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A generalization of Ekeland’s variational principle by using the τdistance with its applications
 AP Farajzadeh^{1},
 S Plubtieng^{2}Email author and
 A Hoseinpour^{2}
https://doi.org/10.1186/s1366001714357
© The Author(s) 2017
 Received: 17 April 2017
 Accepted: 23 June 2017
 Published: 31 July 2017
Abstract
In this paper, a new version of Ekeland’s variational principle by using the concept of τdistance is proved and, by applying it, an approximate minimization theorem is stated. Moreover, by using it, two versions of existence results of a solution for the equilibrium problem in the setting of complete metric spaces are investigated. Finally some examples in order to illustrate the results of this note are given.
Keywords
 τdistance
 Ekeland’s variational principle
 equilibrium problem
 bounded below
 lower semicontinuous function
1 Introduction
Ekeland’s variational principle was first expressed by Ekeland [1, 2] and developed by many authors and researchers. Tataru in [3] defined the concept of Tataru’s distance and using it proved the generalization of Ekeland’s variational principle. Afterwards in 1996, Kada et al. in [4] stated the concept of wdistance and extended Ekeland’s variational principle. The concept of τdistance which is a generalization of wdistance and Tataru’s distance was first introduced by Suzuki. He also improved the concept of Ekeland’s variational principle; see [5]. Over the last few years, several authors have studied Ekeland’s variational principle for equilibrium problems under different conditions; see, for instance, [6, 7]. In these papers, the authors studied the equilibrium version of Ekeland’s variational principle to get some existence results for equilibrium problems in both compact and noncompact domains. In 2007, Bianchi et al. [8] introduced a vector version of Ekeland’s principle for equilibrium problems. They studied bifunctions defined on complete metric spaces with values in locally convex spaces ordered by closed convex cones and obtained some existence results for vector equilibria in compact and noncompact domains. However, several authors have made efforts to get new existence results in the studies of equilibrium problems, for instance, [9–11].
The purpose of this paper is to study equilibrium problem to get some existence results. In fact, we first recall the concept of τdistance on a complete metric space and then by using it a new version of Ekeland’s variational principle by using the concept of τdistance is proved and, by applying it, an approximate minimization theorem is stated. Moreover, as an application, two versions of existence results of a solution for the equilibrium problem in the setting of complete metric spaces are investigated. Finally some examples in order to illustrate the results of this note are given.
2 Ekeland’s variational principle
In this section, we will rely on a new version of Ekeland’s variational principle which improves the corresponding result, replacing a meter by τdistance [5], was obtained by the authors of [6]. We need the following preliminaries in the sequel.
Definition 2.1
[5]
 \(\tau_{1}\)::

\(p(x,z)\leq p(x,y)+p(y,z)\), for all \(x,y,z\in X\),
 \(\tau _{2}\)::

\(\eta (x,0)=0\) and \(\eta (x,t)\geq t\), for all \(x\in X\), \(t\in \mathbb{R}^{+}\),
 \(\tau _{3}\)::

\(\lim_{n} x_{n}=x\) and \(\lim_{n} \sup \{\eta (z _{n},p(z_{n},x_{m}))\mid m\geq n\}=0\) imply \(p(w,x)\leq \liminf_{n} p(w,x _{n})\), for all \(w\in X\),
 \(\tau _{4}\)::

\(\lim_{n} \sup \{p(x_{n},y_{m})\mid m\geq n\}=0\) and \(\lim_{n}\eta (x_{n},t_{n})=0\) imply \(\lim_{n}\eta (y_{n},t_{n})=0\),
 \(\tau _{5}\)::

\(\lim_{n}\eta (z_{n},p(z_{n},x_{n}))=0\) and \(\lim_{n}\eta (z_{n},p(z_{n},y_{n}))=0\) imply \(\lim_{n} d(x_{n},y_{n})=0\).
Example 2.2
The next result plays a key role in proving the theorem concerned with Ekeland’s variational principle.
Proposition 2.3
[5]
Let X be a complete metric space, p be the τdistance on X and \(f:X\rightarrow (\infty,\infty ]\) be proper, i.e., \(f\neq \infty \), lower semicontinuous and bounded from below. Define \(Mx=\{y\in X\mid f(y)+p(x,y)\leq f(x)\}\), for all \(x\in X\). Then, for each \(u\in X\) with \(Mu\neq \emptyset \), there exists \(x_{0}\in Mu\) such that \(Mx_{0}\subset \{x_{0}\}\). In particular, there exists \(y_{0}\in X\) such that \(My_{0}\subset \{y_{0}\}\).
Now we are ready to state a new version of Ekeland’s variational principle.
Theorem 2.4
Proof
The following example illustrates Theorem 2.4.
Example 2.5
Let \(X=[0,\infty ]\) and d be the Euclidean metric on X. Consider the function \(p:X\times X\rightarrow \mathbb{R}\) defined by \(p(x,y)=\vert y\vert \), for all \(x,y\in X\). The function p is a τdistance (see Example 2.2). Define the function \(f:X\times X\rightarrow \mathbb{R}\) by \(f(x,y)=\sin (yx)\), for all \(x,y\in X\). Obviously, f satisfies all the conditions of Theorem 2.4 and so if we take \(x_{0}=0\) then \(\overline{x}=0\) is a candidate which fulfils in the conclusion of Theorem 2.4. Similarly, in Theorem 2.4 for \(x_{0}=\frac{ \pi }{2}\) we can obtain \(\overline{x}=0\).
Remark 2.6
Note that the authors of [6] could generalize Theorem 2.1 in [7] by replacing a norm by a metric. In the following, we also extend the main result of [6] (an extension of Theorem 2.1 in [7]) by substituting a meter by a τdistance. Since each distance (meter) defines a τdistance and Example 2.2 shows that the converse is not generally true, such a generalization is a real extension. Hence, Example 2.5 cannot be solved using the mentioned methods in [6, 7].
As a consequence of Theorem 2.4 we present the next result which is an extension of the corresponding results of [6] and [7], respectively, from metric space and normed space to the τdistance.
Theorem 2.7
Proof
3 Ekeland’s variational principle for equilibrium problems
Let X be a given set and \(f:X\times X\rightarrow \mathbb{R}\) be a given function. An equilibrium problem is finding \(\overline{x}\in X\) such that \(f(\overline{x},y)\geq 0\), for all \(y\in X\). We may abbreviate this problem with EP from now on. In this section, we intend to provide sufficient conditions to solve the EP using the new version of Ekeland’s variational principle and the concept of τdistance.
The following result is a new version of Corollary 2.1 of [6] which guarantees the existence of a solution of EP in complete metric spaces with the notion of τ distance.
Theorem 3.1
Suppose that the assumptions of Theorem 2.4 are satisfied. Moreover, for every \(x\in X\) with \(\inf_{y\in X}f(x,y)<0\), there exists \(z\in X\) such that \(z\neq x\) and \(f(x,z)+ p(x,z)\leq 0\). Then the EP has at least one solution.
Proof
In the following, using some results in [6, 7, 9, 12], we obtain two following theorems stating the existence of solutions of EP in two cases. In the first case, we discuss the existence of solutions of EP in a compact metric space and in the next one, we provide some conditions for the existence of solutions in a noncompact metric space.
Theorem 3.2
Beside the assumptions of Theorem 2.4, assume that X is a compact metric space and for each \(y\in X\) the functions \(x\to f(x,y)\) and \(x\to p(x,y)\) are upper semicontinuous. Then the solution set of EP is nonempty and compact.
Proof
Remark that the result of Theorem 3.2 is valid if we replace the upper semicontinuity of p in the first variable by the bounded above function \(x\to p(x,y)\), for all \(y\in X\).
In the next theorem, we present an existence result for the noncompact case.
Theorem 3.3
Proof
We note that \(F(\overline{x})\cap K\) is nonempty, because of we have two cases, the first case \(\overline{x}\in K\) and so \(\overline{x}\in F( \overline{x})\cap K\), and the second case \(\overline{x}\notin K\). Then by (9) there exists \(y\in K\) such that \(p(y,x_{0})\leq p(x,x _{0})\) and \(f(x,y)\leq 0\), and so \(y\in F(\overline{x})\cap K\). This completes the proof of the assertion.
4 Conclusion
In the present paper, we study the vectorial form of Ekeland’s variational principle by using the concept of τdistance. We obtain some existence results for the equilibrium problems in the setting of complete metric spaces in the cases of compact and noncompact spaces.
Declarations
Acknowledgements
The authors would like to thank Naresuan University, Phitsanulok 65000, Thailand.
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Authors’ Affiliations
References
 Ekeland, I: On the variational principle. J. Math. Anal. Appl. 47, 324353 (1974) MathSciNetView ArticleMATHGoogle Scholar
 Ekeland, I: Nonconvex minimization problems. Bull., New Ser., Am. Math. Soc. 1, 443474 (1979) MathSciNetView ArticleMATHGoogle Scholar
 Tataru, D: Viscosity solutions of HamiltonJacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 163, 345392 (1992) MathSciNetView ArticleMATHGoogle Scholar
 Kada, O, Suzuki, T, Takahashi, W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 44, 381391 (1996) MathSciNetMATHGoogle Scholar
 Suzuki, T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 253, 440458 (2001) MathSciNetView ArticleMATHGoogle Scholar
 AminiHarandi, A, Ansari, QH, Farajzadeh, AP: Existence of equilibria in complete metric spaces. Taiwan. J. Math. 16, 777785 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Bianchi, M, Kassay, G, Pini, R: Existence of equilibria via Ekeland’s principle. J. Math. Anal. Appl. 305, 502512 (2005) MathSciNetView ArticleMATHGoogle Scholar
 Bianchi, M, Kassay, G, Pini, R: Ekeland’s principle for vector equilibrium problems. Nonlinear Anal. 66, 14541464 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Alleche, B, Radulescu, VD: The Ekeland variational principle for equilibrium problems revisited and applications. Nonlinear Anal., Real World Appl. 23, 1725 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Lin, LJ, Du, WS: Ekeland’s variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces. J. Math. Anal. Appl. 323, 360370 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Araya, Y, Kimura, K, Tanaka, T: Existence of vector equilibria Ekeland’s variational principle. Taiwan. J. Math. 12, 19912000 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Zeng, J, Li, SJ: An Ekeland’s variational principle for setvalued mappings with applications. J. Comput. Appl. Math. 230, 477484 (2009) MathSciNetView ArticleMATHGoogle Scholar