Existence and continuity for the εapproximation equilibrium problems in Hadamard spaces
 Pakkapon Preechasilp^{1}Email author
https://doi.org/10.1186/s1366001610735
© Preechasilp 2016
Received: 5 October 2015
Accepted: 19 April 2016
Published: 3 May 2016
Abstract
In this paper, the existence of εapproximate equilibrium points for a bifunction is proved under suitable conditions in the framework of a Hadamard space. We also give the sufficient conditions for the continuity of εapproximate solution maps to equilibrium problems. Then we apply our results to constrained minimization problems and Nashequilibrium problems.
Keywords
MSC
1 Introduction
On the other hand, when we have a mathematical problem, not only the existence of the solution for that problem but also the stability for the solution set of the problem is investigated. Roughly speaking this is about how a slight change in a parameter of a given mathematical problem could affect the solution or solution set of the problem. Hence, stability may be understood as lower (upper) semicontinuity, continuity, and Lipschitz or Hölder continuity. Currently, such a study is, in fact, very important as many useful mathematical problems are usually approximately solved by the problem. Since data of mathematical models of practical problems are obtained by measuring devices or statistical records, the models are also approximations, and hence their exact solutions are the acceptability of the approximate solution, based on a certain allowed error on the parameters of the problem. However, concerning continuity, results of approximate solutions to the parametric equilibrium problems, e.g. [15–17], are established in linear spaces.
Motivated and inspired by the above literature, the aim of this paper is to establish the existence result for (EP) and continuity of the solution mapping to parametric εapproximate parametric equilibrium problem, for short (PEP), in the setting of Hadamard spaces. We also give applications to constrained minimization problems and Nashequilibrium problems.
2 Preliminaries
Let \(A \subseteq X \) be a nonempty subset. Then the convex hull of A, denoted \(co(A) \), is defined as the intersection of all convex subset containing A. It is easy to see that \(co (A ) = \bigcup_{n=0}^{\infty}A_{n}\), where \(A_{0} = A\), and for \(n \geq1 \), the set \(A_{n} \) is the union of all geodesics with start and end in \(A_{n1} \). The convex hull of a finite subset is not necessarily closed, and likewise the closed convex hull of A, denoted \(\overline{co}(A) \), is the intersection of all closed convex subsets containing A. It is shown in [18] that if \(A_{1},\ldots ,A_{n} \) are compact convex subsets in a locally convex Hausdorff space, then the convex hull of their union is also compact.
Definition 1
We recall the KKM mapping in the setting of Hadamard spaces which is used for the existence result for (EP).
Definition 2
[12]
The following concept of the convex hull finite property was first introduced by [12] and used to prove the analog of the KKM lemma in Hadamard spaces.
Definition 3
[12]
We say that a Hadamard space X has the convex hull finite property if the closed convex hull of every nonempty finite family of points of X has the fixed point property.
Lemma 4
[12]
Throughout this paper, if not otherwise specified, let X be a Hadamard space, M be metric space, and \(A\subseteq X \) be nonempty set.
Let \(N(\mu_{0}) \subset M\) be a neighborhood of the considered point \(\mu _{0}\). Let \(K : M \rightrightarrows A\) be a nonempty setvalued mapping and \(f : A \times A \times M \rightarrow\mathbb{R}\).
We collect some concepts and properties of semicontinuity for setvalued mappings in metric spaces.
Definition 5
 (i)
T is said to be upper semicontinuous (u.s.c., for short) at \(x_{0}\in X \) iff for any open set V containing \(T(x_{0}) \), there exists an open set U containing \(x_{0} \) such that \(T(x)\subseteq V \) for all \(x\in U \).
 (ii)
T is said to be lower semicontinuous (l.s.c., for short) at \(x_{0}\in X \) iff for any open set V with \(T(x_{0}) \cap V\), there exists an open set U containing \(x_{0} \) such that \(T(x)\cap V \neq \emptyset\) for all \(x\in U \).
 (iii)
T is said to be continuous at \(x_{0}\in X \) iff it is both l.s.c. and u.s.c.
Proposition 6
 (i)
\(T(\cdot) \) is l.s.c. at \(x_{0} \);
 (ii)if \(\{x_{n}\} \) is any sequence such that \(x_{n}\rightarrow x_{0} \) and \(G\subseteq Y \) an open subset such that \(T(x_{0})\cap G\neq \emptyset\), then$$ \exists N\geq1: T(x_{n})\cap G\neq\emptyset,\quad \forall n\geq N; $$
 (iii)
if \(\{x_{n}\} \) is any sequence such that \(x_{n}\rightarrow x_{0} \) and \(y_{0}\in T(x_{0}) \) arbitrary, then there is a sequence \(\{y_{n}\} \) with \(y_{n}\in T(x_{n}) \) such that \(y_{n} \rightarrow y_{0} \).
Proposition 7
 (i)
\(T(\cdot) \) is u.s.c. at \(x_{0} \);
 (ii)if \(x_{0}\in X \) and \(\{x_{n}\} \) is any sequence such that \(x_{n} \rightarrow x_{0} \) and \(V\subseteq Y \) an open subset such that \(T(x_{0})\subseteq V \), then$$ \exists N\geq1 : F(x_{n})\subseteq V, \quad \forall n\geq N. $$
Remark 8
It follows from (iii) in Proposition 6 that \(T(\cdot) \) is l.s.c. at \(x_{0} \) iff, for every sequence \(x_{n}\rightarrow x_{0} \), we have \(T(x_{0})\subseteq\liminf T(x_{n}) \).
3 Existence and continuity results
In this section, we first present the existence result for an εapproximate solution for (EP). We also study the continuity of εapproximate solution maps for (PEP).
Theorem 9
 (i)
for any \(x\in K \), \(f(x,x) \geq0 \);
 (ii)
for every \(x\in K \), the set \(\{y\in K : f(x,y) + \varepsilon< 0\} \) is convex set;
 (iii)
for every \(y\in K \), \(f(\cdot,y) \) is upper semicontinuous;
 (iv)there exist a compact set \(L\subseteq X \) and a point \(y_{0}\in L\cap K \) such that$$ f(x,y_{0}) + \varepsilon< 0, \quad \forall x\in K\backslash L. $$
Proof
Remark 10
By setting \(L = K \) in Theorem 9, the following corollary is immediately obtained.
Corollary 11
 (i)
for any \(x\in K \), \(f(x,x) \geq0 \);
 (ii)
for every \(x\in K \), the set \(\{y\in K : f(x,y) + \varepsilon< 0\} \) is convex set;
 (iii)
for every \(y\in K \), \(f(\cdot,y) \) is upper semicontinuous.
The following example shows that the εapproximate solution for (EP) depends on ε.
Example 12
The following corollary is a sufficient condition for the existence of the parametric εapproximate solution (PEP).
Corollary 13
 (i)
for each \(\mu\in N(\mu_{0}) \), \(K(\mu) \) is a nonempty, compact and convex valued;
 (ii)
for each \(\mu\in N(\mu_{0}) \) and each \(x\in K(N(\mu_{0})) \), \(f(x,x,\mu) \geq0 \);
 (iii)
for each \(x\in K(N(\mu_{0})) \), the set \(\{y\in K(N(\mu_{0})) : f(x,y,\mu) + \varepsilon< 0\} \) is convex set;
 (iv)
for each \(\mu\in N(\mu_{0}) \) and each \(y\in K(N(\mu_{0})) \), \(f(\cdot,y,\mu) \) is upper semicontinuous on \(K(N(\mu_{0})) \).
Now, we give the sufficient conditions for continuity of an approximate solution S̃ at \((\varepsilon_{0},\mu_{0}) \). Relying on the existence theorem for εapproximation (EP), we assume that the solution of the εapproximation exists for all \((\varepsilon ,\mu)\in \mathbb{R}_{+}\cup\{0\}\times M \).
Theorem 14
 (C_{1}):

K is continuous at \(\mu_{0} \) and \(K(\mu_{0}) \) has compact and convex valued;
 (C_{2}):

there exists a neighborhood \(N(\mu_{0}) \) of \(\mu_{0} \) such that \(f(\cdot,\cdot,\cdot) \) is continuous on \(K(N(\mu_{0}))\times K(N(\mu _{0}))\times\{\mu_{0}\} \);
 (C_{3}):

for each \(y\in K(\mu_{0}) \), \(f(\cdot,y,\mu_{0}) \) is a geodesic concave function on \(K(N(\mu_{0})) \).
Proof
Theorem 15
Assume that the conditions (C_{1})(C_{2}) hold. Then \(\widetilde {S}(\cdot ,\cdot)\) is u.s.c. at \((\varepsilon_{0},\mu_{0}) \).
Proof
Theorem 16
Consider the εapproximate (PEP). We assume that the conditions (C_{1})(C_{3}) hold. Then \(\widetilde {S}(\cdot,\cdot) \) is continuous at \((\varepsilon_{0},\mu_{0}) \).
The following example illustrates that Theorem 16 cannot apply with exact solution maps to (PEP).
Example 17
4 Applications
As mentioned in Section 1, the (EP) contains many optimization related problems as special cases. Therefore, we derive the continuity of the results of Section 3 for such special cases. In this section, we give applications to constrained minimization problems and Nashequilibrium problems.
4.1 Constrained minimization problems
Now, we present the continuity result for (CMP); we assume that \(\widetilde {S}_{CMP}(\varepsilon ,\mu) \) exists for all \((\varepsilon ,\mu)\in\mathbb {R}_{+}\cup\{0\}\times M \).
Corollary 18
 (M_{1}):

K is continuous at \(\mu_{0} \) and \(K(\mu_{0}) \) is compact and convex;
 (M_{2}):

there exists a neighborhood \(N(\mu_{0}) \) of \(\mu_{0} \) such that g is continuous in \(K(N(\mu_{0})) \times\{\mu_{0}\} \);
 (M_{3}):

\(g(\cdot,\mu_{0}) \) is geodesic convex in \(K(N(\mu_{0})) \).
Proof
4.2 Nashequilibrium problems
Corollary 19
 (N_{1}):

\(K_{i}(\cdot) \) is continuous at \(\mu_{0} \) and \(K_{i}(\mu_{0}) \) is compact and convex valued for all i;
 (N_{2}):

there exists a neighborhood \(N(\mu_{0}) \) of \(\mu_{0} \) such that \(f_{i}(\cdot,\cdot,\cdot) \) is continuous on \(K_{i}(N(\mu_{0}))\times K_{i}(N(\mu_{0}))\times\{\mu_{0}\} \) for all i;
 (N_{3}):

for each \(y\in K_{i}(N(\mu_{0})) \), \(f_{i}(\cdot,y,\mu_{0}) \) is a geodetically concave function on \(K_{i}(N(\mu_{0})) \) for all i.
5 Conclusions
In this paper, the classical existence result for εapproximate equilibrium problems is proved in the framework of the Hadamard space (nonlinear space). We also establish the sufficient conditions for the continuity of εapproximate solution maps to equilibrium problems. However, the conclusion of Theorem 16 is not true for an exact solution, in the case \(\varepsilon =0 \). As applications, we apply the continuity results to constrained minimization problems and Nashequilibrium problems.
Declarations
Acknowledgements
This research was supported by Pibulsongkram Rajabhat University grant RDI258312. The author is grateful to the anonymous referees for their helpful comments and suggestions, which improved the presentation of this manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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