A new fixed point theorem is established under the setting of a generalized finitely continuous topological space (GFC-space) without the convexity structure. As applications, a weak KKM theorem and a minimax inequalities of Ky Fan type are also obtained under suitable conditions. Our results are different from known results in the literature.
1. Introduction
In the last decade, the theory of fixed points has been investigated by many authors; see, for example, [1–11] and references therein, which has been exploited in the existence study for almost all areas of mathematics, including optimization and applications in economics. Now, there have been a lot of generalizations of the fixed points theorem under different assumptions and different underlying space, and various applications have been given in different fields.
On the other hand, the weak KKM-type theorem introduced by Balaj [12] has attracted an increasing amount of attention and has been applied in many optimization problems so far; see [12–14] and references therein.
Inspired by the research works mentioned above, we establish a collectively fixed points theorem and a fixed point theorem. As applications, a weak KKM theorem and a minimax inequalities of Ky Fan type are also obtained under suitable conditions. Our results are new and different from known results in the literature.
The rest of the paper is organized as follows. In Section 2, we first recall some definitions and theorems. Section 3 is devoted to a new collectively fixed points theorem under noncompact situation on GFC-space and a new fixed point theorem. In Section 4, we show a new weak KKM theorem in underlying GFC-space, and, by using the weak KKM theorem, a new minimax inequality of Ky Fan type is developed.
2. Preliminaries
Let X be a topological space and C,D⊆X. Let intC and intDC denote the interior of C in X and in D, respectively. Let 〈A〉 denote the set of all nonempty finite subsets of a set A, and let Δn denote the standard n-dimensional simplex with vertices {e0,e1,…,en}. Let X and Y be two topological spaces. A mapping T:X→2Y is said to be upper semicontinuous (u.s.c.) (resp., lower semicontinuous (l.s.c)) if for every closed subset B of Y, the set {x∈X:T(x)∩B≠∅} (resp., {x∈X:T(x)⊆B}) is closed.
A subset A of X is said to be compactly open (resp., compactly closed) if for each nonempty compact subset K of X, A∩K is open (resp., closed) in K.
These following notions were introduced by Hai et al. [15].
Definition 1.
Let X be a topological space, Y a nonempty set, and Φ a family of continuous mappings φ:Δn→X, n∈N. A triple (X,Y,Φ) is said to be a generalized finitely continuous topological space (GFC-space) if and only if for each finite subset N={y0,y1,…,yn} of Y, there is φN:Δn→X of the family Φ.
In the sequel, we also use (X,Y,{φN}) to denote (X,Y,Φ).
Definition 2.
Let S:Y→2X be a multivalued mapping. A subset D of Y is called an S-subset of Y if and only if for each N={y0,y1,…,yn}⊆Y and each {yi0,yi1,…,yik}⊆N∩D, one has φN(Δk)⊂S(D), where Δk is the face of Δn corresponding to {yi0,yi1,…,yik}, that is, the simplex with vertices {ei0,ei1,…,eik}. Roughly speaking, if D is an S-subset of Y, then (S(D),D,Φ) is a GFC-space.
The class of GFC-space contains a large number of spaces with various kinds of generalized convexity structures such as FC-space and G-convex space (see [15–17]).
Definition 3 (see [8]).
Let (X,Y,{φN}) be a GFC-space and Z a nonempty set. Let T:X→2Z and F:Y→2Z be two set-valued mappings; F is called a weak KKM mapping with respect to T, shortly, weak T-KKM mapping if and only if for each N={y0,y1,…,yn}⊆Y, {yi0,yi1,…,yik}⊆N and x∈φN(Δk), T(x)∩⋃j=0kF(yij)≠∅.
Definition 4 (see [8]).
Let X be a Hausdorff space, (X,Y,{φN}) a GFC-space, Z a topological space, T:X→2Z, f:Y×Z→R∪{-∞,+∞}, and g:X×Z→R∪{-∞,+∞}. Let λ∈R. f is called (λ,T,g)-GFC quasiconvex if and only if for each x∈X, z∈T(x), N={y0,y1,…,yn}∈〈Y〉, and Nk={yi0,yi1,…,yik}⊆N, one has the implication f(yij,z)<λ, for all j=0,1,…,k implies that g(x′,z)<λ for all x′∈φN(Δk).
For λ∈R, define β∈R and Hλ:Y→2Z by β=infx∈Xsupz∈T(x)g(x,z) and Hλ(y)={z∈Z:f(y,z)≥λ}, respectively.
Lemma 5 (see [8]).
For λ<β, if f is (λ,T,g)-GFC quasiconvex, then Hλ is a weak T-KKM mapping.
The following result is the obvious corollary of Theorem 3.1 of Khanh et al. [8].
Lemma 6.
Let {(Xi,Yi,{φNi})}i∈I be a family of GFC-spaces and X=∏i∈IXi a compact Hausdorff space. For each i∈I, let Gi:X→2Xi and Fi:X→2Yi be such that the conditions hold as follows:
for each x∈X, each Ni={y0i,y1i,…,ynii}⊆Yi and each {yi0i,yi1i,…,yikii}⊆Ni∩Fi(x), one has φNi(Δk)⊆Gi(x) for all i∈I,
X=⋃yi∈YiintFi-1(yi) for all i∈I.
Then, there exists x-=(x-i)i∈I∈X such that x-i∈Gi(x-) for all i∈I.
3. Fixed Points Theorems
Let I be an index set, Xi topological spaces, X=∏i∈IXi, and Gi:X→2Xi. The collectively fixed points problem is to find x-=(x-i)i∈I∈X such that x-i∈Gi(x-), for all i∈I.
Theorem 7.
Let {(Xi,Yi,{φNi})}i∈I be a family of GFC-spaces and X=∏i∈IXi a Hausdorff space. For each i∈I, let Gi:X→2Xi, Fi:X→2Yi, and Si:Yi→2Xi with the following properties:
for each x∈X, Ni={y0i,y1i,…,ynii}⊆Yi, and {yj0i,yj1i,…,yjkii}⊆Ni∩Fi(x), one has φNi(Δk)⊆Gi(x) for all i∈I,
for each compact subset K of X and each i∈I, K⊆⋃yi∈YiintFi-1(yi);
there exists a nonempty compact subset Ki of Xi and for each Ni∈〈Yi〉, there exists an Si-subset LNi of Yi containing Ni with Si(LNi) being compact such that
(1)S(LN)∖K⊂⋃yi∈LNiintFi-1(yi),
where LN=∏i∈ILNi, K=∏i∈IKi, and S(LN)=∏i∈ISi(LNi).
Then, there exists x-=(x-i)i∈I∈X such that x-i∈Gi(x-) for all i∈I.
Proof.
As K is a compact subset of X, by the condition (ii), there exists a finite set Ni={y0i,y1i,…,ynii}⊆Yi, such that
(2)K⊆⋃k=0niintFi-1(yiki).
By the condition (iii), there exists an Si-subset LNi of Yi containing Ni such that
(3)S(LN)∖K⊂⋃yi∈LNiintFi-1(yi),
and it follows that
(4)S(LN)⊂⋃yi∈LNiintFi-1(yi).
We observe that the family {Si(LNi),LNi,{φNi})}i∈I is a family of GFC-space and Si(LNi) is compact for each i∈I, defining set-valued mapping Gi*:Si(LNi)→2Si(LNi) and Fi*:Si(LNi)→2LNi as follows:
(5)Gi*(x)=Gi(x)∩Si(LNi),Fi*(x)=Fi(x)∩LNi.
We check assumptions (i) and (ii) of Lemma 6 for replaced Gi and Fi by Gi* and Fi*, respectively. By (i) and the definition of S-subset, for each x∈Si(LNi), each Ni={y0i,y1i,…,ynii}⊆LNi and each {yj0i,yj1i,…,yjkii}⊆Ni∩Fi*(x)=Ni∩Fi(x)∩LNi, we have
(6)φNi(Δk)⊆Gi(x)∩Si(LNi)=Gi*(x),
then assumption (i) of Lemma 6 is satisfied.
By (4), we have
(7)S(LN)=(⋃yi∈LNiintFi-1(yi))∩S(LN)=⋃yi∈LNiintS(LN)(Fi-1(yi)∩S(LN)).
On the other hand, for all yi∈LNi,
(8)(Fi*)-1(yi)={x∈X:yi∈Fi(x)}∩S(LN)=Fi-1(yi)∩S(LN).
Hence,
(9)S(LN)=⋃yi∈LNiintS(LN)(F*)-1(yi).
Thus, (ii) of Lemma 6 is also satisfied. According to Lemma 6, there exists a point x-=(x-i)i∈I∈X such that x-i∈Gi(x-) for all i∈I.
Remark 8.
Theorem 7 generalizes Theorem 3.4 of Ding [6] from FC-space to GFC-space, and our condition (iii) is different from its condition (iii). Theorem 7 also extends Theorem 3 in [18]. Note that Theorem 7 is the variation of Theorem 3.2 in [8].
As a special case of Theorem 7, we have the following fixed point theorem that will be used to prove a weak KKM theorem in Section 4.
Corollary 9.
Let X be the Hausdorff space, (X,Y,{φN}) a GFC-space, G:X→2X, F:X→2Y, and S:Y→2X with the following properties:
for each x∈X, N={y0,y1,…,yn}⊆Y, and {yi0,yi1,…,yik}⊆N∩F(x), one has φN(Δk)⊆G(x),
for each compact subset K of X, K⊆⋃y∈YintF-1(y),
for each N∈〈Y〉, there exists an S-subset LN of Y containing N with S(LN) being compact such that
(10)S(LN)∖K⊂⋃y∈LNintF-1(y).
Then, there exists x-∈X such that x-∈G(x-).
4. ApplicationsTheorem 10.
Let X be a Hausdorff space, (X,Y,{φN}) a GFC-space, Z a nonempty set, T:X→2Z, H:Y→2Z, and S:Y→2X; assume that
H is a weak T-KKM mapping,
for each y∈Y, the set {x∈X:T(x)∩H(y)≠∅} is compactly closed,
there exists a compact K of X, and, for any N∈〈Y〉, there exists an S-subset LN of Y containing N with S(LN) being compact such that
(11)S(LN)∖K⊂⋃y∈LNint{x∈X:T(x)∩H(y)=∅}.
Then, there exists a point x-∈X such that T(x-)∩H(y)≠∅ for each y∈Y.
Proof.
Define F:X→2Y and G:X→2X by
(12)F(x)={y∈Y:T(x)∩H(y)=∅},G(x)={x′∈X:∃y∈F(x),T(x′)∩H(y)≠∅}.
Suppose the conclusion does not hold. Then, for each x∈X, there exists a y∈Y such that
(13)T(x)∩H(y)=∅.
It is easy to see that F has nonempty values. By (ii), for each y∈Y,
(14)F-1(y)={x∈X:T(x)∩H(y)=∅}
is compactly open. Then,
(15)X=⋃y∈YintF-1(y).
Since K is a compact subset of X, then there exists N∈〈Y〉 such that
(16)K⊆⋃y∈NintF-1(y).
Then, assumption (ii) of Corollary 9 is satisfied.
It follows from (iii) that there exists a compact K of X and for any N∈〈Y〉, there exists a S-subset LN of Y containing N with S(LN) being compact such that
(17)S(LN)∖K⊂⋃y∈LNintF-1(y).
Therefore, assumption (iii) of Corollary 9 is also satisfied.
Furthermore, G has no fixed point. Indeed, if x∈G(x), then there exists y∈F(x) such that
(18)T(x)∩H(y)≠∅,
which contracts the definition of F. Thus, assumption (i) of Corollary 9 must be violated; that is, there exist an x-∈X, N-={y-0,y-1,…,y-n}⊆Y, and
(19)N-k={y-i0,y-i1,…,y-ik}⊆N-∩F(x-)
such that
(20)φN-(Δk)⊈G(x-).
That is, for each y∈F(x-),
(21)T(x-)∩H(y)=∅.
Hence,
(22)T(x-)∩H(N-k)=∅.
On the other hand, since H is a weak T-KKM mapping and x-∈φN-(Δk), we have
(23)T(x-)∩H(N-k)≠∅,
which is contradict. This completes the proof.
Remark 11.
(1) Theorem 10 extends Theorem 1 in [13] from the G-convex space to GFC-space, and our proof techniques are different. Theorem 10 also generalizes Theorem 4.1 of [8] from the compactness assumption to noncompact situation.
(2) If Z is a topological space, condition (ii) in Theorem 10 is fulfilled in any of the following cases (see [13]):
H has closed values, and T is u.s.c, on each compact subset of X.
H has compactly closed values, and T is u.s.c, on each compact of subset of X and its values are compact.
Theorem 12.
Let X be a Hausdorff space, (X,Y,{φN}) a GFC-space, Z a topological space, T:X→2Z u.s.c., f:Y×Z→R∪{-∞,+∞}. and S:Y→2X; assume that
for each y∈Y, f(y,·) is u.s.c. on each compact subset of Z,
f is (λ,T,g)-GFC quasiconvex for all λ<β sufficiently close to β,
there exists a compact K of X, and, for any N∈〈Y〉, there exists an S-subset LN of Y containing N with S(LN) being compact such that
(24)S(LN)∖K⊂⋃y∈LNint{x∈X:T(x)∩Hλ(y)=∅}.
Let λ<β be arbitrary. By Lemma 5 and condition (ii), Hλ is a weak T-KKM mapping. It follows from condition (i) that Hλ has closed values. Hence, the set {x∈X:T(x)∩Hλ(y)≠∅} is compactly closed for all y∈Y (see Remark 11 (2)). Thus, all the conditions of Theorem 10 are satisfied, and so there exists an x-∈X such that
(26)T(x-)∩Hλ(y)≠∅,∀y∈Y.
This implies that λ≤infy∈Ysupz∈T(x-)f(y,z) and so
(27)λ≤supx∈Xinfy∈Ysupz∈T(x-)f(y,z).
Since λ<β is arbitrary, we get the conclusion. This completes the proof.
Remark 13.
Theorem 12 improves Theorem 4.2 of [8] from the compactness assumption to noncompact situation. Theorem 12 also extends Theorem 4 of [12] from compact G-convex space to noncompact GFC-space. Our result includes corresponding earlier Fan-type minimax inequalities due to Tan [19], Park [20], Liu [21], and Kim [22].
Acknowledgment
This work was supported by the University Research Foundation (JBK120926).
TarafdarE.A fixed point theorem and equilibrium point of an abstract economy199120221121810.1016/0304-4068(91)90010-QMR1081471ZBL0718.90014LanK. Q.WebbJ.New fixed point theorems for a family of mappings and applications to problems on sets with convex sections199812641127113210.1090/S0002-9939-98-04347-0MR1451816ZBL0891.46004AnsariQ. H.YaoJ.-C.A fixed point theorem and its applications to a system of variational inequalities199959343344210.1017/S0004972700033116MR1698009ZBL0944.47037AnsariQ. H.IdzikA.YaoJ.-C.Coincidence and fixed point theorems with applications2000151191202MR1786262ZBL1029.54047DingX. P.TanK.-K.Fixed point theorems and equilibria of noncompact generalized games1992SingaporeWorld Sicence8096MR1190033DingX. P.Continuous selection, collectively fixed points and system of coincidence theorems in product topological spaces20062261629163810.1007/s10114-005-0831-yMR2262419ZBL1117.54032LinL.-J.AnsariQ. H.Collective fixed points and maximal elements with applications to abstract economies2004296245547210.1016/j.jmaa.2004.03.067MR2075176ZBL1051.54028KhanhP. Q.LongV. S. T.QuanN. H.Continuous selections, collectively fixed points and weak Knaster-Kuratowski-Mazurkiewicz mappings in optimization2011151355257210.1007/s10957-011-9889-0MR2851231ZBL1244.90221DingX. P.WangL.Fixed points, minimax inequalities and equilibria of noncompact abstract economies in FC-spaces200869273074610.1016/j.na.2007.06.006MR2426285ZBL1157.47037DingX. P.Collective fixed points, generalized games and systems of generalized quasi-variational inclusion problems in topological spaces20107361834184110.1016/j.na.2010.05.018MR2661364ZBL1229.54054YuanG. X.-Z.1999218New York, NY, USAMarcel DekkerMR1676280BalajM.Weakly G-KKM mappings, G-KKM property, and minimax inequalities2004294123724510.1016/j.jmaa.2004.02.013MR2059883BalajM.O'ReganD.Weak-equilibrium problems in G-convex spaces200857110311710.1007/s12215-008-0005-8MR2420525TangG.-S.ZhangQ.-B.ChengC.-Z.W-G-F-KKM mapping, intersection theorems and minimax inequalities in FC-space200733421481149110.1016/j.jmaa.2007.01.040MR2338675ZBL1123.49002HaiN. X.KhanhP. Q.QuanN. H.Some existence theorems in nonlinear analysis for mappings on GFC-spaces and applications200971126170618110.1016/j.na.2009.06.033MR2566523ZBL1188.49006KhanhP. Q.QuanN. H.Intersection theorems, coincidence theorems and maximal-element theorems in GFC-spaces201059111512410.1080/02331930903500324MR2765472ZBL1185.49007KhanhP. Q.QuanN. H.General existence theorems, alternative theorems and applications to minimax problems20107252706271510.1016/j.na.2009.11.016MR2577830ZBL1192.49015ParkS.Continuous selection theorems in generalized convex spaces1999205-656758310.1080/01630569908816911MR1704961ZBL0931.54017TanK.-K.Comparison theorems on minimax inequalities, variational inequalities, and fixed point theorems198328355556210.1112/jlms/s2-28.3.555MR724726ZBL0497.49010ParkS.Generalized Fan-Browder fixed point theorems and their applications19895177MR1183001LiuF.-C.On a form of KKM principle and Sup Inf Sup inequalities of von Neumann and of Ky Fan type1991155242043610.1016/0022-247X(91)90011-NMR1097292KimI.KKM theorem and minimax inequalities in G-convex spaces200161135142MR1844476