IN TOPOLOGICAL VECTOR SPACES

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FOR SET-VALUED MAPS WITH NONCONVEX OR NONCOMPACT DOMAINS

IN TOPOLOGICAL VECTOR SPACES

KAZIMIERZ WŁODARCZYK AND DOROTA KLIM Received 9 January 2002

A technique, based on the investigations of the images of maps, for obtaining fixed-point and coincidence results in a new class of maps and domains is de- scribed. In particular, we show that the problem concerning the existence of fixed points of expansive set-valued maps and inner set-valued maps on not necessarily convex or compact sets in Hausdorﬀtopological vector spaces has a solution.

As a consequence, we prove a new intersection theorem concerning not necessarily convex or compact sets and its applications. We also give new coincidence and section theorems for maps defined on not necessarily convex sets in Hausdorﬀ topological vector spaces. Examples and counterexamples show a fundamental diﬀerence between our results and the well-known ones.

1. Introduction

Suppose thatEis a Hausdorfftopological vector space,C⊂E,C= ∅. In fixed- point and coincidence theory and its applications, a great part of the vast lit- erature in the last century concerns conditions onC,E,F, andGguaranteeing the existence of fixed points or coincidences of set-valued mapsF:C→2Êand G:C→2Ê. In various methods of investigations, the assumptions that the maps are inner andCare convex compact subsets ofEplay the crucial role (see, e.g., [5,6,7,15,16,17,18,19,20,21,22,23,24,25,27,30,33,41]). The general topic of fixed points and coincidences for set-valued maps on convex compact sets, originating mainly with the work of Kakutani [30], Bohnenblust and Karlin [5], Glicksberg [23], Fan [15,16,17,18,19,20,21,22], and Browder [6,7], has been well developed in various directions.

In the past decade, there was a renewed interest in the fixed-point and coincidence theory of set-valued maps in topological vector spaces (see, e.g., [2, 3,8,9,10,11,12,28,36,37,38,39,40,42,43,44,45,46,47,48]), partially due to new and powerful methods of investigations introduced into it (notably

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based on those introduced by Fan and Browder). Most of the work has centered around the fixed-point and coincidence theory of maps on convex compact sets, but there are also a considerable number of papers devoted to maps on nonconvex and noncompact sets (see, e.g., [8,45]).

There exist a number of introductions to and surveys of fixed-point and coincidence theory. We mention [47] among the more recent ones but also some elder ones [14,49,50]. See also many references therein.

A natural question arises: whether expansive set-valued maps and inner set- valued maps on not necessarily convex or compact sets have fixed points and, as a consequence, theorems of intersection type hold and whether the maps F:C→2Ê andG:C→2Ê on not necessarily convex sets in Hausdorfftopo- logical vector spaces have coincidences. The affirmative answers are given in this paper. Using a technique based on the investigation of the images of maps, we obtain a number of new fixed-point, coincidence, intersection, and section theorems of Fan-Browder type. Examples and counterexamples show a fundamental difference between our results and the known results of the above-mentioned authors.

2. Fixed points and coincidences of expansive set-valued maps on not necessarily convex or compact sets

in topological vector spaces

LetCbe a subset of a Hausdorﬀtopological vector spaceEoverK(K=RorC).

A set-valued mapF:C→E(which will always be denoted by capital letters) is a map which assigns a uniqueF(c)∈2Ê(here 2Êdenotes the family of all subsets ofE) to eachc∈C. We say thatc∈Cis a fixed point ofF:C→2Êifc∈F(c).

We say that a mapF:C→2^Eis expansive ifC⊂F(C) whereF(C)=

c∈CF(c).

Maps in the usual sense will be considered as special (single-valued) set- valued maps and these ordinary maps will always be denoted by small letters

f :C→E.

We prove the following theorem.

Theorem2.1. LetCbe a nonempty subset of a Hausdorﬀtopological vector spaceE overR, letF:C→2^E, and letKbe a convex subset ofE. Assume that the following conditions hold:

(i)C⊂K⊂F(C);

(ii)F(C)is a compact subset ofE;

(iii)for eachc∈C,F(c)is open inF(C);

(iv)for eachy∈K,F⁻¹(y)= {c∈C:y∈F(c)}is nonempty and convex.

Then there existsu∈Csuch thatu∈F(u).

Proof. By (iii), the compact setF(C) is covered by the setsF(c),c∈C, which are open inF(C). Clearly, there exists a finite set{c1,...,cn} ⊂Csuch thatF(ci) are nonempty, 1≤i≤n, andF(C)=_n

i₌1F(ci). Let{ϕ1,...,ϕn}be a partition of

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unity with respect to this cover, that is, a finite family of real-valued nonnegative continuous mapsϕionF(C) such thatϕivanish outsideF(ci) and are less than or equal to one everywhere, 1≤i≤n, andⁿ_i₌1ϕi(y)=1 for ally∈F(C).

Let σ be a simplex spanned by points c1,...,cn and let ϕ:F(C)→σ be a continuous map defined by the formulaϕ(y)=n

i=1ϕi(y)ci,y∈F(C). Clearly, σ⊂K⊂F(C) and henceϕ(σ)⊂ϕ(K)⊂ϕ(F(C))⊂σ.

If y∈K is arbitrary and fixed andϕi(y)=0 for some i∈ {1,...,n}, then y∈F(ci), soci∈F⁻¹(y). As a consequence, for each y∈K,ϕ(y) is a convex linear combination of points ofF⁻¹(y) and by (iv), we get for eachy∈K,

ϕ(y)∈F⁻¹(y), ϕ(y)∈C. (2.1)

From Brouwer’s theorem, we getu=ϕ(u) for someu∈σ and hence, since σ⊂K, by (2.1),u=ϕ(u)∈F⁻¹(u)⊂C, and therefore,u∈F(u) andu∈C, as

required.

By using various setsK, a number of variations ofTheorem 2.1can be obtained, of which the following two are typical.

Theorem2.2. LetCbe a nonempty subset of a Hausdorﬀtopological vector space EoverRand letF:C→2^E. Assume that the following conditions hold:

(i)Fis expansive, that is,C⊂F(C);

(ii)F(C)is convex;

(iii)F(C)is a compact subset ofE;

(iv)for eachc∈C,F(c)is open inF(C);

(v)for eachy∈F(C),F⁻¹(y)= {c∈C:y∈F(c)}is nonempty and convex.

Proof. We useTheorem 2.1forK=F(C).

Theorem2.3. LetCbe a nonempty subset of a Hausdorﬀtopological vector space EoverRand letF:C→2^E. Assume that the following conditions hold:

(i)Fis expansive, that is,C⊂F(C);

(ii)Cis convex;

(iii)F(C)is a compact subset ofE;

(iv)for eachc∈C,F(c)is open inF(C);

(v)for eachy∈C,F⁻¹(y)= {c∈C:y∈F(c)}is nonempty and convex.

Proof. Indeed, ifϕandσ are as in the proof ofTheorem 2.1andCis convex, thenσ⊂C⊂F(C), and we may useTheorem 2.1forK=C.

Example 2.4. (a) LetE=R², letT be a closed triangle with vertices (0,0), (1,0) and (0,1), and let C=C1∪ {(0,0)} ∪C2 whereC1= {c=(c1,0) : 0< c1≤1} andC2= {c=(0,c2) : 0< c2≤1}. Define setsP= {c=(c1,c2) :|c2−c1|<1/2}, H1= {c=(c1,c2) :c2< c1}andH2= {c=(c1,c2) :c2> c1}. IfF(C)=T where

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F(0,0)=T∩P,F(c)=T∩H1 forc∈C1, andF(c)=T∩H2 forc∈C2, then the assumptions ofTheorem 2.2are satisfied,Cis nonconvex and Fix(F)=C.

(b) LetE=R,C=(1; 3),F(C)=[0; 4] whereF(c)=[0; 2) forc∈(1; 2),F(c)= (2; 4] forc∈(2,3) andF(2)=(1; 3). Then the assumptions ofTheorem 2.2are satisfied,Cis noncompact and Fix(F)=C.

If in Theorems2.2or2.3we omit at least one of the assumptions, then we can construct a counterexample.

Example 2.5. (a) The condition onF(C) in (i) cannot be omitted as the following two counterexamples show:

(A1) ifE=R,C=(1; 4),F(C)=[2; 5] where F(c)=[2; 5] for c∈(1; 2) and F(c)=(4; 5] forc∈[2; 4), then assumptions (ii), (iii), (iv), and (v) are satisfied,C⊂F(C),F(C)⊂C, and Fix(F)= ∅;

(A2) if E=R², C= {c=(c1,c2) :|c1| ≤1, |c2| ≤2}, C=C1∪C2∪C3∪ (−C3)∪C4 whereC1= {(0,−2)}, C2= {(0,2)}, C3= {c=(c1,c2) :−1

≤c1<0, |c2| ≤2}, andC4= {c=(c1,c2) :c1=0,|c2|<2},F(C1)=D1, F(C2)=D2, F(C3)=D3, F(−C3)= −D3, F(C4)=D3∪(−D3), and if F(C)= {y=(y1, y2) :|y1| ≤1,|y2| ≤1}whereD1= {y=(y1, y2) :|y1|<

1/2, |y2| ≤1},D2= {y=(y1, y2) : 1/2<|y1| ≤1, |y2| ≤1}, and D3= {y=(y1, y2) : 0< y1≤1,|y2| ≤1}, then assumptions (ii), (iii), (iv), and (v) are satisfied,F(C)⊂C,F(C)=C, and Fix(F)= ∅.

(b) The assumption thatF(C) is convex orCis convex is necessary. Indeed, let E=R²,C=C0∪(−C0) whereC0= {c=(c1,0) :c1∈(1; 2)},F(c)=D0 forc∈

−C0, andF(c)= −D0forc∈C0whereD0= {y=(y1, y2) :(y1, y2)−(3/2,0)≤ 1}. Then assumptions (i), (iii), (iv), and (v) are satisfied, while (ii) and (ii) are not, and Fix(F)= ∅.

(c) The assumption that F(C) is compact is necessary. Indeed, let E=R, C=(0; 1),F(c)=(0;c) for 1/2≤c <1,F(c)=(1/2 +c; 1) for 0< c <1/2. Then assumptions (i), (ii), (iv), and (v) are satisfied and Fix(F)= ∅.

(d) Assumption (iv) is necessary. Indeed, letE=R,C=(1; 4),F(C)=[1; 4]

whereF(c)=[1; 2)∪(3; 4] forc∈(2; 3),F(c)=(2; 5/2) for c∈(1; 2), F(c)= [5/2; 3) forc∈(3; 4),F(2)= {3}andF(3)= {2}. Thus assumptions (i), (ii), (iii), and (v) are satisfied and Fix(F)= ∅.

(e) Assumption (v) (or (v)) is necessary. Indeed, letE=R,C=(1; 5),F(C)= [1; 5] whereF(c)=(2; 3)∪(4; 5] forc∈(1; 2),F(c)=[1; 2)∪(3; 4) forc∈(2; 3), F(c)=[1; 3)∪(4; 5] for c∈(3; 4), F(c)=[1; 2)∪(3; 4] for c∈(4; 5), F(2)= (2; 5],F(3)=[1; 3)∪(3; 5], andF(4)=[1; 4). ThenF⁻¹(4)= {2,3}is nonconvex. Thus assumptions (i), (ii), (iii), and (iv) are satisfied, while (v) is not, and Fix(F)= ∅.

We say that a single-valued map f :C→Eand a set-valued mapF:C→2^E have a coincidence if f(c)∈F(c) for somec∈C.

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The following theorem is a generalization of the above one.

Theorem2.6. LetCbe a nonempty convex subset of a Hausdorﬀtopological vec- tor spaceEoverR, let F:C→2^E be an expansive map, and let f :C→Ebe a single-valued continuous map such that f(C)⊂F(C). Assume that the following conditions hold:

(i)F(C)is a compact subset ofE;

(ii)for eachc∈C,F(c)is open inF(C);

(iii)for eachy∈f(C),F⁻¹(y)= {c∈C:y∈F(c)}is nonempty and convex.

Then there existsu∈Csuch thatf(u)∈F(u).

Proof. Letϕandσ be as in the proof of Theorem 2.1. We haveϕ:F(C)→σ, σ⊂C, and y∈F(ϕ(y)) for each y∈ f(C). On the other hand, ϕ◦f :σ→σ and, by the theorem of Brouwer, (ϕ◦f)(u)=ufor someu∈σ. Consequently,

f(u)∈F(ϕ(f(u)))=F(u).

3. Fixed points and coincidences of set-valued inner maps on not necessarily convex sets in topological

vector spaces

We say that a mapF:C→2^E is inner ifF(C)⊂C. This section is devoted to new fixed-point and coincidence theorems for set-valued inner maps on not necessarily convex sets.

We have the following theorem.

Theorem 3.1. LetC be a nonempty compact subset of a Hausdorﬀtopological vector spaceE overRand let F:C→2^E be an inner map such thatF(C)is a convex subset of E. Assume that the following conditions hold:

(i)for eachc∈C,F(c)is nonempty and convex;

(ii)for eachy∈F(C),F⁻¹(y)= {c∈C:y∈F(c)}is open inC.

Proof. In virtue of (i) and (ii), there exists a finite set{y1,..., yn} ⊂F(C) such thatF⁻¹(yi) are nonempty, 1≤i≤n, andC=_n

i=1F⁻¹(yi). Let{ϕ1,...,ϕn}be a partition of unity with respect to this cover, letσ be a simplex spanned by pointsy1,..., yn, and let a continuous mapϕ:C→σ be defined by the formula ϕ(c)=_n

i=1ϕi(c)yi,c∈C. Note thatσ⊂F(C)⊂Cand, consequently,ϕ(σ)⊂σ.

Ifc∈Candϕi(c)=0 for somei∈ {1,...,n}, thenc∈F⁻¹(yi), thusyi∈F(c).

By (i), for eachc∈C,

ϕ(c)∈F(c). (3.1)

On the other hand, from Brouwer’s theorem, we get thatu=ϕ(u) for someu∈σ and, by (3.1), we haveu=ϕ(u)∈F(u), as required.

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Recall that a mapF:C→2^Eis called upper semicontinuous if, for eachc∈C and any open setVcontainingF(c), there is an open setUcontainingcsuch that F(U∩C)⊂V (for details, see [4]). A mapF:C→2^Eis called upper demicontinuous onC(after Fan [20]) if, for eachc∈Cand any open half-spaceH inE containingF(c), there is a neighbourhoodN(c) ofcinCsuch thatF(x)⊂Hfor eachx∈N(c). It is clear that the condition of upper semicontinuity is stronger than that of upper demicontinuity.

Let Cand D be nonempty sets. The mapsF:C→2^D andG:D→2^C are said to have a coincidence if there exists (u,v)∈C×Dsuch thatv∈F(u) and u∈G(v).

We now establish the following theorem.

Theorem3.2. LetCbe a nonempty compact subset of a Hausdorfflocally convex topological vector spaceEoverR, letF:C→2Êbe an inner map, and letG:C→2Ê be an upper demicontinuous map such thatG(C)⊂F(C). Assume that the follow- ing conditions hold:

(i)F(C)is a closed convex subset of E;

(ii)for eachc∈C,F(c)is nonempty and convex;

(iii)for eachy∈F(C),F⁻¹(y)= {c∈C:y∈F(c)}is open inC;

(iv)for eachc∈C,G(c)is a nonempty closed convex subset of F(C).

Then there exists(u,v)∈C×F(C)such thatu∈F(v)andv∈G(u).

Proof. Letϕandσ be as in the proof ofTheorem 3.1. Sinceσ⊂F(C)⊂Cand ϕ:C→σ, thenG◦ϕ:F(C)→2^F(C)andG◦ϕis upper demicontinuous on the compact convex setF(C). By [20, Theorem 6], there existsv∈F(C) such that v∈G(ϕ(v)). Moreover, in virtue of (3.1),ϕ(v)∈F(v). This implies the assertion

foru=ϕ(v).

4. Intersection theorem with applications on not necessarily convex or compact sets in topological vector spaces

Various intersection theorems concerning convex and compact sets, with their applications, are given in [6,17,18,21,22,33,35]. FromTheorem 2.2, we get the following new intersection theorem.

Theorem4.1. LetEbe a Hausdorﬀtopological vector space overRand letn≥2.

Let C1,...,Cn be nonempty (not necessarily convex or compact) subsets ofE, let K1,...,Knbe compact and convex subsets ofE, letS1,...,Snbe nonempty subsets of Eⁿ, and letC=_n

j=1Cj,K=_n

j=1Kj, andS=_n

j=1Sj. Assume that the following properties hold:

(i)C⊂K=S;

(ii)for eachi,1≤i≤n, and for each point(y1,..., yi−1, yi+1,..., yn)ofⁿ_j₌_iKj, the sectionSi(y1,..., yi−1, yi+1,..., yn), formed by all pointsci∈Cisuch that (y1,..., yi−1,ci, yi+1,..., yn)∈Si, is a nonempty convex subset ofCi;

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(iii)for eachi,0≤i≤n, and for each pointci∈Ci, the sectionSi(ci), formed by all points(y1,..., yi−1, yi+1,..., yn)ofⁿ_j₌_iKjsuch that

y1,..., yi−1,ci, yi+1,..., yn

∈Si, (4.1)

is an open subset of ⁿ_j₌_iKj. ThenC∩_n

i=1Si= ∅.

Proof. DefineF:C→2^K as follows. Fix a pointc∈Cand lety∈K. We say that y∈F(c) if and only if, for eachi∈ {1,...,n}, (y1,..., yi−1,ci, yi+1,..., yn)∈Si.

Writec∈Cin the formc=(c1,...,ci−1,ci,ci+1,...,cn), 1≤i≤n. Using condi- tion (iii) and taking into consideration that, for eachi∈ {1,...,n}, the section Si(ci) is an open subset ofⁿ_j₌_iKj, we obtain thatSi(ci)×Kiis an open subset of K. Therefore, the setF(c)=n

i=1(Si(ci)×Ki) is open inK.

Suppose thaty∈K. Writec=(c1,...,ci₋1,ci,ci+1,...,cn) and note that sincec belongs toF⁻¹(y) if and only if, for eachi∈ {1,...,n},

ci∈Siy1,..., yi−1, yi+1,..., yn, (4.2) we haveF⁻¹(y)=_n

i=1Si(y1,..., yi−1, yi+1,..., yn). But, for eachi∈ {1,...,n}, the sectionsSi(y1,..., yi−1, yi+1,..., yn) are nonempty convex subsets ofCiby condition (ii), and thus,F⁻¹(y) is a nonempty convex subset ofC. We conclude that F(C)=K.

It follows fromTheorem 2.2thatu∈F(u) for someu∈C. This shows that, for each i∈ {1,...,n}, u=(u1,...,ui−1,ui,ui+1,...,un)∈Si, that is, u∈C∩ _n

i=1Si.

Example 4.2. LetE=R²,n=2,K=K1×K2whereK1=K2=[0; 1],C1=C1,0∪ C1,1whereC1,0=[0; 1/3) andC1,1=(2/3; 1],C2=[0; 1],C=C1×C2,S=S1∪S2

whereS1=K∩ {(x1,x2) :x2>−3x1+2}, andS2=K∩ {(x1,x2) :x2<−x1+ 3/2}. Hence C1 is a noncompact and nonconvex subset ofK1, the assumptions of Theorem 4.1are satisfied andC∩2

i₌1Si= ∅.

As an application ofTheorem 4.1we obtain the following theorem.

Theorem4.3. LetEbe a Hausdorﬀtopological vector space overRand letn≥2.

Let C1,...,Cn be nonempty (not necessarily convex or compact) subsets ofE, let K1,...,Knbe convex compact subsets ofE, and letC=_n

j=1Cj,K=_n

j=1Kj. Let f1,..., fnbe real-valued maps defined onK, lett1,...,tn be real numbers, and let the following conditions hold:

(i)C⊂K;

(ii)for eachi,1≤i≤n, and for each point(y1,..., yi−1, yi+1,..., yn)ofⁿ_j₌_iKj, the set{ci∈Ci: fi(y1,..., yi−1,ci, yi+1,..., yn)> ti}is a nonempty convex subset ofCi;

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(iii)for eachi,1≤i≤n, and for each pointci∈Ci, the set

y1,..., yi−1, yi+1,..., yn

∈

j=i

Kj:fi

y1,..., yi−1,ci, yi+1,..., yn

> ti (4.3) is an open subset ofⁿ_j₌_iKj.

Then there is a pointuinCsuch that fi(u)> tifor eachi,1≤i≤n.

Proof. Define the subsetsSiofKto beSi= {y:y∈K, fi(y)> ti},i∈ {1,...,n}. Clearly, (ii) is equivalent to the condition:

(ii) for each i∈ {1,...,n} and for each point (y1,..., yi−1, yi+1,..., yn) of

nj=iKj, the sectionSi(y1,..., yi−1, yi+1,..., yn), formed by all pointsci∈ Cisuch that (y1,..., yi−1,ci, yi+1,..., yn)∈Si, is a nonempty convex subset ofKi,

and (iii) is equivalent to the condition:

(iii) for eachi∈{1,...,n}and for each pointci∈Ci, the sectionSi(ci), formed by all points (y1,..., yi−1, yi+1,..., yn) ofⁿ_j₌_iKjsuch that

y1,..., yi−1,ci, yi+1,..., yn

∈Si, (4.4)

is an open subset ofⁿ_j₌_iKj.

We can applyTheorem 4.1 to obtainC∩n

i=1Si= ∅. Hence, by the defi- nition of Si, the point ufrom this intersection satisfies fi(u)> ti for eachi∈ {1,...,n}.

5. Coincidence theorems for set-valued maps and section theorems on not necessarily convex sets in topological vector spaces

Using his infinite-dimensional version of the KKM theorem as a tool, Fan [16]

established a geometrical “lemma” concerning convex and compact sets. Next, Browder [6] restated it in the more convenient form of a fixed-point theorem.

A weaker form (with a relaxed compactness assumption) of this theorem was afterwards obtained by Fan [21]. Finally, Lassonde [33] extended these results.

He gave a proof of the following interesting coincidence theorem:

Theorem5.1. LetX be a convex space (i.e., a convex set in a vector space with any topology that induces the Euclidean topology on the convex hulls of its finite subsets),Y a topological space, andFthe map of Xinto2^Yfor which the following conditions hold:

(i)for eachx∈X,F(x)is compactly open inY;

(ii)for eachy∈Y,F⁻¹(y)= {x∈X:y∈F(x)}is nonempty and convex;

(iii)for somec-compact setK⊂X, the setY\

x∈KF(x)is compact. Then, for each single-valued continuous map f ofX intoY, there exists anx∈X such thatf(x)∈F(x).

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Our new coincidence theorem does not require convexity.

Theorem5.2. LetCbe a nonempty compact set in a Hausdorﬀtopological vector spaceEoverRand letf :C→Ebe a continuous single-valued map onCsuch that f(C)is a convex set. LetF:C→2^f^(C)be a map such that f(C)=F(C). Suppose that

(i)for eachy∈f(C), the setF⁻¹(y)= {c∈C:y∈F(c)}is open inC;

(ii)for eachc∈C, the set{y∈f(C) :y∈F(c)}is nonempty and convex.

Then there exists a pointu∈Csuch that f(u)∈F(u).

Remark 5.3. If f =IE (the identity map) andC is convex, thenTheorem 5.2 becomes the Browder theorem [6, Theorem 1]. However, his method of proving this fact (based on the partition of unity) is absolutely diﬀerent from ours.

Section theorems concerning convex compact sets in Hausdorﬀtopological vector spaces, with various applications, are given by Fan [18,20]. In the proof ofTheorem 5.2, we need the following two new auxiliary section theorems of Fan type.

Theorem 5.4. Let C be a nonempty compact set (not necessarily convex) in a Hausdorﬀtopological vector spaceEoverK. Let f :C→Eandg:C→Ebe contin- uous maps onC, and let f(C)be convex. LetKbe a subset ofg(C)×f(C)having the following properties:

(i)for each fixedw∈ f(C), the set{t∈C: (g(t),w)∈K}is closed inC;

(ii)for eacht∈C,(g(t), f(t))∈K;

(iii)for any fixedt∈C, the set{w∈ f(C) : (g(t),w)∈/ K}is convex (or empty).

Then there exists a pointc∈Csuch that{g(c)} × f(C)⊂K.

Proof. We use KKM set-valued maps. Define a mapH:f(C)→2^Eas follows:

H(w)=t∈C:g(t),w∈K, w∈f(C). (5.1) Obviously, by (i),H(w) is a compact subset ofCand thus f(H(w)) is a compact subset of f(C) for eachw∈ f(C). Let{w1,...,wm}be any finite and fixed subset off(C). We prove that conv{w1,...,wm} ⊂ f(H(w1))∪ ··· ∪f(H(wm)).

To this goal, we assume that f(s)∈conv{w1,...,wm}but f(s)∈/ f(H(w1))∪

··· ∪f(H(wm)) for somes∈C. Then s /∈H(wi) for all i=1,...,m, that is, (g(s),wi)∈/ K for anyi=1,...,m. Therefore, by (iii), wi,i=1,...,m, are contained in a convex set U = {w ∈ f(C) : (g(s),w)∈/ K}. Consequently, conv{w1,...,wm} ⊂U and, in particular, f(s)∈U, that is, (g(s), f(s))∈/ K, which, by (ii), is impossible. We must have f(s)∈f(H(w1))∪ ··· ∪f(H(wm)).

By virtue of [16, Lemma 1, page 305], this yieldsf(c)∈

{f(H(w)) :w∈f(C)} for somec∈Cand we conclude that{g(c)} ×f(C)⊂Kfor somec∈C.

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Theorem 5.5. Let C be a nonempty compact set (not necessarily convex) in a Hausdorﬀtopological vector spaceEoverK. Let f :C→Eandg:C→Ebe con- tinuous maps onC, and let f(C)be convex. LetBbe a subset ofg(C)×f(C)and suppose that

(i)for each fixedy∈f(C), the set{c∈C: (g(c), y)∈B}is open inC;

(ii)for any fixed c∈C, the set{y∈ f(C) : (g(c), y)∈B}is nonempty and convex.

Then there exists a pointu∈Csuch that(g(u), f(u))∈B.

Proof. Here,Bdenotes a complement of the setK ing(C)×f(C) whereK is

defined inTheorem 5.4

Proof ofTheorem 5.2. We define a setB= {(c, y)∈C×f(C) :y∈F(c)}and ap-

plyTheorem 5.5forg=IE.

6. Coincidences for upper semicontinuous set-valued maps on not necessarily convex sets in locally convex spaces

LetF:C→2^E andG:C→2^E and letΦ:G(c)×F(c)→Efor eachc∈C. We say that mapsF and Ghave aΦ-coincidence if there existc∈Cand (u,v)∈ G(c)×F(c) such thatΦ(u,v)=0; this pointcis called aΦ-coincidence point for FandG. In particular, aΦ-coincidence point is a coincidence point ifΦis of the formΦ(u,v)=u−vfor (u,v)∈G(c)×F(c), andc∈C.

We use these notations in the following theorem.

Theorem6.1. LetCbe a nonempty compact (not necessarily convex) set in a lo- cally convex Hausdorff topological vector spaceEover K. LetΓ be the set of all continuous seminormsponE. LetF:C→2ÊandG:C→2Êbe upper semicontin- uous maps such thatF(c)andG(c)are compact subsets ofEfor eachc∈Cand let, for eachc∈C, the mapΦ:G(c)×F(c)→Ebe continuous onG(c)×F(c).

(a)Then eitherF andGhave aΦ-coincidence or there exist p∈Γandλ >0 such thatp(Φ(u,v))> λfor allc∈Cand all(u,v)∈G(c)×F(c).

(b)Then eitherFandGhave aΦ-coincidence or there existsp∈Γand, for any c∈Cand anyu∈G(c), there existsv∈F(c)such that

0< pΦ(u,v)=MinpΦ(u,w):w∈F(c). (6.1) Proof. (a) IfFandGdo not have aΦ-coincidence inC, then, for allc∈C, the setΦ(G(c)×F(c)) is compact and 0∈/ Φ(G(c)×F(c)).

First, observe that

(i) for eachc∈C, there existpc∈Γandλc>0 such that pc

Φ(u,v)>2λc ∀(u,v)∈G(c)×F(c). (6.2)

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Indeed, for an arbitrary and fixedw∈Φ(G(c)×F(c)), there exists pw∈Γ such thatpw(w)=0 and, by the continuity of pw, there exist a neighbourhood Mw ofwandµw>0 such thatµw=Inf{pw(t) :t∈Mw}. Since the family{Mw: w∈Φ(G(c)×F(c))}is an open cover of a compact set ofΦ(G(c)×F(c)), there exists a finite subset{w1,...,wm}ofΦ(G(c)×F(c)) such that the family{Mwi: i=1,2,...,m}coversΦ(G(c)×F(c)) and we may assume that

pc=Maxpwi:i=1,...,m, λc=1 4

Minµwi:i=1,...,m. (6.3) Now we prove that

(ii) for eachc∈C, there existpc∈Γ,λc>0, and a neighbourhoodWcofc, such that

pc

Φ(u,v)> λc ∀x∈Wc∩C,(u,v)∈G(x)×F(x). (6.4) Indeed, letc∈Cbe arbitrary and fixed and we define open setsAcandBcas follows:

Ac×Bc=(u,v) :pc

Φ(u,v)> λc, (6.5) where pc andλc are as in (i). SinceF(c)⊂Ac,G(c)⊂Bc,F, andG are upper semicontinuous, there exist neighbourhoodsUcandVcofc, such thatF(x)⊂Ac

forx∈Uc∩C, andG(y)⊂Bc for y∈Vc∩C. Consequently, we may assume thatWc=Uc∩Vc.

Finally, for eachc∈C, letpc,λc, andWcbe as in (ii). Since the family{Wc: c∈C}is an open cover of a compact set ofC, there exists a finite subset{c1,..., cn}ofCsuch that the family{Wci:i=1,...,n}coversCand we may assume that p=Maxpci:i=1,...,n, λ=Minλci:i=1,...,n. (6.6) (b) IfFandGdo not have aΦ-coincidence inC, letpandλbe as in (a) and let c∈Cbe arbitrary and fixed. Observe that, for anyu∈G(c), the continuous map p(Φ(u,·)) attains its minimum on a compact setF(c). Letk:G(c)×F(c)→R be a map defined by the formulak(u,v)=p(Φ(u,v))−Min{p(Φ(u,w)) :w∈ F(c)}. Obviously,k(u,v)>0 for each (u,v)∈G(c)×F(c) and, for eachu∈G(c),

there existsv∈F(c) such thatk(u,v)=0.

Example 6.2. LetE=C,U= {c∈E:|c| =1, |Arg(c)| ≤π/4},U1= −U+ 2^1/2, V1= {w:w=tc, 0≤t≤1, c∈U1}. Let C=U1∪(−U1) and letF:C→2^E, G:C→2^E be defined byF(c)= −V1 forc∈U1,F(c)=V1 forc∈ −U1, and G= −F. ThenFandGsatisfy the assumptions ofTheorem 6.1(b) forΦdefined byΦ(u,v)=u−v, (u,v)∈G(c)×F(c),c∈C. The setsC,F(C),G(C),F(c), and G(c) are nonconvex for allc∈C. Moreover,C⊂F(C),C⊂G(C), the setsF(C) andG(C) are not contained inCand anyc∈Cis a coincidence ofFandG.

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7. Coincidences and fixed points for continuous single-valued maps on not necessarily convex sets in locally convex spaces

Two maps f :C→E and g:C→Ehave a Φ-coincidence, where Φ:g(C)× f(C)→E, ifΦ(g(c), f(c))=0 for somec∈C; this pointcis called aΦ-coincidence point for f andg. In particular, aΦ-coincidence point is a coincidence point ifΦis of the formΦ(u,v)=u−vfor (u,v)∈g(C)×f(C). We say that c∈Cis aΦ-fixed point for f :C→EwhereΦ:C×f(C)→E, ifΦ(c, f(c))=0.

In particular, aΦ-fixed point is a fixed point ifΦis of the formΦ(c,v)=c−v for (c,v)∈C×f(C).

For maps defined on convex sets, there are many variations, generalizations, and applications (see, e.g., [1,10,13,24,25,31,32,34,36,38,41,42,43,46]) of the well-known Fan minimax inequality [20], Hartman-Stampacchia variational inequality [26], and Iohvidov theorem [29].

In this section, we will give further applications ofTheorem 5.4. In particular, we derive some minimax theorem (Theorem 7.1), Hartman-Stampacchia type variational inequalities (Theorem 7.2), and a theorem of Iohvidov type (Theorem 7.3(b)) for maps on not necessarily convex sets. One of them will be used later to prove new results concerningΦ-coincidences andΦ-fixed points (in particular, coincidences and fixed points) of continuous single-valued maps on not necessarily convex sets (Theorem 7.4).

A real map ψ, defined on a topological vector spaceE, is said to be lower semicontinuous (upper semicontinuous) onEif, for each real numberµ, the set {x∈E:ψ(x)> µ}({x∈E:ψ(x)< µ}) is open.

A real map,ψdefined on a convex setAof a vector spaceE, is said to be quasi- concave (quasi-convex) onAif, for each real numberµ, the set{a∈A:ψ(a)>

µ}({a∈A:ψ(a)< µ}) is convex.

As a consequence ofTheorem 5.4, we obtain the following theorem.

Theorem 7.1. Let C be a nonempty compact set (not necessarily convex) in a Hausdorﬀtopological vector spaceEoverK. Let f :C→Eandg:C→Ebe con- tinuous maps onCand let f(C)be convex.

(a)LetΨ:g(C)×f(C)→Rbe a map such that(i)for eachv∈ f(C),Ψ(·,v) is a lower semicontinuous map ong(C);(ii)for eachu∈g(C),Ψ(u,·)is a quasi- concave map on f(C). Then there existsc∈Csuch that

SupΨg(c), f(t):t∈C≤SupΨg(s), f(s):s∈C. (7.1) Let, additionally,(iii)Ψ(g(s), f(s))≤0for alls∈C. Then there existsc∈Csuch that

Ψg(c), f(t)≤0 ∀t∈C. (7.2) (b)Let Ω:g(C)×f(C)→Rbe a map such that(iv)for eachv∈ f(C), the mapMin{Ω(·,w) :w∈f(C)} −Ω(·,v)is lower semicontinuous ong(C);(v)for