PROPERTIES OF FIXED POINT SET OF A MULTIVALUED MAP

The study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [17]. Later, an interesting and rich fixed point theory for such maps has been developed. The theory of multivalued maps has applications in control theory, convex optimization, differential inclusions, and economics (see, e.g., [3, 8, 16, 22]). The theory of multivalued nonexpansive mappings is harder than the corresponding theory of single-valued nonexpansive mappings. It is natural to expect that the theory of nonself-multivalued noncontinuous functions would be much more complicated. The concept of a ∗-nonexpansive multivalued map has been introduced and studied by Husain and Latif [9] which is a generalization of the usual notion of nonexpansiveness for single-valued maps. In general, ∗-nonexpansive multivalued maps are neither nonexpansive nor continuous (see Example 3.7). Xu [22] has established some fixed point theorems while Beg et al. [2] have recently studied the interplay between best approximation and fixed point results for ∗-nonexpansive maps defined on certain subsets of a Hilbert space and Banach space. For this class of functions, approximating sequences to a fixed point in Hilbert spaces are constructed by Hussain and Khan [10] and its applications to random fixed points and best approximations in Fréchet spaces are given by Khan and Hussain [12]. In this paper, using the best approximation operator, we (i) establish certain properties of the set of fixed points of a ∗-nonexpansive multivalued nonself-map in the setup of a strictly convex Banach space, (ii) prove fixed point results for ∗-nonexpansive random maps in a Banach space under several boundary conditions, and (iii) provide affirmative answers to the questions posed by Ko [14] and Xu and Beg [24] related to the set of fixed points.


Introduction
The study of fixed points for multivalued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [17].Later, an interesting and rich fixed point theory for such maps has been developed.The theory of multivalued maps has applications in control theory, convex optimization, differential inclusions, and economics (see, e.g., [3,8,16,22]).
The theory of multivalued nonexpansive mappings is harder than the corresponding theory of single-valued nonexpansive mappings.It is natural to expect that the theory of nonself-multivalued noncontinuous functions would be much more complicated.
The concept of a * -nonexpansive multivalued map has been introduced and studied by Husain and Latif [9] which is a generalization of the usual notion of nonexpansiveness for single-valued maps.In general, * -nonexpansive multivalued maps are neither nonexpansive nor continuous (see Example 3.7).
Xu [22] has established some fixed point theorems while Beg et al. [2] have recently studied the interplay between best approximation and fixed point results for * -nonex- pansive maps defined on certain subsets of a Hilbert space and Banach space.For this class of functions, approximating sequences to a fixed point in Hilbert spaces are constructed by Hussain and Khan [10] and its applications to random fixed points and best approximations in Fréchet spaces are given by Khan and Hussain [12].
In this paper, using the best approximation operator, we (i) establish certain properties of the set of fixed points of a * -nonexpansive multivalued nonself-map in the setup of a strictly convex Banach space, (ii) prove fixed point results for * -nonexpansive random maps in a Banach space under several boundary conditions, and (iii) provide affirmative answers to the questions posed by Ko [14] and Xu and Beg [24] related to the set of fixed points.

Notations and preliminaries
Let C be a subset of a normed space X.We denote by 2 X , C(X), K(X), CC(X), CK(X), and CB(X) the families of all nonempty, nonempty closed, nonempty compact, nonempty closed convex, nonempty convex compact, and nonempty closed bounded subsets of X, respectively.
Define d(x,C) = inf y∈C d(x, y).The Hausdorff metric on CB(X) induced by the metric d on X is denoted by H and is defined as (2.1) Tx,T y) < d(x, y) whenever x = y in C, then T is called a strictly nonexpansive mapping [14].
A multivalued map T : C → 2 X is said to be (i) * -nonexpansive if for all x, y ∈ C and u x ∈ Tx with d(x,u x ) = d(x,Tx), there exists u y ∈ T y with d(y,u y ) = d(y,T y) such that d(u x ,u y ) ≤ d(x, y) (see [9,10]), (ii) strictly * -nonexpansive if for all x = y in C and u x ∈ Tx with d(x,u x ) = d(x,Tx), there exists u y ∈ T y with d(y,u y ) = d(y,T y) such that d(u x ,u y ) < d(x, y), (iii ) upper semicontinuous (usc) (lower semicontinuous (lsc) If T is both usc and lsc, then T is continuous, (iv) asymptotically contractive [19] if there exist some c ∈ (0,1) and r > 0 such that where B X is the closed unit ball of X.The map T : C → CB(X) is called (i) H-continuous (continuous with respect to Hausdorff metric H) if and only if for any sequence {x n } in C with x n → x, we have H(Tx n ,Tx) → 0 (the two concepts of set-valued continuity are equivalent when T is compact-valued (cf.[8, Theorem 20.3, page 94])); (ii) demiclosed at 0 if the conditions x n ∈ C, x n converges weakly to x, y n ∈ Tx n and y n → 0 imply that 0 ∈ Tx.An element x in C is called a fixed point of a multivalued map T if and only if x ∈ Tx.The set of all fixed points of T will be denoted by F(T).
For each x ∈ X, let well defined and is called the metric projection.
The space X is said to have the Oshman property (see [18]) if it is reflexive and the metric projection on every closed convex subset is usc.
For the multivalued map T and each x ∈ C, we follow Xu [22] to define the best approximation operator, P T (x) = {u x ∈ Tx : d(x,u x ) = d(x,Tx)}, (possibly empty set).
A single-valued (multivalued) map f :

Abdul Rahim Khan 325
The space X is said to have the Opial condition if for every sequence {x n } in X weakly convergent to x ∈ X, the inequality holds for all y = x.Every Hilbert space and the spaces p (1 ≤ p < ∞) satisfy the Opial condition.
The inward set and γ > 0}.We will denote the closure of C by cl(C).
Let (Ω,Ꮽ) denote a measurable space with Ꮽ sigma algebra of subsets of Ω.A mapping A random operator T : The following results are needed.
) is demiclosed at 0, then the fixed point set function F of T given by F(ω) = {x ∈ C : x ∈ T(ω,x)} is measurable (and hence T has a random fixed point).

* -nonexpansive maps
The properties of the set of fixed points of single-valued and multivalued maps have been considered by a number of authors (see, e.g., Agarwal and O'Regan [1], Browder [4], Bruck [5], Espínola et al. [6], Ko [14], Schöneberg [20], and Xu and Beg [24]).For a wide class of unbounded closed convex sets C in a Banach space, there exist nonexpansive maps T : C → K(C) which fail to have a fixed point (see [13]).We obtain some properties of the set of fixed points of a * -nonexpansive map on a Banach space with values which are not necessarily subsets of the domain.
Markin [17], Xu [22], and Jachymski [11] have utilized "selections;" we employ "nonexpansive selector," P T , of a * -nonexpansive map T to study the structure of the set of fixed points of T. Consequently, we obtain generalized and improved versions of many results in the current literature.
In Theorem 8.2, Browder [4] has established the following result.
Theorem 3.1.Let C be a nonempty closed, convex, subset of a strictly convex Banach space X and let T : C → C be a nonexpansive map.Then the set F(T) of fixed points of T is closed and convex.
Ko [14] pointed out that Theorem 3.1 need not hold for multivalued nonexpansive mappings as follows.(3.1) Note that T is nonexpansive and the norm in R 2 is strictly convex.But the set F(T) = {(x, y) : (x, y) ∈ C and xy = 0} is not convex.
The following generalization of Theorem 3.1 for * -nonexpansive continuous mappings is obtained in [15].Theorem 3.3.Let X be a strictly convex Banach space and C a nonempty weakly compact convex subset of X.Let T : We present a new proof, through the best approximation operator, of Theorem 3.3 without assuming any type of continuity of the map T and obtain the following structure theorem.
Theorem 3.4.Let X be a strictly convex Banach space and C a nonempty weakly compact convex subset of X.Let T : C → CC(C) be a * -nonexpansive map such that F(T) is nonempty.Then the set F(T) is closed and convex.Proof.For each x ∈ C, its image Tx is weakly compact and convex and thus each Tx is Chebyshev.Hence, each u x in P T (x) is unique.Thus by the definition of * -nonexpansiveness of T, there is u y = P T (y) ∈ T y for all y in C such that Hence, P T : C → C is a nonexpansive selector of T (see also [22]).By the definition of P T , we have for each y ∈ C, d y,P T (y) = d y,u y = d(y,T y). ( 3) now implies that F(T) = F(P T ).Thus F(P T ) and hence F(T) is closed and convex by Theorem 3.1.
The following example illustrates our results.
Abdul Rahim Khan 327 Example 3.5.Let T : [0,1] → 2 [0,1] be a multivalued map defined by Then This implies that T is a * -nonexpansive map.Further, T is usc but not lsc (see [8, Remark 15.2, page 71]) and hence T is not continuous according to both definitions as T is compact-valued.Note that If T is a single-valued strictly nonexpansive map, then F(T) is a singleton.In general, this is not true for a multivalued nonexpansive map [17].The set F(T) is said to be singleton in a generalized sense if there exists x ∈ F(T) such that F(T) ⊆ Tx.Ko has given an example of a strictly nonexpansive mapping T : C → CC(C), in a strictly convex Banach space, for which the set F(T) is not singleton in a generalized sense (cf.[17,Example 4]).Ko raised the following question: is F(T) singleton in a generalized sense if T is nonexpansive, I is the identity operator, and I − T is convex?
The following proposition provides an affirmative answer to this question for strictly * -nonexpansive multivalued mappings.Proposition 3.6.Let C be a nonempty closed convex subset of a reflexive strictly convex Banach space X and let T : C → CC(C) be a strictly * -nonexpansive map such that F(T) is nonempty.Then the set F(T) is singleton in a generalized sense.
Proof.Any closed convex subset of a reflexive strictly convex Banach space is Chebyshev, so each Tx is Chebyshev.Thus as in the proof of Theorem 3.4, P T : C → C is a strictly nonexpansive selector of T satisfying (3.3).Hence, F(T) = F(P T ) is singleton in a generalized sense as required.
The following example supports the above proposition.
Example 3.7.Let T : [0,1] → 2 [0,1] be a multivalued map defined by , x = 1 2 . (3.6) Then P T (x) = {1/2} for every x ∈ [0,1].This implies that T is a strictly * -nonexpansive map: This implies that T is not nonexpansive.Obviously, T is compact-valued.Next we show that T is not lsc.Let V 1/4 be any small open neighbourhood of 1/4.Then the set is not open.Thus T is not continuous in the sense of both definitions.Note that F(T) = {1/2} is singleton in a generalized sense.
The conclusion of Proposition 3.6 does not hold for * -nonexpansive maps as follows.
Example 3.8.Let C = [0,∞) and T : C → CK(C) be defined by Then P T (x) = {x} for every x ∈ C.This clearly implies that T is * -nonexpansive but not nonexpansive (cf.[22]).Note that F(T) = C and there does not exist any x in F(T) such that F(T) ⊆ Tx.Thus F(T) is not singleton in a generalized sense.
The above example also indicates that the fixed point set of a * -nonexpansive map need not be bounded in general.However, if T is asymptotically contractive, then we have the following affirmative result.Theorem 3.9.Let X be a uniformly convex Banach space and C a nonempty closed convex subset of X.Let T : C → CC(C) be a * -nonexpansive map which is asymptotically contractive on C. Then F(T) is nonempty closed, convex, and bounded.
Proof.The map T has a nonexpansive selector f which is also asymptotically contractive by the asymptotic contractivity of T. Further, F(T) = F( f ) is nonempty closed bounded and convex (see [19,Corollary 3 and Remark (a)]).
We are now ready to derive a version of the Ky-Fan best approximation theorem [7] (compare the result with [10, Theorem 3.1] and [21,Theorem 4.3]).(3.10)

Theorem 3 . 10 .
Let C be a nonempty closed convex subset of a strictly convex Banach space X with the Oshman property.If T : C → CC(X) is an H-continuous (or a * -nonexpansive) map and T(C) is relatively compact, then there exists y ∈ C such that d(y,T y) = y − f y = d f y,cl I C (y) , for some continuous selector f of T.