Prasit Cholamjiak

Halpern's iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces

We investigate strong convergence for Bregman strongly nonexpansive mappings by Halperns iteration in the framework of reflexive Banach spaces. Using the obtained result, convergence for a family of Bregman strongly nonexpansive mappings is also discussed. As applications, we apply our main result to problems of finding zeros of maximal monotone operators and equilibrium problems in reflexive Banach spaces.

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

The 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.

Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in banach spaces

We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.

A New Hybrid Algorithm for Variational Inclusions, Generalized Equilibrium Problems, and a Finite Family of Quasi-Nonexpansive Mappings

We proposed in this paper a new iterative scheme for finding common elements of the set of fixed points of a finite family of quasi-nonexpansive mappings, the set of solutions of variational inclusion, and the set of solutions of generalized equilibrium problems. Some strong convergence results were derived by using the concept of -mappings for a finite family of quasi-nonexpansive mappings. Strong convergence results are derived under suitable conditions in Hilbert spaces.

Composite iterative schemes for maximal monotone operators in reflexive Banach spaces

In this article, we introduce composite iterative schemes for finding a zero point of a finite family of maximal monotone operators in a reflexive Banach space. Then, we prove strong convergence theorems by using a shrinking projection method. Moreover, we also apply our results to a system of convex minimization problems in reflexive Banach spaces.AMS Subject Classification: 47H09, 47H10

Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions

We first investigate strong convergence of the sequence generated by implicit and explicit viscosity approximation methods for a one-parameter nonexpansive semigroup in a real Banach space E which has a uniformly Gâteaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0, ∞). The main results also improve and extend some known results concerning the normalized duality mapping in the literature.

An iterative method for equilibrium problems and a finite family of relatively nonexpansive mappings in a Banach space

Abstract In this paper, we introduce a new iterative method for finding a common element of the set of fixed points of a finite family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in uniformly convex and uniformly smooth Banach spaces. Then we prove a strong convergence theorem by using the generalized projection.

Existence and iteration for a mixed equilibrium problem and a countable family of nonexpansive mappings in Banach spaces

We prove the existence of a solution of the mixed equilibrium problem (MEP) by using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solutions set of a mixed equilibrium problem and the fixed points set of a countable family of nonexpansive mappings in the frameworks of Banach spaces. By using a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solutions set of MEP and common fixed points set of nonexpansive mappings. Moreover, some applications concerning the equilibrium and the convex minimization problems are obtained.

Strong convergence for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces

We introduce a new iterative algorithm for finding a common fixed point of a countable family of strict pseudocontractions in q-uniformly smooth and uniformly convex Banach spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common fixed point of an infinite family of strict pseudocontractions. Our results mainly improve and extend the results announced by Yao et al. [Y. Yao, Y.-C. Liou, G. Marino, Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl. 2009 (2009) 7 pages. doi:10.1155/2009/279058. Art. ID 279058].

Weak Convergence Theorems for a Countable Family of Strict Pseudocontractions in Banach Spaces

We investigate the convergence of Mann-type iterative scheme for a countable family of strict pseudocontractions in a uniformly convex Banach space with the Fréchet differentiable norm. Our results improve and extend the results obtained by Marino-Xu, Zhou, Osilike-Udomene, Zhang-Guo and the corresponding results. We also point out that the condition given by Chidume-Shahzad (2010) is not satisfied in a real Hilbert space. We show that their results still are true under a new condition.