MÖNCH TYPE RESULTS FOR MAPS WITH WEAKLY SEQUENTIALLY CLOSED GRAPHS

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1 Dynamic Systems and Applications 24 (2015) MÖNCH TYPE RESULTS FOR MAPS WITH WEAKLY SEQUENTIALLY CLOSED GRAPHS DONAL O REGAN School of Mathematics, Statistics and Applied Mathematics National University of Ireland Galay Ireland donal.oregan@nuigalay.ie ABSTRACT. In this paper e present fixed point results of Mönch type and a homotopy result for maps ith eakly sequentially closed graphs. AMS (MOS) Subject Classification. 47H10, 47H04 Keyords: Fixed point theory; Mönch type maps; homotopy 1. INTRODUCTION In this paper e discuss Mönch type maps [1, 6]. We begin in Theorem 2.2 and present a ne fixed point result for Mönch type self maps ith eakly sequentially closed graph. Next e define the notion of an essential map hich is of Mönch type and has eakly sequentially closed graph. We use this notion to present a homotopy type result for this class of maps. 2. FIXED POINT THEORY First e recall the folloing result [7]. Theorem 2.1. Let Q be a nonempty, convex, eakly compact subset of a metrizable locally convex linear topological space E. Suppose F : Q K(Q) has eakly sequentially closed graph; here K(Q) denotes the family of nonempty, convex, eakly compact subsets of Q. Then F has a fixed point in Q. We no prove a result hich ill be needed in Section 3. Theorem 2.2. Let Q be a nonempty, closed, convex subset of a metrizable locally convex linear topological space E and let x 0 Q. Suppose F : Q K(Q) has eakly sequentially closed graph and F takes relatively eakly compact sets into relatively Received June 5, $15.00 c Dynamic Publishers, Inc.

2 130 D. O REGAN eakly compact sets. Also assume the folloing hold: A Q, A = co(x 0 } F(A)) ith A (2.1) = C and C A countable, implies A is eakly compact (2.2) and (2.3) for any relatively compact subset A of E there exists a countable set B A ith B = A Then F has a fixed point in Q. if A is a eakly compact subset of E then co(a) is eakly compact. Remark 2.3. If E is a Banach space then (2.3) holds from the Krein-Šmulian theorem [3, pg. 434, 5 pg. 82]. Note (2.3) holds if a Krein-Šmulian type theorem holds (for example E could be a quasicomplete locally convex linear topological space); for examples see [4 pp 553, 5 pp 82]. Remark 2.4. If K is a eakly compact subset of E and K ith the relative eak topology is metrizable (for example E could be a Banach space hose dual E is separable) then (2.2) holds (recall compact metric spaces are separable). Proof. Let D 0 = x 0 }, D n = co (x 0 } F(D n 1 )) for n = 1, 2,... and D = No for n = 0, 1,... notice D n is convex and D 0 D 1 D n 1 D n Q. Note D is convex and since (D n ) is increasing e have (2.4) D = co (x 0 } F(D n 1 )) = co(x 0 } F(D)). n=1 D n. We claim D n is relatively eakly compact for n 0, 1,... }. Certainly it is true if n = 0. No suppose D k is relatively eakly compact for some k 0, 1,... }. No since F takes relatively eakly compact sets into relatively eakly compact sets then F(D k ) is relatively eakly compact. This together ith (2.3) guarantees D k+1 is relatively eakly compact. No (2.2) implies that for each n 0, 1,... } there exists C n ith C n countable, C n D n, and C n = D n. Let C = C n. No since D n D n D n

3 FIXED POINT AND HOLOTOPY RESULTS 131 e have D n = D n = D and D n = C n = C n = C. Thus C = D so from (2.1) (see (2.4)) e have that D is eakly compact. Consider the map F : D K(D ) given by F (x) = F(x) D. We first sho F (x) for each x D. Note from (2.4) that F(D) D D so D F 1 (D ). No let x D. No since D is eakly compact the Eberlein- Šmulian theorem [4 pg. 549] guarantees that there is a sequence (x n ) in D ith x n x (here denotes eak convergence). Take any y n F(x n ). No since F(D) D e have y n D. Also since D is eakly compact the Eberlein-Šmulian theorem [4 pg. 549] guarantees that e may assume ithout loss of generality that y n y for some y D. Note y n F(x n ), x n x, y n y implies y F(x) since F has eakly sequentially closed graph. Thus y F(x) D so x F 1 (D ). As a result D F 1 (D ) i.e. F (x) for each x D. Note F : D K(D ) has eakly sequentially closed graph. No Theorem 2.1 guarantees a x D ith x F (x) F(x). Remark 2.5. Theorem 2.2 improves [1, Theorem 3.3]. 3. HOMOTOPY RESULTS In this section let E be a metrizable locally convex linear topological space, C a closed convex subset of E, and U a eakly open subset of C ith 0 U. Definition 3.1. F A(U, C) if F : U K(C) has eakly sequentially closed graph, F takes relatively eakly compact sets into relatively eakly compact sets, and F satisfies the folloing condition: if D U and D co(0} F(D)) ith D = C and C D countable, then D is eakly compact. Definition 3.2. We say F A U (U, C) if F A(U, C) ith x / F(x) for x U; here U denotes the eak boundary of U in C. Definition 3.3. A map F A U (U, C) is essential in A U (U, C) if for every G A U (U, C) ith G U = F U there exists x U ith x G(x). Theorem 3.4 (Homotopy Property). Let E be a metrizable locally convex linear topological space, C a closed convex subset of E, U a eakly open subset of C ith 0 U. Suppose F A(U, C) and assume the folloing conditions hold: (3.1) the zero map is essential in A U (U, C)

4 132 D. O REGAN and (3.2) x / λ F x for every x U and λ (0, 1]. Then F is essential in A U (U, C). Proof. Let H A U (U, C) ith H U = F U. We must sho H has a fixed point in U. Consider B = x U : x th(x) for some t [0, 1]}. No B since 0 U. Also B is eakly sequentially closed. To see this let (x n ) be sequence of B hich converges eakly to some x B (in particular x U ) and let (λ n ) be a sequence of [0, 1] satisfying x n λ n Hx n. Then for each n there is a z n Hx n ith x n = λ n z n. By passing to a subsequence if necessary, e may assume that (λ n ) converges to some λ [0, 1] and ithout loss of generality assume λ n 0 for all n. This implies that the sequence (z n ) converges eakly to some z U ith x = λz. Since F has eakly sequentially closed graph then z H(x). Hence x λhx and therefore x B. Thus B is eakly sequentially closed. Let x n } n=1 be a sequence in B. Then there exists a sequence t n} n=1 in [0, 1] ith x n t n Hx n and e may assume ithout loss of generality that t n t [0, 1]. Let C = x n } n=1. Note C is countable and C co(h(c) 0}). Since H A(U, C) then C is eakly compact. The Eberlein-Šmulian theorem [4 pg. 549] guarantees that there is a subsequence N of 1, 2,... } and a x C ith x n x as n in N. No since B is eakly sequentially closed e have x B. Consequently B is eakly sequentially compact, so eakly compact by the Eberlein-Šmulian theorem [3, pg. 430]. No B U = since (3.2) holds; note H U = F U and 0 U. No E = (E, ), the space E endoed ith the eak topology, is completely regular. This there exists a eakly continuous map µ : U [0, 1] ith µ( U) = 0 and µ(b) = 1. Define a map R µ : U K(C) by R µ (x) = µ(x)h(x). Note R µ has eakly sequentially closed graph (since H has eakly sequentially closed graph) and R µ takes relatively eakly compact sets into relatively eakly compact sets. [If A U is eakly compact and y n R µ (A), then y n = µ(x n )z n here z n H(x n ) and x n A. Without loss of generality e may assume there exists x A and z H(A) ith x n x and z n z (recall A and H(A) are eakly compact; in fact from a standard result [7] e note that H : A K(C) has eakly closed graph and from another standard result [1] e have that H(A) is eakly compact). Then z H(x) since H has eakly sequentially closed graph. Let y = µ(x)z. Then y n y and y R µ (A). As a result R µ (A) is eakly compact.] Next suppose D U ith D co(0} R µ (D)) and ith D = C and C D countable. Then since R µ (D) co(0} H(D)) and

5 FIXED POINT AND HOLOTOPY RESULTS 133 0} co(0} H(D)) = co(0} H(D)) e have D co(0} R µ (D)) co (co (0} H(D))) = co(0} H(D)). Then D is eakly compact since H A(U, C). Thus R µ A(U, C) ith R µ U = 0}. No (3.1) guarantees that there exists x U ith x R µ (x). As a result x B, so µ(x) = 1 i.e. x H(x). Next e discuss (3.1). Theorem 3.5. Let E be a metrizable locally convex linear topological space, C a closed convex subset of E, U a eakly open subset of C ith 0 U. Suppose (2.2) and (2.3) hold. Then (3.1) holds. Proof. Let θ A U (U, C) ith θ U = 0}. We must sho θ has a fixed point in U. Let θ(x), x U J(x) = 0}, otherise. Note J : C K(C) has eakly sequentially closed graph. No suppose A C, A = co(0} J(A)) ith A = D and D A countable. Then (3.3) A co(0} θ(u A)) and so (3.4) U A co(0} θ(u A)). Notice D U is countable, D U A U and (3.5) D U = A U since D U A U A U = D U D U (note for sets D 0 and D 1 of C ith D 0 eakly open in C, then D 0 D 1 D 0 D 1 ). No (3.4) and (3.5) and θ A(U, C)) implies that A U is eakly compact. Also since θ A(U, C) (θ takes relatively eakly compact sets into relatively eakly compact sets) e have that θ(a U) is relatively eakly compact. No (2.3) guarantees that co(0} θ(a U) ) is eakly compact. This together ith (3.3) implies that A is eakly compact. No Theorem 2.2 guarantees that there exists x C ith x J(x). If x / U e have x J(x) = 0}, hich is a contradiction since 0 U. Thus x U so x J(x) = θ(x). Combining Theorem 3.4 and Theorem 3.5 yields the folloing nonlinear alternative of Leray-Schauder type.

6 134 D. O REGAN Theorem 3.6. Let E be a metrizable locally convex linear topological space, C a closed convex subset of E, and U a eakly open subset of C ith 0 U. Suppose (2.2) and (2.3) hold. Also suppose F A(U, C) satisfies (3.2). Then F is essential in A U (U, C) (in particular F has a fixed point in U). REFERENCES [1] R. P. Agaral and D. O Regan, Fixed point theory for set valued mappings beteen topological vector spaces having sufficiently many linear functionals, Computers and Mathematics ith Applications 41( 2001), [2] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, Springer Verlag, Berlin, [3] N. Dunford and J. T. Schartz, Linear Operators: Part I, General Theory, Intersci. Publ., Ne York, [4] R. E. Edards, Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, [5] K. Floret, Weakly compact sets, Lecture Notes Math. 801 (1980), [6] D. O Regan, Homototy results for admissible multifunctions, Nonlinear Functional Analysis and Applications 6 (2001), [7] D. O Regan, Homototy results for maps ith eakly sequentially closed graphs, Dynamics of Continuous, Discrete and Impulsive Systems (Ser A: Mathematical Analysis), to appear.