Nonlinear Quasi-variational Inclusions Research Articles

Solution sensitivity of generalized nonlinear parametric ( A , η , m ) -proximal operator system of equations in Hilbert spaces

By using the new parametric resolvent operator technique associated with (A,η,m)-monotone operators, the purpose of this paper is to analyze and establish an existence theorem for a new class of generalized nonlinear parametric (A,η,m)-proximal operator system of equations with non-monotone multi-valued operators in Hilbert spaces. The results presented in this paper generalize the sensitivity analysis results of recent work on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions, and nonlinear mixed quasi-variational inclusion systems in Hilbert spaces.MSC:49J40, 47H05, 90C33.

A Perturbed Algorithm for Completely Generalized Set-Valued Nonlinear Quasi-Variational Inclusions

In this paper, a class of completely generalized set-valued quasi-variational inclusions for fuzzy mappings are introduced and studied. The equivalence between these kinds of variational inclusion and a fixed point problem is established. Based on this equivalence, some algorithms for this problem are suggested and analyzed.

Three-Step Iterative Method for Completely Generalized Set-Valued Strongly Nonlinear Quasi-Variational Inclusions

In this paper, we introduce and study a new class of completely generalized set-valued strongly nonlinear variational inclusions in Hilbert spaces and establish the equivalence between this variational inclusion and the fixed-point problem by using the resolvent operator technique for maximal monotone mapping. We construct a new three-step iterative algorithm and show the existence of solution for this variational inclusion and the convergence of the iterative method generated by the iterative method.

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.

Existence and algorithm of solutions for a new system of generalised variational inclusions involving relaxed Lipschitz mappings

We consider a new system for generalized variational inclusions in Hilbert spaces and define an iterative algorithm for finding the approximate solutions of this class of system of variational inclusions. We also establish that the approximate solutions obtained by our algorithm converge to the exact solution of new system of generalized variational inclusions. One can explore the role of our system of generalized variational inclusions for solving various known equilibrium problem and other related problems. References R. Ahmad and Q. H. Ansari, An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett. 13(5), 23--26, (2000). J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. X. P. Ding and C. L. Luo, Perturbed proximal point algorithms for general quasi-variational-like inclusions, J. Comput. Appl. Math. 113, 153--165, (2000). C. H. Lee, Q. H. Ansari and J. C. Yao, A perturbed algorithm for strongly nonlinear variational inclusions, Bull. Austral. Math. Soc. 62, 417--426, (2000). J. S. Pang, Asymmetric variational inequality problem over product of sets:Applications and iterative methods, Math Prog. 31, 206--219, (1985). R. U. Verma, Partially relaxed monotone mappings and a new system of nonlinear variational inequalities, Nonlinear Funct. Anal. Appl. 5(1), 65--72, (2000). R. U. Verma, Projection method, Algorithm and a new system of nonlinear variational inequalities, Comput. Math. Appl. 41(7-8), 1025--1031, (2001). R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theory Appl. 121(1), 203--210, (2004). R. P. Agarwal, Y. J. Cho and N. J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13(6), 19--24, (2000). R. P. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J. Inequal. Appl. 7(6), 807--828, (2002). R. P. Agarwal, N. J. Huang and M. Y. Tan, Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions, Appl. Math. Lett. 17, 345--352, (2004). Q. H. Ansari and J. C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bull. Austral. Math. Soc. 59(3), 433--442, (1999).

Sensitivity analysis for parametric generalized implicit quasi-variational-like inclusions involving P- η-accretive mappings

In this paper, using proximal-point mapping technique of P- η-accretive mapping and the property of the fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of a parametric generalized implicit quasi-variational-like inclusion involving P- η-accretive mapping in real uniformly smooth Banach space. Further, under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameter. The technique and results presented in this paper can be viewed as extension of the techniques and corresponding results given in [R.P. Agarwal, Y.-J. Cho, N.-J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13 (2002) 19–24; S. Dafermos, Sensitivity analysis in variational inequalities, Math. Oper. Res. 13 (1988) 421–434; X.-P. Ding, Sensitivity analysis for generalized nonlinear implicit quasi-variational inclusions, Appl. Math. Lett. 17 (2) (2004) 225–235; X.-P. Ding, Parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings, J. Comput. Appl. Math. 182 (2) (2005) 252–269; X.-P. Ding, C.L. Luo, On parametric generalized quasi-variational inequalities, J. Optim. Theory Appl. 100 (1999) 195–205; Z. Liu, L. Debnath, S.M. Kang, J.S. Ume, Sensitivity analysis for parametric completely generalized nonlinear implicit quasi-variational inclusions, J. Math. Anal. Appl. 277 (1) (2003) 142–154; R.N. Mukherjee, H.L. Verma, Sensitivity analysis of generalized variational inequalities, J. Math. Anal. Appl. 167 (1992) 299–304; M.A. Noor, Sensitivity analysis framework for general quasi-variational inclusions, Comput. Math. Appl. 44 (2002) 1175–1181; M.A. Noor, Sensitivity analysis for quasivariational inclusions, J. Math. Anal. Appl. 236 (1999) 290–299; J.Y. Park, J.U. Jeong, Parametric generalized mixed variational inequalities, Appl. Math. Lett. 17 (2004) 43–48].

Perturbed Iterative Algorithms for Generalized Nonlinear Set-Valued Quasivariational Inclusions Involving Generalized m-Accretive Mappings

A new class of generalized nonlinear set-valued quasivariational inclusions involving generalized -accretive mappings in Banach spaces are studied, which included many variational inclusions studied by others in recent years. By using the properties of the resolvent operator associated with generalized -accretive mappings, we established the equivalence between the generalized nonlinear set-valued quasi-variational inclusions and the fixed point problems, and some new perturbed iterative algorithms, proved that its proximate solution converges strongly to its exact solution in real Banach spaces.

Sensitivity analysis for relaxed cocoercive nonlinear quasivariational inclusions

Some results on the sensitivity analysis for relaxed cocoercive quasivariational inclusions are obtained, which generalize similar sensitivity analysis results on strongly monotone quasivariational inclusions. Furthermore, some suitable examples of relaxed cocoercive mappings are illustrated.

Perturbed algorithm and stability for strongly nonlinear quasi-variational inclusion involving H-accretive operators

Perturbed algorithm and stability for strongly nonlinear quasi-variational inclusion involving H-accretive operators

Perturbed three‐step approximation process with errors for ageneralized implicit nonlinear quasivariational inclusions

We initiate and develop some new perturbed three‐step approximation process with errors for solving generalized implicit nonlinear quasivariational inclusions. Also, the convergence and stability of the iterative sequences with errors generated by the algorithms are presented.

Sensitivity analysis for parametric completely generalized strongly nonlinear implicit quasi-variational inclusions

In this paper, by using a resolvent operator technique of maximal monotone mappings and the property of a fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity of the solutions of the parametric completely generalized strongly nonlinear implicit quasi-variational inclusion in Hilbert space. Our results extend, improve, and unify the previously many results in this field.

Proximal Methods for Generalized Nonlinear Quasi-Variational Inclusions

We consider the solvability, based on iterative algorithms, of the generalized nonlinear quasi-variational inclusion problems involving the relaxed Lipschitz and relaxed monotone mappings.

Generalized multivalued nonlinear quasivariational inclusions

AbstractIn this paper, we introduce and study a few classes of generalized multivalued nonlinear quasivariational inclusions and generalized nonlinear quasivariational inequalities, which include many classes of variational inequalities, quasivariational inequalities and variational inclusions as special cases. Using the resolvent operator technique for maximal monotone mapping, we construct some new iterative algorithms for finding the approximate solutions of these classes of quasivariational inclusions and quasivariational inequalities. We establish the existence of solutions for this generalized nonlinear quasivariational inclusions involving both relaxed Lipschitz and strongly monotone and generalized pseudocontractive mappings and obtain the convergence of iterative sequences generated by the algorithms. Under certain conditions, we derive the existence of a unique solution for the generalized nonlinear quasivariational inequalities and obtain the convergence and stability results of the Noor type perturbed iterative algorithm. The results proved in this paper represent significant refinements and improvements of the previously known results in this area.

Sensitivity analysis for parametric completely generalized nonlinear implicit quasivariational inclusions

In this paper we introduce a new class of parametric completely generalized nonlinear implicit quasivariational inclusions and study the behavior and sensitivity analysis of the solution set of the parametric completely generalized nonlinear implicit quasivariational inclusion dealing with multivalued and single-valued nonlinear mappings in Hilbert spaces. Our results extend, improve and unify the previously many known results in this area.

Sensitivity analysis for strongly nonlinear quasi-variational inclusions

In this paper, we use the implicit resolvent operator technique to study the sensitivity analysis for strongly nonlinear quasi-variational inclusions. Our results improve and generalize some of the recent ones.

A Perturbated Algorithm for Generalized Nonlinear Quasi-Variational Inclusions

In this paper, a perturbed iterative method for solving a generalized nonlinear quasi-variational inclusions, is presented and a convergence result which generalizes some known results in this field, is given.

Proximal point algorithm with errors for generalized strongly nonlinear quasivariational inclusions

In this paper, a class of generalized strongly nonlinear quasivariational inclusions are studied. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert space, an existence theorem of solutions for generalized strongly nonlinear quasivariational inclusion is established and a new proximal point algorithm with errors is suggested for finding approximate solutions which strongly converge to the exact solution of the generalized strongly nonlinear quasivariational inclusion. As special cases, some known results in this field are also discussed.