Iterative Schemes for Multivalued Quasi Variational Inclusions (original) (raw)
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
In this paper, we suggest and analyze a class of iterative schemes for solving generalized multivalued nonlinear quasi-variational like inclusion problems using the concept of η-subdifferential and η-proximal mappings of a proper functional on Hilbert spaces. A detailed convergence analysis of our method is also included.
Resolvent Iterative Methods for Solving System of Extended General Variational Inclusions
Journal of Inequalities and Applications, 2011
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.
Completely generalized multivalued nonlinear quasi-variational inclusions
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,
Some resolvent methods for general variational inclusions
Journal of King Saud University - Science, 2011
In this paper, we consider and analyze some classes of resolvent-splitting methods for solving the general variational inclusions using the technique of updating the solution. These resolvent-splitting methods are self-adaptive-type methods, where the corrector step size involves the resolvent equation. We prove that the convergence of these new methods only require the pseudomonotonicity, which is a weaker condition than monotonicity. These new methods differ from the previously known splitting and inertial proximal methods for solving the general variational inclusions and related complementarity problems. The proposed methods include several new and known methods as special cases. Our results may be viewed as refinement and improvement of the previous known methods.
An iterative algorithm for generalized nonlinear variational inclusions
Applied Mathematics Letters, 2000
In this paper, we consider the generalized nonlinear variational inclusions for nonclosed and nonbounded valued operators and define an iterative algorithm for finding the approximate solutions of this class of variational inclusions. We also establish that the approximate solutions obtained by our algorithm converge to the exact solution of the generalized nonlinear variational inclusion.
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.
Symmetry, 2021
This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·,·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with H(·,·)-compression XOR-αM-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example.
Iterative algorithm for a system of nonlinear variational-like inclusions
Computers & Mathematics with Applications, 2004
In this paper, we consider a system of nonlinear variational-like inclusions (SNVLI) in Hilbert spaces. In particular, SNVLI reduces to a variational inclusion, an extension of variational inclusion studied by Hassouni and Moudafi . Using fixed-point method, we suggest an iterative algorithm for finding an approximate solution to SNVLI. Further, we prove the existence of solution and discuss convergence criteria for the approximate solution of SNVLI. The theorems presented in this paper improve and unify many known results of variational inclusions and variational inequalities, see for example [1-3]. (~) 2004 Elsevier Ltd. All rights reserved.
Some resolvent iterative methods for variational inclusions and nonexpansive mappings
Applied Mathematics and Computation, 2007
In this paper, we suggest and analyze three-step iterations for finding the common element of the set of fixed points of a nonexpansive mappings and the set of the solutions of the variational inclusions using the resolvent operator technique. We also study the convergence criteria of three-step iterative method under some mild conditions. Our results include the previous results of Noor [M.
New iterative regularization methods for solving split variational inclusion problems
Journal of Industrial and Management Optimization, 2021
The paper proposes some new iterative algorithms for solving a split variational inclusion problem involving maximally monotone multi-valued operators in a Hilbert space. The algorithms are constructed around the resolvent of operator and the regularization technique to get the strong convergence. Some stepsize rules are incorporated to allow the algorithms to work easily. An application of the proposed algorithms to split feasibility problems is also studied. The computational performance of the new algorithms in comparison with others is shown by some numerical experiments.
Parametric generalized mixed multi-valued implicit quasi-variational inclusion problems
Advances in Fixed Point Theory, 2018
In this paper, by using a resolvent operator technique of maximal monotone mappings and the property of a fixed-point set of multi-valued contractive mapping, we study the behavior and sensitivity analysis of a solution set for a parametric generalized mixed multi-valued implicit quasi-variational inclusion problem in Hilbert space. Further, under some suitable conditions, we discuss the Lipschitz continuity (or continuity) of the solution set with respect to the parameter. By exploiting the technique of this paper, one can generalize and improve many known results in the literature.
On a System of General Variational Inclusions
2011
In this paper, we introduce and consider some new systems of general variational inclusions involving five different operators. Using the resolvent operator technique, we show that the new systems of general variational inclusions are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some new iterative methods for this system of general variational inclusions. We also study the convergence analysis of the new iterative method under suitable conditions. Several special cases are also discussed. Results obtained in this paper can be viewed as a significant extension of the known results.
Algorithms for Solving System of Extended General Variational Inclusions and Fixed Points Problems
Abstract and Applied Analysis, 2012
We introduce a new system of extended general nonlinear variational inclusions with different nonlinear operators and establish the equivalence between the aforesaid system and the fixed point problem. By using this equivalent formulation, we prove the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions. We suggest and analyze a new resolvent iterative algorithm to approximate the unique solution of the system of extended general nonlinear variational inclusions which is a fixed point of a nearly uniformly Lipschitzian mapping. Subsequently, the convergence analysis of the proposed iterative algorithm under some suitable conditions is considered. Furthermore, some related works to our main problem are pointed out and discussed.