# Existence of Equilibria and Fixed Points of Set-Valued Mappings on Epi-Lipschitz Sets with Weak Tangential Conditions

Authors: Messaoud Bounkhel
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We prove a new result of existence of equilibria for an u.s.c. set-valued mappingF on a compact set S of Rn which is epi-Lipschitz and satisfies a weak tangential condition. Equivalently this provides existence of fixed points of the set-valued mapping x⇉F(x)-x. The main point of our result lies in the fact that we do not impose the usual tangential condition in terms of the Clarke tangent cone. Illustrative examples are stated showing the importance of our results and that the existence of such equilibria does not need necessarily such usual tangential condition.


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## Body

## 1. Introduction

LetF be a set-valued mapping defined from S⊂Rn into Rm. A familiar result on existence of equilibria on convex compact sets is formulated as follows (see [1]).Theorem 1.
LetF be an u.s.c. set-valued mapping defined on a convex compact set S⊂Rn and suppose that F(x) is nonempty convex compact, ∀x∈S. If(1)Fx∩TS;x≠∅,∀x∈S, then F has equilibria on S; that is, ∃x∗∈S such that 0∈F(x∗).HereT(S;x) is the tangent cone in the sense of convex analysis defined as T(S;x)=cl(R+(S-x)), where R+ denotes the set of all nonnegative real numbers and cl denotes the closure in E. Assume now that S is not necessarily convex and assume that in the tangential condition (1) the tangent cone T(S;x) is replaced by the Clarke tangent cone TC(S;x); that is,(2)Fx∩TCS;x≠∅,∀x∈S. In order to get the same conclusion of Theorem 1 in the nonconvex case, we need one more assumption on S, which is the epi-Lipschitzness of S. The following theorem is an extension of Theorem 1 to the nonconvex case (see [1]).Theorem 2.
LetS be homeomorphic to a convex compact set in Rn and let F be an u.s.c. set-valued mapping with nonempty closed convex values. Assume that S is epi-Lipschitz and (2) holds. Then F has equilibria on S.It is very important to point out that for epi-Lipschitz sets the tangential condition (2) cannot be weakened to(3)Fx∩KS;x≠∅,∀x∈S, where K(S;x) is (generally greater) the contingent cone defined below in Section 2 (see Example 3.1 in [1]).LetS={(x1,x2)∈R2:x1,x2∈[-1,0]} and F(x1,x2)=(2x1+x2;2x1+3x2-1)+B. Clearly S is an epi-Lipschitz compact convex set in R2 and F is u.s.c. on R2 with TC(S;ui)∩F(ui)=∅ for u1=(-1,0), u2=(0,-1), and u3=(-1,-1). However, the point x∗=(0,0)∈S satisfies (0,0)∈F(x∗). This shows that the tangential condition (2) in Theorem 2 is not necessary to get equilibria on S. Our main purpose in this work is to replace TC(S;x) in (2) by a new tangent set which is always larger than TC(S;x) and to prove the existence of equilibria under the new tangential condition.The main result of the paper is read as follows.Theorem 3.
LetS be homeomorphic to a convex compact set in Rn which is epi-Lipschitz and let F be an u.s.c. set-valued mapping with nonempty closed convex values. Let Ω be a convex compact set in Rn with 0∈Ω. Assume that there exists some β≥0 such that for any x∈S(4)Fx∩Tβ,ΩCS;x≠∅.If S is tangentially regular, then there exists x∗∈S such that(5)0∈Fx∗.We point out that the price that we pay for replacing (2) by (4) is the tangential regularity of the set S (i.e., K(S;x)=TC(S;x)) and it is an open question to extend the result of Theorem 3 to the general case without the tangential regularity assumption. The first corollary of Theorem 3 is the following result.Corollary 4.
LetS be homeomorphic to a convex compact set in Rn which is epi-Lipschitz and tangentially regular and let F be an u.s.c. set-valued mapping on S with nonempty closed convex values. Assume that (2) holds. Then F has equilibria on S.Proof.
TakeΩ={0} in Theorem 3. All the assumptions are fulfilled and hence F has equilibria on S.The paper is organised as follows. The next section is devoted to some preliminary concepts and results needed in the development of our approach. In Section3 we prove our main result stated in Theorem 3 and we present an illustrative example showing the importance of our main result in Theorem 3. An application to fixed point results is presented at the end of paper.


## 2. Preliminaries

Throughout this section, we assume thatE is a Hausdorff topological vector space. We will denote by E∗ the topological dual of E and by 〈·,·〉 the pairing between the spaces E and E∗.LetS be a nonempty closed subset of E and let x¯ be a point in S. Let us recall [2, 3] the following classical tangent cones K(S;x¯) of Bouligand and TC(S;x¯) of Clarke.(i)
The Bouligand tangent cone (also called contingent cone)K(S;x¯) to S at x¯ is the set of all h∈E such that, for every neighborhood H of h in E and for every ϵ>0, there exists t∈(0,ϵ) such that (6)x¯+tH∩S≠∅.(ii)
The Clarke tangent coneTC(S;x¯) to S at x¯ is the set of all h∈E such that for every neighborhood H of h in E there exist a neighborhood X of x¯ in E and a real number ϵ>0 such that (7)x+tH∩S≠∅∀x∈X∩S,t∈0,ϵ.(iii)
The Rockefeller hypertangent coneH(S;x¯) to S at x¯ is the set of all h∈E for which there exist a neighborhood H of h in E, a neighborhood X of x¯ in E, and a real number δ>0 such that (8)X∩S+tH⊂S∀t∈0,δ.Clearly H(S;x¯)⊂TC(S;x¯)⊂K(S;x¯). Recall that (see [3, 4]) TC(S;x¯) is a closed convex cone and K(S;x¯) is a closed cone (that may be nonconvex) while H(S;x¯) is an open convex cone.For a closed convex setS and x¯∈S, both Clarke tangent cone and Bouligand tangent cone coincide and they are equal to the convex tangent cone T(S;x¯). The class of nonempty closed sets satisfying the equality TC(S;x¯)=K(S;x¯) is called the class oftangentially regular sets.In [5], we introduced the concept of Ω-epi-Lipschitz sets as follows.Definition 5.
LetS be a closed subset of E and x¯∈S, and let Ω be a bounded set in E. One will say that S is Ω-epi-Lipschitz at x¯ in a direction h¯ if and only if the minimal time function TS,Ω associated with S and Ω is directionally Lipschitz at x¯ in the direction h¯ in the sense of Rockefeller [6], that is, if and only if there exists β≔β(x¯,h¯)≥0 such that TS,Ω♯(x¯,h¯)≤β, where(9)f♯x¯;h¯≔limsupx,α↓fx¯t,h′→0+,h¯⁡t-1fx+th′-α<+∞and (x,α)↓fx¯ means (x,α)∈epif≔{(z,β)∈E×R;f(z)≤β} and (x,α)→(x¯,f(x¯)). Recall that the minimal time function TS,Ω is defined as follows:(10)TS,Ωx≔inf⁡t>0:S∩x+tΩ≠∅.
A geometric characterization ofΩ-epi-Lipschitz sets has been established in [5] saying that a set S is Ω-epi-Lipschitz at x¯ in a direction h¯ if and only if there exist β≔β(x¯,h¯)≥0, δ>0, H∈N(h¯), and X∈N(x¯) such that (11)X∩S+tH⊂S-tβΩ∀t∈0,δ.In our analysis in this work we need the constant β to be uniform with respect to the directions; that is, β≔β(x¯) is only dependent on x¯ and in this case we say that S is (βx¯,Ω)-epi-Lipschitz at x¯ in the direction h¯∈E. Since β does not depend on the directions, we define Hβx¯,Ω(S;x¯) the set of all directions h∈E satisfying TS,Ω♯(x¯,h)≤βx¯; that is,(12)Hβx¯,ΩS;x¯≔h∈E:TS,Ω♯x¯,h≤βx¯.Following the same lines of the proof of Proposition3.4 in [5] we can prove the following result.Proposition 6.
LetS be a closed subset of E and x¯∈S, h¯∈E, and let Ω be a bounded set in E. (i)
If there existβx¯≥0, δ>0, H∈N(h¯), and X∈N(x¯) such that (13)X∩S+tH⊂S-tβx¯Ω∀t∈0,δ,then S is (βx¯,Ω)-epi-Lipschitz at x¯ in the direction h¯.(ii)
Conversely, if there existsβx¯≥0 for which S is (βx¯,Ω)-epi-Lipschitz at x¯ in the direction h¯, then there exist δ>0, H∈N(h¯), and X∈N(x¯) such that (14)X∩S+tH⊂S-tβx¯+1Ω∀t∈0,δ.Remark 7.
Clearly, ifS is (βx,Ω)-epi-Lipschitz at x∈S, then S is (β,Ω)-epi-Lipschitz at x for any β≥βx. Consequently, any epi-Lipschitz set in the sense of Rockefeller [4] is (β,Ω)-epi-Lipschitz for any β≥0 and for any Ω with 0∈Ω and the constant β is uniform for any x∈S. We note that the notion of (β,Ω)-epi-Lipschitz sets recovers some well known concepts in variational analysis. (i)
Obviously a closed set inE is epi-Lipschitz in the sense of Rockefeller [4] if and only if it is (β,{0})-epi-Lipschitz in the sense of Definition 5 and if and only if it is (0,{Ω})-epi-Lipschitz and if and only if it is (β,{Ω})-epi-Lipschitz for any β>0.(ii)
In normed spaces, any compactly epi-Lipschitz setS at x¯∈S with respect to a convex compact set K in the sense of Borwein and Strójwas [7] is (1,Ω)-epi-Lipschitz in the direction k¯ with Ω≔K-k¯, for any k¯∈K. We recall that S is compactly epi-Lipschitz at x¯∈S with respect to a convex compact set K in the sense of [7] provided that there exists r>0 such that (15)x¯+rB∩S+trB⊂S-tK∀t∈0,r.(iii)
Assume thatE is a normed space and K and S are closed sets in E. If S is K-directionally Lipschitz in the sense of [8], then S is (1,Ω)-epi-Lipschitz with Ω=K-k¯, for any k¯∈K.Our main tools in the present work are two tangent sets associated with(β,Ω)-epi-Lipschitz sets. The first tangent set is Hβx¯,Ω(S;x¯) defined above and will be called the (β,Ω)-hypertangent set and it characterizes the class of (β,Ω)-epi-Lipschitz sets by its nonemptiness. The second tangent set will be called the (β,Ω)-Clarke tangent set and is defined as the set of all h¯∈E satisfying that for any neighborhood H of h¯ there exist δ>0 and X∈N(x¯) such that (16)x+tH+βΩ∩S≠∅∀t∈0,δ,x∈S∩X.Observe that for Ω={0} both sets Hβ,Ω(S;x¯) and Tβ,ΩC(S;x¯) coincide, respectively, with the hypertangent cone H(S;x¯) and the Clarke tangent cone TC(S;x¯).Obviously, we always have the following inclusions:H(S;x¯)⊂Hβ,Ω(S;x¯) and TC(S;x¯)⊂Tβ,ΩC(S;x¯). Consequently, any epi-Lipschitz set is (β,Ω)-epi-Lipschitz. In our analysis we need to prove many properties for the tangent sets Hβ,Ω(S;x¯) and Tβ,ΩC(S;x¯). We notice that Hβ,Ω(S;x¯) is an open set in E and Tβ,ΩC(S;x¯) is a closed set in E and both are not necessarily convex.Proposition 8.
LetE be a Hausdorff topological vector space, let S be a nonempty closed subset of E, and let Ω be a convex bounded set in E with 0∈Ω. Let x¯∈S. (1)
S is (βx¯,Ω)-epi-Lipschitz at x¯ if and only if Hβx¯,Ω(S;x¯)≠∅.(2)
Tβx¯,ΩC(S;x¯)+H(S;x¯)⊂Hβx¯,Ω(S;x¯).(3)
Hβx¯,Ω(S;x¯)+H(S;x¯)⊂Hβx¯,Ω(S;x¯).(4)
IfS is epi-Lipschitz at x¯∈S, then Hβx¯,Ω(S;x¯)=int⁡(Tβx¯,ΩC(S;x¯))≠∅.(5)
IfS is epi-Lipschitz at x¯∈S, then cl(Hβx¯,Ω(S;x¯))=Tβx¯,ΩC(S;x¯).Proof.
(1) It follows directly from the definition ofHβx¯,Ω(S;x¯).
(2) Leth1∈H(S;x¯) and h2∈Tβx¯,ΩC(S;x¯). By definition of H(S;x¯) there exist V∈N(0), X1∈N(x¯), and δ1>0 such that(17)X1∩S+th1+V⊂S,∀t∈0,δ1.Choose a symmetric neighborhood W of 0 in E such that W+W⊂V. By definition of Tβx¯,ΩC(S;x¯) there exist X2∈N(x¯) and δ2>0 such that(18)X2∩S+th2+W+tβx¯Ω∩S≠∅,∀t∈0,δ2.Choose now X′∈N(x¯) and δ′>0 such that(19)X′+0,δ′h2+W+βx¯Ω⊂X1.Put X≔X′∩X1∩X2 and δ≔min⁡{δ′,δ1,δ2}. Fix any t∈(0,δ), any w∈W, and any x∈X∩S. Then by (18) there exist w¯∈W and y¯∈Ω such that (20)x+th2+w¯+βx¯y¯∈S.Put x′≔x+t(h2+w¯+βx¯y¯). Then the previous inclusion and (19) ensure that x′∈X1∩S and hence we can use (17) to get (21)x′+th1+w-w¯∈S.Therefore, (22)x+th1+h2+w∈S-tβx¯Ω.Thus, for any x∈X∩S, any t∈(0,δ), and any h∈H we have (23)x+th∈S-tβx¯Ω.This ensures by definition of the (βx¯,Ω)-hypertangent cone that h1+h2∈Hβx¯,Ω(S;x¯).
(3) It is a direct consequence of Part (2) and the inclusionsHβx¯,Ω(S;x¯)⊂Tβx¯,ΩC(S;x¯).
(4) First observe that sinceS is epi-Lipschitz at x¯ we have that H(S;x¯) is nonempty. Hence, there exists some h0∈H(S;x¯). Now, since Hβx¯,Ω(S;x¯)⊂Tβx¯,ΩC(S;x¯) and as Hβx¯,Ω(S;x¯) is open, it is enough to show the inclusion int⁡Tβx¯,ΩCS;x¯⊂Hβx¯,Ω(S;x¯). Consider h∈int⁡[Tβx¯,ΩC(S;x¯)]. Then there exists a positive number λ>0 such that h-λh0∈int⁡[Tβx¯,ΩC(S;x¯)]. Since H(S;x¯) is a cone, we have λh0∈H(S;x¯) and hence by Part (2) we obtain h=(h-λh0)+λh0∈Tβx¯,ΩC(S;x¯)+H(S;x¯)⊂Hβx¯,Ω(S;x¯) and so the proof of Part (4) is finished.
(5) Taking into account the inclusionHβx¯,Ω(S;x¯)⊂Tβx¯,ΩC(S;x¯) and the fact that Tβx¯,ΩC(S;x¯) is closed, it is sufficient to show the inclusion Tβx¯,ΩC(S;x¯)⊂cl[Hβx¯,Ω(S;x¯)].
Fix again some elementh0∈H(S;x¯). For any h∈Tβx¯,ΩC(S;x¯) we have by Part (2) (24)h+1nh0∈Tβx¯,ΩCS;x¯+HS;x¯⊂Hβx¯,ΩS;x¯,∀n≥1.This ensures that h∈cl[Hβx¯,Ω(S;x¯)] because h=limn→∞(h+1/nh0). So the proof is complete.We require the notion of lower semicontinuity for set-valued mappings. LetF be a set-valued mapping on S⊂E, with values that are subsets of another Hausdorff topological vector space G. We say that F:S⇉G is lower semicontinuous (l.s.c.) at (x¯,y¯)∈gphF≔{(x,y)∈E×G:y∈F(x)} provided that, for every V∈N(0), there exists W∈N(0) such that (25)x∈S∩x¯+W⟹y¯∈Fx+V.We say that F:S⇉G is upper semicontinuous (u.s.c.) at (x¯,y¯)∈gphF provided that, for every V∈N(0), there exists W∈N(0) such that (26)x∈S∩x¯+W⟹Fx⊂Fx¯+V.In [6], the author proved the l.s.c. of the set-valued mapping x↦TC(S;x) whenever S is epi-Lipschitz. In the following proposition we extend Theorem 3 in [6] to the set-valued mapping x↦Tβ,ΩC(S;x) with β≥0. This l.s.c. property is very important in the proof of our main result in Theorem 3.Proposition 9.
LetE be a Hausdorff locally convex topological vector space, let S be a nonempty closed subset of E, and let β≥0, Ω be a convex bounded in E with 0∈Ω. If S is epi-Lipschitz at x¯∈S, then the set-valued mapping x↦Tβ,ΩC(S;x) is lower semicontinuous at x¯.Proof.
Letx¯∈S and u¯∈Tβ,ΩC(S;x¯). Assume that S is epi-Lipschitz at x¯. Then there exists h¯∈H(S;x¯). Let U be any neighborhood of zero in G. Since G is locally convex we can choose a balanced neighborhood U0 such that U0⊂U. Set V≔U0∩(-U0). Clearly V⊂U is a symmetric convex neighborhood of zero in G. Fix any v0≔u¯+h¯/ρV(h¯)+1 where ρV is the Minkowski function associated with V. Since H(S;x¯) is a cone, we have h¯/ρV(h¯)+1∈H(S;x¯) and hence by Part (2) in Proposition 8 we obtain (27)v0=u¯+h¯ρVh¯+1∈Tβ,ΩCS;x¯+HS;x¯⊂Hβ,ΩS;x¯.Thus, there exists δ>0, V1∈N(0), and V2∈N(0) such that(28)x¯+V1∩S+tv0+V2⊂S-tβΩ,∀t∈0,δ.Let W∈N(0) such that W+W⊂V1. Fix any x∈(x¯+W)∩S≠∅. Let any x′∈(x+W)∩S. Then x′∈(x¯+W+W)∩S⊂(x¯+V1)∩S and so by (28) we get (29)x′+tv0+V2⊂S-tβΩ,∀t∈0,δ.Therefore, for any x∈(x¯+W)∩S we have v0∈Hβ,Ω(S;x). Hence, u¯∈Hβ,Ω(S;x)-h¯/ρV(h¯)+1. Observe that ρV(h¯/ρV(h¯)+1)=ρV(h¯)/ρV(h¯)+1<1 which implies that h¯/ρV(h¯)+1∈V and hence -(h¯/ρVh¯+1)∈-V=V. Thus, u¯∈Hβ,Ω(S;x)+V⊂Tβ,ΩC(S;x)+V⊂Tβ,ΩC(S;x)+U. Consequently, for any u¯∈Tβ,ΩC(S;x¯) and any U∈N(0) there exists W∈N(0) such that (30)x∈S∩x¯+W⟹u¯∈Tβ,ΩCS;x¯+U.This proves that the set-valued mapping x↦Tβ,ΩC(S;x) is lower semicontinuous at x¯ and the proof is complete.Remark 10.
Following the proof of the previous proposition, we can prove the lower semicontinuity of the set-valued mappingx↦Hβ,Ω(S;x) at x¯ whenever S is epi-Lipschitz set at x¯.Lemma 11.
For anyx¯∈S and any β≥0 one has (31)HS;x¯-βΩ⊂Hβ,ΩS;x¯.Proof.
Letx-∈S. Without loss of generality we assume that H(S;x-)-βΩ≠∅. Then, for any h0∈H(S;x-)-βΩ, there exists h-∈H(S;x-) and w∈Ω such that h0=h--βw. By definition of the hypertangent H(S;x-) there exist V∈N(0), X∈N(x), and δ>0 such that (32)X∩S+th-+V⊂S,∀t∈0,δ.By adding -tβw to both sides we obtain (33)X∩S+th--βw+V⊂S-tβw⊂S-tβΩ,∀t∈0,δ.This ensures by definition of Hβ,Ω(S;x-) that (34)h0=h--βw∈Hβ,ΩS;x-.Therefore, H(S;x-)-βΩ⊂Hβ,Ω(S;x-) and hence the proof is complete.The next lemma establishes an analogue result for the(β,Ω)-Clarke tangent set.Lemma 12.
Assume thatΩ is a convex bounded set with 0∈Ω and that S is epi-Lipschitz at x-. Then (35)TCS;x--βΩ⊂TΩCS;x-.Proof.
Letx-∈S. Since S is epi-Lipschitz at x-, we have by Part (5) in Proposition 8 the equalities TC(S;x)=cl(H(S;x)) and Tβ,ΩC(S;x)=cl(HΩ(S;x)). Therefore, the conclusion follows directly from the previous lemma.It is a natural question to ask whether the inclusion in the previous lemma becomes an equality. The next lemma establishes a positive answer whenever the set is assumed to be tangentially regular. Its proof needs the following proposition which is also needed in the proof of Theorem19. It has been proved in Proposition 5.1 in [5] for Ω-epi-Lipschitz sets. The proof stated below is a direct adaptation of the proof in [5] that we give for the sake of completeness of the paper. To do that, we need the following characterization of K(S;x) in terms of nets (see, e.g., [9]). A vector v∈K(S;x-) if and only if there exist a net (tj)j∈J of positive real numbers converging to zero and a net (vj)j∈J in E converging to v such that (36)x-+tjvj∈S,foreachj∈J.Proposition 13.
LetE be a Hausdorff topological vector space, let S be a nonempty closed subset of E, let Ω be a convex compact set in E with 0∈Ω, and let x-∈S. Assume that for some βx-≥0 one has h-∈Hβx-,Ω(S;x-); then (37)h-∈KS;x--βx-Ω.Proof.
By definition ofHβx-,Ω(S;x-), there exist δ>0, W∈N(0), and X∈N(x-) such that (38)x-+th-+v+tβx-Ω∩S≠∅,∀t∈0,δ,∀v∈W.Choose V∈N(0) such that V+V⊂W and let (tj,vj)j∈J be a net converging to (0,0) in (0,δ)×V. Applying (38) we get, for any j∈J, the existence of wj∈Ω such that (39)x-+tjh-+vj+tjβx-wj∈S.Since Ω is compact, we may extract a subnet (ws(j))j of (wj)j converging to some point w-∈Ω. Put τj≔ts(j), w~j≔ws(j), and hj≔h-+vs(j)+βx-w~j. Then (39) ensures (40)x-+τjhj∈S,∀j∈J.Since hj→h-+βx-w- and τj→0, we deduce from the characterization of the Bouligand cone that h-+βx-w-∈K(S;x-). Thus, (41)h-∈KS;x--βx-Ω.This completes the proof.Lemma 14.
Letβ≥0. Assume that Ω is convex compact with 0∈Ω and that S is epi-Lipschitz at x-. If, in addition, S is assumed to be tangentially regular at x-, then one has (42)Tβ,ΩCS;x-=TCS;x--βΩ.Proof.
Letx-∈S. Since S is epi-Lipschitz at x-, then S is (β,Ω)-epi-Lipschitz at x- and hence Hβ,Ω(S;x-)≠∅. Fix any h-∈Hβ,Ω(S;x-). By Proposition 13, we have h-∈K(S;x-)-βΩ. Consequently, the tangential regularity of S at x- implies that (43)h-∈KS;x--βΩ=TCS;x--βΩ.Since h- is taken to be arbitrary in Hβ,Ω(S;x-), then (44)Hβ,ΩS;x-⊂TCS;x--βΩ.Taking the closure of both sides of the previous inclusion and taking into account the fact that Tβ,ΩC(S;x-)=cl(Hβ,Ω(S;x-)), we obtain (45)Tβ,ΩCS;x-=clHβ,ΩS;x-⊂clTCS;x--βΩ=TCS;x--βΩ.Therefore, the proof of this lemma is finished since the reverse inclusion is always true by Lemma 12.A direct and very important result on the convexity of the setHβ,Ω(S;x-) can be deduced from the previous lemma.Corollary 15.
Letβ≥0, let Ω be convex compact with 0∈Ω, and let S be epi-Lipschitz at x-. If S is tangentially regular at x-∈S, then Hβ,Ω(S;x-) and Tβ,ΩC(S;x-) are both convex.Using Lemma14 we can easily construct many examples of closed sets S and set-valued mappings F for which the tangential condition (2) is not satisfied and the new tangential condition (4) is satisfied.Example 16.
LetE=R2, S={(x1,x2)∈R2:x1,x2∈[-1,0]}, Ω=-S  (see Figure 1), and F(x1,x2)=(2x1+x2,2x2+3x1-1)+B. Let u1≔(-1,0), u2≔(0,-1), and u3≔(-1,-1). Simple computations yield the following: (46)Fu1=-1,-4+B,TCS;u1=R+×R-,Tβ,ΩCS;u1=-β,∞×R-,Fu2=-1,-3+B,TCS;u2=R-×R+,Tβ,ΩCS;u2=R-×-β,∞,Fu3=-2,-6+B,TCS;u3=R+×R+,Tβ,ΩCS;u3=-β,∞2.Hence, F(ui)∩TC(S;ui)=∅, for i=1,2,3. And F(x1,x2)∩Tβ,ΩC(S;(x1,x2))≠∅, for β=5 and for any (x1,x2)∈S. Although the set S is closed convex, the tangential condition (2) is not satisfied while the tangential condition (4) is satisfied.Figure 1
The convex setS and the target set Ω in Example 16.Consider now another example withS that is a nonconvex tangentially regular set.Example 17.
LetE=R2 and S={(x1,x2)∈R2:1≤x12+x22≤4}  (see Figure 2). Clearly S is a closed nonconvex set in R2. Also the set S is epi-Lipschitz and tangentially regular. Let x-=(x-1,x-2)∈S. We can check the following: (47)TCS;x-1,x-2=R2;1<x-12+x-22<4;u,v∈R2:x-2v≤-x-1u;x-12+x-22=4;u,v∈R2:x-2v≥-x-1u;x-12+x-22=1.Let Ω=x1,x2∈R2:x1,x2∈1,0 and β≥0. Then using Lemma 14 we obtain (48)Tβ,ΩCS;x-1,x-2=R2;1<x-12+x-22<4;u,v∈R2:x-2v≤-x-1u+βx-1+x-2;x-12+x-22=4;u,v∈R2:x-2v≥-x-1u+βx-1+x-2;x-12+x-22=1.Also, we can verify that for β=1 and for some points in S we have (49)Tβ,ΩCS;x-1,x-2=TCS;x-1,x-2=R2;1<x-12+x-22<4;u,v∈R2:x-2v≥-x-1u;x-1≤0,x-2≤0,x-12+x-22=4;u,v∈R2:x-2v≥-x-1u;x-1≥0,x-2≥0,x-12+x-22=1.For the rest of points in S the Clarke tangent cone TCS;x-1,x-2 is strictly included in Tβ,ΩC(S;(x-1,x-2)). Define now the set-valued mapping F as follows: (50)Fx1,x2=x1,x2+32B.Clearly F is an u.s.c. set-valued mapping with closed convex values. For this couple F and S we have the following facts:
(i)(51)Fx-1,x-2∩TCS;x-1,x-2⁡=∅;for  anyx-1≥0,x-2≥0,x-12+x-22=4;≠∅;otherwise,
(ii)(52)Fx-1,x-2∩Tβ,ΩCS;x-1,x-2≠∅,for  anyx-1,x-2∈S. Thus, for the nonconvex epi-Lipschitz tangentially regular set S, the tangential condition (2) is not satisfied and the tangential condition (4) is satisfied.Figure 2
The nonconvex setS in Example 17.


## 3. Existence Results

Throughout this sectionE=Rn. Let us recall the following important facts needed in our next proofs.Fact 1 (see [10]).
For any finite covering∪i=1kCi of a compact set S⊂E, with each Ci being open and bounded, there exists a Lipschitz partition of unity {p1,p2,…,pk}; that is, each pi is a Lipschitz function and the following hold: (i)
0≤pi(x)≤1, ∀x∈S, ∀i=1,2,…,k.(ii)
x∉Ci⇒pi(x)=0, ∀i=1,2,…,k.(iii)
∑i=1k‍pi(x)=1, ∀x∈S.Fact 2 (see [11]).
LetS be a compact subset of E and suppose that F is an u.s.c. set-valued mapping on S with images that are nonempty closed convex sets in E. Let γi↓0 be given. Then there exists a sequence of set-valued mappings Fj:S⇉E with closed convex values such that the following hold: (a)
F(x)⊂Fj+1(x)⊂Fj(x)⊂F({x+γjint⁡(B)}), ∀x∈S, ∀j=1,2,…;(b)
Fj is l.s.c.Fact 3 (see [1]).
LetS⊂Rn be homeomorphic to a convex compact set and let F be an upper semicontinuous set-valued mapping on S with nonempty closed convex values in E. Assume that S is epi-Lipschitz and suppose that (2) holds. Then F has a zero on S.In order to prove the first main result we need to prove the following lemma.Lemma 18.
LetS be a compact set. Assume that S is an epi-Lipschitz compact set, ϵ->0, F is l.s.c. on S, and for any x∈S there exists βx≥0 such that (53)Fx∩int⁡Tβx,ΩCS;x≠∅.Then there exists β≥0 (not depending on x nor on ϵ-) and a Lipschitz function fϵ- such that (54)fϵ-x∈Fx+ϵ-int⁡B∩Hβ,ΩS;x≠∅,∀x∈S.Proof.
SinceS is epi-Lipschitz, we have that by Proposition 9 the set-valued mapping Tβx,ΩC(S;·) is l.s.c. at any x∈S. Let ϵ>0 be too small so that ϵ<ϵ-. Thus, for any x∈S and any y∈F(x)∩int⁡(Tβx,ΩC(S;x))≠∅, the l.s.c. of both F and Tβx,ΩC(S;·) imply the existence of δ(x,y,ϵ)>0 such that (55)wehavey∈Fx′+ϵint⁡B∩int⁡Tβx,ΩCS;x′∀x′∈int⁡Bx,δx,y,ϵ.Hence, the family of open balls(56)int⁡Bx,δx,y,ϵ≔x+δx,y,ϵint⁡B;ϵ∈0,ϵ-,x∈S,y∈Fx∩int⁡Tβx,ΩCS;xforms an open covering of S. By compactness of S there exists a finite subcover ∪i=1kint⁡B(xi;δ(xi,yi,ϵi)) of S and let {p1,p2,…,pk} be a Lipschitz partition of unity subordinate to this subcover (by Fact 1). Define now the Lipschitz function (57)fϵ-z≔∑i=1kpizyi.Then (58)piz≠0⟹z∈int⁡Bxi;δxi,yi,ϵi⟹yi∈Fz+ϵiint⁡B∩int⁡Tβxi,ΩCS;z.Therefore, the convexity of both (βxi,Ω)-Clarke tangent sets (by Corollary 15) and images of the set-valued mappings F imply that the selection f satisfies (59)fϵ-z∈Fz+ϵiint⁡B∩int⁡Tβxi,ΩCS;z∀z∈S,∃i∈1,2,…,k.Set β≔max⁡{βxi;i∈{1,…,k}}. Then the fact that ϵi<ϵ- and β≥βxi for any i and the fact that Ω is balanced ensure the inclusion (60)Tβxi,ΩCS;z⊂Tβ,ΩCS;z,Fz+ϵiint⁡B⊂Fz+ϵ-int⁡B.By Proposition 8 we have int⁡(Tβ,ΩC(S;z))=Hβ,Ω(S;z), and consequently we obtain (61)fϵ-z∈Fz+ϵ-int⁡B∩Hβ,ΩS;z,∀z∈S,which completes the proof.The following result can be seen as an approximate equilibria result. It will be used to prove the main result of the paper.Theorem 19.
LetS be homeomorphic to a convex compact set in Rn which is epi-Lipschitz and let F be an u.s.c. set-valued mapping with nonempty closed convex values. Let Ω be a convex compact set in Rn with 0∈Ω. Assume that for any x∈S there exist some βx≥0 such that (62)Fx∩Tβx,ΩCS;x≠∅,∀x∈S.If S is tangentially regular, then there exist β≥0 and x∗∈S such that (63)0∈Fx∗+βΩ.Proof.
Case 1. First assume that F is l.s.c. on S and (64)Fx∩int⁡Tβx,ΩCS;x≠∅,∀x∈S.Then for any ϵ>0 there exists, by the previous lemma, a Lipschitz function fϵ and a constant β≥0 such that (65)fϵz∈Fz+ϵint⁡B∩Hβ,ΩS;z∀z∈S.Let us introduce the ordinary differential equation (66)x˙t=fϵxt.In view of (65) we have for any z∈S(67)fϵz∈Hβ,ΩS;z;that is, for any z∈S the set S is (β,Ω)-epi-Lipschitz at z in the direction fϵ(z). Use now Proposition 13 to deduce that (68)fϵz∈KS;z-βΩ.Since S is tangentially regular, we get (69)TCS;z∩fϵz+βΩ≠∅,∀z∈S.Define the set-valued mapping Fϵ(x)≔fϵ(x)+βΩ. Clearly this set-valued mapping satisfies the hypothesis of Fact 3 and so there exists xϵ∗∈S such that 0∈Fϵ(xϵ∗)≔fϵ(xϵ∗)+βΩ. Clearly, (70)0∈fϵxϵ∗+βΩ⊂Fxϵ∗+ϵint⁡B+βΩ.Case 2. Assume now that F is u.s.c. Let γj↓0 as j→∞ and let Fj be a sequence of l.s.c. approximations of F as in Fact 2. Let ϵj↓0 as j→∞ and let F~j(x)≔Fj(x)+γjB. Clearly F~j is l.s.c. and F~j(x)∩int⁡Tβx,ΩC(S;x)≠∅, ∀x∈S. Indeed, by Part (b) in Fact 2 we have (71)∅≠Fx∩Tβx,ΩCS;x⊂Fjx∩Tβx,ΩCS;x,and so obviously we obtain (72)Fjx+γjB∩int⁡Tβx,ΩCS;x≠∅.And hence (73)F~jx∩int⁡Tβx,ΩCS;x≠∅,∀x∈S.Applying Case 1 for all F~j we obtain a constant β≥0 and an element xj∗∈S such that (74)0∈F~jxj∗+ϵjint⁡B+βΩ.Consequently, from the monotonicity of Fj (by Part (a) in Fact 1), we have (75)0∈F~jxi∗+ϵiint⁡B+βΩ,∀i>j.Since S is assumed to be compact, we can extract a subsequence of xj∗ (still denoted by xj∗) converging to a limit x∗∈S. Let xij∗ be such that ij>j. Then(76)0∈⁡F~jxij∗+ϵijint⁡B+βΩ⊂Fjxij∗+γjB+ϵijint⁡B+βΩ⊂Fjxij∗+2γj+ϵijint⁡B+βΩand therefore by Part (a) in Fact 1 we get (77)0∈Fxij∗+γjint⁡B+2γj+ϵijint⁡B+βΩ.Upon letting j→∞ we obtain 0∈F(x∗)+βΩ and hence the proof is complete.Observe thatβ in the previous theorem cannot be controlled since it depends on the pointwise constants βx in (4). However, if we assume that the tangential condition (4) is satisfied with a uniform β (i.e., β does not depend on x and it is the same for any x∈S), then we get the following first corollary in which the constant is the same satisfying (4).Corollary 20.
LetS be homeomorphic to a convex compact set in Rn and let F be an u.s.c. set-valued mapping with nonempty closed convex values. Let Ω be a convex compact set in Rn with 0∈Ω. Assume that S is epi-Lipschitz and for some β≥0 one has (78)Fx∩Tβ,ΩCS;x≠∅,∀x∈S.If S is tangentially regular, then there exists x∗∈S such that (79)0∈Fx∗+βΩ.Using this corollary we prove our main result establishing an existence result of exact equilibria ofF on S.Theorem 21.
LetS be homeomorphic to a convex compact set in Rn and let F be an u.s.c. set-valued mapping with nonempty closed convex values. Let Ω be a convex compact set in Rn with 0∈Ω. Assume that S is epi-Lipschitz and for some β≥0 one has (80)Fx∩Tβ,ΩCS;x≠∅,∀x∈S.If S is tangentially regular, then there exists x∗∈S such that (81)0∈Fx∗.Proof.
We proceed by approximation. Fix anyϵ>0. First observe the following fact due to the balanced property of Ω: (82)Tϵβ,ΩCS;x=TCS;x-ϵβΩ=TCS;x-βΩ=Tβ,ΩCS;x.Then (80) ensures (83)Fx∩Tϵβ,ΩCS;x≠∅,∀x∈S.Applying now Theorem 19 we obtain for any ϵ>0 the existence of some point xϵ∗∈S with 0∈F(xϵ∗)+ϵβΩ. Using the fact that S is compact we can extract a subsequence of {xϵ∗}ϵ converging to some limit x∗∈S and since obviously our assumptions on F ensure that the limit satisfies 0∈F(x∗), then the proof is complete.We apply this existence result to the following two examples for which we cannot apply the results in [1] because the tangential condition (2) is not satisfied.Example 22.
LetE=R2, S={(x1,x2)∈R2:x1,x2∈[-1,0]}, Ω=-S, and F(x1,x2)=(2x1+x2,2x2+3x1-1)+B. Since F(x1,x2)∩Tβ,ΩC(S;(x1,x2))≠∅ for any (x1,x2)∈S (by Example 16) and since all the assumptions on F and S in Theorem 21 are fulfilled, then there exists some x∗∈S with (0,0)∈F(x∗). In this example the equilibrium is x∗=(0,0)∈S and it is unique.Example 23.
LetE=R2, S={(x1,x2)∈R2:x1,x2≥0,1≤x12+x22≤4}, and F(x1,x2)=(x1,x2)+3/2B. The set S is the intersection of the set given in Example 17 with the first quarter in R2 (see Figure 3). Using the computations presented in Example 17 we can check that, for any (x1,x2)∈S, F(x1,x2)∩Tβ,ΩC(S;(x1,x2))≠∅. Also, the set S is homeomorphic to a convex compact set. Indeed, S can be mapped continuously and with continuous inverse to the segment {(0,x2):x2∈[1,2]} by projection. Thus, all the assumptions of Theorem 21 are satisfied and so there exists x∗∈S such that (0,0)∈F(x∗). In this example we do not have the uniqueness of the equilibria x∗ and there is an infinity of them. Indeed, for any (x1,x2)∈S with x12+x22=1 we have (0,0)∈F(x1,x2).Figure 3
The nonconvex setS in Example 23.Let us now apply our main result in Theorem21 for the existence of fixed points for set-valued mappings on nonconvex sets. It extends Theorems 1.7 and 1.8 in [1] from the case of S and F satisfying the tangential condition (80) with β=0 to the general case with any β≥0.Theorem 24.
LetS be homeomorphic to a convex compact set in Rn and let F be an u.s.c. set-valued mapping with nonempty closed convex values. Let Ω be a convex compact set in Rn with 0∈Ω. Assume that S is epi-Lipschitz and tangentially regular and for some β≥0 one has (84)Gx∩x+Tβ,ΩCS;x≠∅,∀x∈S.Then G has a fixed point in S; that is, there exists x∗∈S such that x∗∈G(x∗).Proof.
It follows directly from Theorem21 by taking F(x)≔G(x)-x.



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