# Set Contractions and KKM Mappings in Banach Spaces

Authors: A. Razani; N. Karamikabir
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Some fixed point theorems for generalized set contraction maps and KKM type ones in Banach spaces are presented. Moreover, a new generalized set contraction is introduced. As an application, some coincidence theorems for KKM type set contractions are obtained.


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## Body

## 1. Introduction

LetE be a Banach space and
(1)𝒫p(E)={A⊂E:Aisanonemptyandhasapropertyp}.
Thus 𝒫bd(E), 𝒫cl(E), 𝒫cv(E), 𝒫cp(E), 𝒫co(E), 𝒫cl,bd(E), 𝒫cp,cv(E), and 𝒫rcp(E) denote the classes of all bounded, closed, convex, compact, connected, closed-bounded, compact-convex, and relatively compact subsets of E, respectively [1]. Let E1 and E2 be two Banach spaces. A multivalued mapping T:E1→𝒫p(E2) is said to be (i)
upper semicontinuous if and only if for every closed subsetB of E2, the set T-(B)={x∈E1:T(x)∩B≠∅} is a closed subset of E1,(ii)
closed if its graphGr(T)={(x,y)∈E1×E2:y∈T(x)} is a closed subset of E1×E2,(iii)
compact ifT(E1)¯ is a compact subset of E2. The first fixed point theorem for multivalued mappings is due to Kakutani in Banach spaces, in 1941 [2]. He proved a generalization of Brouwer’s fixed point theorem to the multivalued mappings.Theorem 1 (see [2]).
LetK be a compact subset of a Banach space E and let F:K→𝒫cv,cp(K) be an upper semicontinuous multivalued operator. Then F has a fixed point.The above theorem has been extended in the literature by generalizing or modifying the domain spaceE, domain set K, and the nature of the multivalued operator F. Here, compactness plays an essential role. The following definition of measure of noncompactness on a bounded subset of the Banach space E is given by Dhage in 2010.Definition 2 (see [1]).
A functionμ:𝒫bd(E)→ℝ+ is called a measure of noncompactness if it satisfies (i)
∅≠μ-1(0)⊂𝒫rcp(E),(ii)
ifA⊆B then μ(A)≤μ(B),(iii)
μ(A¯)=μ(A), where A¯ denotes the closure of A,(iv)
μ(Conv(A))=μ(A), where Conv(A) denotes the convex hull of A,(v)
if{An} is a decreasing sequence of sets in 𝒫cl,bd(E) satisfying limn→∞μ(An)=0, then the limiting set A∞=∩n=1∞An¯ is nonempty.Definition 3 (see [1]).
A multivalued mappingT:E→𝒫cl,bd(E) is called 𝒟-set Lipschitz if there exists a continuous nondecreasing function ψ:ℝ+→ℝ+ such that μ(T(A))≤ψ(μ(A)) for all A∈𝒫cl,bd(E) with T(A)∈𝒫cl,bd(E), where ψ(0)=0. Sometimes we call the function ψ to be a 𝒟-function of T on E. In the spatial case, when ψ(r)=kr, k>0, T is called a k-set Lipschitz mapping and if k<1, then T is called a k-set contraction on E. Further, if ψ(r)<r for r>0, then T is called a nonlinear 𝒟-set contraction on E.Dhage proved a generalization of Theorem1 under weaker upper semicontinuity conditions in 2010 [1].Theorem 4.
LetX be a nonempty, closed, convex, and bounded subset of a Banach space E and let T:X→𝒫cl,cv(X) be a closed and nonlinear 𝒟-set contraction. Then T has a fixed point.Lemma 5 (see [3]).
Ifψ is a 𝒟-function with ψ(r)<r for r>0, then limn→∞ψn(t)=0 for all t∈[0,∞).Recall that a functionφ:ℝ+→ℝ+ is called a comparison function if φ is increasing and limn→∞φn(t)=0 for all t∈ℝ+ [4]. As a consequence, φ(t)<t for any t>0, φ is continuous at 0, and φ(0)=0.A functionφ:ℝ+→ℝ+ is called a (c)-comparison function if φ is increasing and there exist k0∈ℕ, a∈(0,1) and a convergent series of non-negative terms ∑k=1∞‍νk such that
(2)φk+1(t)≤aφk(t)+νk,
for k≥k0 and any t∈[0,∞) [5]. If φ:ℝ+→ℝ+ is a (c)-comparison function, then φ is a comparison function [5]. So, we can define 𝒞-set contraction as follows.Definition 6.
A multivalued mappingT:E→𝒫cl,bd(E) is called 𝒞-set contraction if there exists a continuous (c)-comparison function φ such that μ(T(A))≤φ(μ(A)) for all A∈𝒫cl,bd(E) with T(A)∈𝒫cl,bd(E).Let(X,d) be a metric space. The Hausdorff metric Hd induced by the metric d and defined as follows
(3)Hd(A,B)=max{supa∈Ad(a,B),supb∈Bd(b,A)}.Definition 7 (see [6]).
Letε>0. A multivalued contractive (ε-contractive) map is a map F:X⊸𝒫cl,bd(X) such that for all x,y∈X with x≠y (and for some ε>0, d(x,y)<ε, resp.), Hd(F(x),F(y))<d(x,y).Theorem 8 (see [7]).
LetX be a nonempty compact and connected metric space and let F:X→𝒫cp(X) be a multivalued ε-contractive map, then F has a fixed point.We need the following definitions of KKM theory in the sequel [8].Assume thatX is a convex subset of a topological vector space and Y is a topological space. If G:X→𝒫p(Y) and F:X→𝒫p(Y) are two multivalued maps and for each A∈〈X〉, F(Conv(A))⊂G(A), then G is called generalized KKM mapping with respect to F, where 〈X〉 denote the family of all nonempty finite subsets of X. More generally, if F:X→𝒫p(Y) satisfies the requirement that for any generalized KKM mapping G:X→𝒫p(Y) with respect to F and the family {G(x)¯:x∈X} has the finite intersection property, then F is said to have the KKM property. Let
(4)KKM(X,Y)={F:X→𝒫p(Y):FhastheKKMproperty}.Lemma 9 (see [8]).
LetX be a nonempty convex subset of Hausdorff topological vector space E. Then T|D∈KKM(D,Y) whenever T∈KKM(X,Y) and D is a nonempty convex subset of X.Chen and Chang obtained some fixed point theorems for KKM type set contraction mappings in various spaces [9–12]. In 2010, Amini-Harandi et al. introduced generalized set contraction on topological spaces [13].In Section2, we present some fixed point theorems for generalized set contractions which are ε-contractive (KKM ε-contractive) multivalued maps. In the first step of Section 3, we introduce a new type of generalized set contraction and then prove that the results of Section 2 hold for them. Section 4 is devoted to some KKM coincidence theorems as applications of these results.


## 2. Generalized Set Contractions

In this section by applying Theorem8, we obtain some fixed point theorems for ε-contractive multivalued maps which are either generalized set contraction or KKM type ones. In all cases, the multivalued maps are not necessarily compact values. We consider measurement of noncompactness in Definition 2.Definition 10 (see [13]).
A multivalued mappingF:E→𝒫p(E) is said to be a generalized set contraction, if for each ϵ>0 there exists δ>0 such that for A∈𝒫cl,bd(E) with ϵ≤μ(A)<ϵ+δ, there exists n∈N such that μ(Fn(A))<ϵ.Lemma 11 (see [13]).
LetX be a topological space and let μ be a measure of noncompactness on X. Suppose that F is a generalized set contraction on X. Then for every subset A of X for which F(A)⊂A and μ(A)<∞, one has limn→∞μ(Fn(A))=0.Proposition 12.
LetE be a Hausdorff topological space. If {Xn} is a decreasing sequence of closed and connected sets in 𝒫cl,bd(E) such that limn→∞μ(Xn)=0, then ∩n=1∞Xn is nonempty, compact, and connected.Proof.
Clearly,K=∩n=1∞Xn is a nonempty, closed, and compact subset of E. Let A and B be two nonempty, disjoint, and closed sets so that K=A∪B. We can find disjoint open sets U and V around A and B, respectively. For every n∈ℕ, Xn∖(U∪V) is nonempty. If not, then (U∩Xn) and (V∩Xn) are nonempty and Xn=(U∩Xn)∪(V∩Xn), which cannot happen. The collection of {Xn∖(U∪V)} is also a decreasing sequence of nonempty closed sets. Since μ(Xn∖(U∪V))≤μ(Xn) then μ(Xn∖(U∪V))→0 as n→∞. Hence, ∩n=0∞(Xn∖(U∪V)) is nonempty, that is, K∩(U∪V)c≠∅, which is a contradiction.Theorem 13.
LetX be a nonempty, bounded, closed, and connected subset of Banach space E. If F:X→𝒫cl,co(X) is an ε-contractive and generalized set contraction, then F has a fixed point and the set of fixed points of F is compact.Proof.
LetX0=X and Xn=Fn(X)¯ for all n∈ℕ. Since F is an ε-contractive map, then F is continuous and F(A¯)⊆F(A)¯. On the other hand, Fn+1(X)⊆Fn(X), so we have F(Xn)⊆Xn and Xn+1⊆Xn. Since F is continuous with closed and connected values, then by [14, Lemma 1.6], F(X) is connected. Hence, Xn which is closure of connected set Fn(X) for all n∈ℕ, is connected. But μ(Xn)=μ(Fn(X)), and by Lemma 11 we have μ(Xn)→0. Therefore, by Proposition 12, K=∩n=1∞Xn is nonempty compact and connected. Since F(K)⊆K, then the desired conclusion followed by an application of Theorem 8 to the multivalued map F:K→𝒫cp(K).
LetB={x∈X:x∈F(x)}. We claim that ϵ0=μ(B)=0. If μ(B)≠0 then μ(B)=μ(Fn(B)). Since F is a generalized set contraction, then for ϵ0>0, there exists δ>0 such that for B⊆X with ϵ0≤μ(B)<ϵ0+δ, there exists n∈ℕ such that ϵ0=μ(B)<ϵ0, which is a contradiction. So B is relatively compact, and since F is continuous, then B is a compact subset of X.IfF:E→𝒫p(E) is a k-set contraction, then F is a generalized set contraction, but the converse is not true [13]. Therefore, we have the following result.Corollary 14.
LetX be a nonempty, bounded, closed, and connected subset of Banach space E. If F:X→𝒫cl,co(X) is an ε-contractive and k-set contraction, then F has a fixed point.Corollary 15.
LetX be a nonempty, closed, and bounded subset of Banach space E. If F:X→𝒫cl,bd(X) is an ε-contractive and generalized set contraction, then there exists a compact subset K of X such that F(K)⊆K.Theorem 16.
LetX be a nonempty, closed, and bounded subset of Banach space E. If F∈KKM(X,X) is an ε-contractive and generalized set contraction with nonempty closed and bounded values, then F has a fixed point.Proof.
By Corollary15, there exists a compact subset K of X such that F(K)⊆K. Let Y=F(K)¯. Hence there exists a finite subset A of K such that Y⊆∪x∈AN(x,ε), where N(x,ε)={y∈E:∥x-y∥<ε}. Define a map T:K→𝒫p(Y) by T(x)=Y∖N(x,ϵ) for all x∈K; then T(x) is closed for each x∈K and ∩x∈AT(x)=∅. Since Y⊆K⊆X and F∈KKM(X,X), then by Lemma 9, F|K∈KKM(K,Y). Thus, T is not a generalized KKM map with respect to F|K. Hence, there exists {x0,x1,…,xn} of K such that F(Conv({x0,x1,…,xn}))⊈∪i=0nT(xi). Thus there exists y∈F(Conv({x0,x1,…,xn})) such that y∉∪i=0nT(xi), that is, y∈F(z) for some z∈Conv({x0,x1,…,xn}) and y∈N(xi,ϵ), so xi∈N(y,ϵ) for i∈{0,1,…,n}. Since z∈Conv({x0,x1,…,xn})⊂B(y,ϵ), then y∈F(z)∩B(z,ϵ). Since Y is a compact subset of X, then y converges to some x∈Y as ϵ→0. Consequently, z converges to x as ϵ→0. Since F is continuous, then by [14, Lemma 1.6] we have x∈F(x).


## 3. Asymptotic Generalized Set Contractions

In this section, we define a new type of set contraction in Banach spaces. Then we prove that the results of Section2 hold for them. Also, we conclude some fixed point theorems for nonlinear 𝒟-set contractions.Definition 17.
LetX be a nonempty, closed, and bounded subset of a Banach space E. A multivalued mapping F:X→𝒫p(X) is said to be an asymptotic generalized set contraction, if there exists a sequence {φn} of functions from ℝ+ in to itself satisfies (i)
for eachε>0, there exists δ>0 and m∈ℕ such that φm(t)≤ε for all ε≤t<ε+δ,(ii)
μ(Fn(X))<φn(μ(X)).Theorem 18.
LetX be a nonempty, bounded, closed, and connected subset of Banach space E. If F:X→𝒫cl,co(X) is an ε-contractive and asymptotic generalized set contraction, then F has a fixed point.Proof.
Define a sequence{Xn} of sets in 𝒫cl,bd(X) such that X0=X and Xn=Fn(X)¯ for all n∈ℕ. As the proof of Theorem 13, F(Xn)⊆Xn, Xn+1⊂Xn, and Xn is connected for all n∈ℕ. If there exists an integer N>0 such that μ(XN)=0, then XN is a compact and connected set and invariant under F. Thus Theorem 8 implies that F:XN→𝒫cp(XN) has a fixed point. So we assume that μ(Xn)≠0 for all n∈ℕ. Define εn=μ(Xn) and r=infεn. If r≠0, by Definition 17, there exists n0∈ℕ, δr>0 and m∈ℕ such that φm(t)≤r for all r≤t<r+δr and r≤εn0<r+δr, so
(5)εn0+m=μ(Xn0+m)=μ(Fn0+m(X))<φm(μ(Fn0(X)))=φm(μ(Xn0))≤r,
which is a contradiction. Hence μ(Xn)⇀0 as n→∞. Now by Proposition 12, K=∩n=1∞Xn is nonempty, compact, and connected. Moreover F(K)⊆K. So by Theorem 8, the multivalued map F:K→𝒫cp(K) has a fixed point.Corollary 19.
LetX be a nonempty, closed, and bounded subset of Banach space E. If F:X→𝒫cl,bd(X) is an ε-contractive and asymptotic generalized set contraction, then there exists a compact subset K of X such that F(K)⊆K.The proof of following theorem is similar to that of Theorem16; hence it is omitted.Theorem 20.
LetX be a nonempty, closed, and bounded subset of Banach space E. If F∈KKM(X,X) is an ε-contractive and asymptotic generalized set contraction with nonempty closed and bounded values, then F has a fixed point.Proposition 21.
LetX be a nonempty, closed, and bounded subset of Banach space E. If F:X→𝒫cl,bd(X) is a nonlinear 𝒟-set contraction, then F is an asymptotic generalized set contraction.Proof.
LetF be a nonlinear 𝒟-set contraction with 𝒟-function ψ. Define φn=ψn for all n∈ℕ. Clearly, μ(Fn(X))≤ψn(μ(X))=φn(μ(X)), on the other hand, by Lemma 5 we have φn(t)→0 as n→∞. Thus F is an asymptotic generalized set contraction.Applying Proposition21, Theorems 18 and 20, it is easy to conclude the following results.Corollary 22.
LetX be a nonempty, bounded, closed, and connected subset of Banach space E. If F:X→𝒫cl,co(X) is an ε-contractive and nonlinear 𝒟-set contraction, then F has a fixed point.Corollary 23.
LetX be a nonempty, closed, and bounded subset of Banach space E. If F∈KKM(X,X) is an ε-contractive and nonlinear 𝒟-set contraction with nonempty closed and bounded values, then F has a fixed point.Remark 24.
Since every𝒞-set contraction is a nonlinear 𝒟-set contraction, then Theorem 4, Corollaries 22 and 23 hold for these mappings.


## 4. Some Applications in KKM Theory

In this section we obtain two coincidence theorems for KKM type set contractions.Theorem 25.
LetX be a nonempty, closed, bounded, and convex subset of Banach space E. If F:X→𝒫cl,bd(X) and G:X→𝒫cv(X) are two multivalued mappings satisfying (i)
F∈KKM(X,X),(ii)
F is a generalized (an asymptotic) set contraction and ε-contractive map,(iii)
for each compact subsetC of X and any y∈X, G-(y)∩C is open in C, then, there exists x0,y0∈X such that y0∈F(x0) and x0∈G(y0).Proof.
By Corollary15 (Corollary 19), there exists a compact subset K of X such that F(K)⊂K. Since G(x)≠∅ and K is compact, then X=∪x∈XG-(X) and K=∪i=1n(G-(xi)∩K) for some x1,…,xn∈X. Define a map S:X→𝒫p(K) by S(x)=K∖(G-(x)∩K) for each x∈X, then ∩i=1nS(xi)=∅. Therefore, S is not a generalized KKM map with respect to F. So there exists a finite subset A={a1,…am} of X such that F(Conv(A))⊈S(A). Hence, there exist x0∈Conv(A) and y0∈F(x0) such that y0∉S(A). Thus y0∈G-(ai)∩K and so ai∈G(y0) for i=1,…,m. Since G(y0) is convex, then Conv(A)⊆G(y0) and so x0∈G(y0).By Proposition21, Corollary 19, and slight modification of the proof of Theorem 25, we have the following theorem.Theorem 26.
LetX be a nonempty, closed, bounded, and convex subset of Banach space E. If F:X→𝒫cl,cv(X) and G:X→𝒫cv(X) are two multivalued mappings satisfying (i)
F∈KKM(X,X) is a nonlinear 𝒟-set contraction.(ii)
for each compact subsetC of X and any y∈X, G-(y)∩C is open in C, then, there exists x0,y0∈X such that y0∈F(x0) and x0∈G(y0).



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