Xie Ping Ding

Existence of solutions for quasi-equilibrium problems in noncompact topological spaces

Abstract In this paper, a new Fan-Browder type fixed-point theorem is proved under noncompact setting of general topological spaces. By applying the fixed-point theorem, several new existence theorems of solutions for quasi-equilibrium problems are proved under noncompact setting of topological spaces. These theorems improve and generalize a number of important known results in literature.

Generalized KKM Type Theorems in FC-Spaces with Applications (I)

The class KKM(X,Y) (resp., s-KKM(X,Y,Z)) of set-valued mappings with KKM (resp., s-KKM) property is introduced in FC-spaces without any convexity structure. Some generalized KKM (resp., s-KKM) type theorems are proved in FC-spaces under much weak assumptions. As applications, some new section theorems and coincidence theorems are established in FC-spaces. These theorems generalize many known results in literature. The further applications of these results will be given in a follow-up paper.

Algorithm of Solutions for Mixed Implicit Quasi-Variational Inequalities with Fuzzy Mappings

Abstract In this paper, by applying the auxiliary variational principle technique, some existence theorems of solutions for a class of mixed implicit quasi-variational inequalities with fuzzy mappings are proved in Hilbert spaces. A novel and innovative iterative algorithm to compute approximate solutions is suggested and analyzed. The convergence criteria is also given. As special cases of these results, the open problem put forward by Noor is answered positively. Our results are new and generalize a number of known results to mixed implicity quasi-variational inequalities with fuzzy mappings.

General algorithm of solutions for nonlinear variational inequalities in Banach space

Abstract In this paper, we shall study a class of general nonlinear variational inequalities in reflexive Banach spaces. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the general nonlinear variational inequalities are proved. Next, by applying the auxiliary problem technique, we suggest an innovative iterative algorithm to compute the approximate solutions of the general nonlinear variational inequality. Finally, the convergence criteria is also discussed.

Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces

In this paper, we introduce and study a class of generalized strongly nonlinear mixed variational-like inequalities in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities. First, the existence of solutions of the auxiliary problems for the generalized strongly nonlinear mixed variational-like inequalities is shown. Second, an iterative algorithm for solving the generalized strongly nonlinear mixed variational-like inequalities is constructed by using this existence result. Finally, not only the existence of solutions of the generalized strongly nonlinear mixed variational-like inequalities is shown, but also the convergence of iterative sequences generated by the algorithm is also proven.

Generalized L-KKM type theorems in L-convex spaces with applications☆

Abstract In this paper, a new notion of generalized L-KKM mappings is introduced. Some generalized L-KKM type theorems for generalized L-KKM mappings with finitely closed (or compactly closed) values and with finitely open (or compactly open) values are established in L -convex spaces. As applications, some Ky Fan type matching theorems, fixed-point theorems and minimax inequality are obtained in L -convex spaces. These results generalize a number of known results in recent literature.

The generalized vector quasi-variational-like inequalities☆

Abstract In this paper, we introduce and study a class of Generalized Vector Quasi-Variational-Like Inequality Problem (GVQVLIP) involving set-valued mappings with certain monotonicity. By employing the scalarization technique, several existence results for solutions of the (GVQVLIP) are established under noncompact setting in topological vector spaces. These new existence results improve, unify, and generalize many known results for scalar and vector variational inequalities in recent literature.

Iterative Process with Errors to Locally Strictly Pseudocontractive Maps in Banach Spaces

Abstract Let K be a nonempty subset of a uniformly smooth Banach space X and A : K → X be a locally strickly pseudocontractive map with bounded range. The Mann and Ishikawa iterative sequences with errors, which strongly converge to the unique fixed point of A are given under weaker conditions. These results answer positively the open problems mentioned by Deng and Ding [1, p. 122] and Chidume [2, p. 550]. The related results deal with the problems that the Mann and Ishikawa iterative sequences with errors strongly converge to the unique solution of the equation Tx = f when T : X → X is locally strongly accretive such that the range of I – T is bounded.

Existence of Pareto equilibria for constrained multiobjective games in H-space☆

Abstract In this paper, we introduce and study a new class of constrained multiobjective games in H -spaces without linear structure. An existence theorem of solutions for quasi-equilibrium problems is first proved in noncompact H -spaces. Then, as applications of the quasi-equilibrium existence theorem, several existence theorems of weighted Nash-equilibria and Pareto equilibria for the constrained multiobjective games are established in noncompact H -spaces. These theorem improve, unify, and generalize the corresponding results of the multiobjective games in recent literature.

Generalized implicit quasivariational inclusions with fuzzy set-valued mappings

Abstract In this paper, we introduce and study a new class of generalized implicit quasivariational inclusions with fuzzy set-valued mappings. A existence theorem of solutions is proved without compactness assumptions. A new iterative algorithm is suggested and analyzed. The convergence of iterative sequence generated by the new algorithm is also given. As special cases, some known results are also discussed.