Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansiv
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Research Article Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings Yongfu Su, Meijuan Shang, and Xiaolong Qin Received 1 July 2007; Accepted 3 October 2007 Recommended by Simeon Reich
We show that the general variational inequalities are equivalent to the general WienerHopf equations and use this alterative equivalence to suggest and analyze a new iterative method for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality involving multivalued relaxed monotone operators. Our results improve and extend recent ones announced by many others. Copyright © 2007 Yongfu Su 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. 1. Introduction Variational inequalities introduced by Stampacchia [1] in the early sixties have witnessed explosive growth in theoretical advances, algorithmic development, and applications across all disciplines of pure and applied sciences (see [1, 2] and the references therein). It combines novel theoretical and algorithmic advances with new domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis, and numerical analysis. In recent years, variational inequality theory has been extended and generalized in several directions, using new and powerful methods, to study a wide class of unrelated problems in a unified and general framework. In 1988, Noor [3] introduced and studied a new class of variational inequalities involving two operators, which is known as general variational inequality. We remark that the general variational inequalities are also called Noor variational inequalities. It turned out that oddorder, nonsymmetric obstacle, free, unilateral, nonlinear equilibrium, and moving boundary problems arising in various branches of pure and applied sciences can be studied via Noor variational inequalities (see [3–5]). On the other hand, in 1997, Verma considered the solvability of a new class of variational inequalities involving multivalued
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Journal of Inequalities and Applications
relaxed monotone operators (see [6]). Relaxed monotone operators have applications to constrained hemivariational inequalities. Since in the study of constrained problems in reflexive Banach spaces E the set of all admissible elements is nonconvex but starshaped, corresponding variational formulations are no longer variational inequalities. Using hemivariational inequalities, one can prove the existence of solutions to the following type of nonconvex constrained problems (P): find u in C such that Au − g,v ≥ 0,
∀v ∈ T C (u),
(1.1)
where the admissible set C ⊂ E is a star-shaped set with respect to a certain ball BE (u0 ,ρ), and TC (u) denotes Clarke’s tangent cone of C at u in C. It is easily seen that when C
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