Quasi-variational Inclusions In Banach Spaces Research Articles

Convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces

In this paper, we give the notion of M-η-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space, which is an extension of proximal mappings studied in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; K.R. Kazmi, M.I. Bhat, Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces, Appl. Math. Comput. 113 (2005) 153–165; K.R. Kazmi, M.I. Bhat, N. Ahmad, An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces, J. Comput. Appl. Math. 233 (2009) 361–371]. We prove its existence and Lipschitz continuity in reflexive Banach space. Further, we consider a system of generalized implicit variational-like inclusions in Banach spaces and show its equivalence with a system of implicit equations using the concept of M-η-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational-like inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational-like inclusions and discuss the convergence and stability analysis of the iterative algorithm in the setting of uniformly smooth Banach spaces.

Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces

In order to unify some variational inequality problems, in this paper, a new system of generalized quasivariational inclusion (for short, (SGQVI)) is introduced. By using Banach contraction principle, some existence and uniqueness theorems of solutions for (SGQVI) are obtained in real Banach spaces. Two new iterative algorithms to find the common element of the solutions set for (SGQVI) and the fixed points set for Lipschitz mappings are proposed. Convergence theorems of these iterative algorithms are established under suitable conditions. Further, convergence rates of the convergence sequences are also proved in real Banach spaces. The main results in this paper extend and improve the corresponding results in the current literature. 2000 MSC: 47H04; 49J40.

An iterative algorithm based on [formula omitted]-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces

In this paper, we give the notion of M -proximal mapping, an extension of P -proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener–Hopf equations using the concept of M -proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm.

Algorithm for solving a new class of generalized nonlinear implicit quasi-variational inclusions in Banach spaces

In this paper, we introduce and study a new class of generalized nonlinear implicit quasi-variational inclusions with general H-monotone operator in Banach spaces. By using the proximal mapping and Nadler’s theorem, we suggest and analyze a class of iterative algorithm for solving the generalized nonlinear implicit quasi-variational inclusion. First, an existence theorem of solutions for the generalized nonlinear implicit quasi-variational inclusion is proved under suitable assumptions. Then, the convergence of the iterative sequences generated by the new algorithm is proved.

Generalized nonlinear quasi-variational inclusions in Banach spaces

This paper introduces a new class of generalized nonlinear quasi-variational inclusions involving generalized m -accretive mappings in p -uniformly smooth real Banach spaces. By using the resolvent operator technique for generalized m -accretive mappings due to Huang et al. [N.J. Huang, Y.P. Fang, C.X. Deng, Nonlinear variational inclusions involving generalized m -accretive mappings, in: Proceedings of the Bellman Continuum: International Workshop on Uncertain Systems and Soft Computing, Beijing, China, July, 24–27, 2002, pp. 323–327] and Nadler Theorem [S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475–488], we construct an iterative algorithm for solving generalized nonlinear quasi-variational inclusions with strongly accretive and relaxed accretive mappings in p -uniformly smooth real Banach spaces. Then we prove the existence of solutions for our inclusions without compactness assumption and convergence of the iterative sequences generated by the algorithm in p -uniformly smooth real Banach spaces. Some special cases are also discussed.

Ishikawa type iterative algorithm for completely generalized nonlinear quasi-variational-like inclusions in Banach spaces

In this paper, we consider completely generalized nonlinear quasi-variational-like inclusions in Banach spaces and propose an Ishikawa type iterative algorithm for computing their approximate solutions by applying the new notion of J η -proximal mapping given in [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308]. We prove that the approximate solutions obtained by the proposed algorithm converge to the exact solution of our quasi-variational-like inclusions. The results presented in this paper extend and improve the corresponding results of [R. Ahmad, A.H. Siddiqi, Z. Khan, Proximal point algorithm for generalized multi-valued nonlinear quasi-variational-like inclusions in Banach spaces, Appl. Math. Comput. 163 (2005) 295–308; X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; N.J. Huang, Generalized nonlinear variational inclusions with non-compact valued mappings, Appl. Math. Lett. 9(3) (1996) 25–29; A. Hassouni, A. Moudafi, A perturbed algorithm for variational inclusions, J. Math. Anal. Appl. 185(3) (1994) 706–712]. Some special cases are also discussed.

Parametric generalized strongly nonlinear mixed quasivariational inclusions in Banach spaces

In this paper, by using the concept of the resolvent operator, we study the behavior and sensitivity analysis of the solution set for a class of parametric generalized strongly nonlinear mixed quasivariational inclusion problem in Banach spaces. The results presented in this paper are new and generalize many known results in this field.

Multivalued quasi-variational inclusions and multivalued accretive equations

The purpose of this paper is to introduce and study a class of more general multivalued quasi-variational inclusions in Banach spaces. By using the accretive operator technique, we establish the equivalence between these kinds of multivalued quasi-variational inclusions and a kind of multivalued accretive equations in Banach space. Then, invoking Michael's selection theorem and Nadler's theorem, some existence theorems and iterative algorithms for solving this kind of multivalued quasi-variational inclusions in Banach spaces are established and suggested.

Fuzzy quasivariational inclusions in Banach spaces

The purpose of this paper is to introduce the concept of general fuzzy quasi variational inclusions and to study the existence problem and iterative approximation problem of solutions for some kinds of fuzzy quasivariational inclusions in Banach spaces. By using the resolvent operator technique, Nadler’s fixed point theorem and new analytic technique, some existence theorems of solutions and iterative approximation for solving this kind of fuzzy quasivariational inclusions are established. The results presented in this paper are new, which generalize, improve and unify a number of recent results.

A new class of completely generalized quasi-variational inclusions in Banach spaces

In this paper, a new notion of J-proximal mapping for a nonconvex lower semicontinuous subdifferentiable proper functional on Banach space is introduced. The existence and Lipschitz continuity of J-proximal mapping of a lower semicontinuous subdifferentiable proper functional are proved. By applying the concept, we introduce and study a new class of completely generalized quasi-variational inclusions in reflexive Banach spaces. A novel and innovative iterative algorithm for finding the approximate solutions is suggested and analyzed. The convergence criteria of the iterative sequences generated by the new iterative algorithm is also given. These algorithm and existence result generalize many known results under Hilbert space setting in recent literature to reflexive Banach spaces.