Common Random Fixed Point theorem for compatible random multivalued operators

Transcription

1 IOSR Journal of Mathematics (IOSR-JM) ISSN: Volume, Issue (Sep-Oct. 0), PP9-4 Common Random Fixed Point theorem for compatible random multivalued operators Dr. Neetu Vishwakarma Abstract: The aim of this paper is to prove some common random fixed point theorem for two pairs of compatible random multivalued operators satisfying rational inequality. Keywords: Random fixed point, Compatible maps, Polish space. AMS Mathematics Subject Classification (000): 47H0, 54H5. I. Introduction The systematic study of random equations employing the methods of functional analysis was first initiated by Prague School of Probabilistic in 950's by Spacek [] and Hans [7,8]. In separable metric space, random fixed point theorems for contraction mappings was proved by Spacek [] and Hans [7,8]. Bharucha- Reid [6] generalized Mukherjee's [0] result on general probability measure space. For multivalued mappings Itoh [9] obtained random analogues of corresponding deterministic result for different classes of mappings. Papageoriou [], Beg [,], Beg and Shahzad [5] and Beg and Abbas [4] proved some common random fixed point and random coincidence point of a pair of compatible random operators. Preliminaries: Let (X,d) be a Polish space, that is a separable complete metric space and (, a) be a measurable space. Let x be the family of all subsets of X and CB(X) denote the family of all nonempty bounded closed subsets of X. A mapping T: x, is called measurable, if for any open subset C of X, T - (C) = {: T()C } a. A mapping : X is called measurable selector of a measurable mapping T: X, if is measurable and for any,() T(). A mapping T: X CB(X) is called random multivalued operator, if for every x X, T(.,x) is measurable. A mapping f: X X is called random operator, if for every x X, f(.,x) is measurable. A measurable mapping X, is called the random fixed point of a random multivalued operator T: X CB(X) (f: X X), if for every, () T(()) (f(()) = ()). A measurable mapping : X is a random coincident point of T: X CB(X) and f: X CB(X) if for any, f(, ()) = T(, ()). lim lim Mappings f, g : X X are compatible if n d(fg(x n ), gf(x n )) = 0, provided that n f(x n ) and lim lim lim n g(x n ) exists in X and n f(x n ) = n g(x n ). Random operators S, T: X X are compatible if S(,.) and T(,.) are compatible for each. (See Beg and Shahzad [5]) Main Result. Theorem. Let X be a Polish space and let (S, T) and (Q, T) be two pairs of compatible random multivalued operators from X CB(X) with S(, X) T(, X) and Q(, X) T(, X) for each and d(t(,x),s(,x)) d(t(, y),q(, y)) d(t(,x),s(,x)) d(t(, y),q(, y)) d(t(,x),s(,x)) d(t(, y),q(, y)) H S,x,Q, y ( ) d(t(,x),s(,x)) d(t(, y),q(, y)) ( ) + ()d(t(, x), T(, y)) 9 Page

2 for each x, y X and where,, : (0, ) are measurable mapping such that () + () + () <. If one of the random multivalued operators S, Q and T is continuous, then S, Q and T have unique common random fixed point. (Here H represents the Hausdorff metric on CB(X) induced by the metric d). Proof. Let 0 : X be an arbitrary measurable mapping and choose a measurable mapping : X such that S(, 0 ()) = T(, ()) for each. It further implies that there exists a measurable mapping : X such that for any and Q(, ()) = T(, ()). In general, we can choose measurable mappings n+ and n+ from X such that S(, n ()) = T(, n+ ()) Q(, n+ ()) = T(, n+ ()) for each and n = 0,,,.... Then for each, d(t(, n+ ()), T(, n+ ()) = H(S(, n ()), Q(, n+ ()) d(t(, n( )),S(, n( ))) d(t(, n ( )),Q(, n ( ))) n n n n d(t(, n( )),S(, n( ))) d(t(, n ( )),Q(, n ( ))) ( ) d(t(, ( )),S(, ( ))) d(t(, ( )),Q(, ( ))) ( ) d(t(, ( )),S(, ( ))) d(t(, ( )),Q(, ( ))) + ()d(t(, n ()), T(, n+ ())) n n n n d(t(, n( )),T(, n ( ))) d(t(, n ( )),T(, n ( ))) n n n n d(t(, n( )),T(, n ( ))) d(t(, n ( )),T(, n( ))) ( ) d(t(, ( )),T(, ( ))) d(t(, ( )),T(, ( ))) ( ) d(t(, ( )),T(, ( ))) d(t(, ( )),T(, ( ))) n n n n + ()d(t(, n ()), T(, n+ ())) d(t(, ( )),T(, ( ))) d(t(, ( )),T(, ( ))) n n n n ( ) d(t(, n ( )),T(, n ( ))) d(t(, n ( )),T(, n ( ))) d(t(, n( )),T(, n ( ))) d(t(, n ( )),T(, n( ))) ( ) d(t(, ( )),T(, ( ))) d(t(, ( )),T(, ( ))) n n n n + ()d(t(, n ()), T(, n+ ())) ()[d(t(, n ()), T(, n+ ())+d(t(, n+ ()), T(, n+ ()))] + ()[d(t(, n ()), T(, n+ ())) + d(t(, n+ ()), T(, n+ ()))] + ()d(t(, n ()), T(, n+ ())) (() + () + ()) d(t(, n ()), T(, n+ ())) + (() + ()) d(t(, n+ ()), T(, n+ ())) ( - () - ()) d(t(, n+ ()), T(, n+ ())) (() + () + ())d(t(, n ()), T(, n+ ())) d(t(, n+ ()), T(, n+ ())) kd(t(, n ()), T(, n+ ())) where 40 Page

3 ( ) ( ) ( ) k. ( ) ( ) Similarly, Common Random Fixed Point theorem for compatible random multivalued operators d(t(, n+ ()), T(, n+ ())) kd(t(, n+ ()), T(, n+ ()) In general, k d(t(, n ()), T(, n+ ()) d(t(, n ()), T(, n+ ())) k n d(t(, 0 ()), T(, ())). Furthermore m > n, d(t(, n ()), T(, m ()) d(t(, n ()), T(, n+ ())) + d(t(, n+ ()), T(, n+ ())) d(t(, m- ()), T(, m ())) k n d(t(, 0 ()), T(, ())) d(t(, n ()), T(, m ())) n k + k n+ d(t(, 0 ()), T(, ())) k m- d(t(, 0 ()), T(, ())) ( k) d(t(, 0 ()), T(, ())) 0 as n, m. Thus {T(, n ())} and {T(, m ())} are Cauchy sequence in CB(X), therefore there exists A() CB(X) such that {T(, n ())} A() for some. It further implies that {T(, n+ ())}, {S(, n ())} and {Q(, n+ ())} converges to A() for each. Let : X be a measurable mapping such that for each, () A(). Thus, we have T(, n+ ()) A(), S(, n ()) A() and Q(, n+ ()) A() as n. Now, suppose that T is continuous random multivalued operator, then T(, T(, n+ ())) T(, A()), T(, S(, n ())) T(, A()) for every Since pair (S, T) and (Q, T) are compatible random operator, Since pair (S, T) and (Q, T) are compatible random operator, then for each, we have T(, T(, n+ ())) T(, A()), S(, T(, n+ ())) T(, A()) and Q(, T(, n+ ())) T(, A()). Consider for each H(S(, T(, n ())), Q(, n+ ())) d(t(,t(, n( ))),S(,T(, n( ))) d(t(, n ( ))),Q(, n ( ))) n n n n d(t(,t(, n( ))),S(,T(, n( ))) d(t(, n ( ))),Q(, n ( ))) d(t(,t(, ( ))),S(,T(, ( ))) d(t(, ( ))),Q(, ( ))) ( ) d(t(,t(, ( ))),S(,T(, ( ))) d(t(, ( ))),Q(, ( ))) ( ) + ()d(t(, T(, n ())), T(, n+ ())). On taking limit n both sides, we get d(t(, A()), A()) n n n n 4 Page

4 d(t(,a( )),T(,A( ))) d(t(,a( )),T(,A( ))) d(t(,a( )),T(,A( ))) d(t(,a( )),T(,A( ))) d(t(,a( )),T(,A( ))) d(t(,a( )),T(,A( ))) ( ) d(t(,a( )),T(,A( ))) d(t(,a( )),T(,A( ))) ( ) + () d(t(, A()), A()) (-()) d(t(, A()), A()) 0 d(t(, A()), A()) = 0, T(, A()) = A() for each But () A(). Thus () T(, ()). Now, for any H(S(, A()), Q(, n+ ())) d(t(,a( )),S(,A( ))) d(t(, n ( )),Q(, n ( ))) n n d(t(,a( )),S(,A( ))) d(t(, n ( )),Q(, n ( ))) d(t(,a( )),S(,A( ))) d(t(, ( )),Q(, ( ))) ( ) d(t(,a( )),S(,A( ))) d(t(, ( )),Q(, ( ))) ( ) + () d(t(, A()), T(, n+ ())). Taking limit, we get d(s(, A()), A()) d(a( ),S(,A( ))) d(a( ),A( )) d(a( ),S(,A( ))) d(a( ),A( )) d(a( ),S(,A( ))) d(a( ),A( )) ( ) d(a( ),S(,A( ))) d(a( ),A( )) ( ) n n + () d(a(), A()) d(s(, A()), A()) ()d(a(, S(, A())) + () d(a(), S(, A())) ( - () - ()) d(s(, A()), A()) 0 But () A(). Thus, () S(, A()). Finally, H(S(, T(, A()), Q(, A())) d(s(, A()), A()) 0 S(, A()) = A() for each. d(t(,a( )),S(,A( ))) d(t(,a( )),Q(,A( ))) d(t(,a( )),S(,A( ))) d(t(,a( )),Q(,A( ))) d(t(,a( )),S(,A( ))) d(t(,a( )),Q(,A( ))) ( ) d(t(,a( )),S(,A( ))) d(t(,a( )),Q(,A( ))) ( ) + () d(t(, A()), T(, A())) d(a(), Q(, A())) d(a( ),A( )) d(a( ),Q(,A( ))) ( ) d(a( ),A( )) d(a( ),Q(,A( ))) 4 Page

5 ( ) d(a( ),A( )) d(a( ),Q(,A( ))) d(a( ),A( )) d(a( ),Q(,A( ))) + () d(a(), A()) d(a(), Q(, A())) ()d(a(), Q(, A())) + () d(a(), Q(, A())) ( - () - ()) d(a(), Q(, A())) 0 d(a(), Q(, A())) 0 Q(, A()) = A() for each. But () A(). Thus, () Q(, A()). Hence, () is a random fixed point of random multivalued operator S, Q and T. Uniqueness : To prove uniqueness of common random fixed point of random multivalued operator. Let, : X be two common random fixed point of random multivalued operators S, Q and T such that () = () for each. Consider for each d( (), ()) H(S(, ()), Q(, ())) Thus, d(t(, ( )),S(, ( ))) d(t(, ( )),Q(, ( ))) d(t(, ( )),S(, ( ))) d(t(, ( )),Q(, ( ))) d(t(, ( )),S(, ( ))) d(t(, ( )),Q(, ( ))) ( ) d(t(, ( )),S(, ( ))) d(t(, ( )),Q(, ( ))) ( ) + ()d(t(, ()), T(, ())) d( (), ()) ()d(t(, ()), T(, ())) (- ())d( (), ()) 0 d( (), ()) 0. () = () for each which is a contradiction, so the result follows. Corollary. Let X be a Polish space and let (S, P) and (T, Q) be two pairs of compatible random multivalued operators from X CB(X) with S(, X) Q(, X) and T(, X) P(, X) for each and and H(S(, x), T(, y)) d(p(,x),s(,x)) d(q(,y),t(,y)) ( ) d(p(,x),s(,x)) d(q(,y),t(,y)) ( ) d(p(,x),s(,x)) d(q(, y),t(, y)) d(p(,x),s(,x)) d(q(, y),t(, y)) + ()d(p(, x), Q(, y)) for each x, y X and, where, : (0, ) are measurable mappings such that () + () + () <. If one of the random multivalued operators P, Q, T or S is continuous then P, Q, S and T have unique common random fixed point (where H represents Hausdorff metric on CB(X) induced by metric d). References [.] Badshah, V.H. and Sayyed, Farkhunda, Common random fixed point of random multivalued operator on Polish spaces, Indian J. Pure App. Math. (4), Apr. 00, [.] Beg, I. Random fixed point of random operators satisfying semi-contractivity conditions, Mathematica Japonica 46(997), no., [.] Beg, I., Approximation of random fixed point in normed spaces, Nonlinear Analysis, 5 (00), No. 8, Page

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