Iterative Schemes for Multivalued Quasi Variational Inclusions (original) (raw)
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Three-Step Iterative Algorithms for Multivalued Quasi Variational Inclusions
Journal of Mathematical Analysis and Applications, 2001
In this paper, we suggest and analyze some new classes of three-step iterative algorithms for solving multivalued quasi variational inclusions by using the resolvent equations technique. New iterative algorithms include the Ishikawa, Mann, and Noor iterations for solving variational inclusions (inequalities) and optimization problems as special cases. The results obtained in this paper represent an improvement and a significant refinement of previously known results.
Convergence of Iterative Schemes for Multivalued Quasi-Variational Inclusions
Positivity, 2004
Relying on the resolvent operator method and using Nadler's theorem, we suggest and analyze a class of iterative schemes for solving multivalued quasi-variational inclusions. In fact, by considering problems involving composition of mutivalued operators and by replacing the usual compactness condition by a weaker one, our result can be considered as an improvement and a signi cant extension of previously known results in this eld. 2000 AMS Subject Classi cation: 49J40, 90C33.
General iterative algorithms for solving mixed quasi-variational-like inclusions
Computers & Mathematics with Applications, 2008
a b s t r a c t In this paper, we extend the auxiliary variational inequality technique due to Ding and Yao [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variationallike inclusions in Banach spaces, Comput. Math. Appl. 49 (2005) 857-869] to develop iterative algorithms for finding the approximate solutions of a mixed quasi-variational-like inclusion problem (in short, MQVLIP) in the setting of Banach spaces. We first establish a result on the existence of a solution of the equilibrium problem by virtue of the Fan-KKM lemma. Then by using this result and a result by Ding and Tan [X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (2) (1992) 233-247], we derive the existence of a unique solution of MQVLIP and the existence of approximate solutions generated by the proposed algorithms. Moreover, we also provide the new criteria for convergence of approximate solutions to the exact solution of MQVLIP.
Generalized multivalued nonlinear quasi-variational like inclusions
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We introduce and consider some new systems of extended general variational inclusions involving six different operators. We establish the equivalence between this system of extended general variational inclusions and the fixed points using the resolvent operators technique. This equivalent formulation is used to suggest and analyze some new iterative methods for this system of extended general variational inclusions. We also study the convergence analysis of the new iterative method under certain mild conditions. Several special cases are also discussed.
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International Journal of Mathematics and Mathematical Sciences, 2002
We introduce and study a new class of completely generalized multivalued nonlinear quasivariational inclusions. Using the resolvent operator technique for maximal monotone mappings, we suggest two kinds of iterative algorithms for solving the completely generalized multivalued nonlinear quasi-variational inclusions. We establish both four existence theorems of solutions for the class of completely generalized multivalued nonlinear quasivariational inclusions involving strongly monotone, relaxed Lipschitz, and generalized pseudocontractive mappings, and obtain a few convergence results of iterative sequences generated by the algorithms. The results presented in this paper extend, improve, and unify a lot of results due to Adly,
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Journal of King Saud University - Science, 2011
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An iterative algorithm for generalized nonlinear variational inclusions
Applied Mathematics Letters, 2000
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Computers & Mathematics with Applications, 2010
Extended generalized nonlinear mixed quasi-variational inclusions Perturbed N-step iterative algorithm with mixed errors Resolvent operator technique Variational convergence q-uniformly smooth Banach spaces a b s t r a c t This paper introduces a new system of extended generalized nonlinear mixed quasivariational inclusions involving A-maximal m-relaxed η-accretive (so called (A, η)-accretive (Lan et al. (2006) [37])) mappings in q-uniformly smooth Banach spaces. By using the resolvent operator technique for A-maximal m-relaxed η-accretive mappings due to Lan et al., we establish the existence and uniqueness of solution for this system of extended generalized nonlinear mixed quasi-variational inclusions and construct a new perturbed N-step iterative algorithm with mixed errors for solving the mentioned system. We also prove the convergence of the sequences generated by our algorithms in q-uniformly smooth Banach spaces. The results presented in this paper extend and improve some known results in the literature.
Journal of Inequalities and Applications, 2014
The purpose of this paper is to introduce new approximation methods for solutions of generalized non-accretive multi-valued mixed quasi-variational inclusion systems involving ( A , η ) -accretive mappings in q-uniformly smooth Banach spaces and, by using the new resolvent operator technique associated with ( A , η ) -accretive mappings, Nadler’s fixed point theorem and Liu’s inequality, we prove some existence theorems of solutions for our systems by constructing the new Mann iterative algorithm. Further, we study the stability of the iterative sequence generated by the perturbed iterative algorithms. The results presented in this paper improve and generalize the corresponding results of recent works given by some authors.