On the existence and uniqueness problems of solutions for set‐valued and single‐valued nonlinear operator equations in probabilistic normed spaces

Shih-Sen Chang,Fan Wang,Yeol Je Cho

doi:10.1155/s0161171294000530

Abstract

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 single‐valued nonlinear operator equations in Menger probabilistic normed spaces.

Highlights

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 single-valued nonlinear operator equations in Menger probabilistic normed spaces
Menger PN-space and A be a t-norm of a h-type, let T: Xfx satisfy the following condition: FTx, Ty(t) >_ min{Fx y(O(t)),Fx Tx(o(t)),Fy_Ty(O(t))}
(1) In Theorem 4.2, if we assume that A(t,t)>_t for all rE[0,1], by Remark 3, (4.1) can be weakened as follows: FTx_Ty(t) >_ min{Fx y(O(t)),F_Tx(o(t)),Fy_Ty(o(t)),Fy_Tz(o(t)),Fx Ty((t))}

Summary

Introduction

H(t) ,t 0 if and only if x 0; (-)for (PN 3) Fizz(t) F 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)

Results

Conclusion