Point Problems In Banach Spaces Research Articles

A hybrid iteration scheme for equilibrium problems, variational inequality problems and common fixed point problems in Banach spaces

In this paper, we introduce an iterative process which converges strongly to a common element of a set of common fixed points of finite family of closed relatively quasi-nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for an α -inverse strongly monotone mapping in Banach spaces.

The Shrinking Projection Method for Solving Variational Inequality Problems and Fixed Point Problems in Banach Spaces

We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi-ϕ-nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.

A Hybrid Iterative Scheme for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Banach Spaces

AbstractThe purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of solutions of the equilibrium problem and the set of the variational inequality for an inverse-strongly monotone operator and the set of fixed points of relatively quasi-nonexpansive mappings in a Banach space. Then we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space.

Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings

The purpose of this paper is to introduce a new hybrid projection algorithm for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Our results improve and extend the corresponding results announced by Takahashi and Zembayashi [W. Takahashi, K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. (2008), doi:10.1016/j.na.2007.11.031], Takahashi and Zembayashi [W. Takahashi, K. Zembayashi, Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory Appl. (2008), doi:10.1155/2008/528476], Qin et al. [X. Qin, Y.J. Cho, S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. (2008), doi:10.1016/j.cam.2008.06.011], Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103–115].

Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces

The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of two quasi- ϕ -nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.

On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Let G be a non-empty closed (resp. bounded closed) boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X. Let $$ {\user1{K}}{\left( X \right)} $$ denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance. Moreover, let $$ {\user1{K}}_{G} {\left( X \right)} $$ denote the closure of the set $$ {\left\{ {A \in {\user1{K}}{\left( X \right)}:A \cap G = \emptyset } \right\}} $$ . We prove that the set of all $$ A \in {\user1{K}}_{G} {\left( X \right)}{\left( {{\text{resp}}{\text{.}}{\kern 1pt} A \in {\user1{K}}{\left( X \right)}} \right)} $$ , such that the minimization (resp. maximization) problem min(A,G) (resp. max(A,G)) is well posed, contains a dense G δ-subset of $$ {\user1{K}}_{G} {\left( X \right)}{\left( {{\text{resp}}{\text{.}}{\kern 1pt} {\kern 1pt} {\user1{K}}{\left( X \right)}} \right)} $$ , thus extending the recent results due to Blasi, Myjak and Papini and Li.