Quasi-variational Inclusion Problems Research Articles

Ekeland type variational principle with applications to quasi-variational inclusion problems

In this paper, we study the Ekeland type variational principle, a Caristi–Kirk type fixed point theorem and a maximal element theorem in the setting of uniform spaces. By using these results, we establish some existence results for solutions of quasi-variational inclusion problems, quasi-optimization problems and equilibrium problems defined on separated and sequentially complete uniformly spaces.

Some results on systems of quasi-variational inclusion problems and systems of generalized quasi-variational inclusion problems

In this paper, we study systems of quasi-variational inclusion problem and systems of quasi-variational disclusion problem. From the existence theorems of solution for these two types of problems, we study various types of systems of quasi-variational inclusion problems, systems of quasi-equilibrium problems, systems of quasi-KKM theorem, abstract economics and system of KKM theorem. We also show their equivalent relations. We study further existence theorems of solution for generalized quasi-variational inclusion problem. Our results are different from any existence result in the literature.

Systems of generalized vector quasi-variational inclusions and systems of generalized vector quasi-optimization problems in locally FC-uniform spaces

In this paper, we introduce some new systems of generalized vector quasi-variational inclusion problems and system of generalized vector ideal (resp., proper, Pareto, weak) quasi-optimization problems in locally FC-uniform spaces without convexity structure. By using the KKM type theorem and Himmelberg type fixed point theorem proposed by the author, some new existence theorems of solutions for the systems of generalized vector quasi-variational inclusion problems are proved. As to its applications, we obtain some existence results of solutions for systems of generalized vector quasi-optimization problems.

On the existence of solutions to quasivariational inclusion problems

We establish sufficient existence conditions for general quasivariational inclusion problems, which contain most of variational inclusion problems and quasiequilibrium problems considered in the literature. These conditions are shown to extend recent existing results and sharpen some of them even for particular cases.

SYSTEMS OF GENERALIZED VECTOR QUASI-VARIATIONAL INCLUSION PROBLEMS AND APPLICATION TO MATHEMATICAL

In this paper, we introduce and study some new systems of generalized vector quasi-variational inclusion problems involving condensing mappings in locally $FC$-uniform spaces. These systems contain many known systems of generalized vector quasi-variational inclusion problems, systems of generalized vector quasi-equilibrium problems and systems of vector quasi-optimization problems as special cases. By applying an existence theorem of maximal elements of a family of set-valued mappings involving condensing mapping due to author, we prove some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems. As applications, some existence results of solutions of the mathematical programs with systems of generalized vector quasi-variational inclusion constraints are established in noncompact locally $FC$-uniform

From quasivariational inclusion problems to Stampacchia vector quasiequilibrium problems, Stampacchia set-valued vector Ekeland’s variational principle and Caristi’s fixed point theorem

In this paper, we first study existence theorems of solution for quasivariational inclusion problems. We apply existence theorems of solution for quasivariational inclusion problem to study the existence theorems of solution for the Stampacchia generalized vector quasiequilibrium problems and Stampacchia vector quasiequilibrium problems. Our results and approach to the Stampacchia generalized vector quasiequilibrium problem are different from any existence results of this type of problem. Finally, we apply the existence theorems of solution for the Stampacchia vector quasiequilibrium problems to study simultaneous solutions of Caristi’s fixed point theorem and the Stampacchia set-valued vector Ekeland’s variational principle. Our results on Stampacchia set-valued vector Ekeland’s variational principle are different from any existence results of these types of problems.

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

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.

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.

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 of generalized set-valued quasi-variational inclusion in Banach spaces

The sensitivity analysis for a class of generalized set-valued quasi-variational inclusion problems is investigated in the setting of Banach spaces. By using the resolvent operator technique, without assuming the differentiability and monotonicity of the given data, equivalence of these problems to the class of generalized resolvent equations is established.

Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems

In this paper, we study an existence theorem of systems of generalized quasivariational inclusions problem. By this result, we establish the existence theorems of solutions of systems of generalized equations, systems of generalized vector quasiequilibrium problem, collective variational fixed point, systems of generalized quasiloose saddle point, systems of minimax theorem, mathematical program with systems of variational inclusions constraints, mathematical program with systems of equilibrium constraints and systems of bilevel problem and semi-infinite problem with systems of equilibrium problem constraints.

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.

On the Existence of Solutions of Quasivariational Inclusion Problems

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 general vector ideal, proper, Pareto, weak quasioptimization problems, quasivariational inequalities, and vector quasiequilibrium problems. Further, we prove theorems on the existence for solutions of systems of these inclusions. As a corollary, we obtain an ideal minimax theorem concerning vector functions.

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.