Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 152978, 16 pages
doi:10.1155/2010/152978
Research Article

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

Yeol Je Cho,1 Heng-you Lan,2 and Nan-jing Huang3

1Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea
2Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, China
3Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received 8 January 2010; Accepted 23 March 2010

Academic Editor: Yuming Xing

Copyright © 2010 Yeol Je Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.