AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 959612 10.1155/2014/959612 959612 Research Article Noncompact Equilibrium Points for Set-Valued Maps Chebbi Souhail http://orcid.org/0000-0002-8380-9424 Samet Bessem Ben-El-Mechaiekh Hichem Department of Mathematics College of Science King Saud University P.O. Box 2455, Riyadh 11451 Saudi Arabia ksu.edu.sa 2014 142014 2014 23 08 2013 17 01 2014 25 3 2014 2014 Copyright © 2014 Souhail Chebbi and Bessem Samet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove a generalized result on the existence of equilibria for a monotone set-valued map defined on noncompact domain and take its values in an order of topological vector space. As consequence, we give a new variational inequality.

1. Introduction

In the literature, the notion of an equilibrium point (or equilibrium problem) has been firstly introduced by Karamardian in  and Allen in . By using the well-known KKM principle, they proved that for a real valued function f defined on a product of two sets X and Y, there exists an element x¯ of X, which will be called an equilibrium point, satisfying for all yX: (1)f(x¯,y)0.

The classical hypothesis used to prove this type of equilibrium result concerns the convexity and compactness of the domain X, the monotonicity, the convexity, and the continuity of f and all extensions of this result obtained in the literature are about these hypotheses. In a recent work (see ), this result was extended to the noncompact case by using a coercivity type condition on a bifunction f. In this context the function f is supposed to take its values in a topological vector space endowed with an order defined by a cone C in the same way that has been used by . Note that the result on the existence of equilibrium points proved in  was obtained via a result on the existence of what we called weak equilibrium points, that is, a point x¯X, satisfying the following condition: (2)f(x¯,y)-intC,yX, where intC denotes the interior of the cone C in Y.

In this paper, we investigate the extension of equilibrium points to set-valued maps F in the same context. Generally, we have many choices to formulate the notion of equilibrium point. In fact, if P is a closed convex cone of a topological vector space Y with nonempty interior, (PY), and F:X×XY is a set-valued map, then the equilibrium point for a set-valued map can be extended in several possible ways (see [8, 9]) as follows: F(x,y)P; F(x,y)-intP=; F(x,y)-intP; F(x,y)P. In this paper, we select the one that will be more adapted technically to our arguments. We will put a “moving” order on Y by a cone and the notions of convexity and continuity are naturally extended in our setting. We will use the pseudomonotonicity condition on F borrowed from . As an application, we prove a variational inequality. The results obtained in this paper generalize the corresponding one in [9, 10].

2. Preliminaries

We extend the notions of convexity, monotonicity, and continuity given previously to set-valued maps. If X and Y are two sets, a set-valued map F:X2Y, where 2X denotes the family of all subsets of X, is a map that is assigned to each xX, a subset F(x)Y. Note that for the notation of set-valued maps, we will simply write F:XY instead of F:X2Y.

We firstly need to define an order on the codomain of set-valued maps as it has done for single valued maps. If X is a subset of some real topological vector space E, let Y be another real topological vector space, and let CY be a closed convex cone (not necessarily pointed) with nonempty interior and CY. Then C defines an ordering “” on Y by means of (3)y0yC,y0yintC. We extend this notation to arbitrary subset SY by setting (4)S0SC,S0SintC,S0S-C,S0S-intC.

By using this order, we naturally extend the notion of convexity for set-valued maps as follows.

Definition 1.

Given a set-valued map F:XY defined on a vector space X with values in a vector space Y endowed with an order defined by a convex cone CY, we say that F is convex with respect to C if for all x,yX and α[0,1]: (5)F(αx+(1-α)y)αF(x)+(1-α)F(y), which means that (6)F(αx+(1-α)y)αF(x)+(1-α)F(y)-C.

Note that in particular, if X=Y= and C=+, we obtain the standard definition of convex set-valued maps.

As in the case of single valued maps, we can find many kinds of monotonicity for set-valued maps in the literature. We will use the notion of pseudomonotonicity defined in  which in turn extends the corresponding one defined in  for single valued maps.

Definition 2.

Let E and Y be two real topological vector spaces, let XE be a nonempty closed and convex set, and let C:XY be a set-valued map such that for every xX, C(x) is a closed and convex cone in Y with intC(x). Consider a set-valued map F:X×XY. F is said to be pseudomonotone if, for any given x,yX, (7)F(x,y)-intC(x)F(y,x)-C(y).

We recall the classical notions of continuity for set-valued maps as follows.

Definition 3.

Given a set-valued map F:XY defined on a vector space X with values in a vector space Y. Then

F  is said to be lower semicontinuous (l.s.c) at x0X if, for every open set VY with F(x0)V, there exists a neighborhood UX of x with F(x)V for all xU. F is said to be l.s.c. on X if F is l.s.c. at every xX.

F is said to be upper semicontinuous (u.s.c) at x0X if, for every open set VY with F(x0)V, there exists a neighborhood set UX of x with F(x)V for all xU. F is said to be u.s.c. on X if F is u.s.c. at every xX.

A set-valued map which is both lower and upper semicontinuous is called continuous.

In this paper, we will use the definition of coercing family borrowed from .

Definition 4.

Consider a subset X of a topological vector space and a topological space Y. A family {(Ci,Ki)}iI of pair of sets is said to be coercing for a set-valued map F:XY if and only if

for each iI, Ci is contained in a compact convex subset of X and Ki is a compact subset of Y;

for each i,jI, there exists kI such that CiCjCk;

for each iI, there exists kI with xCkF(x)Ki.

Remark 5.

Definition 1 can be reformulated by using the “dual” set-valued map F*:YX defined for all yY by F*(y)=XF-1(y). Indeed, a family {(Ci,Ki)}iI is coercing for F if and only if it satisfies conditions (i), (ii) of Definition 4, and the following one: (8)iI,kI,yYKi,F*(y)Ck.

Note that in the case where the family is reduced to one element, condition (iii) of Definition 4 and in the sense of Remark 5 appeared first in this generality (with two sets K and C) in  and generalized condition of Karamardian  and Allen . Condition (iii) is also an extension of the coercivity condition given by Fan . For other examples of set-valued maps admitting a coercing family that is not necessarily reduced to one element, see .

The following generalization of KKM principle obtained in  will be used in the proof of the main result of this paper.

Proposition 6.

Let E be a Hausdorff topological vector space, Y a convex subset of E, X a nonempty subset of Y, and F:XY a KKM map with compactly closed values in Y (i.e., for all xX, F(x)C is closed for every compact set C of Y). If F admits a coercing family, then xXF(x)ϕ.

3. The Main Result

As it is mentioned in the introduction, at an abstract level all possible extension of equilibria can be handled equally well. But there are great practical differences if we try to replace the resulting abstract conditions by simpler, verifiable hypotheses like convexity or semicontinuity. This is even more so if we admit a “moving” ordering cone P(x) (see ). For these reasons we choose to consider here the following generalized equilibrium problem.

Definition 7.

Let X be a nonempty convex subset of some real topological vector space E, Y a real topological vector space, and P:XY a set-valued map such that for any xX, P(x) is a closed convex cone with intP(x) and P(x)Y. Let F:X×XY be a set-valued map. The generalized equilibrium problem is to find x¯X such that (9)F(x¯,y)-intP(x¯)yX; in this case, x¯ is said to be an equilibrium point.

Theorem 8.

Let E and Y be real topological vector spaces (not necessarily Hausdorff). Let a nonempty, convex set XE and three set-valued mappings F:X×XY, C:XY, and D:XY be given. Suppose that the following conditions are satisfied.

For all x,yX, F(x,y)C(x) implies F(y,x)D(y) (pseudomonotonicity).

For all yX, {xX:F(y,x)D(y)} is closed in X.

For all xX, {yX:F(x,y)C(x)} is convex.

For all xX, F(x,x)C(x).

There exists a family {(Ci,Ki)}iI satisfying conditions (i) and (ii) of Definition 4 and the following one: for each iI, there exists kI such that (10){xX:F(y,x)D(y),yCk}Ki.

Then there exists x¯X such that F(y,x¯)D(y) for all yX.

Proof.

Let us consider a set-valued map S:XY defined for every yX by (11)S(y):={xX:F(y,x)D(y)}.

Then we can see firstly that S is a KKM map; that is, for every finite subset {y1,,yn} of X there holds (12)co{y1,,yn}i=1nS(yi).

In fact, let zco{y1,,yn} and assume by contradiction that zi=1nS(yi); it means that z=iIλiyi with λi0, iIλi=1 and zS(yi) for all i. Then F(yi,z)D(yi) for all i, hence from condition (1)F(z,yi)C(z) for all i. It follows from condition (3) that F(z,iIλiyi)C(z), and then F(z,z)C(z), which contradicts condition (4); thus S is a KKM map.

It is also clear from condition (2) that, for all yX, S(y) is closed.

In addition, we can verify that condition (5) implies that the family {(Ci,K)}iI satisfies the following condition: for all iI there exists kI with (13)yCkS(y)Ki.

We deduce that S satisfies all hypothesis of Proposition 6, so we have (14)yXS(y). Therefore there exists x¯X such that for any yX, x¯S(y). Hence (15)F(y,x¯)D(y),  yX.

Theorem 9.

Let E, Y, X, F, C, and D satisfy the assumptions of Theorem 8 and the additional following conditions.

For all x,yX with yx and u(x,y) if F(u,x)D(u) and F(u,y)C(u), then F(u,v)C(u) for all v(x,y).

For all x,yX with yx, {u[x,y]:F(u,y)C(u)} is open in [x,y].

Then there exists x¯X such that F(x¯,y)C(x¯) for all yX.

Proof.

By Theorem 8, there exists x¯X with F(y,x¯)D(y) for all yX. Assume that F(x¯,y)C(x¯) for some yX; then yx¯ by (6) and from (7) there exists u(x¯,y) such that F(u,y)C(u). Since F(u,x¯)D(u), we deduce that F(u,u)C(u), but this contradicts (6) and the theorem is proved.

The following result, which corresponds to Theorem  1 in , can be deduced from the two previous theorems.

Corollary 10.

Let F, C, D satisfy hypothesis (14) of Theorem 8, (6,7) of Theorem 9 and the following condition.

There exists a nonempty compact set AX and a compact convex set BX such that for every xXA there exists yB with F(x,y)C(x).

Then there exists x¯A such that F(x¯,y)C(x) for all yX.

Proof.

By taking for all iI, Ci=B, which is convex compact set, and Ki=A, which is compact set, and by using hypothesis (5), we can see that S admits a coercing family in the sense of Remark 5; that is, for all xXA, S*(x)B. Suppose, per absurdum, that there exists x0XA with S*(x0)B=. Hence for all yB, yS*(x0). This means that for all yB, yS-1(x0) and so x0S(y). Therefore, there exists x0XA such that for all yB, we have (16)F(y,x0)D(y).

Then by Theorem 9, we deduce that there exists x0XA such that for all yB(17)F(x0,y)C(x0),

but this contradicts hypothesis (5′).

Corollary 11.

Let F:X×XY be a set-valued map satisfy the following conditions.

For all x,yX, F(x,y)-intP(x) implies F(y,x)-P(y).

For all yX, F(y,·) is lower semicontinuous.

For all xX, F(x,·) is convex with respect to P(x).

The map intP(x) has open graph in X×Y.

For all x,yX, F(·,y) is upper semicontinuous and compact valued on [x,y].

For all xX, F(x,x)-intP(x).

There exists a family {Ci,Ki}iI satisfying conditions (i) and (ii) of Definition 4 coercing and the following one. For each iI, there exists kI such that (18){xX:F(y,x)-P(y),yCk}Ki.

Then there exists x¯X such that F(x¯,y)-intP(x¯) for all yX.

Proof.

Following , if the map F(y,·) is lower semicontinuous and D(y) is closed, then condition (7) of Theorem 9 is satisfied. Furthermore and also by , condition (7) of Theorem 9 is fulfilled, if for all xX, the map F(·,x) is upper semicontinuous along line segments [x,y]X with compact values, and the map C(·) has open graph in X×Y.

Now let L:XZ denote the space of all continuous linear operators XZ. For ϕL(X,Z), we write ϕ,x:ϕ(x) and for ΦL(X,Z), we write Φ,x:={ϕ,x:ϕΦ}. The following result is a variational inequality formulation of our main result.

Corollary 12.

Let a map Φ:KL(X,Z) be given such that for all xK,Φ(x) is nonempty. Suppose the following.

For all x,yK, Φ(x),y-x-intP(x) implies Φ(y),x-y-P(y).

The map intP(·) has open graph in K×Z.

For all x,yK, Φ(·),y-x is upper semicontinuous on [x,y] and compact valued.

There exists a family {Ci,Ki}iI satisfying conditions (i) and (ii) of Definition 4 and the following one: for each iI, there exists kI such that (19){xX:Φ(y),x-y-P(y),yCk}Ki.

Then there exists x¯X such that Φ(x¯),y-x¯)-intP(x¯) for all yK.

Proof.

Take F(x,y):=Φ(x),y-x, C(x):=-intP(x), and D(x):=-P(x). Then conditions (1) and (5) of Theorem 9 are clearly satisfied. (2) holds since each member of Φ(y) is continuous and D(y) is closed. (4) is satisfied since F(x,x)={0} and P(x)Z. (3) and (6) hold since for all α[0,1]: (20)F(x,αy1+(1-α)y2)αF(x,y1)+(1-α)F(x,y2).

To verify hypothesis (7), we have to show that R={u[x,y]:Φ(u),y-u} is closed in [x,y]. Let {ui} be a net in R converging to u[x,y]; we may assume uy, since yR, and we may assume uiy for all i as well. Thus y-u=λ(y-x) with λ0 and y-ui=λi(y-x) with λi0. For every i, there exists wiΦ(ui),y-ui with wi-intP(ui); then zi=λi-1wiΦ(ui),y-x. We conclude as above that there is a subnet zj converging to some zΦ(u),y-x. The corresponding wj converges to w=λzΦ(u),y-u, since -intP(·) has open graph; we obtain w-intP(u); hence uR.

Note that Corollaries 11 and 12 extend, respectively, Corollaries 1 and 2 in  obtained in noncompact case since our coercivity condition is more general.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group Project (RGP-VPP 237).

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