cap T\left(x\right)$ is nonempty, for each x in C, and

$$\forall z\in T\left(x\right)\setminus D,\exists w\in T\left(x\right)\cap D:F(w,z)>0.$$

(9)

Then, the solution set of MQEP is nonempty and relatively compact.

Proof. 

The proof is similar to the proof of Theorem 3 by applying Theorem 7 for the set-valued map $T_{{\mid }_D}:D\to 2^D$ defined by $T_{{\mid }_D}\left(x\right)=T\left(x\right)\cap D$. □

The following theorem solves MEP for the sum of two bifunctions in the setting of metric spaces.

Theorem 10.

Let $r\in [0,1]$, $(X,d)$ be a complete metric space, $T:X\to X$ a map, and $F,G:X\times X\to \mathbb{R}$ be bifunctions. Let μ be a mean on $l^{\infty }$ and $\left\{x_n\right\}$ a bounded sequence. Assume that the following conditions hold

(i) 

there exists $\widehat{x}\in X$ such that

(a) $y\to F(\widehat{x},y)$ is a lower semicontinuous from above and bounded above function,

(b) $F(\widehat{x},y)-F(z,y)+G(y,z)\le F(\widehat{x},z),\forall y,z\in X$.

(ii) 

$F(x,y)+F(y,x)\le 0,\forall x,y\in X$;

(iii) 

if $d(x,Tx)\le (1+r)d(x,y)$, then ${\mu }_{n}d(x_n,Ty)\le F(x,y)+G(x,y)$.

Then there exists $\overline{x}\in X$ such that

$$(F+G)(z,\overline{x})\le 0,\forall z\in X,$$

and the solution set of (MEP) for $F+G$ is closed.

Furthermore, if $d(Tx,T^{2}x)\le (1+r)d(x,Tx),\forall x\in X$, then $\overline{x}$ is the unique solution of problem (GQEP) for $F+G$, i.e.,

$$(F+G)(z,\overline{x})\le 0\le (F+G)(\overline{x},z),\forall z\in X$$

Proof. 

Since $d(x,Tx)\le (1+r)d(x,T(x\left)\right)$, then it follows from (iii), (ii) and (i)(b) that

$${\mu }_{n}d(x_n,T^{2}x)\le F(x,Tx)+G(x,Tx)\le -F(Tx,x)+G(x,Tx)\le F(\widehat{x},Tx)-F(\widehat{x},x)$$

Repeating this process and using the properties of μ, we obtain

$$0\le {\mu }_{n}d(x_n,T^{k+1}x)\le G(T^{k-1}x,T^{k}x)-F(T^{k}x,T^{k-1}x)\le F(\widehat{x},T^{k}x)-F(\widehat{x},T^{k-1}x)$$

(10)

for each $k\in \mathbb{N}$. So, $\left\{F(\widehat{x},T^{k}x)\right\}$ is nondecreasing sequence. It follows from (i)(a) that ${lim}_{k\to \infty }F(\widehat{x},T^{k}x)$ exists. By virtue of (10), we obtain

$$\underset{k\to \infty }{lim}{\mu }_{n}d(x_n,T^{k+1}x)=0.$$

(11)

Hence, Lemma 3 assures that $\left\{T^{k}x\right\}$ is a Cauchy sequence, hence, since X is complete metric space, there exists $\overline{x}\in X$ such that $T^{k}x\to \overline{x}$ as $k\to \infty$. By (11) and Lemma 1 we get

$${\mu }_{n}d(x_n,\overline{x})=0.$$

(12)

Thus, for any $u\in X$, there exists $\overline{u}\in X$ such that ${lim}_{k\to \infty }{T}^{k}u=\overline{u}$ and ${\mu }_{n}d(x_n,\overline{u})=0$. By Lemma 2, $\overline{u}=\overline{x}$. Therefore, we have $\overline{x}={lim}_{k\to \infty }{T}^{k}z,\forall z\in X$.

For each $k\in \mathbb{N}\cup \left\{0\right\}$, put $u_k=T^{k}x$. Take any $z\in X\setminus \left\{\overline{x}\right\}$ and let z be fixed. Since $\overline{x}\ne z$ and $u_k\to \overline{x}$ as $k\to \infty$, there exists $N\in \mathbb{N}$ such that for each $k\ge N$, we get

$${\displaystyle \frac{1}{r+1}}d(u_k,u_{k+1})\le d(u_k,u_{k+1})\le d(u_k,z)$$

(13)

By (13), (iii), (ii) and (i)(b), for $k\ge N$, we have

$${\mu }_{n}d(x_n,Tz)\le F(u_k,z)+G(u_k,z)\le G(u_k,z)-F(z,u_k)\le F(\widehat{x},z)-F(\widehat{x},u_k).$$

(14)

Hence, $F(\widehat{x},u_1)\ge F(\widehat{x},u_2)\ge \dots \ge F(\widehat{x},u_k)\ge \dots$ and (14) via (i)(a), we get

$$\begin{array}{c}{\mu }_{n}d(x_n,Tz)+F(\widehat{x},\overline{x})\le {\mu }_{n}d(x_n,Tz)+\underset{k\to \infty }{lim}F(\widehat{x},u_k)\\ \le \underset{k\to \infty }{lim\; sup}({\mu }_{n}d(x_n,Tz)+F(\widehat{x},u_k))\\ \le F(\widehat{x},z).\end{array}$$

(15)

Hence, (i)(b) implies

$${\mu }_{n}d(x_n,Tz)\le F(\widehat{x},z)-F(\widehat{x},\overline{x})\le F(\widehat{x},z)-G(z,\overline{x})$$

(16)

Therefore,

$$G(z,\overline{x})\le G(z,\overline{x})+{\mu }_{n}d(x_n,Tz)\le F(\overline{x},z),\forall z\in X\setminus \left\{\overline{x}\right\}.$$

On the other hand, (i)(b) gets $G(\overline{x},\overline{x})\le F(\overline{x},\overline{x})$. Consequently,

$$G(z,\overline{x})\le F(\overline{x},z),\forall x\in X,$$

(17)

and so, (ii) gives $G(z,\overline{x})\le -F(z,\overline{x}),\forall z\in X$. This means

$$(F+G)(z,\overline{x})\le 0,\forall z\in X.$$

(18)

Hence, $\overline{x}$ is a solution of MEP. To see that the solution set of MEP is closed, we suppose that $\left\{u_n\right\}$ is a sequence of the solutions of MEP which converges to $\overline{u}$ as $n\to \infty$. It follows from the lower semicontinuity from above F in the second argument that

$$F(z,\overline{u})\le \underset{n\to \infty }{lim}F(z,{\overline{u}}_n)\le 0,\forall z\in X.$$

So, $\overline{u}$ is a solution of $\mathrm{MEP}$, i.e., $\mathrm{MEP}$ is closed.

By (ii) we get $F(\overline{x},\overline{x})\le 0$. Also, it follows from $d(Tx,T^{2}x)\le (1+r)d(x,Tx),\forall x\in X$, that

$$d(T\overline{x},T\left(T\overline{x}\right))\le (1+r)d(T\overline{x},\overline{x}).$$

By (iii) and (ii), we have

$${\mu }_{n}d(x_n,T\overline{x})+F(\overline{x},T\overline{x})\le G(T\overline{x},\overline{x}).$$

(19)

Hence, (17) and (19) imply that

$$F(\overline{x},T\overline{x})\le G(T\overline{x},\overline{x})\le F(\overline{x},T\overline{x}).$$

Thus, $F(\overline{x},T\overline{x})\le G(T\overline{x},\overline{x})$. By (19), we obtain

$$0\le {\mu }_{n}d(x_n,T\overline{x})\le G(T\overline{x},\overline{x})-F(\overline{x},T\overline{x})=0.$$

This means

$${\mu }_{n}d(x_n,T\overline{x})=0.$$

(20)

By (12) and (20) via Lemma 2, we obtain $T\overline{x}=\overline{x}$. This means $\overline{x}$ is a fixed point of T and by (12) the fixed points set of T is singleton and equals to $\overline{x}$. Consequently, $G(\overline{x},\overline{x})=F(\overline{x},\overline{x})\le 0$.

We are going to show that $0\le (F+G)(\overline{x},z),\forall z\in X$. Suppose z is an arbitrary element of X, since $\overline{x}$ is fixed point of T, we have $d(\overline{x},T\overline{x})\le (1+r)d(\overline{x},\overline{x})\le (1+r)d(\overline{x},z)$. Hence, by (iii), we have

$$0\le {\mu }_{n}d(x_n,Tz)\le F(\overline{x},z)+G(\overline{x},z).$$

Thus,

$$0\le (F+G)(\overline{x},z),\forall z\in X.$$

(21)

It follows from (18) and (21) that

$$(F+G)(z,\overline{x})\le 0\le (F+G)(\overline{x},z),\forall z\in X,$$

and the proof is completed. □

The next result, using Theorem 10, establishes an existence theorem of a solution for MEP and unique solution for QEP and MQEP.

Theorem 11.

Let $r\in [0,1]$, $(X,d)$ be a complete metric space, $T:X\to X$ a map, and $F:X\times X\to \mathbb{R}$ a bifunction. Let μ be a mean on $l^{\infty }$ and $\left\{x_n\right\}$ a bounded sequence. Suppose the following assertions are satisfied

(i) 

there exists $\widehat{x}\in X$ such that

(a) $y\to F(\widehat{x},y)$ is a lower semicontinuous from above and bounded above function,

(b) $F(\widehat{x},y)+F(y,z)\le F(\widehat{x},z),\forall y,z\in X$,

(ii) 

if $d(x,Tx)\le (1+r)d(x,y)$, then ${\mu }_{n}d(x_n,Ty)\le F(x,y)$.

Then, there exists $\overline{x}\in X$ such that

$$F(z,\overline{x})\le 0,\forall z\in X,$$

and the set of solution of (MEP) for F is closed.

Furthermore, if $d(Tx,T^{2}x)\le (1+r)d(x,Tx),\forall x\in X$, then $\overline{x}$ is the unique solution of problem (QEP) and (MQEP), i.e.,

$$F(z,\overline{x})\le 0\le F(\overline{x},z),\forall z\in X$$

Proof. 

By (i)(b), we have

$F(y,z)\le F(\widehat{x},z)-F(\widehat{x},y)$ and $F(z,y)\le F(\widehat{x},y)-F(\widehat{x},z)$, $\forall y,z\in X$

By combining the last inequalities, we get

$$F(y,z)+F(z,y)\le 0,\forall y,z\in X.$$

(22)

Therefore, the map F fulfils all the conditions of Theorem 10 when G equals to the zero map. Hence, the result follows from Theorem 10. □

Remark 5.

It follows from (i)(b) of Theorem 11 that $F(\widehat{x},y)+F(y,z)\le F(\widehat{x},z),\forall y,z\in X$. If we define $g\left(x\right)=F(\widehat{x},x)$, for all $x\in X,$ then it follows from condition (i)(b) of Theorem 11 that $F(y,z)\le g\left(z\right)-g\left(y\right),\forall y,z\in X$, hence F is cyclically monotone.

It is easy to see that the result of Theorem 11 is still valid if the conditions (i)(a) and (i)(b)are replaced with the following:

(i) 

for every $\widehat{x}\in X$ such that

(a) $y\to F(\widehat{x},y)$ is a lower semicontinuous from above and bounded above function.

(b) F is cyclically monotone.

Remark 6.

Theorem 10 improves Theorem 3.1 in [12] because:

(a) 

The lower semicontinuity and bounded above $F(x,·)$, for some $x\in X$ is replaced by the lower semicontinuity from above and bounded above function $F(x,·)$, for some $x\in X$.

Also Theorem 11 and Remark 5 improve and extend Corollary 3.2 in [12], because

(a) 

The lower semicontinuity and bounded above of $F(x,·)$, for some $x\in X$ are replaced by the lower semicontinuity from above and bounded above of $F(x,·)$, for some $x\in X$;

(b) 

The lower semicontinuity and bounded above $F(x,·)$, for some $x\in X$ and the triangle inequality property $-F$ are replaced by the lower semicontinuity from above and bounded above $F(x,·)$, for every $x\in X$ and the cyclically monotonicity of F, respectively.

Definition 7

([13,23]). Let X be a real linear space, $f:X\to \mathbb{R}\cup \{\infty \}$ a proper and bounded from below function, and $\mathrm{dom}\left(\mathit{f}\right)$ a convex set. The function f is called

• quasiconvex, if for any $x,y\in X$, $f(\lambda x+(1-\lambda \left)y\right)\le max\left\{f\right(x),f(y\left)\right\}$, where $0\le \lambda \le 1$, or equivalently for any $\alpha \in \mathbb{R}$, $\{x\in X:f(x)\le \alpha \}$ is convex.
• nearly quasiconvex function, if for any $\epsilon >m=inff$ there exists $\delta \left(\epsilon \right)\in (m,\epsilon ]$ such that for any $x,y\in X$, $f\left(x\right)<\delta \left(\epsilon \right)$ and $f\left(y\right)<\delta \left(\epsilon \right)$ imply that $f(\lambda x+(1-\lambda \left)y\right)<\epsilon$, where $0\le \lambda \le 1$.

Definition 8

([13]). Let X be a topological space, $f:X\to \mathbb{R}\cup \{\infty \}$ be an extended real-valued function and $m=inff$. We say that f is

• regular-global-inf (rgi) at $x_0\in X$ if $f\left(x_0\right)>inff$ implies there exist neighborhood U of $x_0$ and $c>inff$ such that $inff\left(U\right)>c$. If f is rgi at each point $x\in X$, then we say that f is rgi on X.
• quasi-regular-global-inf (qrgi) at $x_0\in X$ if there exists neighborhood U of $x_0$ such that $inff\left(U\right)>m$ whenever $m\notin f\left(V\right)$ for some neighborhood V of $x_0$. If f is qrgi at each point $x\in X$, then we say that f is qrgi on X.

Proposition 1

([13]). Let X be a topological space, and let $f:X\to \mathbb{R}\cup \{\infty \}$ be bounded from below and let $x\in X$.

(i) 

If f is lower semicontinuous at $x\in X$, then f is rgi at x.

(ii) 

If f is rgi at $x\in X$, then f is qrgi at x.

(iii) 

If f is qrgi at $x\in X$, then f is transfer weakly lower continuous at x.

Definition 9

([13]). Let X be a topological space and $f:X\to \mathbb{R}\cup \{\infty \}$ be bounded from below. A sequence (net) $\left\{x_n\right\}$ such that $f\left(x_n\right)\to inff$ is called a minimizing sequence (net).

Corollary 1

([13]). Let X be a topological space, and let $f:X\to \mathbb{R}\cup \{\infty \}$ be bounded from below. Assume that $arg{\min }_{\mathit{X}}\mathit{f}=\varnothing$. Then, the following statements are equivalent:

(i) 

f is rgi;

(ii) 

there are not converging minimizing nets;

(iii) 

if $\left\{x_{\alpha }\right\}$ is a converging net, then ${lim\; inf}_{\alpha }f\left(x_{\alpha }\right)>inff$.

Lemma 4

([13]). Let X be a topological space, and let $f:X\to \mathbb{R}\cup \{\infty \}$ be bounded from below. Then, the following statements are equivalent:

(i) 

f is transfer weakly lower continuous at $x_0\in X$;

(ii) 

there exists a $s\in X$ such that for all nets $\left\{x_{\alpha }\right\}$ which converge to $x_0$

$$\underset{\alpha }{lim\; inf}f\left(x_{\alpha }\right)\ge f\left(s\right)$$

Theorem 12

([13]). Let X be a real linear space, and let $f:X\to \mathbb{R}\cup \{\infty \}$ be bounded from below and nearly quasiconvex. Then, there exists a quasiconvex function $g:X\to \mathbb{R}\cup \{\infty \}$ such that $f\le g$, $inff=infg$, and $arg{\min }_{\mathit{X}}\mathit{f}=arg{\min }_{\mathit{X}}\mathit{g}$.

Theorem 13.

Let C be a compact convex subset of a real topological vector space and $f:C\to \mathbb{R}\cup \{\infty \}$ be a bounded from below and nearly quasiconvex function. If f is transfer weakly lower continuous such that $arg{\min }_{\mathit{C}}\mathit{f}$ is closed, then there exists a transfer lower continuous quasiconvex function $g:X\to \mathbb{R}\cup \{\infty \}$ such that $f\le g$, $inff=infg$, and $arg{\min }_{\mathit{C}}\mathit{f}=arg{\min }_{\mathit{C}}\mathit{g}$.

Proof. 

We take $g:C\to \mathbb{R}\cup \{\infty \}$ as the quasiconvex function provided by Theorem 12. If there is no converging minimizing net for f on C, then $arg{\min }_{\mathit{C}}\mathit{f}=\varnothing$. Hence, by Corollary 1, f is rgi, and by Proposition 1 the function f is transfer weakly lower continuous. Since f is bounded below, there is sequence $\left\{x_n\right\}$ in C such that $inff={lim\; inf}_{n}f\left(x_n\right)$ and by the compactness of C there exists subsequence $\left\{x_{n_k}\right\}$ which converges to element x of C. Now it follows from Lemma 4 that there exists $s\in C$ such that $inff={lim\; inf}_{\alpha }f\left(x_{n_k}\right)\ge f\left(s\right)$. This means $s\in arg{\min }_{\mathit{C}}\mathit{f}=arg{\min }_{\mathit{X}}\mathit{g}$ which is a contradiction. Consequently, $arg{\min }_{\mathit{C}}\mathit{g}\ne \varnothing$. Now, since $arg{\min }_{\mathit{C}}\mathit{g}\ne \varnothing$ and the $arg{\min }_{\mathit{C}}\mathit{f}$ is closed, then it follows from Theorem 1 that g is transfer lower continuous. This completes the proof. □

Let X be a Hausdorff topological space and $2^X$ denote the class of all subsets of X. A measure of noncompactness on X associates numbers to subsets of X in such a way that compact sets all get the measure zero and the other sets get measures that are bigger according to how far they are removed from compactness.

Definition 10

([13]). A measure of noncompactness on X is simply any function $\gamma :2^X\to \mathbb{R}\cup \{\infty \}$ such that

(i) 

$\gamma \left(\overline{A}\right)=\gamma \left(A\right),\forall A\in 2^X$;

(ii) 

$\gamma \left(A\right)=0$ if and only if A is relatively compact;

(iii) 

$\gamma (A\cup B)=max\{\gamma \left(A\right),\gamma \left(B\right)\},\forall A,B\in 2^X$.

As a consequence, $A\subseteq B$ implies $\gamma \left(A\right)\le \gamma \left(B\right)$.

Corollary 2

([13]). Let $(X,\parallel ·\parallel )$ be a Banach space and β be the measure of noncompactness (with respect to the weak topology). Assume that $f:X\to \mathbb{R}\cup \{\infty \}$ is proper, transfer weakly lower continuous and quasiconvex and that $m=inff\in \mathbb{R}$ and ${lim}_{c\to m^{+}}\beta \left([f\le c]\right)=0$. Then, $arg{\min }_{\mathit{X}}\mathit{f}$ is nonempty and relatively weakly compact.

The next result generalizes Corollary 2 from quasiconvex functions to nearly quasiconvex functions.

Theorem 14.

Let $(X,\parallel ·\parallel )$ be a Banach space and β be the measure of (weak) noncompactness. Assume that $f:X\to \mathbb{R}\cup \{\infty \}$ is proper, transfer weakly lower continuous, and nearly quasiconvex and that $m=inff\in \mathbb{R}$ and ${lim}_{c\to m^{+}}\beta \left([f\le c]\right)=0$. Then, $arg{\min }_{\mathit{X}}\mathit{f}$ is nonempty and relatively weakly compact.

Proof. 

By Theorem 13, there exists a quasiconvex and transfer lower continuous function $g:X\to \mathbb{R}\cup \{\infty \}$ such that $f\le g$, $inff=infg$, and $arg{\min }_{\mathit{X}}\mathit{f}=arg{\min }_{\mathit{X}}\mathit{g}$. Since $[g\le c]\subseteq [f\le c]$ for all $c\in \mathbb{R}$ and ${lim}_{c\to m^{+}}\beta \left([f\le c]\right)=0$, then ${lim}_{c\to m^{+}}\beta \left([g\le c]\right)=0$. By Corollary 2, $arg{\min }_{\mathit{X}}\mathit{g}$ is nonempty and relatively weakly compact, and so $arg{\min }_{\mathit{X}}\mathit{f}$ is nonempty and relatively weakly compact. □

As applications of Theorem 14, the next results are given.

Theorem 15.

Let $(X,\parallel ·\parallel )$ be a Banach space and $F:X\times X\to \mathbb{R}$ a bifunction. Suppose that

(i) 

there exists $\widehat{z}\in X$ such that $F(\widehat{z},·)$ satisfies in the conditions of Theorem 14;

(ii) 

F satisfies the triangle inequality property.

Then, the solution set of EP is nonempty. Also, if the function F is restricted to a compact and convex subset C of X, then, there exists a bifunction $G:C\to \mathbb{R}$ such that G is bounded from below, quasi-convex and transfer lower continuous with $F(\widehat{z},·)\le G$, $infF(\widehat{z},·)=infG$, and $arg{\min }_{\mathit{C}}\mathit{F}(\widehat{\mathrm{z}},·)=arg{\min }_{\mathit{C}}\mathit{G}$.

Proof. 

By Theorem 14, the $arg{\min }_{\mathit{X}}\mathit{F}(\widehat{z},·)$ is nonempty. Hence, there exists $\tilde{x}\in X$ which is a minimum point of $F(\widehat{z},·)$ on X. Now, since F satisfies the triangle inequality property, we have

$$F(\tilde{x},y)\ge F(\widehat{z},y)-F(\widehat{z},\tilde{x})\ge 0,\forall y\in X.$$

Thus, the solution set of EP is nonempty. The rest of proof follows from Theorem 13. □

Theorem 16.

Let $(X,\parallel ·\parallel )$ be a Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that the following conditions hold

(i) 

there exists a function $f:X\to \mathbb{R}\cup \{\infty \}$ satisfies in the conditions of Theorem 14;

(ii) 

F is cyclically antimonotone.

Then, the solution set of EP is nonempty. Also, if the function f is restricted to a compact and convex subset C of X, then there exists a bounded from below and quasiconvex and transfer lower continuous function $g:C\to \mathbb{R}\cup \{\infty \}$, such that

$f\le g$, $inff=infg$, and $arg{\min }_{\mathit{C}}\mathit{f}=arg{\min }_{\mathit{C}}\mathit{g}$.

Proof. 

All the conditions of Theorem 14 for function $f:X\to \mathbb{R}$ hold; therefore, Theorem 14 concludes that the set $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$. Therefore, there exists $\tilde{x}\in X$ which is a minimum point of f on X. Now, since F is cyclically antimonotone, we have

$$F(\tilde{x},y)\ge f\left(y\right)-f\left(\tilde{x}\right)\ge 0,\forall y\in X.$$

Hence, the solution set of EP is nonempty. The rest of the proof follows from Theorem 13. □

In the next part of this section, using the following basic facts, some evidence of a solution for EP and MEP in a metric-space setting is given.

Condition 1

($\mathcal{N}$ [11]). Let C be a subset of the topological space X. The bifunction $F:C\times C\to \mathbb{R}$ is said to satisfy in condition $\left(\mathcal{N}\right)$ if for all $x,y\in C$ and neighborhood $U_x$ of x, there exist $z\in U_x$ and $t\in (0,1]$ such that $tF(z,y)+(1-t)F(z,x)\ge 0$.

Remark 7.

It is easy to see that, as a special case of the condition $\left(\mathcal{N}\right)$, if C is a subset of a Banach space and $F:C\times C\to \mathbb{R}$ is a bifunction, then F satisfies in the condition $\left(\mathcal{N}\right)$ when for all $x,y\in C$ and $r>0$, there exist $z\in C\cap (x+rB)$ and $t\in (0,1]$ such that $tF(z,y)+(1-t)F(z,x)\ge 0$, where B is the open unit ball.

Theorem 17

([11]). Let C be a topological space and $F:C\times C\to \mathbb{R}$ be a bifunction satisfying condition $\left(\mathcal{N}\right)$. If for any $y\in C$ the set $\mathcal{A}\left(y\right)=\{x\in C:F(x,y)<0\}$ is open, then the solution set of MEP is a subset of the solution set of EP.

Lemma 5

([14]). Let $(X,d)$ be a metric space and let $f:X\to \mathbb{R}$ be a bounded from below function. Then f is transfer weakly lower continuous if and only if either $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$ or f has no convergent minimizing sequence.

Proposition 2.

Let $(X,d)$ be a metric space and $F:X\times X\to \mathbb{R}$ be a bifunction. Suppose that

(i) 

there exists $\widehat{z}\in X$ such that $F(\widehat{z},·)$ is transfer weakly lower continuous,

(ii) 

$F(\widehat{z},·)$ has at least a convergent minimizing sequence,

(iii) 

F satisfies the triangle inequality property.

Then the solution set of EP is nonempty.

Proof. 

By (ii) there exists sequence $\left\{x_n\right\}$ in X which converges to $\overline{x}$ and $limF(\widehat{z},x_n)=infF(\widehat{z},·)$. Hence, there exists $\tilde{x}\in X$ such that $infF(\widehat{z},·)=lim\; infF(\widehat{z},x_n)\ge F(\widehat{z},\tilde{x})$. Therefore $\tilde{x}\in arg{\min }_{\mathit{X}}\mathit{F}(\widehat{z},·)$. Now by (iii) we get

$$F(\tilde{x},y)\ge F(\widehat{z},y)-F(\widehat{z},\tilde{x})\ge 0,\forall y\in X.$$

This means $\tilde{x}$ is a solution of EP. □

Proposition 3.

Let $(X,d)$ be a metric space and $F:X\times X\to \mathbb{R}$ be a bifunction. If

(i) 

F is cyclically antimonotone,

(ii) 

g is transfer weakly lower continuous,

(iii) 

g has at least a convergent minimizing sequence,

then the solution set of EP is nonempty.

Proof. 

By the similar proof given in Proposition 2, the $arg{\min }_{\mathit{X}}\mathit{g}$ is nonempty. Now the result follows from (i). □

Proposition 4.

Let $(X,d)$ be a metric space and $F:X\times X\to \mathbb{R}$ be a bifunction. Suppose that the following conditions hold

(i) 

there exists $\widehat{z}\in X$ such that $F(\widehat{z},·)$ is transfer weakly lower continuous,

(ii) 

$F(\widehat{z},·)$ has at least a convergent minimizing sequence,

(iii) 

$-F$ satisfies the triangle inequality property.

Then the solution set of MEP is nonempty.

Proof. 

The proof is similar to the proof of Proposition 2. □

Proposition 5.

Let $(X,d)$ be a metric space and $F:X\times X\to \mathbb{R}$ be a bifunction. Assume that the following hypotheses are satisfied

(i) 

F is cyclically monotone,

(ii) 

g is transfer weakly lower continuous,

(iii) 

g has at least a convergent minimizing sequence.

Then the solution set of MEP is nonempty.

Proof. 

The proof is similar to the proof of Proposition 3. □

Corollary 3.

Let $(X,d)$ be a metric space and $F:X\times X\to \mathbb{R}$ be a bifunction. Suppose that

(i) 

there exists a transfer weakly lower continuous and bounded from below function $f:X\to \mathbb{R}$ such that F is cyclically monotone,

(ii) 

f has at least a convergent minimizing sequence.

Then, the solution set of EP is nonempty.

Proof. 

Lemma 5 ensures that $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$ and the existence of equilibria follows from that fact that $arg{\min }_{\mathit{X}}\mathit{f}$ is a subset the solution set of EP. □

Corollary 4.

Let $(X,d)$ be a metric space and $F:X\times X\to \mathbb{R}$ a bifunction. Assume that

(i) 

there exist a transfer weakly lower continuous and bounded from below function $f:X\to \mathbb{R}$ such that F is cyclically monotone,

(ii) 

f has at least a convergent minimizing sequence.

Then, the solution set of MEP is nonempty.

Proof. 

Lemma 5 ensures that $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$ and now, the existence of a solution for the Minty equilibria follows from that fact that $arg{\min }_{\mathit{X}}\mathit{f}$ is a subset of the solution set of MEP. □

Proposition 6.

Let $(X,d)$ be a metric space and $F:X\times X\to \mathbb{R}$ a bifunction. Suppose that the following conditions hold

(i) 

there exist a transfer weakly lower continuous and bounded from below function $f:X\to \mathbb{R}$ such that F is cyclically monotone;

(ii) 

f has at least a convergent minimizing sequence;

(iii) 

F satisfies the condition $\left(\mathcal{N}\right)$;

(iv) 

for every $y\in X$, the function $F(·,y)$ is upper semicontinuous.

Then, the solution set of EP is nonempty.

Proof. 

The proof follows from Proposition 1 and Theorem 17. □

Remark 8.

Proposition 6 generalizes Theorem 5.27 in [20] from compact metric spaces to metric spaces.

Remark 9

([13], p. 2847). Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $f:X\to \mathbb{R}\cup \{\infty \}$ be sequentially weakly lower semicontinuous (that is sequentially lower semicontinuous with respect to the weak topology) and the sublevel sets of f be nonempty and bounded. Then, the set $arg{\min }_{\mathit{C}}\mathit{f}$ is nonempty.

Theorem 18.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

there exists a sequentially weakly lower semicontinuous function $f:X\to \mathbb{R}\cup \{\infty \}$ such that F is cyclically antimonotone,

(ii) 

the sublevel sets of f are nonempty and bounded.

Then the solution set of EP is nonempty.

Proof. 

Remark 9 ensures that $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$. Hence, the result follows from the cyclically antimonotonicity of F. □

Remark 10.

Theorem 18 generalizes and improves Theorem 3.4 in [9].

Theorem 19.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

there exists a sequentially weakly lower semicontinuous function $f:X\to \mathbb{R}\cup \{\infty \}$ such that F is cyclically monotone, i.e.,

$F(x,y)\le f\left(y\right)-f\left(x\right)$, $\forall x,y\in X;$

(ii) 

sublevel sets of f are nonempty and bounded.

Then the solution set of MEP is nonempty.

Proof. 

Remark 9 ensures that $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$ and the result follows from the cyclically monotonicity of F. □

Remark 11.

Theorem 19 improves and generalizes Theorem 3.1 in [11]

Corollary 5.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

there exists a sequentially weakly lower semicontinuous function $f:X\to \mathbb{R}\cup \{\infty \}$ such that F is cyclically monotone, i.e.,

$F(x,y)\le f\left(y\right)-f\left(x\right)$, $\forall x,y\in X;$

(ii) 

sub level sets of f are nonempty and bounded;

(iii) 

F satisfying condition $\left(\mathcal{N}\right)$;

(iv) 

for every $y\in X$, $F(·,y)$ is upper semicontinuous.

Then the solution set of EP is nonempty.

Proof. 

The proof follows from Theorem 19 and Theorem 17. □

Theorem 20

([15]). Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $f:X\to \mathbb{R}\cup \{\infty \}$ be a sequentially weakly lower semicontinuous function. Then following properties are equivalent.

(i) 

$arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$;

(ii) 

(A) For any unbounded minimizing sequence $\left\{x_n\right\}$, there are scalar sequence $\left\{{\alpha }_k\right\}$ and vector sequences $\left\{y_k\right\}$ and $\left\{z_k\right\}$ such that, for all enough large k;

(a) $\parallel y_k\parallel \le \parallel x_{n_k}\parallel$, ${\alpha }_k\in (0,\parallel y_k\parallel ]$, and $\parallel z_k-{\displaystyle \frac{y_k}{\parallel y_k\parallel }}\parallel <1$,

(b) $f(y_k-{\alpha }_k{z}_k)\le f\left(x_{n_k}\right)$.

(iii) 

(R) For any unbounded minimizing sequence $\left\{x_n\right\}$, there is a sequence $\left\{w_k\right\}$ such that, for all enough large k, $\parallel w_k\parallel \le \parallel x_{n_k}\parallel$ and $f\left(w_k\right)\le f\left(x_{n_k}\right)$.

Proposition 7.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ a bifunction. Suppose that

(i) 

F satisfies the triangle inequality property;

(ii) 

there exists $\widehat{z}\in X$ such that $F(\widehat{z},·)$ is sequentially weakly lower semicontinuous and at least one of the conditions (A) or (R) for $F(\widehat{z},·)$ holds.

Then the solution set of EP is nonempty.

Proof. 

The result follows from Theorem 20 and condition (i). □

Remark 12.

Proposition 7 generalizes Theorem 5.1 in [18].

Proposition 8.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

$-F$ satisfies the triangle inequality property;

(ii) 

there exists $\widehat{z}\in X$ such that $F(\widehat{z},·)$ is sequentially weakly lower semicontinuous and at least one of conditions (A) and (R) for $F(\widehat{z},·)$ holds;

Then the solution set of MEP is nonempty.

Proof. 

The result follows from Theorem 20 and (i). □

Proposition 9.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

$-F$ satisfies the triangle inequality property;

(ii) 

there exists $\widehat{z}\in X$ such that $F(\widehat{z},·)$ is sequentially weakly lower semicontinuous and at least one of conditions (A) and (R) for $F(\widehat{z},·)$ holds;

(iii) 

F satisfies condition $\left(\mathcal{N}\right)$;

(iv) 

for every $y\in X$, $F(·,y)$ is upper semicontinuous.

Then the solution set of EP is nonempty.

Proof. 

The result is the combination of Proposition 8 and Theorem 17. □

Proposition 10.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

F is cyclically antimonotone;

(ii) 

f is sequentially weakly lower semicontinuous and at least one of conditions (A) or (R) holds.

Then the solution set of EP is nonempty.

Proof. 

By using Theorem 3 and Proposition 7 the proof is obvious. □

Remark 13.

Proposition 10 improves Theorem 3.5 in [10] in the following ways

(a) 

We replace the triangle inequality property of F by cyclically antimonotonity assumption of F.

(b) 

We replace the lower semicontinuous from above $F(x,·)$, for all $x\in X$ by the sequentially weakly lower semicontinuous $F(x,·)$ for all $x\in X$.

(c) 

We replace the closedness of the subset D of metric space by one of conditions (A) or (R).

Proposition 11.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

F is cyclically monotone;

(ii) 

for all $\widehat{z}\in X$, $F(\widehat{z},·)$ is sequentially weakly lower semicontinuous and at least one of conditions (A) and (R) for $F(\widehat{z},·)$ holds.

Then the solution set of MEP is nonempty.

Proof. 

The proof follows from Theorem 3 and Proposition 8. □

Corollary 6.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

F is cyclically monotone;

(ii) 

for all $\widehat{z}\in X$, $F(\widehat{z},·)$ is sequentially weakly lower semicontinuous and at least one of conditions (A) and (R) for $F(\widehat{z},·)$ holds;

(iii) 

F satisfies condition $\left(\mathcal{N}\right)$;

(iv) 

for every $y\in X$, $F(·,y)$ is upper semicontinuous.

Then the solution set of EP is nonempty.

Proof. 

By combining Proposition 11 and Theorem 17 the proof is obvious. □

Corollary 7.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ a bifunction. If the following assertions hold

(i) 

there exists a sequentially weakly lower semicontinuous function $f:X\to \mathbb{R}\cup \{\infty \}$ such that F is cyclically antimonotone,

(ii) 

at least one of conditions (A) and (R) holds.

Then, the solution set of EP is nonempty.

Proof. 

Theorem 20 ensures that $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$, now the result follows from that fact that $arg{\min }_{\mathit{X}}\mathit{f}$ is a subset the solution set of EP. □

Remark 14.

Corollary 7 generalizes and extends Theorem 3.8 in [9] in reflexive Banach spaces in their respective ways below.

(a) 

We replace the nonempty weakly closed subset of reflexive Banach space $(X,\parallel ·\parallel )$ by any reflexive Banach space $(X,\parallel ·\parallel )$.

(b) 

We replace the following coercivity condition

there exists a nonempty weakly compact subset K of weakly closed subset C such that

$$\forall x\in C\setminus K,\exists y\in C,\parallel y\parallel <\parallel x\parallel :f\left(y\right)\le f\left(x\right),$$

by one of conditions (A) and (R).

Corollary 8.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that

(i) 

there exists a sequentially weakly lower semicontinuous function $f:X\to \mathbb{R}\cup \{\infty \}$ such that F is cyclically monotone;

(ii) 

at least one of conditions (A) and (R) holds.

Then the solution set of MEP is nonempty.

Proof. 

Theorem 20 ensures that $arg{\min }_{\mathit{X}}\mathit{f}\ne \varnothing$ and the existence of Minty equilibria follows from that fact that $arg{\min }_{\mathit{X}}\mathit{f}$ is a subset the solution set of MEP. □

Remark 15.

Corollary 4 generalizes and extends Theorem 3.2 in [11] in reflexive Banach spaces the ways below, respectively.

(a) 

We replace the nonempty weakly closed subset of reflexive Banach space $(X,\parallel ·\parallel )$ by any reflexive Banach space $(X,\parallel ·\parallel )$.

(b) 

We replace the following coercivity condition

there exists a nonempty weakly compact subset K of weakly closed subset C such that

$$\forall x\in C\setminus K,\exists y\in C,\parallel y\parallel <\parallel x\parallel :F(y,x)\ge 0,$$

by one of conditions (A) and (R).

Corollary 9.

Let $(X,\parallel ·\parallel )$ be a reflexive Banach space and $F:X\times X\to \mathbb{R}\cup \{\infty \}$ be a bifunction. Suppose that the following assumptions hold:

(i) 

There exists a sequentially weakly lower semicontinuous function $f:X\to \mathbb{R}\cup \{\infty \}$ such that F is cyclically monotone;

(ii) 

aAt least one of conditions (A) and (R) holds;

(iii) 

F satisfying condition $\left(\mathcal{N}\right)$;

(iv) 

For every $y\in X$, $F(·,y)$ is upper semicontinuous.

Then, the solution set of EP is nonempty.

Proof. 

By combining Corollary 4 and Theorem 17 is obvious. □

Remark 16.

Corollary 9 generalizes and improves Corollary 3.3 in [11] as follows.

(a) 

We replace the nonempty weakly closed subset of reflexive Banach space $(X,\parallel ·\parallel )$ by any reflexive Banach space $(X,\parallel ·\parallel )$.

(b) 

the following coercivity condition

there exists a nonempty weakly compact subset K of weakly closed subset C such that

$$\forall x\in C\setminus K,\exists y\in C,\parallel y\parallel <\parallel x\parallel :F(y,x)\ge 0,$$

is replaced by one of conditions (A) or (R).

4. Conclusions

In this paper, some existence theorems for a solution of an equilibrium problem, quasi-equilibrium problem, scalar quasi-equilibrium problem, and scalar generalized quasi-equilibrium problem for bifunctions by using the assumptions of the triangle inequality property, cyclically monotone, and cyclically antimonotone on the maps in the setting of Hausdorff topologicalvector spaces are proved. In the proof of the main results the notions transfer lower continuous, transfer weakly lower continuous, lower semicontinuous from above and sequentially weakly lower semicontinuous have played important role. The results of the paper can be considered as generalizations, improvements and extensions of the corresponding results presented in [6,9,10,11,12,18,19,20] from reflexive Banach spaces to locally convex topological vector spaces.