Common fixed point theorems and minimax inequalities in locally ...

Applicable Analysis Vol. 88, No. 12, December 2009, 1691–1699

Common fixed point theorems and minimax inequalities in locally convex Hausdorff topological vector spaces Ravi P. Agarwala*, Mircea Balajb and Donal O’Reganc a

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA; bDepartment of Mathematics, University of Oradea, Oradea, Romania; cDepartment of Mathematics, National University of Ireland, Galway, Ireland Communicated by R.P. Gilbert (Received 1 September 2009; final version received 5 September 2009) In this article, we introduce the concept of a family of set-valued mappings generalized Knaster–Kuratowski–Mazurkiewicz (KKM) w.r.t. other family of set-valued mappings. We then prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and T and S are two families of self set-valued mappings of X such that S is generalized KKM w.r.t. T , under some natural conditions, the set-valued mappings S 2 S have a fixed point. Other common fixed point theorems and minimax inequalities of Ky Fan type are obtained as applications. Keywords: family of set-valued mappings generalized KKM; fixed point; minimax inequality AMS Subject Classifications: 47H10; 49J35

1. Introduction and preliminaries Let X be a convex subset of a vector space, Y be a nonempty set and S, T : X ( Y be two set-valued mappings. Recall that S is said to be a generalized Knaster– Kuratowski–Mazurkiewicz (KKM) mapping w.r.t. T if for any nonempty finite subset {x1, . . . , xn} of X there is {u1, . . . , un}  X such that T ðcofui : i 2 I gÞ  S Sðx i Þ for each nonempty subset I of {1, . . . , n}, where co A denotes the convex i2I hull of A. If ui ¼ xi (i ¼ 1, . . . , n) for any nonempty finite subset {x1, . . . , xn} of X, we say that S is a KKM mapping w.r.t. T. This concept proved very useful in fixed point theory, equilibrium problems and minimax inequalities [1–7]. The following close concept is due to Lin and Chang [8]: if X is a nonempty set, Y is a convex subset of a vector space and S S, T : X ( Y S are two set-valued mappings, S is called a T-KKM mapping if coð ni¼1 Tðxi ÞÞ  ni¼1 Sðxi Þ, for any nonempty finite subset {x1, . . . , xn} of X. Inspired by these concepts we introduce a new one, concerning two families of set-valued mappings.

*Corresponding author. Email: [email protected] ISSN 0003–6811 print/ISSN 1563–504X online ß 2009 Taylor & Francis DOI: 10.1080/00036810903331874 http://www.informaworld.com

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Definition 1 Let X be a nonempty set, Y be a convex subset of a vector space and T and S are two families of set-valued mappings with nonempty values from X into Y. We say that S is generalized KKM w.r.t. T if for any nonempty finite subfamily  S S {S1, . . . , Sn} of S there exist T1, . . . , Tn 2 T such that coð i2I Ti ðxÞ  i2I Si ðxÞ, for each nonempty subset I of {1, . . . , n} and for all x 2 X. Example 1 For  2 (0, 1) and n 2 N define the set-valued mappings S : [0, 1] ( R and Tn : [0, 1] ( R by    þ x þ 2 , Tn ðxÞ ¼ ½an ðxÞ, bn ðxÞ, S ðxÞ ¼   x  2, 1 where   an ðxÞ ¼ min ð1Þn , 1n þ ð1Þn ðx þ 2Þ

  bn ðxÞ ¼ max ð1Þn , 1n þ ð1Þn ðx þ 2Þ :

Consider the families of set-valued mappings S ¼ {S :  2 (0, 1)} and T ¼ {Tn : n 2 N}. One can easily check that T (a) 2ð0,1Þ S ðxÞ 6¼ ; for S all x 2 [0, 1]. Consequently, for every nonempty finite subset D of (0, 1), fS : 2Dg  is an interval. (b) For each  2 (0, 1) put n ¼ 1 . Then Tn(x)  S(x) for all x 2 [0, 1]. By S(a) and (b), S it follows that if D is a nonempty finite subset of (0, 1), coð 2D Tn ðxÞÞ  2D S ðxÞ, for all x 2 [0, 1], hence S is generalized KKM w.r.t. T . Remark 1 If Y is a convex subset of a topological vector space and S is generalized KKM w.r.t. T , then for each x 2 X, fSðxÞ : S2Sg has the finite intersection property. Proof Let {S1, . . . , Sn}  S and T1, . . . , Tn 2 T be the companion mappings in the previous definition. For x 2S X arbitrarily fixed choose y1 2 T1(x), . . . , yn 2 Tn(x). Then S cofyi : i2I g  i2I Si ðxÞ  i2I STi ðxÞ for each nonempty subset I of {1, . . . , n}. By the g KKM principle, it follows that ni¼1 Si ðxÞ 6¼ ;. In this article we shall prove that if X is a nonempty compact convex subset of a locally convex Hausdorff topological vector space and T and S are two families of self set-valued mappings of X such that S is generalized KKM w.r.t. T , under some natural conditions, the set-valued mappings S 2 S have a fixed point. If X and Y are topological spaces, a set-valued mapping T : X ( Y is said to be: (i) upper semicontinuous (in short, u.s.c) if for every closed subset B of Y the set {x 2 X : T(x) \ B 6¼ ;} is closed; (ii) closed if its graph (i.e. the set GrT ¼ {(x, y) 2 X  Y : y 2 T(x), x 2 X }) is a closed subset of X  Y; (iii) compact if T(X ) is contained in a compact subset of Y. Concerning the above concepts, recall the following. LEMMA 1

[9]

(i) If Y is regular and T is u.s.c. with closed values, then T is closed. (ii) If Y is compact and T is closed, then T is u.s.c. Since the topological vector spaces are regular, by (i) and (ii) we infer that, if Y is a compact subset of a topological vector space, a closed-valued mapping T : X ( Y is u.s.c. if an only if it is closed.

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For a set-valued mapping of two variables S : X  D ( Y and x 2 X,  2 D, denote by S : X ( Y and Sx : D ( Y the set-valued mappings defined by S(x) ¼ S(x, ), Sx() ¼ S(x, ). THEOREM 1 Let X, D and  be three nonempty sets and Y be a nonempty convex set in a vector space. Let T : X   ( Y, R : X   ( Y and S : X  D ( Y be three set-valued mappings such that: (i) for each  2 D there exists  2  such that R(x, )  S(x, ), for all x 2 X; (ii) for each x 2 X, Rx is a Tx-KKM mapping. Then the family of set-valued mappings S ¼ {S}2D is generalized KKM w.r.t. T ¼ {T }2. Proof Let {1, . . . , n}  D. By (i), there exist  1, . . . ,  n 2  such that R(x,  i)  S(x, i) for all i 2 {1, . . . , n} and all x 2 X. Then, for each I  {1, . . . , n} and x 2 X we have ! ! [ [ [ [ [ Ti ðxÞ ¼ co Tx ði Þ  Rx ði Þ  Sx ði Þ ¼ Si ðxÞ: g co i2I

i2I

i2I

i2I

i2I

In order to prove the main result, we need the following two lemmas. LEMMA 2 [5] Let X be a topological space, Y be a topological vector space and S, T : X ( Y be two set-valued mappings. If S is u.s.c. with nonempty compact values and T is closed, then S þ T is a closed set-valued mapping. LEMMA 3 Let X be a topological space and Y be a Hausdorff topological vector space. If f : X ! R is a continuous function and T : X ( Y a compact closed set-valued mapping, then the mapping f T : X ( Y defined by ( f T )(x) ¼ f (x)T(x) is closed. Proof Let ðx, yÞ2Grð f T Þ. Then there exists a net {(x, y)}2D in Gr( f T ) converging to (x, y). For each  2 D we have y ¼ f (x)z for some z 2 T(x). Since TðXÞ is compact, there is a subnet {z} of {z} converging to a point z2TðXÞ. Since the mapping T is closed, z 2 T(x). Hence, y ! f (x)z 2 ( f T )(x). The space Y being Hausdorff, y ¼ f (x)z. It follows that (x, y) 2 Gr( f T ), hence the mapping f T is closed. g THEOREM 2 Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space E and T and S be two families of self set-valued mappings of X. Suppose that (i) S is generalized KKM w.r.t. T ; (ii) for each S 2 S, {x 2 X : x 2 S(x)} is closed; (iii) each T 2 T is u.s.c. with nonempty closed convex values. T Then there exists x0 2 X such that x0 2 S2S Sðx0 Þ. Proof By (ii), for each S 2 S the set S GS ¼ {x 2 X : x 2= S(x)} is open. Suppose that the conclusion is not true. Then X ¼ S2S G SS . Since X is compact, there exists a finite family {S1, . . . , Sn}  S such that X ¼ ni¼1 Gi , where, for the sake of simplicity, instead of GSi we put Gi. Let {1, . . . , n} be a partition of unity on X subordinated to

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the open cover {G1, . . . , Gn}. Recall that this means that 8 i : X ! ½0, 1 is continuous for each i2f1, . . . , ng; > > > < i ðxÞ40 ) x2Gi ; n P > > > : i ðxÞ ¼ 1 for each x2X: i¼1

Let {T1, . . . , Tn} be the subfamily of T which correspond, according to Definition 1, to {S1, . . . , Sn}. Define the mapping P : X ( X by PðxÞ ¼ 1 ðxÞT1 ðxÞ þ    þ n ðxÞTn ðxÞ: It is clear that P has nonempty compact convex values. Combining Lemmas 2 and 3, we infer that P is u.s.c. By the Kakutani–Fan–Glicksberg fixed point theorem, there exists e x2X such that e x2Pðe x Þ. Let x Þ 4 0g: I ¼ fi2f1, . . . , ng : i ðe Then e x2Pðe x Þ2co

[

! Ti ðe xÞ 

i2I

[

Si ðe x Þ:

i2I

On the other hand, for each i 2 I, since e x 2 Gi , we have e x 2S = i ðe x Þ, hence e x 2= The obtained contradiction completes the proof.

Sk

i¼1

Si ðe x Þ: g

Remark 2 It can be easily seen that condition (ii) in Theorem 2 is fulfilled if each set-valued mapping S 2 S is closed. COROLLARY 1 Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space and S be a family of self set-valued mappings of X u.s.c. with nonempty closedSconvex values. Suppose that for any nonempty finite . . . , Sn} of S, ni¼1 Si ðxÞ is convex for all x 2 X. Then there exists x0 2 X subfamily {S1,T such that x0 2 S2S Sðx0 Þ. Proof

g

Apply Theorem 2 for the case T ¼ S.

COROLLARY 2 Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space, Y be a nonempty set and S : X  Y ( X be a set-valued mapping satisfying the following conditions: (i) for each y 2 Y, the set {x 2 X : x 2 S(x, y)} is closed; (ii) for each nonempty finite subset {y1, . . . , yn} of Y there exist {u1, . . . , un}  X such that forSany nonempty subset I of {1, . . . , n} and for all x 2 X, cofui : i2I g  i2I Sðx, yi Þ. T Then there exists x0 2 X such that x0 2 y2Y Sðx0 , yÞ. Proof

For y 2 Y and u 2 X define the set-valued mappings Sy, Tu : X ( X by Sy ðxÞ ¼ Sðx, yÞ,

Tu ðxÞ ¼ fug

for all x2X:

One readily verifies that the families of set-valued mappings S ¼ {Sy : y 2 Y} and T ¼ {Tu : u 2 X } satisfy all the requirements of Theorem 2. The desired conclusion follows from this theorem. g

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THEOREM 3 Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space E and T and P be two families of self set-valued mappings of X satisfying condition (iii) in Theorem 2 and the following conditions: (a) for each x 2 X there exists P 2 P such that x 2 P(x); (b) for each P 2 P, {x 2 X : x 2 P(x)} is open in X. Then there exists a finite subfamily T {P1, . . . , Pn} of P such that for each {T1, . . . , Tn}  T , cofTi ðxÞ : i2I g \ i2I Pi ðxÞ 6¼ ; for some I  {1, . . . , n} and for some x 2 X. Proof Associate to each P 2 P the set-valued mapping Pc : XT ( X defined by Pc(x) ¼ X n P(x) and set S ¼ {Pc : P 2 P}. By (a), it follows that x 2= P2P Pc ðxÞ for all x 2 X. By Theorem 2 we infer that the family S is not generalized KKM w.r.t. T . Hence, there exists a finite subfamily {P1, . . . , Pn} of P such that for each S {T1, . . . , Tn}  T , cofTi ðxÞ : i2I g¯ i2I Pci ðxÞ for some I  {1, . . . , n} and for some x 2 X, and thereby the desired conclusion follows. g

3. Applications Let us recall that if E is a locally convex topological vector space, its topology is generated by a directed family of seminorms P on E and the family of all sets B( p, ) ¼ {x 2 X : p(x)5"}, p 2 P, 05" 2 R forms a neighbourhood basis of the origin of E. LEMMA 4 Let E be a locally convex topological vector space whose topology is generated by a directed family of seminorms P. Let A be a closed subset of E and x 2 E such that dp(x, A) ¼ 0 for each p 2 P, where dp(x, A) ¼ inf{ p(x  u) : u 2 A}. Then x 2 A. Proof If V is an arbitrary neighbourhood of x, then there exists p 2 P and "40 such that {u 2 X : p(x  u)5"}  V. Since dp(x, A) ¼ 0, there exists u 2 A such that g p(x  u)5", hence u 2 V \ A. Thus x2A ¼ A. THEOREM 4 Let E be a locally convex Hausdorff topological vector space whose topology is generated by a directed family of seminorms P, X be a nonempty compact convex subset of E and T be a family of self set-valued mappings of X with nonempty closed convex values. Suppose that for any finite subfamily {T1, . . . , Tn} of T , for each x 2 X and p, p0 2 P we have ! n [ 0 Tðxi Þ : ð1Þ min dp ð y, Ti ðxÞÞ p ð y  xÞ for all y2co 1 i n

i¼1

Then there exists x0 2 X such that x0 2 Proof

T

T2T

Tðx0 Þ.

For each (T, p) 2 T  P define the set-valued mapping ST,p : X ( X by ST,p ðxÞ ¼ fy2X : dp ð y, TðxÞÞ pð y  xÞg:

Put S ¼ {ST,p}(T,p)2T P . We prove that for (T, p) 2 T  P, the set M ¼ {x 2 X : x 2 ST,p(x)} ¼ {x 2 X : dp(x, T(x)) ¼ 0} is closed. Let us consider an arbitrary x2M and

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{x}2D a net in M converging to x. Then, for each  2 D there exists y 2 T(x) such that p(y  x) ¼ 0. Since X is compact, we may suppose that {y}2D converges to a y 2 X. The set-valued mapping T is closed, hence y 2 T(x). Since p is continuous function, it follows that p( y  x) ¼ 0, hence dp(x, T(x)) ¼ 0. Thus x 2 M and M is closed. We show that S is generalized KKM w.r.t. T . Let {(T1, p1), . . . , (Tn, pn)}  T  P. We have to prove that for I  {1, . . . , n} and x 2 X ! [ [ Ti ðxÞ  STi ,pi ðxÞ: ð2Þ co i2I

S

i2I

 Let y2co i2I Ti ðxÞ . Since P is a directed family of seminorms, there exists p 2 P such that pi(u) p(u) for each i 2 I and u 2 X. It can be easily shown that dpi( y, A) dp( y, A) for each A  X and i 2 I. Let j 2 I such that dp ð y, Tj ðxÞÞ ¼ min dp ð y, Ti ðxÞÞ: i2I

By (1), we have min dp ð y, Ti ðxÞÞ pj ð y  xÞ: i2I

It follows that dpj ð y, Tj ðxÞÞ dp ð y, Tj ðxÞÞ ¼ min dp ð y, Ti ðxÞÞ pj ð y  xÞ: i2I

Thus y 2 STj pj(x), hence (2) holds. T By Theorem 2, there exists x0 2 X such that x0 2 ðT,pÞ2TP ST,p ðx0 Þ. Fix, arbitrarily, T 2 T . For each p 2 P we have dp(x0, T(x0)) ¼ 0. By Lemma 4, it g follows that x0 2 T(x0) and the proof is complete. When E is a normed space, Theorem 4 reduces to the following corollary. COROLLARY 3 Let X be a nonempty compact convex set in a normed space and T be a family of self set-valued mappings of X with nonempty closed convex values. Suppose that for any finite subfamily {T1, . . . , Tn} of T and for each x 2 X, ! n [ Tðxi Þ : min d ð y, Ti ðxÞÞ k y  xk for all y2co 1 i n

Then there exists x0 2 X such that x0 2

T

i¼1 T2T

Tðx0 Þ.

Definition 2 Let X be a nonempty set, Y be a convex set in a vector space and F and G be two families of real functions defined on X  Y. We say that G is F -quasiconvex (respectively, F -quasiconcave) in y if for each nonempty subfamily {g1, . . . , gn} of G there exists { f1, . . . , fn}  F such that for any {y1, . . . , yn} 2 Y and x 2 X, mini2I gi ðx, yÞ maxi2I fi ðx, yi Þ (respectively, maxi2I gi ðx, yÞ mini2I fi ðx, yi Þ for each I  {1, . . . , n} and y 2 co{yi : i 2 I }). Remark 3

G is F -quasiconvex in y whenever the following conditions are satisfied:

(i) for each g 2 G there exists f 2 F such that g(x, y) f (x, y) for all (x, y) 2 X  Y; (ii) for any finite subfamily {gi : i 2 I } of G and each x 2 X the function y ° min{gi(x, y) : i 2 I } is quasiconvex.

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Proof Let {g1, . . . , gn}  G. By (i), there exists f1, . . . , fn 2 F such that gi (x, y) fi (x, y) for 1 i n and all (x, y) 2 X  Y. For x 2 X and I  {1, . . . , n}, arbitrarily fixed, let h : X ! R be the function defined by hð yÞ ¼ mini2I gi ðx, yÞ. For each j 2 I, hð yj Þ gj ðx, yj Þ fj ðx, yj Þ maxi2I fi ðx, yi Þ. Since h is quasiconvex, for g y 2 co{yi : i 2 I } we have hð yÞ maxi2I hð yi Þ maxi2I fi ðx, yi Þ. THEOREM 5 Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space E and F and G be two families of real functions defined on X  X and  2 R. Suppose that (i) (ii) (iii) (iv)

G is F -quasiconvex in y; each f 2 F is continuous on X  X; for each f 2 F and x 2 X the set {y 2 X : f (x, y) } is nonempty and convex; for each g 2 G, {x :2 X : g(x, x) } is closed.

Then there exists x0 2 X such that g(x0, x0)  for all g 2 G. Proof

For f 2 F and g 2 G define the set-valued mappings Tf, Sg : X ( X by Tf ðxÞ ¼ fy2X : f ðx, yÞ g, Sg ðxÞ ¼ fy2X : gðx, yÞ g

for x2X:

We intend to apply Theorem 2 to the families of set-valued mappings T ¼ {Tf : f 2 F }, S ¼ {Sg : g 2 G}. By (ii) and (iii), it readily follows that each Tf is u.s.c. with nonempty closed convex values. By (iv), for each g 2 G, {x 2 X : x 2 Sg(x)} is closed. We now claim that the family S is generalized KKM w.r.t. T . Now let {g1, . . . , gn}  G and f1, . . . , fn 2 F be the companion functions in Definition 2. We show that for each nonempty subset I of {1, . . . , n} and for x 2 X we have ! [ [ Tfi ðxÞ  Sfi ðxÞ: ð3Þ co i2I

i2I

 Let y2co i2I Tfi ðxÞ . Since the sets Tfi(x) are convex, there exist points yi 2 Tfi(x) (i.e. fi (x, yi) ) such that y 2 co{yi : i 2 I }. By (i), mini2I gi ðx, yÞ maxi2I fi ðx, yi Þ . Consequently, for some i 2 I, gi (x, y) , that is y 2 Sgi(x). Thus, (3) is proved. So all the requirements of Theorem 3 are fulfilled and the desired conclusion follows from this theorem. g S

THEOREM 6 Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space E and F and G be two families of real functions defined on X  X satisfying conditions (i), (ii) and (iv) in Theorem 2 and the following conditions: (iii) for each f 2 F and any x 2 X, f (x, ) is quasiconvex; (v) each function g 2 G is l.s.c. on DX ¼ {(x, x) : x 2 X }. Then infx2X supg2G gðx, xÞ supf2F maxx2X miny2X f ðx, yÞ. Proof

We may suppose that sup max min f ðx, yÞ 5 1: f2F x2X y2X

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For 4 supf2F maxx2X miny2X f ðx, yÞ arbitrarily chosen, it readily follows that each of conditions (iii) and (v) implies the condition similarly denoted in Theorem 5. By Theorem 5, there exists x0 2 X such that inf sup gðx, xÞ sup gðx0 , x0 Þ ,

x2X g2G

and the proof is complete.

g2G

g

Remark 4 If all the functions f 2 F are equal and all the functions g 2 G are equal too, then G is F -quasiconvex in y means that g is f quasiconvex in y in the sense of [3, p. 232]. Thus, Theorem 6 can be regarded as a multiplied version of some two-function minimax inequalities of Ky Fan type, obtained in Theorem 3 of [10] and Theorem 4.1 of [11]. THEOREM 7 Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space E and F and G be two families of real functions defined on X  X. Suppose that (i) G is F -quasiconvex in y and F -quasiconcave in x; (ii) each f 2 F is continuous on X  X; (iii) for each f 2 F and x, y 2 X the sets {u 2 X : f (x, u) 0} and {u 2 X : f (u, y) 0} are nonempty and convex; (iv) each function g 2 G is continuous on DX. Then there exist x1, x2 2 X such that for each g 2 G there exists x 2 co{x1, x2} satisfying g(x, x) ¼ 0. Proof By Theorem 5, there exists x1 2 X such that g(x1, x1) 0 for all g 2 G. Associate to each f 2 F (respectively, g 2 G) the function f 0 (x, y) ¼ f ( y, x) (respectively, g0 (x, y) ¼ g( y, x)) for all (x, y) 2 X  X and denote by F 0 (respectively, G0 ) the family of all functions f 0 (respectively, g0 ). Applying Theorem 5 to the families F 0 and G0 , we get a point x2 2 X such that g(x2, x2) 0 for all g 2 G. Since each g 2 G is g continuous on DX, the conclusion follows easily.

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[7] L.J. Lin and W.P. Wan, KKM type theorems and coincidence theorems with applications to the existence of equilibria, J. Optim. Theory Appl. 123 (2004), pp. 105–122. [8] L.J. Lin and T.H. Chang, S-KKM theorems, saddle points and minimax inequalities, Nonlinear Anal. 34 (1998), pp. 73–86. [9] J.P. Aubin and A. Cellina, Differential Inclusions, Springer Verlag, Berlin, 1994. [10] M. Balaj, Two minimax inequalities in G-convex spaces, Appl. Math. Lett. 19 (2006), pp. 235–239. [11] Q. Zhang and C. Cheng, Some fixed-point theorems and minimax inequalities in FC-space, J. Math. Anal. Appl. 328 (2007), pp. 1369–1377.