ON THE EXISTENCE AND UNIQUENESS PROBLEMS OF SOLUTIONS FOR SET-VALUED AND SINGLE-VALUED NONLINEAR OPERATOR EQUATIONS IN PROBABILISTIC NORMED SPACES

In this paper, we introduce the concept of more general probabilistic contractors in probabilistic normed spaces and show the existence and uniqueness of solutions for set-valued and

A triplet (X,,A) is called a Menger probabilistic normed space (briefly, a Menger PN-space) if X is a real vector space, is a mapping from X into (for X, the distribution function (z) is denoted by F and Fx(t) is the value of F at tR) and A is a t-norm satisfying the following conditions: (PN-1) Fx(O)=0; (PN 2) F x(t H(t) for all _> 0 if and only if x 0; F (-)for all R,c, # 0; (PN 3) Fizz(t) (PN-4) F x + y(t + t2) _> (AFx(tl),Fy(t2)) for all z,y X and tl,t 2 +.
Note that if (X,,A) is a Menger PN-space with the t-norm A satisfying the following condition: sup A(t,t) 1, (2.1) 0<t<l then (X,q,A) is a real metrizable Hausdorff vector topological space with the topology induced by the family of neighborhoods, where {Uy(, A):y x, > 0,A > 0}, ( Uy(,A) {x X:Fz_y()> Let (X,,A) be a Menger PN-space with the t-norm A satisfying the condition (2.1) and fX be a family of all nonempty probabilistically bounded r-closed subsets of X.For any given Then, from the definitions of FA, B(t and FA(t), we have the following: LEMMA 2.1.Let (X,'Y,A) be a Menger P/V-space (resp., an N.A. Menger P/V-space) with the t-norm A satisfying the condition (2.1) and A ft X.Then we have the following: (1) FA(0) 0; (2) FA(t for all > 0 if and only if e A; (3) FAA(t FA(---V) for all e ,$ # 0; (4) For any A,B X and e B, FA(t FA, B(t) for all R; (5) If the t-norm A is continuous, then we have FA+z(t +t2) A(Fz(tl),FA(t2) (resp., FA+z(maz{tl,t2}) A(Fz(tl), FA(t2))) for M1 tl, t2q + d zX.
Recl hat a sequence {n} in x converges o a poin 6 X in he opology r (denoted by lira Fn_ (t) n(t) for all > O.
A sequence {zn} in X is called a r-Cauchy sequence in x if llm /"x (t) H(t) for all > 0.
The ,i)a(' X is ,all to 1), r-COml)lcte if ,vcrv r-Cam hv sequence' in x converges to a point in the topology DEFINITION 2.1.Let (.\', ,A) and (t',ff.A) be two Menger PN-spaces with the t-norm A satisfying the condition (2.1).Let and be the topologies induced by the family of neighborhoods of the type (2.2) on (X,,A) and (I',,A), respectively.A set-valued mapping P:D(P) C XFt t. (resp.a single-valued mapping P:D(P) C X---,Y) is said to be r-closed if for any Zn D(P) and ln P(xn) (resp.yn= P(gn)), whenever znz and n-, we have z D(P) and u P(z)(resp.u P(z)).DEFINITION 2.2.A function 0:[0, + .)-.[0, + ) is said to satisfy the condition (el,) if it is nondecreasing, (0) 0 and h,,, vn(t) +o for all > 0.
In this section, assume that (X, ,A) is a rl-complete Menger PN-space, (Y,,A) is a r 2- complete Menger PN-space, A is a t-norm of h-type, and f. is a nonempty family of probabilistically bounded r:2-closed subsets of Y.
For the single-vahw,1 ,nalpiug P, we have the similar inequalities (3.2) and (3.3) which are equal to (3.1 b).Now we are ready to show the existence and uniqueness of solutions for the set-valued nonlinear operator equation P(,).
(3.4) THEOREM 3.1.Let (X,',A) be a rl-complete N.A. Menger PN-space, (Y,q,A) be a 2- complete Menger PN-space and A be of a h-type.Let P:D(P)C X-ray be a r-closed set-valued mapping.Suppose that F:X--L(Y,X)satisfies the following conditions: + F(x)y E D(P) for all D(P) and Y; F is a probabilistic contractor of P with respect to u, i.e., F satisfies the condition (i) (2) (3.1a); ( There exists a constant M > 0 such that, for any E D(P) and v E Y, F(z)y (t)>_ Fy ()for all t> 0; For any A, B Dy and a A, there exists a point e B snch that Fa_b(t > FA, B(t for all > 0. Then the nonlinear set-valued operator equation (3.4) has a solution x* in D(P).Further, the sequence {xn} defined by Zn + Xn F(zn)y n converges to the solution z* in the topology r 1.
PROOF.Suppose that the conclusion is not true.Then there exists a number O > 0 such that.
THE PROOF OF THEOREM 3.1.