RANDOM FIXED POINTS AND RANDOM APPROXIMATIONS IN NONCONVEX DOMAINS

Stochastic generalizations of some fixed point theorems on a class of nonconvex sets in a locally bounded topological vector space are established. As applications, Brosowski-Meinardus type theorems about random invariant approximation are obtained. This work extends or provides stochastic versions of several well known results. Random Fixed Point, Random Approximation, Nonexpansive


Introduction
Probabilistic functional analysis is an important mathematical discipline because of its applications to probabilistic models in applications.Random operator theory is needed for the study of various classes of random equations.The study of random fixed point theorems was initiated by the Prague school of probabilists in the 1950s.Random fixed point theorems and random approximations are stochastic generalizations of classical fixed point and approximation theorems, and find their applications in probability theory and nonlinear analysis.The random fixed point theory for self-maps and nonself-maps has been developed by various authors (see e.g., [2-3, 13, 19]).Recently, this theory has been further extended for 1-set contractive mappings that include condensing, nonexpansive, semicontractive and completely continuous random maps, etc.
Brosowski [4] initiated the study of invariant approximations using the fixed point theory and subsequently various generalizations of Brosowski's results have appeared in the literature (see for example, [7,12]).
In this paper, we prove some random fixed point theorems for nonexpansive random operators defined on a class of nonconvex sets containing the subclass of starshaped sets in a locally bounded topological vector space.As applications of our results, we obtain Brosowski-Meinardus type results on random invariant approximation (cf.[4,14]).These results are obtained in Section 3, while in Section 2 we recall certain technical deinitions and known results.

Preliminaries
A subset of a (real or complex) linear space is called if there exists at least W \ starshaped one point such that for all , ; is called a  known that the topology of any Hausdorff locally bounded topological vector space is given by some -norm.In the sequel, unless mentioned specifically otherwise, denotes a : \ Hausdorff separable complete locally bounded topological vector space (not necessarily locally convex) whose topology is generated by a -norm.Of course, if , then will : : oe " \ stand for a separable Banach space.
Let and a family of functions from into such that . Following Dotson [6], the family is said to be if there exists a α Y − W contractive function such that for all and all , we have The family is said to be jointly continuous (resp.
) if in and in (resp. in and jointly weakly continuous

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We observe that and it has the additional property that it is contractive, jointly [ Y © continuous and jointly weakly continuous.
If for a subset of , there exists a contractive jointly continuous family of functions W \ Y oe Ö0 × W α α−W (resp.a family of functions satisfying (*)), then we say that has the property of contractiveness and joint continuity (resp.property (*)).
Example 2.1: Any subspace, a convex set with , a starshaped subset with center and a !!cone of a -normed space have the family of functions associated with them which satisfy : condition (*).
Let be a random operator.If we restrict it to the family , then the , is a nonempty and compact set in .quasi-Chebyshev Q Every finite dimensional subspace of a Banach space is pseudo-Chebyshev.Moreover, \ every pseudo-Chebyshev subspace is quasi-Chebyshev but not conversely, in general (see [15] for more details).
We say that (the dual of is if for each nonzero , there exists . In this case, the weak topology of is Hausdorff ( [11]).We shall need the following results. (

Main Results
We begin with the following "random fixed point theorem" for nonexpansive operators which extends Theorem 1 of Dotson [6].
has a random fixed point .For each , define where is the set of all nonempty compact subsets of .
and hence it has a measurable selector .We now show that is the random fixed point of .0 0 X Fix .Since , therefore there exists a sub . The continuity of implies and hence, using and the joint continuity, we get As is Hausdorff, we get X Ð ß Ð ÑÑ oe Ð Ñ = 0 = 0 = .As an immediate consequence of the above theorem, we have the following Brosowksi-Meinardus type theorem on best random approximation and random fixed points.
As is nonexpansive, so : : .So for each .Therefore, is -invariant for each .Also, being Hence, each has a random fixed point (see [9]) and so for each , therefore there exists a subsequence of that = 0 = 0 = = .The weak continuity of implies and hence, using the joint weak continuity, we get . By the Hausdorff property of the weak topology, we get the required result X Ð ß Ð ÑÑ oe Ð ÑÞ = 0 = 0 = The following result generalizes Theorem 8 of Habinaik [7].Theorem 3.6: Let be a weakly continuous nonexpansive random opera-X À ‚ \ Ä \ H tor with deterministic fixed point .Assume that leaves a weakly compact subset of is compact and has the property .Then the point has a The proof is analogous to that of Theorem 3.2 and is therefore omitted.An affine continuous map is weakly continuous so we deduce from Theorem 3.5, the following random version of Corollary 1.10 of Veeramani [18].
Corollary 3.7: Let be a weakly compact convex subset of a separable Banach space .W \ If is an affine nonexpansive random operator, then has a random fixed Let be a weakly compact convex subset of a strictly convex separable Corollary 3.8: W Banach space .If is an isometric random operator, then has a random \ XÀ ‚W ÄW X H fixed point.
Proof: It is sufficient to show that is affine for each .Let and By the strict convexity of , we get , for each in \ XÐ ß DÑ oe >XÐ ß BÑ  Ð"  >ÑXÐ ß CÑ > Ò!ß "Ó = = = and .= H − An application of Corollary 3.7 leads to the following random analog of a result of Brosowski [4].
Similar to the proof of Theorem 3.10 with the exception that we apply Theorem B instead of Theorem A to show that has a fixed point for each .