Mixed Variational-Like Inequality for Fuzzy Mappings in Reflexive Banach Spaces
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Research Article Mixed Variational-Like Inequality for Fuzzy Mappings in Reflexive Banach Spaces Poom Kumam1, 2 and Narin Petrot2, 3 1
Department of Mathematics, Faculty of Science, King Mongkuts University of Technology Thonburi, Bang Mod, Bangkok 10140, Thailand 2 Centre of Excellence in Mathematic, CHE, Sriayudthaya Road, Bangkok 10400, Thailand 3 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Narin Petrot, [email protected] Received 21 April 2009; Accepted 24 July 2009 Recommended by Vy Khoi Le Some existence theorems for the mixed variational-like inequality for fuzzy mappings FMVLIP in a reflexive Banach space are established. Further, the auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution. Consequently, not only the existence of solutions of the FMVLIP is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results. Copyright q 2009 P. Kumam and N. Petrot. 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.
1. Introduction The concept of fuzzy set theory was introduced by Zadeh 1. The applications of the fuzzy set theory can be found in many branches of mathematical and engineering sciences including artificial intelligence, control engineering, management sciences, computer science, and operations research 2. On the other hand, the concept of variational inequality was introduced by Hartman and Stampacchia 3 in early 1960s. These have been extended and generalized to study a wide class of problems arising in mechanics, physics, optimization and control, economics and transportation equilibrium, and so forth. The generalized mixed variational-like inequalities, which are generalized forms of variational inequalities, have potential and significant applications in optimization theory 4, 5, structural analysis 6, and economics 4, 7. Motivated and inspired by the recent research work going on these two different fields, Chang 8, Chang and Huang 9, Chang and Zhu 10 and Noor 11 introduced and studied the concept of variational inequalities and complementarity problems for fuzzy mappings in different contexts.
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Journal of Inequalities and Applications
It is noted that there are many effective numerical methods for finding approximate solutions of various variational inequalities e.g., the projection method and its variant forms, linear approximation, descent and Newton’s methods, and there are very few methods for general variational-like inequalities. For example, among the most effective numerical technique is the projection method and its variant forms; however, the projection type techniques cannot be extended for constructing iterative algo
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