Fuzzy Sets and Systems 160 (2009) 3166 – 3174 www.elsevier.com/locate/fss On random variational inclusions with random fuzzy mappings and random relaxed cocoercive mappings Rais Ahmad a , ∗ , A.P. Farajzadeh b a Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India b Department of Mathematics, Razi University, Kermanshah 67149, Iran Received 26 February 2008; received in revised form 9 January 2009; accepted 9 January 2009 Available online 17 January 2009 Abstract In this paper, we introduce and study the random variational inclusions with random fuzzy and random relaxed cocoercive mappings. We define an iterative algorithm for finding the approximate solutions of this class of variational inclusions and establish the convergence of iterative sequences generated by proposed algorithm. Our results improve and generalize many known corresponding results. © 2009 Elsevier B.V. All rights reserved. Keywords: Variational inclusions; Algorithm; Random fuzzy mappings; Random relaxed cocoercive mappings 1. Introduction It is well known that fuzzy set theory, which was introduced by Zadeh [25] in 1965, has gained importance in analysis, from both theoretical and practical point of view. Fuzzy sets are distinguished from ordinary or crisp sets in that the degree of membership of an element in a fuzzy set can be any number in the unit interval, [0,1] as opposed to a number from the binary pair {0, 1} for crisp sets. This property of fuzzy sets enables us to represent realistically, imprecise concepts in which the transition from nonmembership to membership is gradual. The theory of variational inequality describes a broad spectrum of interesting and important developments involving a link among various fields of mathematics, physics, economics and engineering sciences, etc. This theory is widely used as a mathematical programming tool in modeling many optimization and decision making problems. However, facing uncertainty is a constant challenge for optimization and decision making. Treating uncertainty by fuzzy mathematics results in the study of fuzzy optimization and decision making. In 1989, Chang and Zhu [4] first introduced the concept of variational inequalities for fuzzy mappings. Since then several classes of variational inequalities with fuzzy mappings were considered by Chang and Huang [7], Noor [22], Huang [16], Park and Jeong [23,24], Ding and Park [13] and Ding [12], etc. in Hilbert spaces. Recently, Huang and Lan [17] considered nonlinear equations with fuzzy mapping in fuzzy normed spaces and Lan and Verma [21] considered fuzzy variational inclusion problems in Banach spaces. ∗ Corresponding author. Tel.: +919358973995. E-mail addresses: raisain_123@rediffmail.com (R. Ahmad), ali-ff@sci.razi.ac.ir (A.P. Farajzadeh). 0165-0114/$-see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.01.002