Equilibrium Problems and Applications

By Gábor Kassay and Vicențiu Rădulescu

Equilibrium Problems and Applications develops a unified variational approach to deal with single-valued, set-valued and quasi-equilibrium problems. The authors promote original results in relationship with classical contributions to the field of equilibrium problems. The content evolved in the general setting of topological vector spaces and it lies at the interplay between pure and applied nonlinear analysis, mathematical economics, and mathematical physics.

This abstract approach is based on tools from various fields, including set-valued analysis, variational and hemivariational inequalities, fixed point theory, and optimization. Applications include models from mathematical economics, Nash equilibrium of non-cooperative games, and Browder variational inclusions. The content is self-contained and the book is mainly addressed to researchers in mathematics, economics and mathematical physics as well as to graduate students in applied nonlinear analysis.

• A rigorous mathematical analysis of Nash equilibrium type problems, which play a central role to describe network traffic models, competition games or problems arising in experimental economics
• Develops generic models relevant to mathematical economics and quantitative modeling of game theory, aiding economists to understand vital material without having to wade through complex proofs
• Reveals a number of surprising interactions among various equilibria topics, enabling readers to identify a common and unified approach to analysing problem sets
• Illustrates the deep features shared by several types of nonlinear problems, encouraging readers to develop further this unifying approach from other viewpoints into economic models in turn

• Language: English
• Publisher: Academic Press
• Release date: Oct 9, 2018
• ISBN: 9780128110300

Gábor Kassay

Gábor Kassay received his Ph.D. thesis at the Babes-Bolyai University in Cluj-Napoca, Romania, under the supervision of József Kolumbán in 1994. He is a Professor in Mathematics at the same University, with more than 75 published research papers, several books and book-chapters in the larger area of nonlinear analysis, and more than 1500 citations. Gábor Kassay delivered many invited and plenary talks, was session organizer and guest of honor at prestigious international conferences. He has more than 35 coauthors and collaborators from all over the world. He is currently the supervisor of the Research Group of Analysis and Optimization accredited by the Faculty of Mathematics and Computer Science of the Babes-Bolyai University. Between 2002 and 2004 he was an associate professor of Eastern Mediterranean University in Famagusta, Cyprus.

Book preview

2018

Chapter 1

Preliminaries and Basic Mathematical Tools

Mathematics is the most beautiful and most powerful creation of the human spirit.

Stefan Banach (1892–1945)

Abstract

The aim of this chapter is to introduce the most important concepts needed in the sequel. This includes both topological (as continuity, or semicontinuity) and algebraic (convexity, quasi-convexity, etc.) concepts. We also recall some basic properties and facts known from the literature. Part of this chapter is self-contained in the sense that proofs for these basic properties are also provided (we refer here to Sperner's and KKM lemmata, and, as well, to Brouwer's fixed point theorem).

Keywords

Semicontinuity; Set-valued mapping; Selection; Sperner lemma; KKM lemma; Brouwer fixed point theorem; Kakutani fixed point theorem; Michael selection theorem; Convex analysis

Chapter Outline

1.1  Elements of Functional Analysis

1.1.1  Continuity of Functions

1.1.2  Semicontinuity of Extended Real-Valued Functions

1.1.3  Hemicontinuity of Extended Real-Valued Functions

1.2  KKM Lemma and the Brouwer's Fixed Point Theorem

1.2.1  The Sperner's Lemma

1.2.2  KKM Lemma

1.2.3  Brouwer's Fixed Point Theorem

1.3  Elements of Set-Valued Analysis

1.3.1  Semicontinuity of Set-Valued Mappings

1.3.2  Selections of Set-Valued Mappings

1.3.3  Elements of Convex Analysis

Chapter points

•  Some basic topological notions are introduced, as continuity and semicontinuity for single and set-valued mappings.

•  Some basic algebraic notions are introduced, as convexity, convexly quasi-convexity (which generalizes both the convexity of set-valued mappings and the quasi-convexity of real single-valued mappings), and concavely quasi-convexity (which generalizes both the concavity of set-valued mappings and the quasi-convexity of real single-valued mappings).

•  We provide a proof for Brouwer's fixed point theorem by Sperner's and KKM lemmata.

1.1 Elements of Functional Analysis

.

, and with the usual operations involving +∞ and −∞. For a subset A of a Hausdorff topological space X, we denote by cl A, the closure of A and by int A, the interior of A.

A subset K of a topological space is called compact if every open cover of K . A topological space X is called compact space if X is a compact set. A family of subsets has the finite intersection property if every finite subfamily has a nonempty intersection.

Let us recall the following characterization of compact spaces.

Proposition 1.1

A topological space is compact if and only if every family of closed subsets with the finite intersection property has a nonempty intersection.

A subset L of a real vector space X . The subset M of X is defined by

while its affine hull by

For each nonempty affine set M (cf. [152]).

A subset K . The conical hull of a subset S of a vector space is given by

(1.1)

, we mean a mapping F from a set X to the collection of nonempty subsets of a set Y, we talk about extended real single-valued or extended real set-valued mappings.

For a subset S of a real vector space, convS will denote the convex hull of S.

If X the dual space of X and Xis defined by

1.1.1 Continuity of Functions

Let X and Y be Hausdorff topological spaces. A function f if for every open subset V of Yis continuous on a subset S of X if it is continuous at every point of S.

1.1.2 Semicontinuity of Extended Real-Valued Functions

In the investigation about solving equilibrium problems, the notions of semicontinuity and hemicontinuity on a subset play an important role. Various results on the existence of solutions of equilibrium problems have been obtained without the semicontinuity and the hemicontinuity of the bifunction on the whole domain, but just on the set of coerciveness.

Let X is said to be lower semicontinuous , there exists an open neighborhood U such that

is said to be upper semicontinuous if −f .

We have considered extended real-valued functions in the above definitions because such functions are more general and convenient in our study. As pointed out by Rockafellar and Wets [155], considering such definitions for extended real-valued functions is also convenient for many purposes of the variational analysis.

is said to be lower (resp., upper) semicontinuous on a subset S of X if it is lower (resp., upper) semicontinuous at every point of S. Obviously, if f is lower (resp., upper) semicontinuous on a subset S of Xof f on S is lower (resp., upper) semicontinuous on S. The converse does not hold true in general.

Proposition 1.2

Let X be Hausdorff topological space, a function and let S be a subset of X. If the restriction of f on an open subset U containing S is upper (resp., lower) semicontinuous on S, then any extension of to the whole space X is upper (resp., lower) semicontinuous on S.

Proposition 1.3

Let X be a Hausdorff topological space, a function and S a subset of X. Then,

1.  The following conditions are equivalent

(a)  f is lower semicontinuous on S;

(b)  for every ,

(c)  for every ,

In particular, if f is lower semicontinuous on S, then the trace on S of any lower level set of f is closed in S and the trace on S of any strict upper level set of f is open in S.

2.  The following conditions are equivalent

(a)  f is upper semicontinuous on S;

(b)  for every ,

(c)  for every ,

In particular, if f is upper semicontinuous on S, then the trace on S of any upper level set of f is closed in S and the trace on S of any strict lower level set of f is open in S.

If X is a metric space (or more generally, a Fréchet-Urysohn space), then f in X converging to x, we have

.

1.1.3 Hemicontinuity of Extended Real-Valued Functions

Let X is said to be hemicontinuous .

defined by

, but not continuous.

1.2 KKM Lemma and the Brouwer's Fixed Point Theorem

The equivalent statements of Knaster,¹ Kuratowski,² and Mazurkiewicz³ (KKM lemma) and Brouwer's⁴ fixed point theorem represent two of the most important existence principles in mathematics. They are also equivalent to numerous, apparently completely different, cornerstone theorems of nonlinear analysis (see, for instance, [174], Chapter 77). In the sequel we provide a proof of the KKM lemma by using Sperner's⁵ lemma a combinatorial analogue of Brouwer's fixed point theorem, which is equivalent to it. Then, Brouwer's fixed point theorem will be deduced by KKM lemma (see [175]).

1.2.1 The Sperner's Lemma

Let X be a real vector space. By an N-simplex are linear independent. By a k-face .

By a triangulation of Nsuch that:

;

is a common k.

is the so-called barycentric subdivision is called the barycenter , where b . By induction, the barycentric subdivision of an N-simplex with barycenter b is the collection of all N.

be associated with each vertex v introduced above, according to the following rule: if

(1.2)

should be associated with vis called a Sperner simplex .

Lemma 1.1

(E. Sperner [163]) For every triangulation satisfying the rule (1.2), the number of Sperner simplices is odd.

Proof

is a 1-simplex (namely, a segment). A 0-face is called distinguished if and only if it carries the number 0. We have exactly the following two possibilities:

is a Sperner simplex).

is not a Sperner simplex).

But since the distinguished 0-faces occur twice in the interior and once on the boundary, the total number of distinguished 0-faces is odd. Hence, the number of Sperner simplices is odd.

. The distinguished 1-faces occur twice in the interior. By is odd. Thus, the total number of distinguished 1-faces is odd, and hence the number of Sperner 1-simplices is also odd.

. Then it is also true for N. □

1.2.2 KKM Lemma

Now we can prove the following result.

Lemma 1.2

(B. Knaster, C. Kuratowski, and S. Mazurkiewicz [109]) Let be an N-simplex in a finite dimensional normed space X, where . Suppose that we are given closed sets in X such that

(1.3)

for all possible systems of indices and all . Then .

Proof

.

. Let v , where

(1.4)

By .

We associate the number k with the vertex k. It follows from .

.

Following the reasoning above, there are points:

(1.5)

such that

(1.6)

.

By .

. □

1.2.3 Brouwer's Fixed Point Theorem

. One of the most famous fixed point theorems for continuous functions was proven by Brouwer and it has been used across numerous fields of mathematics. This property is stated in the following theorem.

Theorem 1.1

Every continuous function f from a nonempty convex compact subset C of a finite dimensional normed space X to C itself has a fixed point.

Proof

is homeomorphic to some Nin X (see, for instance, Zeidler is an N.

, then apply the intermediate-value theorem to conclude that the continuous real function g .

is a triangle. Each point u has the representation

(1.7)

where

(1.8)

of the point u are uniquely determined by u and depend continuously on u, by [175], Section 1.12, Proposition 5. We set

.

and f is closed. Furthermore, the crucial condition (1.3) of Lemma 1.2 is satisfied, i.e.,

(1.9)

, i.e.,

(1.10)

This is a contradiction to