A System of Nonlinear Operator Equations for a Mixed Family of Fuzzy and Crisp Operators in Probabilistic Normed Spaces

Yeolje Cho,Nan-Jing Huang,Heng-You Lan

doi:10.1155/2010/152978

Abstract

By using a random version of the theory of contractor introduced by Altman, we introduce and study a system of nonlinear operator equations for a mixed family of fuzzy and crisp operators in probabilistic normed spaces. We construct some new iterative algorithms for solving this kind of nonlinear operator equations. We also prove some new existence theorems of solutions of a new system of nonlinear operator equations for a mixed family of fuzzy and crisp operators and some new convergence results of sequences generated by iterative algorithms under joint orbitally complete conditions.

Highlights

Altman 1, 2 introduced the theory of contractor and contractor direction, which has a very strong significant for the study of existence and uniqueness for solving nonlinear operator equations in Banach spaces
Cho et al 10 introduced the concept of probabilistic contractor couple in probabilistic normed spaces and discuss the solution for nonlinear equations of fuzzy mappings and the convergence of sequences generated by the algorithms in Menger probabilistic normed spaces
Hadzicand Pap 16 introduced some new classes of probabilistic contractions in probabilistic metric spaces

Summary

Introduction

Altman 1, 2 introduced the theory of contractor and contractor direction, which has a very strong significant for the study of existence and uniqueness for solving nonlinear operator equations in Banach spaces. Cho et al 10 introduced the concept of probabilistic contractor couple in probabilistic normed spaces and discuss the solution for nonlinear equations of fuzzy mappings and the convergence of sequences generated by the algorithms in Menger probabilistic normed spaces. Hadzicand Pap 16 introduced some new classes of probabilistic contractions in probabilistic metric spaces They obtained a new fixed point theorem for the ψ-probabilistic contraction and gave some applications to random operators. We prove some new existence theorems of solution for the system of nonlinear operator equations for a mixed family of fuzzy and crisp operators and new convergence results of sequences generated by the iterative algorithms under joint orbitally complete condition. The results presented in this paper improve and generalize corresponding results of 9, 15–17

Preliminaries

Some Countable t-Norms

The Main Results