# Monch type results for maps with weakly sequentially closed graphs.

1. INTRODUCTION

In this paper we discuss Monch type maps [1, 6]. We begin in Theorem 2.2 and present a new fixed point result for Monch type self maps with weakly sequentially closed graph. Next we define the notion of an essential map which is of Monch type and has weakly sequentially closed graph. We use this notion to present a homotopy type result for this class of maps.

2. FIXED POINT THEORY

First we recall the following result .

Theorem 2.1. Let Q be a nonempty, convex, weakly compact subset of a metrizable locally convex linear topological space E. Suppose F : Q [right arrow] K(Q) has weakly sequentially closed (graph; here K(Q) denotes the family of nonempty, convex, weakly compact subsets of Q. Then F has a fixed point in Q.

We now prove a result which will 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.sub.0] [member of] Q. Suppose F : Q [right arrow] K(Q) has weakly sequentially closed graph and F takes relatively weakly compact sets into relatively weakly compact sets. Also assume the following hold:

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then F has a fixed point in Q.

Remark 2.3. If E is a Banach space then (2.3) holds from the Krein-Smulian theorem [3, pg. 434, 5 pg. 82]. Note (2.3) holds if a Krein-Smulian 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 weakly compact subset of E and K with the relative weak topology is metrizable (for example E could be a Banach space whose dual [E.sup.*] is separable) then (2.2) holds (recall compact metric spaces are separable).

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now for n = 0, 1, ... notice [D.sub.n] is convex and

[D.sub.0] [subset or equal to] [D.sub.1] [subset or equal to] ... [D.sub.n-1] [subset or equal to] [D.sub.n] ... [subset or equal to] Q.

Note D is convex and since (Dn) is increasing we have

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We claim [D.sub.n] is relatively weakly compact for n [member of] {0, 1, ...}. Certainly it is true if n = 0. Now suppose [D.sub.k] is relatively weakly compact for some k [member of] {0, 1, ...}. Now since F takes relatively weakly compact sets into relatively weakly compact sets then F([D.sub.k]) is relatively weakly compact. This together with (2.3) guarantees [D.sub.k+1] is relatively weakly compact.

Now (2.2) implies that for each n [member of] {0, 1, ...} there exists [C.sub.n] with [C.sub.n] countable, [C.sub.n] [member of] [D.sub.n], and = [bar.[C.sub.n.sup.w]] = [bar.[D.sub.n.sup.w]] Let C = [[union].sup.[infinity].sub.n=0] [C.sub.n]. Now since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus [bar.[C.sup.w]] = [bar.[D.sup.w]] so from (2.1) (see (2.4)) we have that [bar.[D.sup.w]] is weakly compact. Consider the map [F.sup.*] : [bar.[D.sup.w]] [right arrow] K([bar.[D.sup.w]]) given by

[F.sup.*](x) = F(x) [intersection] [bar.[D.sup.w]].

We first show [F.sup.*](x) 0 for each x [member of] [bar.[D.sup.w]]. Note from (2.4) that F(D) [subset or equal to] D [subset or equal to] [bar.[D.sup.w]] so D [subset or equal to] [F.sup.-1]([bar.[D.sup.w]]). Now let x [member of] [bar.[D.sup.w]]. Now since [bar.[D.sup.w]] is weakly compact the Eberlein-Smulian theorem [4 pg. 549] guarantees that there is a sequence ([x.sub.n]) in D with [x.sub.n] [??] x (here [?/] denotes weak convergence). Take any [y.sub.n] [member of] F([x.sub.n]). Now since F(D) [subset or equal to] D we have [y.sub.n] [member of] D. Also since [bar.[D.sup.w]] is weakly compact the Eberlein-Smulian theorem [4 pg. 549] guarantees that we may assume without loss of generality that [y.sub.n] [??] y for some y [member of] [bar.[D.sup.w]]. Note [y.sub.n] [member of] F([x.sub.n]), [x.sub.n] [??] x, [y.sub.n] [??] y implies y [member of] F(x) since F has weakly sequentially closed graph. Thus y [member of] F(x) [intersection] [bar.[D.sup.w]] so x [member of] [F.sup.-1] ([bar.[D.sup.w]]). As a result [bar.[D.sup.w]] [subset or equal to] [F.sup.-1]([bar.[D.sup.w]]) i.e. [F.sup.*](x) [not equal to] 0 for each x [member of] [bar.[D.sup.w]].

Note [F.sup.*] : [bar.[D.sup.w]] [right arrow] K([bar.[D.sup.w]]) has weakly sequentially closed graph. Now Theorem 2.1 guarantees a x [member of] [bar.[D.sup.w]] with x [member of] [F.sup.*](x) [subset or equal to] 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 weakly open subset of C with 0 G U.

Definition 3.1. F [member of] A([bar.[U.sup.w]], C) if F : [bar.[U.sup.w]] [right arrow] K(C) has weakly sequentially closed graph, F takes relatively weakly compact sets into relatively weakly compact sets, and F satisfies the following condition: if D [subset or equal to] [bar.[U.sup.w]] and D [subset or equal to] [bar.co]({0} [union] F(D)) with [bar.[D.sup.w]] = [bar.[D.sup.w]] and C [subset or equal to] D countable, then [bar.[D.sup.w]] is weakly compact.

Definition 3.2. We say F [member of] [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C) if F [not member of] A([bar.[U.sup.w]], C) with x [not member of] F{x) for x [member of] [partial derivative]U; here [partial derivative]U denotes the weak boundary of U in C.

Definition 3.3. A map F [member of] [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C) is essential in [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C) if for every G [member of] [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C) with G[|.sub.[partial derivative]U] = F[|.sub.[partial derivative]U] there exists x [member of] U with x [member of] 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 weakly open subset of C with 0 [member of] U. Suppose F [member of] A([bar.[U.sup.w]], C) and assume the following conditions hold:

(3.1) the zero map is essential in [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C)

and

(3.2) x [not member of] A F x for every x [member of] [partial derivative]U and [lambda] [member of] (0,1].

Then F is essential in [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C).

Proof. Let H [member of] [A.sub.[partial derivative]U]([bar.[U.sup.w]], C) with [H.sub.|[alpha]U] = F[|.sub.[partial derivative]U]. We must show H has a fixed point in U. Consider

B = {x [member of] [bar.[U.sup.w]] : x [member of] tH(x) for some t [member of] [0,1]}.

Now B [not equal to] 0 since 0 [member of] U. Also B is weakly sequentially closed. To see this let ([x.sub.n]) be sequence of B which converges weakly to some x [member of] [bar.[B.sup.w]] (in particular x [member of] [bar.[U.sup.w]]) and let ([[lambda].sub.n]) be a sequence of [0,1] satisfying [x.sub.n] [member of] [[lambda].sub.n]H[x.sub.n]. Then for each n there is a [z.sub.n] [member of] H[x.sub.n] with [x.sub.n] = [[lambda].sub.n][z.sub.n]. By passing to a subsequence if necessary, we may assume that ([[lambda].sub.n]) converges to some [lambda] [member of] [0,1] and without loss of generality assume [[lambda].sub.n] [not equal to] 0 for all n. This implies that the sequence ([z.sub.n]) converges weakly to some z [member of] [bar.[U.sup.w]] with x = [lambda]z. Since F has weakly sequentially closed graph then z [member of] H (x). Hence x [member of] [lambda]Hx and therefore x [member of] B. Thus B is weakly sequentially closed.

Let [{[x.sub.n]}.sup.[infinity].sub.n=1] be a sequence in B. Then there exists a sequence [{[t.sub.n]}.sup.[infinity].sub.n=1] in [0,1] with [x.sub.n] [member of] [t.sub.n]H[x.sub.n] and we may assume without loss of generality that [t.sub.n] [right arrow] t [member of] [0,1]. Let C = [{[x.sub.n]}.sup.[infinity].sub.n=1]. Note C is countable and C [subset or equal to] co(H(C) [union] {0}). Since H [member of] A([bar.[U.sup.w]], C) then [bar.[C.sup.w]] is weakly compact. The Eberlein-Smulian theorem [4 pg. 549] guarantees that there is a subsequence N of {1, 2, ...} and a x [member of] [bar.[C.sup.w]] with [x.sub.n] [??] x as n [right arrow] [infinity] in N. Now since B is weakly sequentially closed we have x [member of] B. Consequently B is weakly sequentially compact, so weakly compact by the Eberlein-Smulian theorem [3, pg. 430].

Now B [intersection] [partial derivative]U = 0 since (3.2) holds; note H[|.sub.[partial derivative]U] = F[|.sub.[partial derivative]U] and 0 G U. Now E = (E, w), the space E endowed with the weak topology, is completely regular. This there exists a weakly continuous map [mu] : [bar.[U.sup.w]] [right arrow] [0, 1] with [mu]([partial derivative]U) = 0 and [mu](B) = 1. Define a map [R.sub.[mu]] : [bar.[U.sup.w]] [right arrow] K(C) by [R.sub.[mu]](x) = [mu](x)H(x). Note [R.sub.[mu]] has weakly sequentially closed graph (since H has weakly sequentially closed graph) and takes relatively weakly compact sets into relatively weakly compact sets. [If A [subset or equal to] [bar.[U.sup.w]] is weakly compact and [y.sub.n] [member of] [R.sub.[mu]](A), then [y.sub.n] = [mu]([x.sub.n])[z.sub.n] where [z.sub.n] [member of] H([x.sub.n]) and [x.sub.n] [member of] A. Without loss of generality we may assume there exists x [member of] A and z [member of] [bar.H[(A).sup.w]] with [x.sub.n] [??] x and [z.sub.n] [??] z (recall A and [bar.H[(A).sup.w]] are weakly compact; in fact from a standard result  we note that H : A [right arrow] K(C) has weakly closed graph and from another standard result  we have that H(A) is weakly compact). Then z [member of] H(x) since H has weakly sequentially closed graph. Let y = [mu](x)z. Then [y.sub.n] [??] y and y [member of] [R.sub.[mu]](A). As a result [R.sub.[mu]](A) is weakly compact.] Next suppose D [subset or equal to] [bar.[U.sup.w]] with D [subset or equal to] [bar.co]({0} [union] [R.sub.[mu]](D)) and with [bar.[D.sup.w]] = [bar.[C.sup.w]] and C [subset or equal to] D countable. Then since [R.sub.[mu]](D) [subset or equal to] co({0} [union] H{D)) and {0} [union] co({0} [union] H(D)) = co({0} [union] H(D)) we have

D [subset or equal to] [bar.co] ({0} [union] [R.sub.[mu]](D)) [subset or equal to] [bar.co] (co ({0} [union] H(D))) = [bar.co] ({0} [union] F(D)).

Then [bar.[D.sup.w]] is weakly compact since H [member of] A([bar.[U.sup.w]], C). Thus [R.sub.[mu]] [member of] A([bar.[U.sup.w]], C) with [R.sub.[mu]/[partial derivative]U] = {0}. Now (3.1) guarantees that there exists x [member of] U with x [member of] [R.sub.[mu]](x). As a result x [member of] B, so [mu](x) = 1 i.e. x [member of] H(x).

Next we 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 weakly open subset of C with 0 [member of] U. Suppose (2.2) and (2.3) hold. Then (3.1) holds.

Proof. Let [theta] [member of] [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C) with [theta][|.sub.[partial derivative]U] = {0}. We must show [theta] has a fixed point in U. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note J : C [right arrow] K(C) has weakly sequentially closed graph. Now suppose A [subset or equal to] C, A = co({0} [union] J(A)) with [bar.[A.sup.w]] = [bar.[D.sup.w]] and D [subset or equal to] A countable. Then

(3.3) A [subset or equal to] [bar.co]({0} [union] [theta](U [intersection] A))

and so

(3.4) U [intersection] A [subset or equal to] W({0} [union] [theta](U [intersection] A)).

Notice D [intersection] U is countable, D [intersection] U [subset or equal to] A [intersection] U and

(3.5) [bar.D [intersection] [U.sup.w]] = [bar.A [intersection] [U.sup.w]]

since

D [intersection] U C A [intersection] U C [bar.[A.sup.w]] [intersection] U = [bar.[D.sup.w]] [intersection]

(note for sets [D.sub.0] and [D.sub.1] of C with [D.sub.0] weakly open in C, then [D.sub.0] [intersection] [bar.[D.sub.1.sup.w]] [subset or equal to] [bar.[D.sub.0] [intersection] [D.sub.1.sup.w]]). Now (3.4) and (3.5) and 9 G A(TN, C)) implies that Af]Uw is weakly compact. Also since [theta] [member of] A([bar.[U.sup.w]], C) (9 takes relatively weakly compact sets into relatively weakly compact sets) we have that 9(A n U) is relatively weakly compact. Now (2.3) guarantees that [bar.co]({0} [intersection] [bar.[theta][(A [intersection] U).sup.w]]) is weakly compact. This together with (3.3) implies that [bar.[A.sup.w]] is weakly compact.

Now Theorem 2.2 guarantees that there exists x [member of] C with x [member of] J(x). If x [member of] U we have x [member of] J(x) = {0}, which is a contradiction since 0 [member of] U. Thus x [member of] U so x [member of] J(x) = [theta](x).

Combining Theorem 3.4 and Theorem 3.5 yields the following nonlinear alternative of Leray-Schauder type.

Theorem 3.6. Let E be a metrizable locally convex linear topological space, C a closed convex subset of E, and U a weakly open subset of C with 0 [member of] U. Suppose (2.2) and (2.3) hold. Also suppose F [member of] A(UW,C) satisfies (3.2). Then F is essential in [A.sub.[partial derivative]U] ([bar.[U.sup.w]], C) (in particular F has a fixed point in U).

Received June 5, 2014

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DONAL O'REGAN

School of Mathematics, Statistics and Applied Mathematics National University of Ireland

Galway Ireland

donal.oregan@nuigalway.ie
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