Generalized nonlinear implicit quasivariational inclusions
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We introduce and study a new class of generalized nonlinear implicit quasivariational inclusions involving relaxed Lipschitzian mappings. We prove the existence of solution for the generalized nonlinear implicit quasivariational inclusions and construct some new stable perturbed iterative algorithms with errors. We also give an application to a class of generalized nonlinear implicit variational inequalities. 1. Introduction Variational inequality theory and complementarity problem theory are very powerful tools of the current mathematical technology. In recent years, classical variational inequality and complementarity problems have been extended and generalized to study a wide class of problems generated in mechanics, physics, optimization and control, nonlinear programming, economics and transportation equilibrium, and engineering sciences, and so forth. A useful and important generalization of variational inequalities is a variational inclusion. Using the resolvent operator technique, many authors have studied various variational inequalities and inclusions with applications (see [1, 2, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20, 21] and the references therein). In 1997, Verma [19] studied the solvability, based on an iterative algorithm, of a class of generalized nonlinear variational inequalities involving relaxed Lipschitz and relaxed monotone operators. Recently, Huang [9, 10] introduced and studied the Mann- and Ishikawa-type perturbed iterative sequence with errors for the generalized nonlinear implicit quasivariational inequalities and inclusions. On the other hand, Huang et al. [12] and Shim et al. [16] proved some existence theorems of solutions for the generalized nonlinear mixed quasivariational inequalities (inclusions) and convergence theorems of the iterative sequences generated by the perturbed algorithms with errors. Inspired and motivated by the recent papers [1, 9, 10, 11, 12, 16, 19], in this paper, we introduce and study a new class of generalized nonlinear implicit quasivariational inclusions involving relaxed Lipschitz mappings and construct some new perturbed iterative algorithms with errors. We discuss the convergence and stability of perturbed iterative sequences with errors generated by the algorithms for solving the generalized nonlinear Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:3 (2005) 261–275 DOI: 10.1155/JIA.2005.261
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Generalized implicit quasivariational inclusions
implicit quasivariational inclusions. We also give an application to a class of generalized nonlinear implicit variational inequalities. 2. Preliminaries Let H be a real Hilbert space endowed with a norm · and an inner product ·, ·, respectively. For given mappings f ,g, p : H → H, and N : H × H → H. Let M : H × H → 2H be a set-valued mapping such that,for each fixed t ∈ H, M(·,t) : H → 2H is a maximal monotone mapping and Range(p) Dom(M(·,t)) = ∅. We consider the following problem. Find u ∈ H such that
p(u) ∈ Dom M ·,g(u) ,
(2.1)
0
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