On the existence of ɛ-fixed points

Cent. Eur. J. Math. • 12(9) • 2014 • 1320-1329DOI: 10.2478/s11533-014-0418-x

Central European Journal of Mathematics

On the existence of ε-fixed points

Research Article

Tiziana Cardinali1∗

1 Department of Mathematics and Informatics, University of Perugia, via L. Vanvitelli, 1, 06123 Perugia, Italy

Received 30 June 2013; accepted 17 September 2013

Abstract: In this paper we prove some approximate fixed point theorems which extend, in a broad sense, analogous resultsobtained by Brânzei, Morgan, Scalzo and Tijs in 2003. By assuming also the weak demiclosedness property westate two fixed point theorems. Moreover, we study the existence of ε-Nash equilibria.

MSC: 47H10, 47H04

Keywords: Partially closed • β-w-partially closed • Weakly demiclosed • ε-fixed point • Fixed point • ε-Nash equilibrium© Versita Sp. z o.o.

1. Introduction

Fixed point theorems find its application in many applied fields among others in game theory and mathematical economics.Let us recall that the Nash equilibrium, introduced in [9], is one of widely used tool for predicting the outcome of astrategic interaction between players. It is known that existence of Nash equilibria is equivalent to the existence offixed points of suitable multifunctions (e.g. see [3, 7, 9]). The existence of such equilibria usually requires a compactnesscondition on the strategy set of each player. Without this property a Nash equilibrium need not exist. This fact hasrecently been studied by Brânzei, Morgan, Scalzo and Tijs in [1] where the existence of approximate fixed points formultimaps in a Banach space is obtained. The aforementioned authors introduce (for fixed ε > 0) the notion of ε-equi-librium (i.e. a ε-strategy profile such that the unilateral deviation of one of the players does not increase his payofffunction more than ε) and show that the existence of Nash approximate equilibria is equivalent to the existence ofapproximate fixed points for a suitable multifunction.We present new approximate fixed point results for multimaps that extend, in a broad sense, those proved in [1]. Toachieve these goals, we introduce the notion of a β-partially closed multimap and use the classic Glebov fixed pointtheorem [6] for partially closed multimaps. The basic idea of the new definition is that in order to obtain an ε-fixedpoint it is sufficient to require an “approximate partial closeness” (see Theorems 3.1, 3.7 and Corollaries 3.2, 3.8). Weremark that the multimaps in this article are defined on a not necessarily bounded set, in contrast to [1]. Moreover, weprove that even in the class of multimaps defined on bounded and convex domains, there exist examples of multimaps∗ E-mail: [email protected]

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satisfying all conditions of Corollary 3.2 but which are not w-closed, as required in [1, Theorem 2.1]. By assuming thatthe multimap values are also closed we are able to remove the separability assumption on the reflexive Banach spacethat is required in [1, Theorem 2.2] (see Corollary 3.5). Moreover, in Corollary 3.8 we obtain the existence of approximatefixed points without any assumptions on the Banach space.Next by using the weakly demiclosed property we prove the existence of fixed points as a natural consequence ofCorollaries 3.2 and 3.8. It is important to point out that the multimaps considered in Theorems 4.2 and 4.5 (which areβε-w-partially closed, for every ε > 0) are not necessarily w-partially closed (see Example 4.1). Therefore, in the classof the multimaps examined in our fixed point propositions, the Glebov theorem does not apply. Moreover, by using akey proposition proved in [1] we describe how our results can be applied to obtain the existence of approximate Nashequilibria for n-person strategic games. We end with an open problem (see Problem 6.3).2. Preliminaries

Let V be a Hausdorff topological linear space. For every A ⊂ V we denote the convex hull of A by coA . We say thatthe set A is totally bounded if for every ε > 0, there exist x1, . . . , xpε ∈ V such that A ⊂ ⋃i∈{1,...,pε} B(xi, ε). For anonempty convex subset X of V , let P(X ) be the collection of all nonempty subsets of X ; let Pc(X ) be the collection ofall nonempty and convex subsets of X and Pf,c(X ) be the collection of all nonempty closed and convex subsets of X .Recall the following definitions (see, for example [4]). A map F : X → P(V ) is said to be upper semicontinuous in x ∈ Xif for every open set A in V such that F (x) ⊂ A, there exists a neighborhood U of x such that F (x) ⊂ A, for all x ∈ U∩X .Put GrF = {(x, y) ∈ X×V : y ∈ F (x)}, we say that F : X → P(V ) is closed if the set GrF is closed in X×V , i.e.if (xδ )δ∈∆ , xδ ∈ X , xδ → x, x ∈ X , and (yδ )δ∈∆ , yδ ∈ F (xδ ), yδ → y, then y ∈ F (x). Next, as in [6], we say that amap F : X → P(V ) is partially closed if the following property holds: if (xδ )δ∈∆ , xδ ∈ X , xδ → x, x ∈ X , and (yδ )δ∈∆ ,yδ ∈ F (xδ ), yδ → y, then F (x) ∩ L(x, y) 6= ∅, where L(x, y) = {x + λ(y − x) : λ ≥ 0}. Let us introduce the followingproperty.Definition 2.1.For fixed β ∈ ]0, 1[, we say that a map F : X → P(V ) is β-partially closed if for every net (xδ )δ∈∆ , xδ ∈ X , xδ → x ∈ Xand for every net (yδ )δ∈∆ , yδ ∈ F (xδ ), yδ → y, F (x) ∩ L(x/(1−β), y) 6= ∅.By considering V with the weak topology Tw , we say that F is respectively w-upper semicontinuous, w-closed,w-partially closed , β-w-partially closed. If (V , ‖ ·‖) is a normed space and A ⊂ V , put d(x, A) = infa∈A ‖x − a‖.For fixed F : X → P(V ), let us denote by WF(X ) the set of all points x ∈ X such that there exists at least onesequence (xn)n in X which weakly converges to x and such that limn→∞ d(xn, F (xn)) = 0. The map I − F is said to beweakly demiclosed in x ∈WF(X ) if for every sequence (xn)n in X which weakly converges to x (xn ⇀ x) and such thatlimn→∞ d(xn, F (xn)) = 0, we have x ∈ F (x) (in [5] the strongly demiclosed property for multimaps is introduced).Remark 2.2.One can easily check that the map I − F is weakly demiclosed on WF(X ) if it is weakly demiclosed at every point ofthe set WF(X ) and this definition is well posed if the set WF(X ) is nonempty.Finally we recall the following definition (see [1]).Definition 2.3.Fixed ε > 0, a point x ∈ X is said an ε-fixed point for the map F : X → P(V ) if d(x, F (x)) ≤ ε.For every ε > 0, we denote by Fixε(F ) the set of all ε-fixed points for F ; and by Fix(F ) the set of all fixed points for F .

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3. Approximate fixed points

First, we give a result on the existence of approximate fixed points for multimaps in reflexive Banach spaces.Theorem 3.1.Let V be a reflexive real Banach space, let X be a nonempty convex subset of V and ε > 0. Assume F : X → Pc(X ) isa map satisfying the following properties:(i) coF (X ) has nonempty interior,(ii) F (X ) is bounded in V ,(iii) there exists βε ∈ ]0, 1[ , αβε ≤ ε, such that F is βε-w-partially closed, where α = sup

x∈F (X ) ‖x‖.Then, there exists xε ∈ coF (X ) ∩ Fixε(F ).Proof. From (i) and (ii) we have that the convex set H = coF (X ) is also bounded and with nonempty interior. Supposewithout loss of generality that 0 ∈ intH. Put

Hε = (1− βε)H, (1)where βε is the positive number introduced in (iii) and H is the closure of H in the Banach space V , we prove thefollowing inclusion:

Hε ⊂ H. (2)Fixed z∗ ∈ Hε = (1 − βε)H, there exists z ∈ H such that z∗ = (1 − βε)z. Obviously, if z ∈ H we have z∗ ∈ H. Now,consider the case z ∈ H \H. If we show that the segment [0, z [ = {tz : t ∈ [0, 1[} ⊂ H then z∗ ∈ H again. Suppose,contrary our claim, that there exists ω ∈ [0, z [ such that ω /∈ H. By convexity of H we have that [0, z] ⊂ H and soω ∈ H. Then we note that, for every n ∈ N, there exists a point xn ∈ B(ω, 1/n) such that

xn /∈ H. (3)Put ρ = ‖ω‖ > 0 and r > 0 such that B(0, r) ⊂ H, there exists ρ∗ > 0 with ρ = r + ρ∗. Next, for every n ∈ N, consider

in = inft≥0 ‖z + t(xn − z)‖. (4)

Clearly, we havein ≤ t‖xn − ω‖+ ‖z + t(ω− z)‖ for all t ≥ 0.

Therefore, for fixed t ≥ 0 such that 0 = z + t(ω − z), we can deduce that the sequence (in)n converges to 0 by thefollowing inequality: 0 ≤ in ≤ t‖xn − ω‖ ≤ tn for all n ∈ N.

Hence there exists n ∈ N such that 1/n < ρ∗ and in < r. Therefore, by (4) we can find tn ≥ 0 such that the pointyn = z + tn(xn − z) ∈ B(0, r). If tn ≤ 1 we deduce

‖z − ω‖+ ρ = ‖z‖ ≤ ‖yn‖+ ‖z − yn‖ < r + tn‖xn − z‖ ≤ r + ‖xn − ω‖+ ‖ω− z‖,and so we get the contradiction: ρ < r + ‖xn − ω‖ < r + ρ∗ = ρ.Conversely, suppose that tn > 1, the convexity of H implies

xn = yntn

+ (1− 1tn

)z ∈ H,

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which is contrary to (3). Therefore, we can conclude that the segment [0, z [ is included in H and then z∗ ∈ H again.Consequently, (2) is true. Next, let us consider the map Gε : Hε → Pc(V ) defined byGε(x) = (1− βε)F (x), x ∈ Hε. (5)

By virtue of (1) and taking into account that the values of F are convex, this multimap assumes values in the familyPc(Hε). The closed and convex set Hε in the Banach space V is also closed in the Hausdorff locally convex topologicallinear space (V ,Tw ). The reflexivity of the Banach space V guarantees that the set Hε is w-compact, being Hε normedbounded and w-closed (see [4, Theorem 3.6.4]). Moreover, the map Gε is w-partially closed. Indeed, for a fixed net(xδ )δ∈∆ , xδ ∈ Hε , xδ ⇀ x, x ∈ Hε , and fixed (yδ )δ∈∆ , yδ ∈ Gε(xδ ), yδ ⇀ y, we can prove that Gε(x)∩ L(x, y) 6= ∅. Notice,for every δ ∈ ∆, there exists zδ ∈ F (xδ ) such that yδ = (1−βε)zδ . The convergence of the net (yδ )δ∈∆ in (V ,Tw ) impliesthat, for every W ∈ W (0) (where W (0) is the family of all neighbourhoods of 0 in the weak topology) we have that thereexists δ∗ ∈ ∆ such that yδ ∈ y+ (1− βε)W for all δ ∈ ∆, δ∗ � δ, and so the net (zδ )δ∈∆ ,

zδ = yδ1− βε ∈ y1− βε +W for all δ ∈ ∆, δ∗ � δ,

weakly converges to the point z = y/(1− βε). Thus, by (iii) we conclude thatF (x) ∩ L( x1− βε , z

)6= ∅,

so there exists a point v ∈ F (x) and a number λ ≥ 0 characterized byv = x1− βε + λ

(z − x1− βε

).

By (5) we can write (1− βε)v = x + λ(y− x) ∈ Gε(x) ∩ L(x, y).Since the map Gε on the topological linear space (V ,Tw ) satisfies all the conditions of the Glebov theorem [6], thereexists a point xε ∈ Hε such that xε ∈ Gε(xε) = (1−βε)F (xε), and hence there exists a point yε ∈ F (xε), xε = (1−βε)yε.By (iii) we have

d(xε, F (xε)) ≤ ‖xε − yε‖ = ‖(1− βε)yε − yε‖ = βε‖yε‖ ≤ αβε ≤ ε,i.e. xε ∈ X is a ε-fixed point for F . Moreover, by (2) we also have xε ∈ coF (X ). Therefore, the desired result isestablished.Let us next claim the followingCorollary 3.2.Let V be a reflexive real Banach space, X be a nonempty convex subset of V . Assume F : X → Pc(X ) is a map satisfyingthe following properties:(i) coF (X ) has nonempty interior,(ii) F (X ) is bounded in V ,(iii)∗ for every ε > 0, there exists βε ∈ ]0, 1[ , αβε ≤ ε, such that F is βε-w-partially closed, where α = sup

x∈F (X ) ‖x‖.Then, for every ε > 0, there exists xε ∈ coF (X ) ∩ Fixε(F ).Remark 3.3.The following example shows that Corollary 3.2 extends, in a broad sense, [1, Theorem 2.1].

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Example 3.4.Put X = ]−2, 2] ⊂ V = R, let F : X → P(V ) be the map defined by

F (x) =

[−x2 , 0

], x ∈ [0, 2[,]0,−x2], x ∈ ]−2, 0[,

{0}, x = 2.It is easy to see that F satisfies hypotheses (i) and (ii) of Corollary 3.2. Moreover, for fixed ε > 0 assuming thatβε ∈ ] 0,min{1, ε}[ we can say F verifies also (iii)∗ of Corollary 3.2. On the other hand, F is not w-closed, required in[1, Theorem 2.1]. To this end it is sufficient to note that the sequence (xn)n, xn = 2− 1/n, xn → x = 2 and the sequence(zn)n, zn = −1 + 1/n ∈ F (xn), zn → z = −1, but we have z = −1 /∈ F (2) = {0}.Next by using Corollary 3.2 we provide an extension of [1, Theorem 2.2].Corollary 3.5.Let V be a real reflexive Banach space and let X be a nonempty convex subset of V . Assume F : X → Pf,c(X ) is a mapsatisfying the following properties:(i) coF (X ) has nonempty interior,(ii) F (X ) is a bounded subset in V ,(iii) F is w-upper semicontinuous.

Then, for every ε > 0, there exists xε ∈ coF (X ) ∩ Fixε(F ).Proof. Consider the Hausdorff locally convex topological linear space (V ,Tw ). Since F has closed and convex valuesin V , we can say that its values are closed in (V ,Tw ). By (ii) they are also compact in (V ,Tw ). By [4, Proposition 4.1.14],and (iii), we conclude that the graph of F is closed in (V ,Tw )× (V ,Tw ). Next we show that F satisfies the hypotheses(iii)∗ of Corollary 3.2. Put α = supx∈F (X ) ‖x‖ and fix ε > 0, let βε ∈ ]0, 1[ be such that αβε ≤ ε. Fix a net (xδ )δ∈∆ suchthat xδ ∈ X , xδ ⇀ x ∈ X . For every net (zδ )δ∈∆ , zδ ∈ F (xδ ), zδ ⇀ z, by the ω-closedness of the graph of F , we deducez ∈ F (x) and so F (x) ∩ L(x/(1 − βε), z) 6= ∅ holds. Finally we are in a position to apply Corollary 3.2 to the map F .Hence, for every ε > 0, there exists in coF (X ) an ε-fixed point for F .Remark 3.6.In Corollary 3.5, by assuming that the values of F are closed, we can drop the separability assumption on the reflexiveBanach space V , required in [1, Theorem 2.2]. Moreover, Corollary 3.5 considers multimaps defined in a set that is notnecessarily bounded, whereas boundedness is required in [1].Now we turn our attention to the existence of approximate fixed points in a Banach space that is not necessarily reflexive.Theorem 3.7.Let V be a real Banach space and let X be a nonempty convex subset of V . Let ε > 0. Assume F : X → Pc(X ) is a mapsatisfying the following properties:(i) coF (X ) has nonempty interior,(ii) F (X ) is totally bounded in V ,(iii) there exists βε ∈ ]0, 1[ , βε(ε+ hε) ≤ ε, such that F is βε-partially closed, where hε = max{‖xi‖ : i ∈ {1, . . . , pε}}

for x1, . . . , xpε ∈ V with F (X ) ⊂ ⋃i∈{1,...,pε} B(xi, ε).Then there exists xε ∈ coF (X ) ∩ Fixε(F ).

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Proof. Repeating the arguments from the proof of Theorem 3.1 we can say that the convex set H = coF (X ) satisfiesthe following properties: 0 ∈ intH, Hε = (1− βε)H ⊂ H, (6)where βε is the positive number introduced in (iii). Clearly, the set Hε is nonempty and convex. By (ii) we deduce thatcoF (X ) is a totally bounded subset of V , so Hε is totally bounded too. Since Hε is also closed, we have that the metricspace Hε is complete, with the metric induced by the norm in V . Therefore, it is also compact in the Banach space V(see [4, Theorem 1.4.26]).As in the proof of Theorem 3.1, we consider the multimap Gε : Hε → Pc(V ) defined in (5). The map Gε is partially closed.Hence, it satisfies in the Banach space V all the conditions of the Glebov Theorem of [6]. Therefore, there exists a pointxε ∈ Hε such that xε ∈ Gε(xε) = (1 − βε)F (xε). It follows that the point yε ∈ F (xε) ⊂ H such that xε = (1 − βε)yεsatisfies the following property:

d(xε, F (xε)) ≤ ‖xε − yε‖ = βε‖yε‖ ≤ βε(‖yε − xk‖+ ‖xk‖) ≤ βε(ε + hε) ≤ ε,

where k ∈ {1, . . . , pε} is such that yε ∈ B(xk , ε) (see (iii)). Moreover, (6) implies xε ∈ coF (X ). Therefore the setcoF (X ) ∩ Fixε(F ) is nonempty.By using Theorem 3.7 we can immediately deduce the following proposition, which extends [1, Theorem 2.3].Corollary 3.8.Let V be a real Banach space and let X be a nonempty convex subset of V . Assume F : X → Pc(X ) is a map satisfyingthe following properties:(i) coF (X ) has nonempty interior,(ii) F (X ) is totally bounded in V ,(iii) for every ε > 0, there exists βε ∈ ]0, 1[ , βε(ε + hε) ≤ ε, such that F is βε-partially closed, where hε = max{‖xi‖ :

i ∈ {1, . . . , pε}} for x1, . . . , xpε ∈ V with F (X ) ⊂ ⋃i∈{1,...,pε} B(xi, ε).Then for every ε > 0, there exists xε ∈ coF (X ) ∩ Fixε(F ).4. Fixed points

First we note here that there exist multimaps satisfying all the hypotheses of Corollary 3.2 but which are not w-partiallyclosed, as the following example illustrates.Example 4.1.Let X = [1, 2] ⊂ V = R and let F : X → P(X ) be the map defined by F (x) = ]1, 2], x ∈ X . For every ε > 0, putβε = ε/2. Then F is βε-partially closed. On the other hand, for a fixed sequence (xn)n converging to x = 1 andyn = 1 + 1/n ∈ F (xn), we have that the sequence (yn)n converges to y = 1 and clearly the set F (x) ∩ L(x, y) is empty.Hence, in the setting of Corollary 3.2 we emphasize that the Glebov theorem is not usable. Now we present two fixedpoint theorems, which are a natural outgrowth of our approximate fixed point results.Theorem 4.2.Let V be a reflexive real Banach space and let X be a nonempty closed and convex subset of V . Assume F : X → Pc(X )is a map satisfying the following properties:(i) coF (X ) has nonempty interior,(ii) F (X ) is bounded in V ,

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(iii) for every ε > 0, there exists βε ∈ ]0, 1[ , αβε ≤ ε such that F is βε-w-partially closed, where α = supx∈F (X ) ‖x‖,(iv) the multimap I − F is weakly demiclosed on WF(X ).Then, Fix(F ) is nonempty.

Proof. Set ε = 1/n, n ∈ N, by Corollary 3.2, we can choose xn ∈ coF (X ) ⊂ X so that d(xn, F (xn)) < 1/n. In viewof (ii) we have that the approximate fixed points sequence (xn)n is bounded in the reflexive Banach space V . Therefore,there exists a subsequence (xnk )nk of (xn)n weakly convergent to a point x ∈ coF (X ) (see [4, Corollary 3.6.8]). Since Xis closed and convex, x ∈ X . The subsequence (xnk )nk weakly converges to x and satisfies limk→∞ d(xnk , F (xnk )) = 0;therefore, x ∈WF(X ). By (iv), x ∈ F (x).Remark 4.3.Note that there exist multimaps satisfying all hypotheses of Theorem 4.2 but not having closed values (for example,consider F (x) = ]0, 1[ , x ∈]0, 1[). Moreover, the following example shows that the closedness of X in the above fixedpoint theorem cannot be removed.Example 4.4.Let X = ]0, 1[⊂ V = R and let F : X → P(X ) be the map defined by

F (x) = { xx + 1

}, x ∈ X.

Note that the set X is not closed. Moreover, since F satisfies the hypotheses (i), (ii) and (iii)∗ of Corollary 3.2, thereexists an approximate fixed points sequence (un)n having the property limn→∞ d(un, F (un)) = 0. On the other hand, iteasy to see that for every x ∈ ]0, 1[ , any sequence (xn)n converging to x does not satisfy limn→∞ d(xn, F (xn)) = 0, so theset WF(X ) is empty (i.e., in this setting, (iv) of Theorem 4.2 is not well posed).Next we study the existence of fixed points in a Banach space that is not necessarily reflexive.Theorem 4.5.Let V be a real Banach space and let X be a nonempty closed and convex subset of V . Assume F : X → Pc(X ) is amap satisfying the following properties:(i) coF (X ) has nonempty interior,(ii) F (X ) is totally bounded in V ,(iii) for every ε > 0, there exists βε ∈ ]0, 1[ , βε(ε + hε) ≤ ε, such that F is βε-partially closed, where hε = max{‖xi‖ :

i ∈ {1, . . . , pε}} for x1, . . . , xpε ∈ V with F (X ) ⊂ ⋃i∈{1,...,pε} B(xi, ε),(iv) the multimap I − F is weakly demiclosed on WF(X ).Then Fix(F ) is nonempty.

Proof. Fix ε = 1/n, n ∈ N. By Corollary 3.8, we can choose xn ∈ coF (X ) ⊂ X so that d(xn, F (xn)) < 1/n. Hencelimn→∞ d(xn, F (xn)) = 0. Since by (ii) we have that the set F (X ) is compact in V , the Krein–Smulian theorem impliesthat the set coF (X ) is weakly compact (see [4, Theorem 3.5.15]). Now, by virtue of the Eberlein–Smulian theorem (see[4, Theorem 3.5.3]) there exists a subsequence (xnk )nk of (xn)n weakly converging to a point x ∈ coF (X ) ⊂ X . Therefore,x ∈WF(X ) and the weak demicloseness of I − F implies that x ∈ F (x).

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5. An application: ε-Nash equilibria

In this section we turn our attention to the existence of approximate Nash equilibria for strategic games with noncompactstrategy sets. We consider an n-person strategic game G = {Xi;ui}i∈I , where the elements of I = {1, . . . , n} represent theplayers and, for every i ∈ I, Xi denotes the set of strategies of player i (i.e. profile of strategies) and ui : X =∏i∈I Xi → Rits payoff function. If any player i ∈ I chooses a strategy xi ∈ Xi, the profile x = (x1, . . . , xn) is made up and the outcomeof the game is: u1(x) for the player 1, . . . , un(x) for the player n. Put x−i = (x1, . . . , xi−1, xi+1, . . . , xn) and denote by(xi, x−i) the vector x. Assume that every player wishes to maximize his payoff. Recall that for fixed ε > 0, a profile ofstrategies x∗ = (x∗1 , . . . , x∗n) ∈ X is called an ε-Nash equilibrium if (see [1])ui(x∗) + ε ≥ ui(xi, x∗−i) for all xi ∈ Xi, i ∈ I.

(If ε = 0 a profile of strategies x∗ is said to be a Nash equilibrium.) For every i ∈ I, put X−i = ∏j∈I\{i} Xj , the payofffunction ui will be viewed as realizing the following property:

supti∈Xi

ui(ti, x−i) <∞ for all xi ∈ Xi, i ∈ I . (7)Moreover, we define the ε-best response map Bεi : X−i → P(Xi) as follows:

Bεi (x−i) = {xi ∈ Xi : ui(xi, x−i) ≥ supti∈Xi

ui(ti, x−i)− ε}.The ε-best multiresponse map Bε : X → P(X ) is defined as

Bε(x) =∏i∈I

Bεi (x−i), x ∈ X.

Clearly, another way of stating (7) is to say that the values of the multimap Bε are nonempty. By using Corollary 3.2the following proposition holds.Theorem 5.1.Let G = {Xi;ui}i∈I be the n-person strategic game where, for every i ∈ I, Xi is a bounded and convex subset of areflexive real Banach space Vi, di is a metric which induces the weak topology on Xi and ui : X = ∏

i∈I Xi → R is afunction such that(γ) ui is uniformly continuous on the metric space (X, d), where d is the metric defined by d(x, y) = ∑

i∈I di(xi, yi),x, y ∈ X .

Moreover, for every ε > 0 the ε-best multiresponse map Bε : X → Pc(X ), X ⊂ V = ∏i∈I Vi, satisfies the following

assumptions:(γγ) coBε(X ) has nonempty interior,(γγγ) for every δ > 0 there exists βδ ∈ ]0, 1[ , αβδ ≤ δ, such that Bε is βδ-w-partially closed, where α = supx∈Bε (X ) ‖x‖.Then, for every ε > 0, there exists an ε-Nash equilibrium for the strategic game G.

Proof. By Corollary 3.2, we can say that Fixδ (Bε) is nonempty, for each δ > 0 and ε > 0. Therefore, the strategicgame G = {Xi;ui}i∈I satisfies all conditions required in the key proposition of [1], so we can conclude the existence ofan ε-Nash equilibrium for the strategic game G, for every ε > 0.Remark 5.2.Similarly, by using Corollary 3.5 or Corollary 3.8, we can deduce again the existence of ε-Nash equilibria for a n-strategicgame G.

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6. Final considerations

We state an immediate consequence of [6, Theorem].Corollary 6.1.Let V be a reflexive real Banach space and let X be a nonempty closed and convex subset of V . Assume F : X → Pc(X )is a map satisfying the following properties:(ν) F (X ) is bounded in V ,(νν) F is w-partially closed.

Then, Fix(F ) is nonempty.

Proof. First, by (ν) we have that the closed and convex set S = coF (X ) is also bounded. Therefore, S is w-compact.Since F�S satisfies all conditions of [6, Theorem] in the Hausdorff locally convex topological linear space (V ,Tw ) wehave that the set Fix(F ) is nonempty.Remark 6.2.Note that Example 4.4 shows that the closedness of the set X cannot be removed in Corollary 6.1. Moreover, Corollary 6.1obviously extends Theorem 4.2. Vice versa, Example 4.1 also shows that there exist multimaps satisfying the property(iii) of Theorem 4.2 but not the hypothesis (νν) of Corollary 6.1.Now, as a natural outgrowth of the previous discussion we suggest the following problem for further investigation.Problem 6.3.Let V be a reflexive real Banach space and let X be a closed and convex subset of V . Assume F : X → Pc(X ) is a mapsatisfying the following properties:(j) coF (X ) has nonempty interior,(jj) F (X ) is bounded in V ,(jjj) for every ε > 0, there exists βε ∈ ]0, 1[ , αβε ≤ ε, such that F is βε-w-partially closed, where α = sup

x∈F (X ) ‖x‖.Then, Fix(F ) is nonempty.We hope that the results presented in this paper can help the study of all games having an intractability proving theexistence of a Nash equilibrium (see for example [2, 8], etc.) and support the research on Cournot–Nash equilibrium inthe dynamic nonlinear oligopoly theory, recently studied by Puu in [10].Acknowledgements

The author wishes to thank the referee for helpful comments.

References

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T. Cardinali

[2] Chien S., Sinclair A., Convergence to approximate Nash equilibria in congestion games, In: Proceedings of theEighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, January 7–9, 2007, Society forIndustrial and Applied Mathematics, Philadelphia, 2007, 169–178[3] Debreu G., A social equilibrium existence theorem, Proc. Nat. Acad. Sci. U.S.A., 1952, 38, 886–893[4] Denkowski Z., Migórski S., Papageorgiou N.S., An Introduction to Nonlinear Analysis: Theory, Kluwer, Boston,2003[5] García-Falset J., Llorens-Fuster E., Suzuki T., Fixed point theory for a class of generalized nonexpansive mappings,J. Math. Anal. Appl., 2011, 375(1), 185–195[6] Glebov N.I., On a generalization of the Kakutani fixed point theorem, Soviet Math. Dokl., 1969, 10(2), 446–448[7] Glicksberg I.L., A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points,Proc. Amer. Math. Soc., 1952, 3(1), 170–174[8] Morgan J., Raucci R., Lower semicontinuity for approximate social Nash equilibria, Internat. J. Game Theory, 2002,31(4), 499–509[9] Nash J.F. Jr., Equilibrium points in n-person games, Proc. Nat. Acad. Sci. U.S.A., 1950, 36, 48–49[10] Puu T., On the stability of Cournot equilibrium when the number of competitors increases, J. Econom. BehaviorOrganization, 2008, 66(3-4), 445–456

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