Existence theorems in nonlinear analysis and applications to optimization related models

31 2 Nonlinear existence theorems for mappings on product GFC-spaces and applications.. The existence theorems, including theorems about various points like fixedpoints, coincidence poin

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VIETNAM NATIONAL UNIVERSITY - HCMC

UNIVERSITY OF SCIENCE

Nguyen Hong Quan

EXISTENCE THEOREMS IN NONLINEAR ANALYSIS AND APPLICATIONS TO OPTIMIZATION-RELATED MODELS

Major: Mathematical Optimization

Codes: 62 46 20 01

Referee 1: Assoc.Prof Dr Nguyen Dinh

Referee 2: Dr Huynh Quang Vu

Referee 3: Assoc.Prof Dr Lam Quoc Anh

Independent Referee 1: Prof D.Sc Nguyen Dong Yen

Independent Referee 2: Assoc.Prof Dr Mai Duc Thanh

SCIENTIFIC SUPERVISOR

Professor Phan Quoc Khanh

Ho Chi Minh City - 2013

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Preface IV

1 Existence theorems in nonlinear analysis and applications 1

1.1 Notions and definitions 1

1.2 Existence theorems in GFC-spaces 6

1.3 Applications 23

1.3.1 Variational inclusions 23

1.3.2 Minimax theorems 31

2 Nonlinear existence theorems for mappings on product GFC-spaces and applications 37

2.1 Existence theorems on product GFC-spaces 37

2.2 Applications 42

3 Topological characterizations of existence in nonlinear analysis and optimization-related problems 52

3.1 Topological existence theorems 52

3.2 The existence of solutions to optimization-related problems 70

3.2.1 Variational relation problems 70

3.2.2 Invariant-point theorems 74

3.2.3 Equilibrium problems of the Stampacchia and Minty types 77 3.2.4 Minimax theorems 80

3.2.5 Nash equilibria 88

4 Generic stability and essential components of generalized KKM points and applications 92

4.2 Generic essentialness of T -KKM mappings 94

4.3 Essential components of sets of T -KKM points 98

4.4 Applications to maximal elements and variational inclusions 99

4.4.1 Essential components of T0-maximal elements 100 4.4.2 Essential components of solutions to variational inclusions 100

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Contents III

4.4.3 Essential components of particular cases of variational

inclusions 101

Conclusion 104

List of the author’s papers related to the thesis 105

List of the author’s conference reports related to the thesis 106

References 107

Glossary of Symbols 111

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The existence theorems, including theorems about various points like fixedpoints, coincidence points, intersection points, maximal elements, and other re-sults as KKM theorems, minimax theorems, etc., constitute one of the most im-portant parts of mathematics They are crucial tools in the solution existence study

of wide-ranging problems of optimization and applied mathematics

The existence theorems have a long history of development passing more than

a century with the following major milestones: the Brouwer fixed point theorem(1912, [14]), Classical KKM principle (1929, [57]), Kakutani fixed point theorem(1941, [50]), KKM-Fan principle (1961, [33]) From 80s of the 20th century, tomeet demands of practical situations, many classes of problems in optimizationhave appeared One of the first and most important issues of such a class is toknow if solutions exist or not This requires more new effective mathematicaltools Hence, existence theorems have been intensively developed to responsethat requirement Especially, in recent years the theory of existence theorems hasobtained many significant achievements

Like other mathematical theories, the existence theorems have been built fromsimple basic results by generalization and abstraction methods In early forms ofthese fundamental results, the convexity played a central role in formulating re-sults Therefore, most of the later results mainly focus on improving assumptions

on convexity or replacing them by purely topological assumptions According

to our observations, for the last three decades the existence theorems have beendeveloped in three ways First, some researchers renovated classical notions ofconvexity based on linear structures For instance, the KKM-Fan mapping (Fan[33]) was extended into the S-KKM mapping (e.g., Chang and Zhang [15], Changand Yen [16], Chang, Huang, Jeng and Kuo [17]) In terms of these notions,new existence results were achieved Second, many authors replaced the classi-cal convexity by abstract convexity notions, not using linear structures, and theyextended earlier existing notions and results to these structures In the frame-work of this research direction, types of spaces with generalized convexity struc-tures were proposed and studied Started with Lassonde ([60]) where a convexspace was created, the following spaces (in the chronological order) were in-

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Preface V

troduced: S-contractible spaces (Horvath [45]), H-spaces (Horvath [46-47]), convex spaces (Park and Kim [79-85]), and FC-spaces (Ding [24-32]) Thesespaces have been used in studying existence theorems and nonlinear problems Inthe third approach, a number of authors (e.g., Wu [93], Tuy [91-92], Geraghty andLin [38], Kindler and Trost [54], Kindler [55-56], Konig [59], Tarafdar and Yuan[90]) proved existence results which did not require any convexity structures,where convexity conditions were replaced by connectedness conditions whichare purely topological conditions

G-One of the main purposes of this thesis is to develop further the theory ofexistence theorems, focusing on the last two approaches Based on analyzing theearlier notions and results, we introduce several new structures and use them toformulate new definitions and establish new or more general results By providingillustrative examples, we show the existence of our structures and their usefulness

in many applications Moreover, our notions and results in this thesis improve orinclude as special cases a number of known notions and results

It is worthwhile noticing that the equivalence of mathematical theorems ismeaningful in applications because it allows us to approach to a problem fromdifferent angles Therefore, researchers pay much attention on proving equiva-lence relations between existence results For examples, in Ha [40], an extension

of Kakutani fixed-point theorem was proved to be equivalent to a section rem and a minimax result, an equivalence between the KKM-Fan theorem and

theo-a Browder-type fixed-point theorem wtheo-as shown in Ttheo-artheo-afdtheo-ar [89] Ltheo-ater on mtheo-anyresearchers discovered similar equivalence relations for other kinds of results likecoincidence theorems, matching theorems, intersection theorems, maximal-pointtheorems, section theorems and some geometric results One of our attempts inthis thesis is to show the equivalence between many of our existence theorems

On the other hand, any mathematical result needs be applicable for certain uations Typical applications of existence theorems are in optimization problems.Therefore, using our existence theorems to establish solution existence resultsfor optimization-related models is also a purpose of this thesis The harvestedresults include many new solution existence theorems for various problems asminimax problems, equilibrium problems, generalized inclusions, variational re-lation problems, or practical problems as traffic networks, Nash equilibria, ab-stract economies, etc

sit-One of the important topics in nonlinear optimization, which have been tracting many mathematicians recently, is properties of solution sets and solu-tion maps The properties of solution sets such as closedness, connectedness,convexity, etc, were studied in many papers (e.g., Fort [37], Jones and Gowda[49], Khanh and Luc [53], Papageorgiou and Shahzad [78], Rapcsak [86], Zhong,Huang and Wong [105]) The properties of solution maps, like semicontinuities,continuity, differentiability, etc., which are commonly called the stability prop-erties, have also been intensively investigated during recent years (e.g., Anh andKhanh [1-7], Khanh and Luc [53], Xiang, Liu and Zhou [94], Yang and Yu [96],

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at-Preface VI

Yu and Xiang [98], Yu, Yang and Xiang [99], Zhou, Xiang and Yang [101]).Based on relationships between sets of particular points of set-valued maps andsolution sets of optimization problems, we propose solution map notions for thesepoints, consider their stability and apply the obtained results to optimization prob-lems Stability issues considered in this thesis are included in the generic stabilitystudy

The thesis consists of four chapters and contains the results of 10 papers (fromthe list of 14 related papers of the author):

Chapter 1: “Existence theorems in nonlinear analysis and applications” isbased on the papers (Q2), (Q3), (Q4), (Q6), (Q7);

Chapter 2: “Nonlinear existence theorems for mappings on product spaces and applications” is based on the paper (Q5);

GFC-Chapter 3: “Topological characterizations of existence in analysis and mization related problems” is based on the papers (Q10), (Q11), (Q12);

opti-Chapter 4: “Generic stability and essential components of generalized KKMpoints and applications” is based on the paper (Q8)

Acknowledgments

I express my deep gratitude to Professor Phan Quoc Khanh, my supervisor, for

a continuous guidance, encouragement and valuable suggestions I would like tothank very much the University of Science of Hochiminh City for providing meall conditions and facilities for my work I am also indebted to the Vietnam Insti-tute for Advanced Study in Mathematics (VIASM) and its members During mystay there as a visiting young researcher, they facilitated me with both a financialsupport and a perfect research environment for the completion of a part of thisthesis

Ho Chi Minh City, October 2013 Nguyen Hong Quan

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to a general variational inclusion problem, which contains most of the existingresults of this type, and develop in detail general types of minimax theorems.Examples are given to explain advantages of our results.

1.1 Notions and definitions

We recall notions used in the whole thesis Let Y be a nonempty set, hY i standsfor the set of all finite subsets of Y For n ∈ N, the set of all natural numbers, ∆n

stands for the n-simplex with the vertices being the unit vectors e0= (1, 0, , 0),

e1= (0, 1, , 0), , en= (0, 0, , 1) of a basis of Rn+1 For N = {y0, y1 , yn} ∈

hY i and M = {yi0, yi1, , yik} ⊂ N, let ∆|N| ≡ ∆n, and ∆M ≡ ∆k be the face of

∆|N| corresponding to M, i.e., ∆M = co{ei0, ei1 , eik} If A, B ⊂ X, X being atopological space, then A (or clA), AB (or clBA), intA, intBA and Ac signify theclosure, closure in B, interior, interior in B and complement X \ A, respectively(shortly, resp), of A Let X , Y be nonempty sets and F : X ⇒ Y be a set-valuedmap For x ∈ X and y ∈ Y , the sets F(x), F−1(y) = {x ∈ X | y ∈ F(x)} and F∗(y) =

X\ F−1(y) are called an image, a fiber (or inverse image) and a cofiber, resp Themap F−1 (F∗) is called the inverse map (dual map, resp) of F The graph of F

is GphF := {(x, y) ∈ X × Y | y ∈ F(x)} Now let X and Y be topological spaces,

F : X ⇒ Y , and f : X → R F is called closed (open, resp) if its graph is closed(open, resp) F is said to be upper semicontinuous (usc, for short) (resp, lowersemicontinuous (lsc)) if for each open (resp, closed) subset U of Y , the set {x ∈

X | F(x) ⊂ U} is open (resp, closed) F is said to be continuous if it is both uscand lsc f is said to be usc (resp, lsc), if, for all α ∈ R, the set {x ∈ X | f (x) ≥ α}(resp, {x ∈ X | f (x) ≤ α}) is closed

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The following concepts are taken from [22, 25] A subset A of a topologicalspace X is called compactly open (compactly closed, resp) if, for each nonemptycompact subset K of X , A ∩ K is open (closed, resp) in K The compact interiorand compact closure of A are defined by, resp,

cintA =S

{B ⊂ X : B ⊂ A and B is compactly open in X},cclA =T

{B ⊂ X : B ⊃ A and B is compactly closed in X}

F : Y ⇒ X is called transfer open-valued (transfer closed-valued, resp) if ∀y ∈ Y ,

∀x ∈ F(y) (∀x /∈ F(y), resp), ∃y0∈ Y such that x ∈ int(F(y0) (x /∈ cl(F(y0), resp)

F is termed transfer compactly open-valued (transfer compactly closed-valued)

if ∀y ∈ Y , ∀K ⊂ X : nonempty and compact, ∀x ∈ F(y) ∩ K (∀x /∈ F(y) ∩ K),

∃y0∈ Y such that x ∈ cintF(y0) (x /∈ cclF(y0), resp) Of course intA ⊂ cintA (clA ⊃cclA), and transfer open-valuedness (transfer closed-valuedness) implies transfercompact open-valuedness (transfer compact closed-valuedness, resp) Moreover,

a set-valued mapping has open values (closed values) then it is transfer valued (transfer closed-valued) We will use these notions in order to compareour results directly with many known existing ones

open-Lemma 1.1.1 (e.g., [22]) Let Y be a set, X be a topological space and F : Y ⇒ X.The following statements are equivalent

(i) F is transfer compactly closed-valued (transfer compactly open-valued, resp);(ii) for each compact subset K ⊂ X ,

(K ∩ intKF(y)), resp

We propose the following definition of a GFC-space to unify a number of lier existing notions of spaces with generalized convexity structures, but withoutlinear structures This notion is proposed based on observing that although theabstract convexity structures associated with the earlier existing spaces such asconvex spaces ([60]), H-spaces ([46-47]), G-convex spaces ([79-85]), FC-spaces([24-32]) are different, all of them use the image of a simplex through a continu-ous map

ear-Definition 1.1.1 Let X be a topological space, Y be a nonempty set and Φ be afamily of continuous mappings ϕ : ∆n→ X, n ∈ N Then a triple (X,Y, Φ) is said

to be a generalized finitely continuous topological space (GFC-space in short) iffor each finite subset N ∈ hY i, there is ϕN : ∆|N| → X of the family Φ (we alsouse (X ,Y, {ϕN}) to denote (X,Y, Φ))

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The above mentioned existing spaces are the examples for GFC-space In ticular, a convex subset A of a topological vector space is a GFC-space, where

par-X = Y = A and each N = {a0, a1, , an} ∈ hAi, there is ϕN : ∆|N|→ A which isdefined by ϕN(e) = ∑ni=0λiaifor all e = ∑ni=0λiei∈ ∆|N| GFC-spaces are properlymore general than known existing spaces Therefore it is reasonable and valuable

to study existence theorems and nonlinear problems in GFC-spaces without linearstructure

The following example shows that GFC-spaces are properly more generalthan G-convex spaces Recall that, a G-convex space is [79-85] a triple (X ,Y,ϒ ),where X and Y are as Definition 1.1.1 and ϒ : hY i ⇒ X is such that, for each

N ∈ hY i, there exists a continuous map ϕN : ∆|N| → ϒ (N) such that, for each

M ∈ hNi, ϕN(∆M) ⊂ ϒ (M) A G-convex space (X ,Y,ϒ ) is called trivial iff, forall N ∈ hY i, ϒ (N) = X Of course, any above-mentioned space can be made into

a trivial G-convex space, but a trivial G-convex space has no use

Example 1.1.1 Let Y =]0, +∞[, X = {(x, x) | x ∈]0, +∞[} ⊂ R2 and, for each

N∈ hY i, ϕN(e) = (α(N), α(N)) ∈ X for all e ∈ ∆|N|, where α(N) = min N max N.Then, (X ,Y, {ϕN}) is a GFC-space Suppose there exists ϒ : hY i ⇒ X such that(X ,Y,ϒ ) is a nontrivial G-convex space, and (X ,Y, {ϕN}) can be made into(X ,Y,ϒ ) Then, since (X ,Y,ϒ ) is a G-convex space, we have, for all N ∈ hY iand y ∈ Y , ϕN∪{y}(∆|N|) = (α(N ∪ {y}), α(N ∪ {y})) ∈ ϒ (N) It follows that, forall N ∈ hY i,

Next, we define several concepts in a GFC-space

Definition 1.1.2 Let (X ,Y, Φ) be a GFC-space, D,C ⊂ Y and S : Y ⇒ X be given.(i) D is called an S-subset of Y (S-subset of Y wrt C) if for all N ∈ hY i and for all

M⊂ N ∩ D (for all M ⊂ N ∩C, resp), ϕN(∆M) ⊂ S(D)

(ii)If in addition to (i), S−1(ϕN(∆M)) ⊂ D, then D is called an SGFC-subset of

Y The GFC-hull wrt S of C is defined by GFCS(C) =T

{D ⊂ Y | D is SGFCsubset of Y containing C}

-Clearly, if D is an SGFC-subset of Y , D must be an S-subset of Y Roughlyspeaking, if D is an S-subset of Y then (S(D), D, Φ) is a ”pre” GFC-space, and if

Dis an SGFC-subset of Y then (S(D), D, Φ) is a ”full” GFC-space When X = Y ,i.e, (X ,Y, Φ) = (X , Φ) is an FC-space, and S = I is the identity map, then being

an SGFC-subset or an S-subset of X coincides with being an FC-subspace of X([30]) Moreover, the notion of a GFC-hull wrt S of a set extends the notions of

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an FC-hull of a set in a FC-space ([25]) and a convex hull of a set in a convex space ([64]) The extension of concepts of subspaces and convex hulls asabove also shows the rationality of the GFC-spaces The following Lemma 1.1.2extends Lemma 2.1 of [95] from FC-spaces to GFC-spaces

G-Lemma 1.1.2 Let (X ,Y, Φ) be a GFC-space, C a nonempty subset of Y and

Proof (a) and (b) are obvious

(c) For every N ∈ hCi, it is clear that GFCS(N) ⊂ GFCS(C) Hence,

i=1Ni), one has further M0 ⊂ GFCS(Sl

i=1Ni) Since each GFC - hullwrt S is SGFC-subset, by Definition 1.1.2, S−1(ϕN0(∆M0)) ⊂ GFCSSl

i=1Ni

⊂ B.Hence, again by this definition, B is an SGFC-subset of Y The notion of a GFC-space helps us also to extend the notion of generalizedKKM maps which plays an essential role in the theory of existence

Definition 1.1.3 Let (X ,Y, Φ) be a GFC-space, Z be a topological space, F :

Y ⇒ Z and T : X ⇒ Z be set-valued mappings F is said to be a generalized KKMmapping with respect to (shortly, w.r.t) T (T -KKM mapping in short) if, for each

N∈ hY i and each M ⊂ N, one has T (ϕN(∆M)) ⊂S

y∈MF(y)

The definition of T -KKM mappings was introduced for X being a convex set of a topological vector space in [16] and extended to FC-spaces in [24] Defini-tion 1.1.3 includes these definitions as particular cases It encompasses also manyother kinds of generalized KKM mappings We mention here some of them Let(X , {ϕN}) be an FC-space, Y be a nonempty set and s : Y → X be a mapping Wedefine a GFC-space (X ,Y, {ϕN}) by setting ϕN= ϕs(N)for each N ∈ hY i Then, ageneralized s-KKM mapping wrt T introduced in [25] becomes a T -KKM map-ping by Definition 1.1.3 A multivalued mapping F : Y ⇒ X, being an R-KKMmapping as defined in [20], is a special case of T -KKM mappings on GFC-spacewhen X = Z and T is the identity map The definition of generalized KKM map-pings wrt to T in [61] is as well a particular case of Definition 1.1.3

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sub-1.1 Notions and definitions 5

Definition 1.1.4 Let (X ,Y, Φ) be a GFC-space and Z be a topological space

A multivalued mapping T : X ⇒ Z is called better admissible if T is usc andcompact-valued such that for each N ∈ hY i and each continuous mapping ψ :

T(ϕN(∆|N|)) → ∆|N|, the composition ψ ◦ T |ϕN(∆

|N| )◦ ϕN: ∆|N|⇒ ∆|N|has a fixedpoint The class of all such better admissible mapping from X to Z is denoted byB(X,Y,Z)

The class of better admissible mappings generalizes classes of admissiblemappings proposed in [24, 82, 85]

Definition 1.1.5 Let (X ,Y, Φ) be a GFC-space We say that a set-valued ping T : X ⇒ Z has the generalized KKM property if, for each T-KKM mapping

map-F: Y ⇒ Z, the familyF(y) : y ∈ Y has the finite intersection property, i.e all nite intersections of sets of this family are nonempty By KKM(X,Y,Z) we denotethe class of all mappings T : X ⇒ Z which enjoy the generalized KKM property.Lemma 1.1.3 Let (X ,Y, Φ) be a GFC-space, Z be a topological space, S :

fi-Y ⇒ X, D be an S-subset of Y , and T ∈ KKM(X,Y, Z) Then, T |S(D) ∈ KKMS(D), D, T (S(D))

Proof Assume that R : D⇒ T (S(D)) is a T |S(D)-KKM mapping Then, for each

N∈ hDi ⊂ hY i and each M ⊂ N,

Clearly F is a T-KKM mapping Since T ∈ KKM(X ,Y, Z), the familyF(y) : y ∈

Y has the finite intersection property It follows that the family R(y) : y ∈ D has this property too Thus, the lemma is proved Lemma 1.1.4 Let (X ,Y, Φ) be an GFC-space and S : Y ⇒ X Assume that Ω :

Z ⇒ Y has nonempty values such that Ω−1 is transfer compactly open-valued.LetGFCS◦ Ω : Z ⇒ Y be defined by (GFCS◦ Ω )(z) = GFCS(Ω (z)) (GFCS◦ Ω

is called a GFC-hull mapping wrt S of Ω ) Then (GFCS◦ Ω )−1 is also transfercompactly open-valued

Proof Let K ⊂ Z be any nonempty compact subset, y ∈ Y and z ∈ (GFCS◦

Ω )−1(y) ∩ K Then y ∈ GFCS(Ω (z)) By Lemma 1.1.2(c), there is N ∈ hΩ (z)isuch that y ∈ GFCS(N) We have z ∈ Ω−1(y0) ∩ K for all y0∈ N Since Ω−1 istransfer compactly open-valued, there exists bN ∈ hY i with |N| = |N| such thatb

z∈ intK Ω−1( ˆy) ∩ K ⊂ Ω−1( ˆy) ∩ K for all ˆy∈N Then, bN∈ hΩ (z)i and hence

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1.2 Existence theorems in GFC-spaces

Using two elementary topological tools: the finite intersection property of pact sets and the existence, for a finite covering of a compact set, of a partition ofunity associated with this covering, we first prove the following three versions ofresults

com-Theorem 1.2.1 (Intersection) Let (X ,Y, {ϕN}) be a GFC-space, Z be a logical space Let T ∈B(X,Y,Z) and F : Y ⇒ Z satisfy the following conditions(i) for each y ∈ Y , F(y) is compactly closed;

topo-(ii) F is T-KKM;

(iii) either of the following three conditions holds

(a) there are N0∈ hY i and a compact subset K of Z withT

y∈N 0F(y) ⊂ K;(b) there is S : Y ⇒ X such that for each N ∈ hY i, there exists an S-subset LN of Y,containing N, so that S(LN) is a compact subset and, for some nonempty andcompact subset K of Z, T(S(LN)) ∩T

Theorem 1.2.2 (Coincidence points) Let (X ,Y, {ϕN}) be a GFC-space and Z

be a topological space Let S: Y ⇒ X, T : X ⇒ Z and F : Z ⇒ Y be multivaluedmappings with T ∈B(X,Y,Z) Assume that

(i) for each x ∈ X and each z ∈ T (x), F(z) is an S-subset of Y;

(ii) for each y ∈ Y, F−1(y) contains a compactly open Oy(some Oymay be empty)

of Z such that K:=S

y∈YOy is nonempty and compact;

(iii) either of the following three conditions holds:

(a) there is N0∈ hY i such thatT

so that S(LN) is compact

Then, a point( ¯x, ¯y, ¯z) ∈ X ×Y × Z exists such that ¯x∈ S( ¯y), ¯y ∈ F(¯z) and ¯z ∈ T ( ¯x)

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Theorem 1.2.3 (Maximal-elements) Let (X ,Y, {ϕN}) be a GFC-space, Z be atopological space and K ⊂ Z be nonempty and compact Let F : Z ⇒ Y and

T : X ⇒ Z be multivalued mapping such that T ∈ B(X,Y, Z) and the followingassumptions are satisfied

(i) for each y ∈ Y , F−1(y) includes a compactly open subset Oy (some Oy may

be empty) of Z such thatS

y∈Y(Oy∩ K) =S

y∈Y(F−1(y) ∩ K);

(ii) for any N ∈ hY i and M ⊂ N, T (ϕN(∆M)) ∩T

y∈MOy= /0;

(iii) either of the following conditions hold:

(a) there is N0∈ hY i such that Z\K ⊂S

y∈N0Oy;(b) there is a multivalued map S : Y ⇒ X such that, for each N ∈ hY i, there is anS-subset LN of Y containing N so that S(LN) is compact and T (S(LN))\K ⊂

S

y∈LNOy

Then, a point ¯z ∈ K exists such that F(¯z) = /0

Theorem 1.2.4 (Maximal-elements) Let (X ,Y, {ϕN}), Z, F and T be defined as inTheorem 1.2.3 such that (ii) is satisfied and (i) and (iii) are replaced respectivelyby

(i’) for each y ∈ Y, F−1(y) contains a compactly open subset Oy, which may beempty, of Z such that,S

Then, an element ¯z ∈ Z exists with F(¯z) = /0

Remark 1.2.1 Our Theorem 1.2.1 contains Theorem 3 of [83] and Theorems 1-3

of [22] as special cases for the case of G-convex spaces Theorem 1.2.2 izes Theorem 1 of [80], Theorem 1.2.3 extends Theorem 2.2 of [24], and Theorem1.2.4 includes Theorem 2.1 of [24] as a special case for the FC-space setting.Proof To prove the above theorems, we first prove the following lemma

general-Lemma 1.2.1 Let (X ,Y, {ϕN}) be a GFC-space and Z be a topological space.Let F : Y ⇒ Z and T : X ⇒ Z be multivalued mappings Assume that

(i) for each y ∈ Y , F(y) is compactly closed;

(ii) T ∈B(X,Y,Z) and F is T-KKM

Then, for each N ∈ hY i, T (ϕN(∆|N|)) ∩T

y∈NF(y) 6= /0

Proof of Lemma 1.2.1 Suppose to the contrary that N = {y0, y1, , yn} ∈ hY iexists such that

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T(ϕN(∆|N|)) = [

y i ∈N

(Z\F(yi)) ∩ T (ϕN(∆|N|)),

i.e the family(Z\F(yi)) ∩ T (ϕN(∆|N|)) ni=0is an open covering of the compactset T (ϕN(∆|N|)) Let {ψi}n

i=0 be a continuous partition of unity associated withthis covering and ψ : T (ϕN(∆|N|)) → ∆|N| be defined by ψ(z) = ∑ni=0ψi(z)ei.Then ψ is continuous Since T is better admissible, there is a fixed point of ψ ◦

T|ϕN(∆|N|)◦ϕN, i.e there is z0∈ T (ϕN(∆|N|)) such that z0∈ T (ϕN(ψ(z0))) SettingJ(z0) = j ∈ {0, 1, , n} : ψj(z0) 6= 0 and M(z0) = {yj∈ N | j ∈ J(z0)}, We have

Hence, there exists j ∈ J(z0), z0∈ F(yj)

On the other hand, by the definitions of J(z0) and of the partition {ψi}n

F(y) By (i) and (a),U(y) y∈Y is a family of sets which are closed in K For each

N∈ hY i, setting M = N ∪ N0, by Lemma 1.2.1 we have

Case of (b) By (i),K ∩ T (X) ∩ F(y)

y∈Y is a family of sets which are closed in

K (K is given in (b)) Suppose that

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i.e.,

T(X ) ∩ \

y∈N

F(y) ⊂ Z\K

In view of the assumption (b), T (S(LN)) ∩T

y∈L NF(y) ⊂ K On the other hand,

com-y∈L NF(y) 6= /0, a contradiction

Case of (c) If K is compact, then K ∩ T (X) ∩ F(y)

y∈Y is a family of closedsubsets of K Suppose that K ∩ T (X ) ∩T

y∈YF(y) = /0 Then there is N ∈ hY isuch that

Now we can argue similarly as for the case (b) to get a contradiction

Proof of Theorem 1.2.2 Define a new multivalued mapping G : Y ⇒ Z by ting, ∀y ∈ Y, G(y) = Ocy, which is compactly closed by (ii), i.e assumption (i) ofTheorem 1.2.1 for G in the place of F is fulfilled It is clear that (a), (b), and (c)imply the corresponding assumptions of Theorem 1.2.1 for G By the definition

This in turn is equivalent to the existence of ¯x∈ ϕN(∆M) and ¯z ∈ T ( ¯x) such that

¯z ∈ Oy, for all y ∈ M Since Oy⊂ F−1(y) by (ii), y ∈ F(¯z), which is an S-subset

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of Y Hence ¯x∈ ϕN(∆M) ⊂ S(F(¯z)), which means that there is ¯y∈ F(¯z) such that

¯

x∈ S( ¯y)

Proof of Theorem 1.2.3 We check the assumptions of Theorem 1.2.1 in order

to apply it for, instead of F, a new multivalued mapping G : Y ⇒ Z defined byG(y) = Ocy Assumption (i) is clearly fulfilled For (ii), with arbitrary N ∈ hY i and

M⊂ N, we have, by (ii) of Theorem 1.2.3,

/0 6= K ∩ T (X ) ∩\

y∈Y

G(y)

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(ii) for each (y, z) ∈ Y × T (S(y)), y /∈ GFCS(Ω (z));

(iii) for each compact subset D ⊂ X , T (D) is compact;

(iv) there is a nonempty compact subset K of Z such that for each N ∈ hY i, there

is an S-subset LN of Y containing N such that S(LN) is compact and for each

z∈ T (S(LN)) \ K, there exists y ∈ LN such that z∈ cint(GFCS◦ Ω )−1(y).Then, there is ˆz ∈ Z such that Ω (ˆz) = /0

Theorem 1.2.6 (Coincidence points) Let (X ,Y, {ϕN}) be a GFC-space, Z a logical space, T ∈ KKM(X,Y, Z) and S : Y ⇒ X a multivalued mapping suchthat Y is an S-subset of itself For multivalued mappings H and Ω from Z into Y ,impose the following conditions

topo-(i) H has nonempty-values and H−1is transfer compactly open-valued;

(ii) GFCS(H(z)) ⊂ Ω (z) for all z ∈ Z;

(iii) for each compact subset D ⊂ X , T (D) is compact;

(iv) there is a nonempty compact subset K of Z such that for each N ∈ hY i, there

is an S-subset LN of Y containing N such that S(LN) is compact and for each

z∈ T (S(LN))\K, there exists y ∈ LN with z∈ cint(GFCS◦ H)−1(y)

Then, a point( ˆx, ˆy, ˆz) ∈ X ×Y × Z exists such that ˆx∈ S( ˆy), ˆz ∈ T ( ˆx) and ˆy ∈ Ω (ˆz)

Theorem 1.2.7 (Nonempty intersection) Let (X ,Y, {ϕN}) be a GFC-space, Z atopological space, T ∈ KKM(X,Y, Z) and S : Y ⇒ X a multivalued mapping suchthat Y is an S-subset of itself Assume P, Q : Y ⇒ Z satisfy the following conditions(i) P is transfer compactly closed-valued;

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(ii) for all y ∈ Y , T (S(y)) ⊂ Q(y);

(iii) for all N ∈ hY i, Q(GFCS(N)) ⊂ P(N);

(iv) for each compact subset D ⊂ X , T (D) is compact;

(v) there is a nonempty compact subset K of Z such that, for each N ∈ hY i, there is

an S-subset LNof Y containing N such that S(LN) is compact and T (S(LN)) ∩

The following example illustrates Theorem 1.2.5, showing in particular thatits assumptions are relaxed and not hard to be checked

Example 1.2.1Let X = [−2, 2], Y = [−2, 0[ and, for all N = {y0, y1, , yn} ∈ hY i,

ϕN(e) = ∑ni=0λiyi, where e = ∑ni=0λiei∈ ∆|N|, eibeing the ith unit vector of Rn+1.Then, (X ,Y, {ϕN}) is a GFC-space Let Z = [0, 2] and S : Y ⇒ X, T : X ⇒ Z and

Ω : Z⇒ Y be defined by S(y) = {y}, T (x) = [0, |x|] and Ω (z) = ]−z, 0[ Then, it isnot hard to see that T ∈ KKM(X ,Y, Z) and, for each z ∈ Z, Ω (z) is an SGFC-subset

of Y , i.e GFCS(Ω (z)) = Ω (z) It is also clear that, for all (y, z) ∈ Y × T (S(y)) =

Y × [0, |y|], y /∈ ] − z, 0[ = GFCS(Ω (z)) One has Ω−1(−2) = /0 and Ω−1(y) ={z ∈ Z| − z < y < 0} = ] − y, 2] for y ∈ ] − 2, 0[ Then, Ω−1(y) is open in Z Hence,

Ω−1 is transfer compactly open-valued Thus (i) and (ii) of Theorem 1.2.5 aresatisfied (iii) is obviously fulfilled (iv) is checked with K = Z and LN = Y foreach N ∈ hY i Theorem 1.2.5 implies the existence ofbz∈ Z such that Ω (bz) = /0.One easily sees directly thatbz= 0 is a maximal-element of Ω

Proof We first prove Theorem 1.2.5

Proof of Theorem 1.2.5 Define F : Y ⇒ Z by F(y) = Z \ (GFCS◦ Ω )−1(y) for

y∈ Y By Lemma 1.1.2 and the transfer compact open-valuedness of Ω−1, F

is transfer compactly closed-valued We prove that F is T -KKM Take arbitrary

N ∈ hY i, M ⊂ N and suppose that T (ϕN(∆M)) ⊂S

y 0 ∈MF(y0) is not satisfied.Then, there are x ∈ ϕN(∆M) and z ∈ T (x) such that z /∈ F(y0) for every y0∈ M, i.e.,

z∈ (GFCS◦ Ω )−1(y0) for every y0∈ M Hence y0∈ GFCS(Ω (z)) for every y0∈ M.Since GFCS(Ω (z)) is an S-subset of Y , ϕN(∆M) ⊂ S GFCS(Ω (z)) Therefore,

y∈ GFCS(Ω (z)) exists such that x ∈ S(y) and hence z ∈ T (S(y)) Thus, there

is (y, z) ∈ Y × T (S(y)) such that y ∈ GFCS(Ω (z)), which contradicts assumption(ii)

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Now suppose ad absurdum that, for all z ∈ Z, Ω (z) 6= /0 Then, for all z ∈ K,(GFCS◦ Ω )(z) 6= /0 By the transfer compact open-valuedness of (GFCS◦ Ω )−1and the compactness of K, there is N ∈ hY i such that

K=S

y∈Y

(GFCS◦ Ω )−1(y) ∩ K

=S

y∈Y

cint(GFCS◦ Ω )−1(y) ∩ K

=S

y∈N

cint(GFCS◦ Ω )−1(y) ∩ K

y∈ LN with z ∈ cint(GFCS◦ Ω )−1(y), we have

by ˆF(y) = cclF(y) ∩ T (S(LN)) As F is T -KKM, by Lemma 1.1.3 we see that

T|S(LN)∈ KKM S(LN), LN, T (S(LN)) and ˆF is T |S(LN)-KKM Hence the family{cl ˆF(y) : y ∈ LN} = {cclF(y) ∩ T (S(LN)) : y ∈ LN} has the finite intersectionproperty Since this is a family of closed subsets of compact set T (S(LN)), onehas

\

y∈LN

cclF(y) ∩ T (S(LN))= \

y∈LN

ˆF(y) 6= /0

It follows that there is ¯z ∈ T (S(LN)) such that ¯z ∈T

y∈LNcclF(y), which dicts (1.2.2)

contra-Theorem 1.2.5 implies contra-Theorem 1.2.6 Suppose the conclusion of contra-Theorem 1.2.6

is false Then, for all (y, z) ∈ Y × T (S(y)), y /∈ Ω (z) By (ii), y /∈ GFCS(H(z)).Thus, the assumptions of Theorem 1.2.5 are satisfied for H Hence, we have ˆz ∈ Ksuch that H(ˆz) = /0, contradicting (i)

Theorem 1.2.6 implies Theorem 1.2.7 Define H, Ω : Z⇒ Y by H(z) = Y \ P−1(z)and Ω (z) = Y \ Q−1(z) We verify the assumptions of Theorem 1.2.6 For y ∈ Y ,

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H−1(y) = {z ∈ Z : y ∈ Y \ P−1(z)} = {z ∈ Z : z /∈ P(y)} = Z \ P(y) By (i), H−1

is transfer compactly open-valued as required in (i) of Theorem 1.2.6

To check (ii) of Theorem 1.2.6 suppose to the contrary the existence of

z ∈ Z and y ∈ GFCS(H(z)) such that y /∈ Ω (z) = Y \ Q−1(z), i.e z ∈ Q(y)

By Lemma 1.1.2(c), there exists N ∈ hH(z)i with y ∈ GFCS(N) Then, by(iii), z ∈ Q(y) ⊂ Q(GFCS(N)) ⊂ P(N) Hence, ¯y ∈ N ⊂ H(z) exists such that

z∈ P( ¯y) = Z \ H−1( ¯y), i.e ¯y∈ H(z), a contradiction (iii) of Theorem 1.2.6 is/assumption (iv)

To see (iv) of Theorem 1.2.6, for each N ∈ hY i, by (v) one has an S-subset LN

of Y containing N such that S(LN) is compact and T (S(LN)) ∩T

y∈L NcclP(y) ⊂ K.This means that for each z ∈ T (S(LN)) \ K, there is y ∈ LN such that z ∈ Z \ cclP(y)

= cint(Z \ P(y)) = cintH−1(y) ⊂ cint(GFCS◦ H)−1(y) as required

Now, by (ii), for all (x, y, z) ∈ X × Y × Z with x ∈ S(y) and z ∈ T (x), one has

z∈ Q(y) = Z \ Ω−1(y), i.e y /∈ Ω (z) Then, the conclusion of Theorem 1.2.6

is not true for H and Ω Thus, assumption (i) of Theorem 1.2.6 must be lated Consequently, there exists ˆz ∈ Z such that /0 = H(ˆz) = Y \ P−1(ˆz) Hence

Theo-Y and z ∈ T (S(y)), from (ii) of Theorem 1.2.5 we see that z /∈ (GFCS◦ Ω )−1(y).Then, z ∈ Z \ (GFCS◦ Ω )−1(y) = Q(y) This means that T (S(y)) ⊂ Q(y) To see(v) of Theorem 1.2.7, by (iv) of Theorem 1.2.5 we have a nonempty compactsubset K ⊂ Z and for each N ∈ hY i, one has an S-subset LN of Y containing

N such that S(LN) is compact Take any T (S(LN)) ∩T

y∈L NcclP(y) If z /∈ K,then z ∈ T (S(LN)) \ K and by (iv) of Theorem 1.2.5, there is y ∈ LN such that

z ∈ cint(GFCS◦ Ω )−1(y), i.e., z /∈ Z \ cint(GFCS◦ Ω )−1(y) ⊂ Z \ cintΩ−1(y)

= cclP(y) which is a contraction Thus, by Theorem 1.2.7,T

y∈YP(y) 6= /0 Hence,there is ˆz ∈ Z such that ˆz /∈ Ω−1(y) for every y ∈ Y , i.e Ω (ˆz) = /0 The following consequence is an extension of Theorem 1 in [11]

Corollary 1.2.1 Let (X ,Y, {ϕN}) be a GFC-space,Z be a nonempty set, Se : Y ⇒

X , Ω : X ⇒ Y , F : X ⇒ eZ and G: Y ⇒ eZ Assume that

(i) G(Y ) = eZ and F−1is onto and transfer compactly open-valued;

(ii) for all x ∈ X , Ω (x) is an SGFC-subset of Y ;

(iii) for each x ∈ X ,y ∈ Y : F(x) ∩ G(y) 6= /0 ⊂ Ω (x);

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(iv) there is a nonempty compact subset A of X such that for each N ∈ hY i, there is

an S-subset LNof Y containing N such that S(LN) is compact and S(LN))\A ⊂

S

y∈LNcintx ∈ X : F(x) ∩ G(y) 6= /0

Then, a point( ˆx, ˆy) ∈ X ×Y exists such that ˆx∈ S( ˆy) and ˆy ∈ Ω ( ˆx)

Proof Define H : X ⇒ Y by H(x) = {y ∈ Y : F(x) ∩ G(y) 6= /0} We check theassumptions of Theorem 1.2.6 for S, T = I and Ω (with Z = X ) Let eK ⊂ X beany nonempty compact subset, y ∈ Y and x ∈ H−1(y) ∩ eK Then, there is z ∈

F(x) ∩ G(y) such that x ∈ F−1(z) ∩ eK By (i), there exists z0∈Zesuch that

From the Theorems 1.2.5-1.2.7 we derive equivalent forms of a general ternative theorem Alternative theorems of this type have been rather rarelystudied but they are also important For H : X ⇒ Y we define Hc : X ⇒ Y by

al-Hc(x) = Y \ H(x) for x ∈ X The mentioned forms of a general alternative rem are stated as follows

theo-Theorem 1.2.8 Let (X ,Y, {ϕN}) be a GFC-space,Z be a nonempty set, Se : Y ⇒ X,

F : X⇒ eZ and G: Y ⇒ eZ Assume that

(i) F−1 is transfer compactly open-valued;

(ii) for all x ∈ X ,y ∈ Y : F(x) 6⊂ Gc(y) is an SGFC-subset of Y ;

(iii) for each (y, x) ∈ grS, F(x) ⊂ Gc(y);

S

y∈L Ncintx ∈ X : F(x) 6⊂ Gc(y)

Then, at least one of the following assertions holds:

(a) there exists ¯x∈ X such that F( ¯x) = /0;

(b) there exists ¯z ∈ eZ such that G−1(¯z) = /0

Theorem 1.2.9 Let (X ,Y, {ϕN}) be a GFC-space,Z be a nonempty set, Se : Y ⇒ X,

P: X ⇒ eZ and Q: Y ⇒ eZ, impose that

(i) P−1is transfer compactly closed-valued;

(ii) for all x ∈ X ,y ∈ Y : Qc(y) 6⊂ P(x) is an SGFC-subset of Y ;

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(iii) for each (x, y) ∈ grS, Qc(y) ⊂ P(x);

S

y∈L Ncintx ∈ X : Qc(y) 6⊂ P(x)

(a)T

z∈e ZP−1(z) 6= /0

(b)T

y∈YQ(y) 6= /0

Theorem 1.2.10 Let (X ,Y, {ϕN}) be a GFC-space,Z be a nonempty set, Se : Y ⇒

X , F: X ⇒ eZ and Q: Y ⇒ eZ, assume that

(i) F−1 is transfer compactly open-valued;

(ii) for all x ∈ X ,y ∈ Y : F(x) 6⊂ Q(y) is an SGFC-subset of Y ;

(iii) for each (x, y) ∈ grS, F(x) ⊂ Q(y);

S

y∈LNcintx ∈ X : F(x) 6⊂ Q(y)

(a) there existsbx∈ X such that F(bx) = /0;

(b)T

y∈YQ(y) 6= /0

Theorem 1.2.11 Let (X ,Y, {ϕN}) be a GFC-space,Z be a nonempty set, Se : Y ⇒

X , P: X⇒ eZ and G: Y ⇒ eZ, impose that

(i) P−1is transfer compactly closed-valued;

(ii) for all x ∈ X ,y ∈ Y : G(y) 6⊂ P(x) is an SGFC-subset of Y ;

(iii) for each (x, y) ∈ grS, G(y) ⊂ P(x);

S

y∈L Ncintx ∈ X : G(y) 6⊂ P(x)

(a)T

z∈e ZP−1(z) 6= /0;

(b) there exists ¯z ∈ eZ such that G−1(¯z) = /0

Remark 1.2.3 If X = Y and S = I, Theorem 1.2.8 implies Theorem 3 of [11]and different to Theorem 7 of [10]

Proof Theorem 1.2.5 implies Theorem 1.2.8 Suppose to the contrary that, forall (x, z) ∈ X × eZ, F(x) 6= /0 and G−1(z) 6= /0 Then G(Y ) = eZ Define Ω : X⇒ Y by

Ω (x) =y ∈ Y : F(x) 6⊂ Gc(y) We check the assumptions of Theorem 1.2.5 for

Z= X and T = I By (iv) assumptions (iii) and (iv) of Theorem 1.2.5 are obvious.(ii) and (iii) imply assumption (ii) of Theorem 1.2.5 Arguing for Ω similarly asfor H in the proof of Corollary 1.2.1 we see that Ω−1is transfer compactly open-valued as required in assumption (i) of Theorem 1.2.5 Applying this theorem

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we have ¯x∈ X such that Ω ( ¯x) = /0 Then, F( ¯x) ∩ G(y) = /0 for all y ∈ Y Hence,

Theorem 1.2.12 Let (X ,Y, Φ) be a GFC-space, Z be a topological space, S :

Y ⇒ X and F : Y ⇒ Z be set-valued mappings and T ∈ KKM(X,Y, Z) such that

Y is an S-subset of itself Let the following conditions hold

(i) T (S(Y )) is a compact subset (of Z);

(ii) F is T-KKM and transfer compactly closed-valued

As T ∈ KKM (X ,Y, Z), the family { bF(y) : y ∈ Y } = { bF(y) : y ∈ Y } has the nite intersection property Since this is a family of closed subsets of compact set

fi-T(S(Y )), by Lemma 1.1.1 one has

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Remark 1.2.4 Theorem 1.2.12 is proved based on the finite intersection erty of compact sets To see advantages of our Theorem 1.2.12, we discuss itsparticular cases Let (X , Φ) be an FC-space, Z be a topological space, Y be aset, F : Y ⇒ Z, s : Y → X and T ∈ KKM(X,Y, Z) be given Following [25],

prop-F is called an s-KKM wrt T if for each N = {y0, y1, , yn} ∈ hY i and each

So Theorem 1.2.12 includes properly Theorem 3.1 of [25] and Theorem 3.1 of[31] When (X ,Y, Φ) = (X ,Y,Γ ) is a G-convex space [79] and S ≡ I, our Theo-rem 1.2.12 implies Theorem 1 of [61], where the assumption corresponding to ourcondition (ii) is more stringent Assume that X is a convex space, co(·) is the usualconvex hull operator in this convex space In [17] F : Y⇒ Z is called a generalizedS-KKM mapping wrt T if, for any N = {y0, y1, , yn} ∈ hY i, T (coS(N)) ⊂ F(N)

We define a GFC-space (X ,Y, {ϕN}) as follows Take s : Y → X which is any fixedselection of S and, for any N = {y0, y1, , yn} ∈ hY i, take ϕN : ∆|N| → X by thedefinition ϕN(e) = ∑ni=0λis(yi) for all e = ∑ni=0λiei∈ ∆|N| Then (X ,Y, {ϕN}) isclearly a GFC-space It is equally obvious that a set-valued mapping F is S-KKMwrt T only if F is T-KKM by our Definition 1.1.3 Consequently Theorem 4.3 of[17] is a true special case of our Theorem 1.2.12 with S(·) replaced by co(S(·))

Using a coercivity condition to replace the compactness condition (i) in orem 1.2.12, we obtain the following result

The-Theorem 1.2.13 Let (X ,Y, Φ) be a GFC-space, Z be a topological space, S :

Y ⇒ X and F : Y ⇒ Z be set-valued mappings and T ∈ KKM(X,Y, Z) Assumethat

(i1) for each compact subset D ⊂ X , T (D) is compact;

(i2) there is a compact subset K of Z such that for each N ∈ hY i, there is an subset LN of Y, containing N with either S(LN) or S(LN) being compact and

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Then, the compact set K (in (i2)) has an open covering K ∩ T (S(Y ))c, {K ∩

F(y)c}y∈Y and hence there is N ∈ hY i such that

by F1(y) = T (S(LN)) ∩ F(y) for all y ∈ LN We see that LN is an S|LN-subset ofitself Moreover, clT(S(L

N ))T|S(LN)(S|LN(LN)) is compact by (i1) and (i2), where

T|S(LN)(ϕN∗(∆M∗)) ⊂ T (ϕN∗(∆M∗))⊂ T (S(LN)) ∩S

y∈M ∗F(y) =S

y∈M ∗F1(y),i.e., F1is T |S(LN)-KKM Making use of Theorem 1.2.12 yields

(b) If F is transfer compactly closed-valued, we consider bF defined by bF(y) =cclF(y) and apply part (a) together with Lemma 1.1.1 Remark 1.2.5 Since our Definition 1.1.3 of a T -KKM mapping includes manydefinitions of KKM type mappings as discussed after Theorem 1.2.12, it is easy

to see that Theorem 1.2.13 has true special cases as follows When applied to theparticular case, where X = Y and S ≡ I, Theorem 1.2.13 improves Theorem 3.3 of[31] If (X ,Y, Φ) = (X ,Y,Γ ) is a G-convex space and S ≡ I then Theorem 1.2.13implies Theorem 3 of [61] and Theorem 5.1 of [19]

For S-KKM mappings with respect to T and the class S-KKM(X ,Y, Z) defined

in [17] we have the following consequence which is Theorem 5.1 of [17]

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Corollary 1.2.2 Let X, Y be convex spaces and Z be a Hausdorff topologicalspace Let S : Y ⇒ X, F : Y ⇒ Z and T ∈ S-KKM(X,Y, Z) Let the followingconditions hold

(a) S(C) is compact and convex for each compact convex subset C of Y ;

(b) for each compact subset D ⊂ X, T (D) is compact;

(c) F is an S-KKM mapping wrt T and compactly closed-valued;

(d) there is a nonempty compact convex subset L of Y and a compact subset K of

The condition that F is T-KKM, imposed in Theorems 1.2.12 and 1.2.13, can

be replaced by other assumptions which may be easier to check in some cases asfollows

Theorem 1.2.14 Let (X ,Y, Φ) be a GFC-space, Z be a topological space, S : Y⇒

X and F, G, M : Y ⇒ Z be set-valued mappings and T ∈ KKM(X,Y, Z) Assumethat Y is an S-subset of itself Assume further that

(i) T (S(Y )) is compact;

(ii1) F is transfer compactly closed-valued;

(ii2) for each y ∈ Y , G(y) ⊂ F(y) and T (S(y)) ⊂ M(y);

(ii3) for each z ∈ Z, Y \M−1(z) is an S-subset of Y wrt Y \G−1(z)

By (ii3), ϕN(∆k) ⊂ S(Y \M−1(z)) and hence there exists y0∈ Y \M−1(z) such that

x∈ S(y0) Consequently, z ∈ T (S(y0)) ⊂ M(y0), contradicting the fact that y0∈

Theorem 1.2.15 (X ,Y, Φ), Z, S, F, G, M, T , (ii1), (ii2), (ii3) as in Theorem 1.2.14and the following conditions hold

(i1) for each compact subset D of X, T (D) is compact;

(i2) there is compact subset K of Z such that for each N ∈ hY i, there is anS-subset LN of Y containing N so that S(LN) is compact and T (S(LN)) ∩

T

y∈L cclF(y) ⊂ K

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Theorem 1.2.16 (Section theorem) Let (X ,Y, Φ) be a GFC-space, Z be a logical space, S : Y ⇒ X be such that Y is an S-subset of itself and T ∈KKM(X,Y, Z) be such that T (S(Y )) is compact Let A, B and C be subsets of

topo-Y× Z with B ⊂ A Let the following conditions hold

(i) eF is transfer compactly open-valued, where eF : Y ⇒ Z is defined by, for all

y∈ Y ,F(y) = {z ∈ Z : (y, z) /e ∈ A};

(ii) for each y ∈ Y , {y} × T (S(y)) ⊂ C;

(iii) for each z ∈ Z,y ∈ Y : (y, z) /∈ C is an S-subset of Y wrt y ∈ Y : (y, z) /∈ B Then, there existsbz∈ T (S(Y )) such that Y × {bz} ⊂ A

Theorem 1.2.17 (Coincidence point theorem) Let (X ,Y, Φ), Z, S and T be as

in Theorem 1.2.16 Let H, P, Q : Z ⇒ Y be given Let the following conditionshold

(i) H−1is transfer compactly open valued and H(z) 6= /0 for each z ∈ T (S(Y ));(ii) for each z ∈ Z, H(z) ⊂ P(z);

(iii) for each z ∈ Z, Q(z) is an S-subset of Y wrt P(z)

Then, there exists a coincidence point(x,by,bbz) for S, Q, T, i.e.xb∈ S(by),by∈ Q(bz)and

bz∈ T (bx)

in Theorem 1.2.16 Let P, Q : Z ⇒ Y and F : Y ⇒ Z be given Let the followingconditions hold

(i) F is transfer compactly closed-valued and F−1(z) 6= Y for each z ∈ T (S(Y ));(ii) for each y ∈ Y , Z\P−1(y) ⊂ F(y);

(iii) for each z ∈ Z, Q(z) is an S-subset of Y wrt P(z)

Then, there is a coincidence point(bx,by,bz) for S, Q, T

Remark 1.2.7 Note that when applied to the particular case where Y = X (and

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we have an FC-space) and S ≡ I, Theorems 1.2.16 - 1.2.18 slightly improve orems 4.1 - 4.3 of [25]

The-Proof Theorem 1.2.14 implies Theorem 1.2.16 Define set-valued mappings

F, G, M : Y ⇒ Z by, for y ∈ Y ,

F(y) =z ∈ Z : (y, z) ∈ A ,G(y) =z ∈ Z : (y, z) ∈ B ,M(y) =z ∈ Z : (y, z) ∈ C

By (i), F(y) is transfer compactly closed-valued G(y) ⊂ F(y) as B ⊂ A and

T(S(y)) ⊂ M(y) for each y due to (ii) The last assumption (ii3) of Theorem 1.2.14

is nothing else than (iii) Now that all the conditions hold, this theorem confirmsthe existence ofbz∈ T (S(Y )) such thatbz∈ F(y) for each y ∈ Y , i.e Y × {bz} ⊂ A.Theorem 1.2.16 implies Theorem 1.2.17 Set A = (Y × Z)\graphH, B = (Y ×Z)\graphP and C = (Y × Z)\graphQ We check the assumptions of Theorem1.2.16 For y ∈ Y , eF(y) =z ∈ Z : (y, z) /∈ A = H−1(y) and hence eF is transfercompactly open-valued by (i) of Theorem 1.2.17 (ii) implies that B ⊂ A The set{y ∈ Y : (y, z) /∈ C} = Q(z) is an S-subset of Y wrt the set {y ∈ Y : (y, z) /∈ C} =P(z) Now suppose, to the contrary of the conclusion, that for each y ∈ Y and

z∈ T (S(y)), y /∈ Q(z), i.e (y, z) ∈ C This means {y} × T (S(y)) ⊂ C and sequently all the assumptions of Theorem 1.2.16 are fulfilled So, bz∈ T (S(Y ))exists such that Y × {bz} ⊂ A, i.e H(bz) = /0 contradicting assumption (i)

con-Theorem 1.2.17 implies con-Theorem 1.2.18 Choose H : Z ⇒ Y by setting H(z) =

Y\ F−1(z) to see that all the assumptions of Theorem 1.2.17 are satisfied

Theorem 1.2.18 implies Theorem 1.2.14 Define P, Q : Z⇒ Y by, for z ∈ Z,

P(z) = Y \ G−1(z), Q(z) = Y \ M−1(z)

By (ii2), Z \ P−1(y) = G(y) ⊂ F(y) Assumption (ii3) implies (iii) of Theorem1.2.18 Now suppose to the contrary of the conclusion of Theorem 1.2.14 that,for each z ∈ T (S(Y )), z /∈T

y∈YF(y), i.e F−1(z) 6= Y and all the assumptions

of Theorem 1.2.18 are satisfied For a coincidence point (bx,by,bz) existing by thistheorem we see thatbz∈ T (S(y)) andb by∈ Q(bz) = Y \ M−1(bz), i.e.bz∈ M(/ by), con-

in Theorem 1.2.17 except the compactness of T(S(Y )) Let (ii), (iii) of Theorem1.2.17 and the following conditions hold

(i) H−1is transfer compactly open valued and H(z) 6= /0 for each z ∈ Z;

(iv1) for each compact subset D of X, T (D) is compact;

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1.3 Applications 23

(iv2) there is a compact subset K of Z such that for each N ∈ hY i, there is an subset LN of Y containing N such that S(LN) is compact and T (S(LN)) \ K ⊂

S-S

y∈LNcintH−1(y)

Then, coincidence points exist for S, Q and T

Proof By (i) and the compactness of K we have N ∈ hY i such that

y∈LN

intT(S(LN))H−1(y)∩T(S(LN)).Define new set-valued mappings H1, P1, Q1: T (S(LN)) ⇒ LN by

H1(z) = H(z) ∩ LN, P1(z) = P(z) ∩ LN, Q1(z) = Q(z) ∩ LN

It is easy to see that all the assumptions of Theorem 1.2.17 are satisfied with(S(LN), LN, Φ), T (S(LN)), S|LN, T |S(LN), H1, P1, Q1 in the places of (X ,Y, Φ), Z,

S, T, H, P and Q, respectively By this theorem a point (bx,by,bz) ∈ S(LN) × LN×

T(S(LN)) exists such thatbx∈ S|LN(by),by∈ Q1(bz) andbz∈ T |S(LN)(x) This point isbalso a required point of Theorem 1.2.19

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inten-1.3 Applications 24

relaxed The most powerful and frequently used tool is existence theorems, whichhave been generalized by many authors This subsection aims at applying aboveexistence theorems to establish sufficient conditions for the solution existence of

a general variational inclusion, which encompasses most of problems of this type

in the literature Let Y , Z, Ω and D be nonempty sets Let S1: Z⇒ Z, S2: Z⇒ Y ,

T : Z ×Y ⇒ Ω , f : Ω × Z ×Y ⇒ D and g : Ω × Z ⇒ D be multifunctions One ofthe often-considered general variational inclusion consists in

finding ¯z ∈ S1(¯z) such that, ∀y ∈ S2(¯z), and ∀t ∈ T (¯z, y),

f(t, ¯z, y) ⊂ g(t, ¯z) (1.3.3)However, in some practical situations, the term ”for all t ∈ T (¯z, y)” should bereplaced by ”there exists t ∈ T (¯z, y)” Similarly, (1.3.3) may be replaced by therequest of the negation relation 6⊂, or of ”f(t, ¯z, y) ∩ g(t, ¯z) 6= /0” or its negation(with = /0) Thus, we have eight problems, which may have quite different prac-tical meanings but may be mathematically treated very similarly In [1-7, 41,43-44] a general problem setting was proposed to encompass these problems, butstill not all of them Now we modify this setting to include all eight problems asfollows For any sets U and V we use the notations

r1(U,V ) means U ⊂ V ; r2(U,V ) means U ∩V 6= /0;

r3(U,V ) means U 6⊂ V ; r4(U,V ) means U ∩V = /0;

α1(x,U ) means ∀x ∈ U ; α2(x,U ) means ∃x ∈ U ,

and the conventions that r5= r1, r6 = r2, α3 = α1 and, for i ∈ {1, 2, 3, 4} and

j∈ {1, 2}, ri= ri+2, αj= αj+1 For each r ∈ {r1, r2, r3, r4} and α ∈ {α1, α2},the general inclusion problem is as follows

(IPrα) Find ¯z ∈ S1(¯z) such that, ∀y ∈ S2(¯z), one has

α t, T ( ¯z, y), r f (t, ¯z, y), g(t, ¯z)

This setting includes the encountered eight problems Although it looks ingly a purely mathematical formalism and not explicit, but it helps to shortenremarkably the presentation of results This problem encompasses most of op-timization - related problems, which were introduced and studied recently, such

seem-as equilibrium problems, variational inequalities, vector minimization problems,etc., (see [1-7, 41-44, 51-52, 63-71]) Note further that several authors con-sider also quasivariational inclusion problems (or corresponding special cases

of quasiequilibrium problems and quasivariational inequalities), where a mon point ¯t ∈ T (¯z) is required for all y ∈ S2(¯z) as follows Let Z, Ω , D, S1, S2, f , g

com-be as above Let Y = Z and T : Z ⇒ Ω The problem is to find ¯z ∈ S1(¯z) suchthat ∃¯t ∈ T (¯z), ∀y ∈ S2(¯z), r f (¯t, ¯z, y), g(¯t, ¯z) To express this seemingly differ-ent problem as a special case of (IPrα) we set Z0= Ω × Z, S10 : Z0⇒ Z0 defined

by S01(z0) = T (z) × S1(z), S02 : Z0⇒ Y by S02(t, z) = S2(z), T0: Z0× Y ⇒ Ω by

T0(t, z, y) = {t} (then T0 can be omitted from the setting), f0: Z0× Y ⇒ D by

f0(z0, y) = f (t, z, y) and g0 : Z0⇒ D by g0(z0) = g(t, z) Then this problem is

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con-T : Z ×Y ⇒ Ω , h : Ω × Z ×Y ⇒ D and k : Ω × Z ⇒ D be multivalued mappings.For r ∈ {r1, r2, r3, r4} and α ∈ {α1, α2} define another multifunction hrα : Y ⇒ Z

by hrα(y) =z ∈ Z | α t, T (z, y), r h(t, z, y), k(t, z)

Definition 1.3.1 Multifunction h is called (k, T, rα)-quasiconvex wrt H iff, for all

N ∈ hY i, M ⊂ N and z ∈ H(ϕN(∆M)), there exists y ∈ M such that α t, T (z, y),

r h(t, z, y), k(t, z)

Such a definition is very general If X , Y and Z are convex subsets of cal vector spaces and H = I, then this definition collapses to the k-quasiconvexitywrt T or k-quasiconvexlikeness wrt T (depending on r,α) defined in Section

topologi-2 of [41] If, furthermore, T (z, y) = {z} and k(t, z) := k(z), it becomes thestrong k-diagonal quasiconvexity or strong k-diagonal quasiconcavity proposed

in [67] If Y = X (the GFC-space becomes an FC-space), Ω = Z, T (z, y) = {z},h(t, z, y) = h(z, y) and k(t, z) = k(z), then Definition 2.3 contains the k-diagonalquasiconvexity wrt H of type I, II and III stated in Definition 4.2 of [27] Asexplained in [27, 41, 67], all these definitions generalize the classical quasicon-vexity

Proposition 1.3.1 h is (k, T, rα)-quasiconvex wrt H if and only if hrα is H-KKM.Proof “Only if” For each N ∈ hY i, each M ⊂ N and each z ∈ H(ϕN(∆M)),there is y ∈ M such that

α t, T (z, y), r h(t, z, y), k(t, z) (1.3.4)

If hrα was not H-KKM, there would be N ∈ hY i, M ⊂ N such that H(ϕN(∆M)) 6⊂

S

¯

y∈Mhrα(y) This means the existence of z ∈ H(ϕN(∆M)) such that, ∀ ¯y∈ M,

z∈ h/ rα(y), i.e α t, T (z, y), r h(t, z, y), k(t, z), which contradicts (1.3.4)

“If” Assume the H-KKM property of hrα, i.e for each N ∈ hY i, each M ⊂ N,one has

H(ϕN(∆M)) ⊂S

y∈Mhrα(y) (1.3.5)Suppose that h be not (k, T, rα)-quasiconvex wrt H Then we easily use the defi-nition of this quasiconvexity to get a contradiction to (1.3.5) Note that, if X = Y (and a GFC-space becomes a FC-space), Ω = Z, T (z, y) ={z}, h(t, z, y) = h(z, y) and k(t, z) = k(z), this proposition collapses to Proposition

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1.3 Applications 26

4.1 of [27]

Proposition 1.3.2 If a multifunction S : Y ⇒ Z exists such that Y \S−1(H−1(z)) is

a S-subset of Y wrt Y\h−1rα(z) for each z ∈ Z, then h is (k, T, rα)-quasiconvex wrtH

Proof We show that hrα is H-KKM Suppose, to the contrary, the existence of

N ∈ hY i and M ⊂ N such that H(ϕN(∆M)) 6⊂S

y∈Mhrα(y) This means the istence of x ∈ ϕN(∆M) and z ∈ H(x) with z /∈ hrα(y) for all y ∈ M, i.e M ⊂N∩ (Y \h−1rα(z)) As Y \S−1(H−1(z)) is an S-subset of Y wrt Y \h−1rα(z), one has x

ex-∈ ϕN(∆k) ⊂ S(Y \S−1(H−1(z))) Consequently, y ∈ Y \S−1(H−1(z)) exists suchthat z ∈ H(x) ⊂ H(S(y)), i.e y ∈ S−1(H−1(z)), which is impossible Applyingnow Proposition 1.3.1 completes the proof For the special case as afore-mentioned for Proposition 1.3.1, Proposition1.3.2 is reduced to Proposition 4.2 of [27]

Proposition 1.3.3 If a multifunction S : Y ⇒ Z exists such that

(i) for each y ∈ Y and z ∈ H(S(y)), z ∈ hrα(y);

(ii) for each z ∈ Z, Y \h−1rα(z) is a S-subset of Y ,

then h is(k, T, rα)-quasiconvex wrt H

Proof Suppose, ad absurdo, the existence of N ∈ hY i, M ⊂ N and z ∈ H(ϕN(∆k))such that, for all y ∈ M, α t, T (z, y), r h(t, z, y), k(t, z), i.e y ∈ Y \h−1

rα(z) Sincethe last set is a S-subset of Y , z ∈ H(S(Y \h−1rα(z))) Hence, z ∈ H(S(y)) forsome y ∈ Y \h−1rα(z) By (i), z ∈ hrα(y) and we arrive at the contradiction that

For the above-encountered special case, this proposition collapses to tion 4.3 of [27]

Proposi-Proposition 1.3.4 Let S : Y ⇒ Z and M : Ω × Z ×Y ⇒ D Assume that

(i) for all y ∈ Y , H(S(y)) ⊂ Mrα(y) :=z ∈ Z | α t, T (z, y), r M(t, z, y), k(t, z)) ;(ii) either of the following conditions hold

(ii1) h be (k, T, rα)-quasiconvex wrt Mrα∗ : X⇒ Z defined by

Mrα∗ (x) =z ∈ Z | z ∈ Mrα(S−1(x)) ;(ii2) for all z ∈ Z, Y \Mrα−1(z) be an S-subset of Y wrt Y \h−1rα(z)

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1.3 Applications 27

(ii2) By (i), for all z ∈ Z, S−1(H−1(z)) ⊂ Mrα−1(z) Then, Y \Mrα−1(z) ⊂ Y \S−1(H−1(z)).Hence Y \S−1(H−1(z)) is also an S-subset of Y wrt Y \h−1rα(z) Applying Proposi-

Using Theorem 1.2.13 we establish a sufficient condition for (IPrα) as lows Besides (X ,Y, Φ), Z, Ω , D, H, T , h, k, hrα defined as above, let S1: Z⇒ Z,

fol-S2: Z⇒ Y , f : Ω × Z ×Y ⇒ D and g : Ω × Z ⇒ D be given additionally Let Estand for the set of fixed points of S1, i.e., E = {z ∈ Z | z ∈ S1(z)}

Theorem 1.3.1 Assume for problem (IPrα) that

(i) for each compact subset X0⊂ X, clH(X0) be compact;

(ii) E be nonempty, compactly closed; for each z ∈ Z\E, S2(z) 6= /0 and for each

∀z ∈ clH(S(LN))\K, ∃y ∈ LN, z∈ cintS−12 (y); (b) ∀z ∈ clH(S(LN))\K ∩ E,

∃y ∈ LN, z∈ cint(h∗rα(y))

Then,(IPrα) has solutions

Proof Since E 6= /0, reasoning ad absurdum, suppose that ∀z ∈ E, ∃y ∈ S2(z),

α t, T (z, y), r f (t, z, y), g(t, z) This and (iii) imply that ∀z ∈ E, ∃y ∈ S2(z),

α t, T (z, y), r h(t, z, y), k(t, z) Then, for any z ∈ E, there exists y ∈ S2(z)such that z /∈ hrα(y)) Hence, for this y, z /∈ (Z\S−12 (y)) ∪(E ∩ hrα(y)) := Q(y).Now, for any z ∈ Z \ E, since S2(z) 6= /0 (by (ii)), there exists y ∈ Y such that

z∈ (Z\S/ 2−1(y)) ∪ E, hence z /∈ Q(y) Thus, for each z ∈ Z, there is y ∈ Y such that

z∈ Q(y), i.e.,/ T

y∈YQ(y) = /0

We check the assumptions of Theorem 1.2.13 for Q in the place of F Itsassumption (i1) is obvious To show that Q is H-KKM, let N ∈ hY i, M ⊂ N and

z∈ H(ϕN(∆M)) If z ∈ E, by (iv) and Proposition 1.3.1 there is y ∈ M such that

z∈ E ∩ hrα(y) ⊂ Q(y) and we are done If z ∈ (Z\E) ∩ S2−1(y), for all y ∈ M, then

y∈ S2(z) According to assumption (ii) of this theorem, z ∈ H(ϕN(∆M)) ⊂ S1(z),i.e z ∈ E, a contradiction Thus there is y ∈ M such that z ∈ Z\S−12 (y) ⊂ Q(y).Hence Q is H-KKM

To prove (ii) of Theorem 1.2.13 it is more convenient to show that the plement Z\Q(·) is transfer compactly open-valued For any y ∈ Y , one has

com-Z\Q(y) =S−12 (y) ∩ (Z\E)∪S−12 (y)) ∩ (Z\hrα(y))

=S−12 (y) ∩ (Z\E)∪ h∗rα(y)

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1.3 Applications 28

Now consider any nonempty compact subset K ⊂ Z and any z ∈ Z\Q(y) ∩ K

If z ∈ (Z\E) ∩ S−12 (y) ∩ K, by the compact closedness of E and transfer compactopen-valuedness of S−12 , there is y0∈ Y such that

z∈ (Z\E) ∩ intK S−12 (y0) ∩ K = intK (Z\E) ∩ S−12 (y0) ∩ K

con-z∈ cinth∗rα(y) ⊂ cint Z\Q(y) = Z\cclQ(y),another contradiction

Now Theorem 1.2.13 yields thatT

y∈YQ(y) 6= /0, contradicting the first part of

Remark 1.3.1 In assumption (vi), if K = Z then (a) and (b) are satisfied thermore, in (v) we can replace the condition on h∗rα by the transfer compactclosed-valuedness of hrα If in (IPrα), X = Y = Z is a convex subset of a topo-logical vector space, the four problems considered in [41, 44] become specialcases of four cases among the eight ones of (IPrα) Taking H and S as the identitymaps, these four special cases of Theorem 1.3.1 still have assumptions weakerthan the corresponding ones in the main theorems of [41, 44] (assumptions aboutconvexity, openness, closedness, coercivity, etc) Hence this theorem generalizesand improves also the corresponding results in [51, 52, 67, 68] Consider nowTheorem 1.3.1 for the special case, where X = Y (and a GFC-space becomes aFC-space), S1(z) = Z, S2(z) = Y , Ω = Z, T (z, y) = {z} If assumption (iv) is re-placed by the equivalent assertion given in Proposition 1.3.3 and assumption (vi)

Fur-is imposed particularly with S = I, then Theorem 1.3.1 collapses to Theorem 4.2

of [31]

Taking Proposition 1.3.4 (and some facts from Propositions 4.1-4.6 of [27] fordetailed checking, if needed) into account we obtain the following consequences,for the particular case of a FC-space with S1(z) = Z, S2(z) = Y , Ω = Z, T (z, y) ={z} (hence f (t, z, y) = f (z, y), g(t, z) = g(z)) investigated in [27] Let us impose

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1.3 Applications 29

in Theorem 1.3.1 assumption (vi) with S = I With (iv) replaced by (i) and (ii1) ofProposition 1.3.4, Theorem 1.3.1 is reduced to Theorem 4.1 of [27] While with(iv) replaced by (i) and (ii2) we receive Theorem 4.2 of [27]

Note that one might criticize that, in Theorem 1.3.1, f , g and condition (iii)are too technical It would be better to omit them and state the conclusion for rand k This is in fact the case However we choose this version of statement inorder to compare it directly with existing results as above

Consider now an example, where we apply Theorem 1.3.1

Example 1.3.1Let Y = [-2, 0], Z = [0, 2], Ω = D = R Let S1: Z⇒ Z, S2: Z⇒ Y ,

T : Z ×Y ⇒ Ω , f : Ω × Z ×Y ⇒ D and g : Ω × Z ⇒ D be given by

S1(z) = [0, z] if 0 ≤ z ≤ 1,

]z, 2] if 1 < z ≤ 2,, S2(z) = ]-z, 0[,

T(z, y) = ]-1, 1[,

f(t, z, y) = [min{tz, ty}, max{tz, ty}], g(t, z) = ]-∞, z]

Find ¯z ∈ S1(¯z) such that, for all y ∈ S2(¯z) and all t ∈ T (¯z, y), one has [min {t ¯z,ty},max{t ¯z,ty}] ⊂ ]-∞, ¯z]

This is a variational inclusion problem (IPr1α1) To apply Theorem 1.3.1,choosing X = [0, 1], we define a GFC-space (X ,Y, {ϕN}) as follows ∀N ∈ hY i,take ϕN : ∆|N| → [0, 1] as the canonical projection on the first coordinate axis(among |N| + 1 axes) Next we choose H ∈ KKM(X ,Y, Z) by setting H(x) = {2},

∀x ∈ X (then clearly H ∈ KKM(X,Y, Z)) Now we verify the assumptions ofTheorem 1.3.1 (i) and (iii) are obvious For (ii) we see that E = [0, 1] ∪ {2}

is closed and hence compactly closed Further, ∀z ∈ Z\E = ]1, 2[, ∀N ∈ hY i,

∀M ⊂ N ∩ S2(z), z ∈ H(ϕN(∆M)) = {2}, we see that, ∀y ∈ M and ∀t ∈ T (z, y)

= ]-1, 1[,

f(t, 2, y) = [min{2t,ty}, max{2t,ty}]

= [ty, 2t] if 0 ≤ t < 1,[2t,ty] if − 1 < t < 0

⊂ [ty, 2] if 0 ≤ t < 1,[2t, 2] if − 1 < t < 0

⊂ ]-∞, 2]

= g(t, 2),i.e f is (g, T, r1α1)-quasiconvex wrt H Passing to (v) we first compute S−12 (y).Clearly S−12 (y) = /0 for y = 2 and y = 0 For other y ∈ [−2, 0], one has

S−12 (y) = {z ∈ Z | − z < y < 0} = ]-y, 2]

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h∗r1α1(y) ⊂ S−12 (y) ∩ [0, −y[ = /0.

Thus, both S−12 and h∗r1α1 are transfer compactly open-valued in Y Finally for(vi) we simply take K = Z and S defined by S(y) = X, for all y ∈ Y According

to Theorem 1.3.1, problem (IPr1α1) in question has solutions By direct checking

we see that z = 12 is a solution

The results in [41, 44] cannot be applied to Example 1.3.1, since many sumptions are not satisfied, e.g S1is not closed, coS2(z) 6⊂ S1(z) for z ∈ Z\E, etc.The problem given in next example can hardly be considered by making use

as-of known results in FC-spaces, G-convex spaces, but is easily investigated byTheorem 1.3.1

Example 1.3.2Let N be the set of the natural numbers, Y = N∪(−N), Z = [−1, 1],

D = R, f (z, y) = [zy, +∞[ and g(z) = [z, +∞[ Find z ∈ Z such that f (z, y) ⊂[z, +∞[ for all y ∈ Y

This is a special case of (IPr1α1), but α1 needs not be included as S1(z) = Z,

S2(z) = Y , Ω = Z, T (z, y) = {z} in this case If one uses a FC-space (Y, {ϕN}), onecan hardly choose a suitable topology on Y and the corresponding ϕN: ∆|N|→ Y Now we define a GFC-space by taking X = R and, ∀N = {y0, y1, , yn} ∈ hY i,

ϕN(e) = ∑ni=0λiyi, where e = ∑ni=0λiei∈ ∆|N| We choose H : X ⇒ Z as H(x) ={0}, ∀x ∈ X Clearly H ∈ KKM(X,Y, Z) and assumptions (i)-(iii) of Theorem1.3.1 are satisfied For (iv) (with h = f and k = g), with any N ∈ hY i, any M ⊂ Nand z ∈ H(ϕN(∆M)) = {0} we have f (z, y) = [0, +∞[ = g(z) for all y ∈ M Hence

f is (g, r1)-quasiconvex wrt H To check (v) we compute

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1.3 Applications 31

which is open in Z, for each y ∈ Y Therefore, fr∗1 is transfer compactly valued For (vi) we take K = Z, S : Y ⇒ X as S(y) = [−|y|, |y|], ∀y ∈ Y and LN= Nfor all N ∈ hY i Then it is not hard to see that LN is an S-subset of Y and S(LN)) iscompact Furthermore, cl H(S(LN))\K = /0 Then (vi) is fulfilled Theorem 1.3.1allows one to conclude that the considered problem has solutions We easily seedirectly that z = 0 is a solution

open-1.3.2 Minimax theorems

As particular applications to minimax theory, we obtain saddle-point theoremsand other new results in various problem settings

Definition 1.3.2 Let (X ,Y, {ϕN}) be a GFC-space, Z a topological space, H :

X ⇒ Z, h : Z × Y → R = R ∪ {±∞} and λ ∈ R h is called λ quasiconvex (λ quasiconcave, resp.) wrt H in y if for all N ∈ hY i, M ⊂ N, z ∈ H(ϕN(∆M)),miny∈Mh(z, y) ≤ λ (maxy∈Mh(z, y) ≥ λ , resp.) If Z = X and H = I we omit ”wrtH” in the terminology

-The concept of λ -quasiconvexity in Definition 1.3.2 generalizes the ing notions in [22, 69, 88] The following notion is defined by Ding

correspond-Definition 1.3.3 (see [22], correspond-Definition 2.6) Let Y be a nonempty set, Z be a logical space, h : Z × Y → R and λ ∈ R h is called λ -transfer compactly lower(upper, resp.) semicontinuous in z if for each compact subset K ⊂ Z and z ∈ K,

topo-“∃y, h(z, y) > λ ( < λ , resp.)” ⇒ “∃U (z), open neighborhood, ∃y0∈ Y , ∀z0∈

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y∈L Ncclx ∈ X | h(x, y) ≤ λ ⊂ X0;(iv) there exist a compact subset Y0⊂ Y and S0: X ⇒ Y such that, ∀N0∈ hXi,there is an S0-subset LN0 of X containing N0 with S0(LN0) being compact and

S0(LN0) ∩T

x∈LN0ccly ∈ Y | k(x, y) ≥ γ ⊂ Y0.Then, a saddle point (x, y) of f exists, i.e f (x, y) ≤ f (x, y) ≤ f (x, y), for all(x, y) ∈ X ×Y Hence minx∈Xsupy∈Yf(x, y) = maxy∈Yinfx∈Xf(x, y)

Corollary 1.3.1 Let (X ,Y, {ϕN}) and (Y, X, {ϕN0 }) be GFC-spaces, X and Y ing Hausdorff topological spaces, S: Y⇒ X,S0: X⇒ Y , f : X ×Y → R and µ ∈ R.Assume that

be-(i) the function f be µ-quasiconvex, µ-transfer compactly and upper uous in y, and be µ-quasiconcave, µ-transfer compactly and lower semicon-tinuous in x;

semicontin-(ii) there exist compact subsets X0⊂ X and Y0⊂ Y such that, ∀N ∈ hY i, ∀N0∈ hXi,there are S-subset LN of Y containing N and S0-subset LN0 of X contain-ing N0 with S(LN) and S0(LN0) being compact and S(LN) ∩T

y∈L Nccl{x ∈

X | f (x, y) ≤ µ} ⊂ X0, S0(LN0) ∩T

x∈LN0ccly ∈ Y | f (x, y) ≥ µ ⊂ Y0.Then, there exists a saddle (x, y) of f , i.e., f (x, y) ≤ f (x, y) ≤ f (x, y), for all(x, y) ∈ X ×Y Hence minx∈Xsupy∈Yf(x, y) = maxy∈Yinfx∈Xf(x, y)

Remark 1.3.2Theorem 1.3.2 includes properly Theorems 4.1-4.2 of [22], rem 9 of [69], Theorem 3.3 of [88] and the well-known Ky Fan minimax inequal-ity Corollary 1.3.1 includes and improves Theorems 4.3 - 4.5 of [22], Theorems4.1-4.2 and 4.4 of [88], Theorem 10 of [69] and of course also the celebrated vonNeumann minimax theorem

Theo-The following minimax inequalities are established by using Theo-Theorems 1.2.7

1.2.5-Theorem 1.3.4 Let (X ,Y, {ϕN}) be a GFC-space, Z a topological space, C (Z)the family of the compact subsets of Z, S : Y ⇒ Z with S(Y ) being compact,

f : Y × Z → R and g : X × Z → R Assume that

(i) for each (y, x, z) ∈ Y × S(Y ) × Z, f (y, z) ≤ g(x, z);

(ii) any λ ∈ R and z ∈ Z, the set {y ∈ Y : f (y, z) > λ } is an SGFC-subset of Y ;(iii) for each λ ∈ R, f is λ -transfer compactly lower semicontinuous in z;

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1.3 Applications 33

(iv) for each K ∈C (Z), there is λ∗≤ supy∈Yinfz∈Ksupx∈S(y)g(x, z) such that themap x7→z ∈ K : g(x, z) ≤ λ∗ belongs to KKM(X,Y, Z)

Then,infz∈Zsupy∈Yf(y, z) ≤ infK∈C (Z)supy∈Yinfz∈Ksupx∈S(y)g(x, z)

Proof Suppose to the contrary that

infz∈Zsupy∈Yf(y, z) > infK∈C (Z)supy∈Yinfz∈Ksupx∈S(y)g(x, z)

Then there are λ , λ∗∈ R and (nonempty) K ∈ C (Z) such that

infz∈Zsupy∈Yf(y, z) > λ ≥ supy∈Yinfz∈Ksupx∈S(y)g(x, z) ≥ λ∗

and the map x 7→z ∈ K : g(x, z) ≤ λ∗ belongs to KKM(X,Y, Z) Since λ ≥ λ∗,the map x 7→z ∈ K : g(x, z) ≤ λ also belongs to KKM(X,Y, Z)

We define T : X ⇒ Z and Ω : Z ⇒ X by

T(x) = z ∈ K : g(x, z) ≤ λ if x ∈ S(Y ),

K if x /∈ S(Y )

Ω (z) =y ∈ Y : f (y, z) > λ Clearly, T ∈ KKM(X ,Y, Z), T (x) 6= /0 for all x ∈ X and Ω (z) 6= /0 for all z ∈ Z

We check the assumptions of Theorem 1.2.6 for S, T and Ω = H By (iii), Ω−1

is transfer compactly open-valued By assumption (ii) one sees that, for each

z∈ Z, Ω (z) is SGFC-subset of Y , i.e., Ω (z) = GFCS(Ω (z)) Thus, assumptions (i),(ii) of Theorem 1.2.6 are satisfied Finally for (iii)-(iv) of Theorem 1.2.6 Since

T(x) ⊂ K and K is compact, T is compact Now, for each N ∈ hY i, we simplytake LN = Y and assumptions (iii)-(iv) of Theorem 1.2.6 are satisfied According

to Theorem 1.2.6 there is a point (bx,by,bz) ∈ X ×Y × Z such thatbx∈ S(y),b bz∈ T (x)bandby∈ Ω (bz), i.e., g(bx,bz) ≤ λ < f (by,bz), contradicts assumption (i) Corollary 1.3.2 Let (X , {ϕN}) be an FC-space, Z be a compact topological spaceand f : X × Z → R ∪ {±∞} Assume that

(i) any λ ∈ R and z ∈ Z, the setx ∈ X : f (x, z) > λ is an FC-subspace of X;(ii) f (x, ) is lower semicontinuous on Z;

(iii) there is λ∗≤ supx∈Xminz∈Zf(x, z) such that the map x 7→z ∈ Z : f (x, z) ≤

λ∗ belongs to KKM(X, X, Z)

Then,minz∈Zsupx∈Xf(x, z) = supx∈Xminz∈Zf(x, z)

Remark 1.3.3(a) If the following condition holds then assumption (iv) of orem 1.3.4 is satisfied:

The-(iv’) g(x, z) is lower semicontinuous on X × Z such that for each K ∈C (Z) there

is λ∗≤ supy∈Yinfz∈Ksupx∈S(y)g(x, z) such that the map x 7→z ∈ K : g(x, z) ≤

λ∗ has acyclic nonempty values

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minz∈[0,1]zx = 0 if x> 0,

1

x if x< 0,supx∈R\{0}minz∈[0,1]xz = 0

λ, 0[ ∪ ]0, +∞[ if λ < 0, z > 0,]0, +∞[ if λ = 0, z > 0,]0, z

[0, 1] if λ = 0, x < 0,{0} if λ = 0, x > 0

The map x 7→ {z ∈ Z | f (x, y) ≤ 0} clearly belongs to KKM(X ,Y, Z) Thus, plying Corollary 1.3.2 yields

ap-minz∈[0,1]supx∈R\{0}z

x = supx∈R\{0}minz∈[0,1]z

x.Since

supx∈R\{0}zx = 0 if z= 0,

+ ∞ if 0 < z ≤ 1,this value is 0

Theorem 1.3.5 Let (X ,Y, {ϕN}) be a GFC-space, S : Y ⇒ X with S(Y ) beingcompact, f, g : X ×Y → R ∪ {±∞} and λ ∈ R Assume that

(i) for each x ∈ X ,y ∈ Y : g(x, y) < λ is an SGFC-subset of Y ;

(ii) g is λ -transfer compactly upper semicontinuous in x;

(iii) for all (x, y) ∈ X ×Y , f (x, y) < λ implies g(x, y) < λ ;

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