Quasi-variational Inclusions Research Articles

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 Theorems of Quasivariational Inclusion Problems with Applications to Bilevel Problems and Mathematical Programs with Equilibrium Constraint

In this paper, we establish existence theorems of quasivariational inclusion problems; from them, we establish existence theorems of mathematical programs with quasivariational inclusion constraint, bilevel problems, mathematical programs with equilibrium constraint and semi-infinite problems.

Iterative approximation of solutions for a class of completely generalized set-valued quasi-variational inclusions

In this paper we study a new class of completely generalized set-valued quasi-variational inclusion problems in Banach spaces. This inclusion problem is a generalization of the generalized set-valued variational inclusion problem studied by Chang et al. We show that Chang et al.’s algorithm can be modified for finding approximate solutions to this class of quasi-variational inclusion problems. Moreover, the modified algorithm generalizes and improves a number of results in the literature.

Semicontinuity of Solution Sets to Parametric Quasivariational Inclusions with Applications to Traffic Networks II: Lower Semicontinuities Applications

We propose some notions related to semicontinuity of a multivalued mapping and provide a clear insight for various semicontinuity-related definitions. We establish verifiable sufficient conditions for solution sets of general quasivariational inclusion problems to have these semicontinuity-related properties. Our results are proved to include and improve recent ones in the literature by corollaries and examples. Part I is devoted to upper semicontinuity properties of solution sets. Part II discusses lower semicontinuities of these sets and applications, where we discuss in details a traffic network problem as a sample for employing the main results in practical situations

Generalizations of vector quasivariational inclusion problems with set-valued maps

Existence theorems are given for the problem of finding a point (z 0,x 0) of a set E × K such that $$(z_0,x_0)\in B(z_0,x_0)\times A(z_0,x_0)$$ and, for all $$\eta\in A(z_0,x_0), (F(z_0,x_0,x_0,\eta), C(z_0,x_0,x_0,\eta))\in \alpha$$ where ? is a relation on 2 Y (i.e., a subset of 2 Y × 2 Y ), $$A : E\times K\longrightarrow 2^K,$$ $$B : E\times K\longrightarrow 2^E, C : E\times K\times K\times K\longrightarrow 2^Y$$ and $$F : E\times K\times K\times K\longrightarrow 2^Y$$ are some set-valued maps, and Y is a topological vector space. Detailed discussions are devoted to special cases of ? and C which correspond to several generalized vector quasi-equilibrium problems with set-valued maps. In such special cases, existence theorems are obtained with or without pseudomonotonicity assumptions.

Semicontinuity of Solution Sets to Parametric Quasivariational Inclusions with Applications to Traffic Networks I: Upper Semicontinuities

We propose some notions related to semicontinuity of a multivalued mapping and provide a clear insight for various semicontinuity-related definitions. We establish verifiable sufficient conditions for solution sets of general quasivariational inclusion problems to have these semicontinuity-related properties. Our results are proved to include and improve recent ones in the literature by corollaries and examples. Part I is devoted to upper semicontinuity properties of solution sets. Part II discusses lower semicontinuities of these sets and applications, where we discuss in details a traffic network problem as a sample for employing the main results in practical situations

Stability of iterative processes with errors for a system of nonlinear [formula omitted]-accretive variational inclusions in Banach spaces

In this paper, by using the concept of ( A , η ) -accretive mappings and the new resolvent operator technique associated with ( A , η ) -accretive mappings, we introduce and study a system of general mixed quasivariational inclusions involving ( A , η ) -accretive mappings in Banach spaces, and construct a new perturbed iterative algorithm with mixed errors for this system of nonlinear ( A , η ) -accretive variational inclusions in q -uniformly smooth Banach spaces. Our results improve and generalize the corresponding results of recent works.

Sensitivity analysis for a system of generalized nonlinear mixed quasi-variational inclusions in q-uniformly smooth Banach spaces

In this paper, we study the behavior and sensitivity analysis of the solution set for a new system of parametric generalized nonlinear mixed quasi-variational inclusions with H-accretive mappings in q-uniformly smooth Banach spaces. The present results improve and extend many known results in the literature.

Systems of nonempty intersection theorems with applications

In this paper, we study the systems of nonempty intersection theorems. By the systems of nonempty intersection theorems, we can study the existence theorem of systems of quasi-KKM problems, the existence theorems of systems of quasi-variational inclusions problems, the existence theorems of systems of quasi-variational disclusions problems, and fixed point theorem. We also establish some of their equivalent relations.

System of vector quasi-variational inclusions with some applications

In this paper, we consider systems of vector quasi-variational inclusions which include systems of vector quasi-equilibrium problems for multivalued maps, systems of vector optimization problems and several other systems as special cases. We establish existence results for solutions of these systems. As applications of our results, we derive the existence results for solutions of system vector optimization problems, mathematical programs with systems of vector variational inclusion constraints and bilevel problems. Another application of our results provides the common fixed point theorem for a family of lower semicontinuous multivalued maps. Further applications of our results for existence of solutions of systems of vector quasi-variational inclusions are given to prove the existence of solutions of systems of Minty type and Stampacchia type generalized implicit quasi-variational inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

The p-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with [formula omitted]-accretive operators in q-uniformly smooth banach spaces

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with ( A , η ) -accretive operators in q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with ( A , η ) -accretive operators, we construct a new p-step iterative algorithm for solving this system of generalized mixed quasi-variational inclusions in real q-uniformly smooth Banach spaces. We also prove the existence of solutions for the generalized mixed quasi-variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.

PARAMETRIC GENERALIZED MIXED IMPLICIT QUASI-VARIATIONAL INCLUSIONS

An existence theorem for a new class of parametric generalized mixed implicit quasi-variational inclusion problems is established in Hilbert spaces. Further, we study the behavior and sensitivity analysis of the solution set in this class of parametric variational inclusion problems.

Nonlinear parametric multi-valued variational inclusion systems involving [formula omitted]-accretive mappings in Banach spaces

In this paper, by using the new parametric resolvent operator technique associated with ( A , η ) -accretive mappings, we analyze and establish an existence theorem for new nonlinear parametric multi-valued variational inclusion systems involving ( A , η ) -accretive mappings in Banach spaces. Our results generalize sensitivity analysis results of other recent works on strongly monotone quasi-variational inclusions, nonlinear implicit quasi-variational inclusions and nonlinear mixed quasi-variational inclusion systems.

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).

Systems of Set-Valued Quasivariational Inclusion Problems

We propose four kinds of systems of set-valued quasivariational inclusion problems in product spaces, which include many known systems of equilibrium problems and systems of variational inequalities as well as inclusion problems. Sufficient conditions for the solution existence are established. When applied to special cases, these conditions improve many existing results in the literature. To ensure the generality of our problem setting and results, applications in fixed-point theory and quasioptimization problems are included.

On quasivariational inclusion problems of type I and related problems

The quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of a general vector ideal (proper, Pareto, weak) quasi-optimization problems, of quasivariational inequalities, and of vector quasi-equilibrium problems. Further, we prove theorems on the existence for solutions of the sum of these inclusions. As corollaries, we shall show several results on the existence of solutions to another problems in the vector optimization problems concerning multivalued mappings.

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].

Existence of solutions and convergence of iterative algorithms for a system of generalized nonlinear mixed quasi-variational inclusions

In this paper, we introduce and study a new system of generalized nonlinear mixed quasi-variational inclusions in q -uniformly smooth Banach spaces. We prove the existence and uniqueness of solutions for this system of generalized nonlinear mixed quasi-variational inclusions. We also prove the convergence of several new two-step iterative algorithms with or without errors for this system of generalized nonlinear mixed quasi-variational inclusions. The results in this paper extend and improve some known results in the literature.

ALGORITHM OF SOLUTIONS FOR A SYSTEM OF GENERALIZED MIXED IMPLICIT QUASI-VARIATIONAL INCLUSIONS INVOLVING h-η-MAXIMAL MONOTONE MAPPINGS

In this paper, we introduce a new class of $h$-$\eta$-maximal monotone mappings and a new system of generalized mixed implicit quasi-variational inclusions involving set-valued mappings and $h$-$\eta$-maximal monotone mappings. By using resolvent operator technique of $h$-$\eta$-maximal monotone mappings, A new iterative algorithm to compute approximate solutions of the system is suggested and analyzed. The convergence of the iterative sequence generated by the new algorithm is also proved. These results generalize many known results in literature.

General framework for sensitivity analysis to a class of nonlinear relaxed cocoercive quasivariational inclusions

General framework for sensitivity analysis to a class of nonlinear relaxed cocoercive quasivariational inclusions