Somyot Plubtieng

A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings

Abstract In this paper, we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for α -inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main theorem extends a recent result of Yao and Yao [Y. Yao, J.-C. Yao, On modified iterative method for nonexpansive mappings and monotone mappings, Applied Mathematics and Computation 186 (2) (2007) 1551–1558].

Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space

In this paper, we establish strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Our results extend and improve the recent ones announced by Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266], Matinez-yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411], and many others.

Hybrid Iterative Methods for Convex Feasibility Problems and Fixed Point Problems of Relatively Nonexpansive Mappings in Banach Spaces

The convex feasibility problem (CFP) of finding a point in the nonempty intersection is considered, where is an integer and the s are assumed to be convex closed subsets of a Banach space . By using hybrid iterative methods, we prove theorems on the strong convergence to a common fixed point for a finite family of relatively nonexpansive mappings. Then, we apply our results for solving convex feasibility problems in Banach spaces.

Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces

Let C be a nonempty closed convex subset of real Hilbert space H and S={T(s):[emailxa0protected]?s [emailxa0protected]?. For a contraction f on C, and [emailxa0protected]?(0,1), let x[emailxa0protected]?C be the unique fixed point of the contraction [emailxa0protected]?tf(x)+(1-t)[emailxa0protected][emailxa0protected]!0^@l^^tT(s)xds, where {@lt} is a positive real divergent net. Consider also the iteration process {xn}, where x[emailxa0protected]?C is arbitrary and xn+[emailxa0protected]nf(xn)[emailxa0protected]nxn+([emailxa0protected][emailxa0protected]n)1s[emailxa0protected]!0^s^^nT(s)xnds for n>=0, where {@an},{@bn}@?(0,1) with @a[emailxa0protected]n<1 and {sn} are positive real divergent sequences. It is proved that {xt} and, under certain appropriate conditions on {@an} and {@bn}, {xn} converges strongly to a common fixed point of S.

Fixed point result and applications on a b-metric space endowed with an arbitrary binary relation

In this paper, we introduce the concept of q-set-valued α-quasi-contraction mapping and establish the existence of a fixed point theorem for this mapping in b-metric spaces. Our results are generalizations and extensions of the result of Aydi et al. (Fixed Point Theory Appl. 2012:88, 2012) and some recent results. We also state some illustrative examples to claim that our results properly generalize some results in the literature. Further, by applying the main results, we investigate a fixed point theorem in a b-metric space endowed with an arbitrary binary relation. At the end of this paper, we give open problems for further investigation.MSC:47H10, 54H25.

Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space

We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. The result obtained in this paper improves and extends the recent ones announced by Moudafi (2009),Iemoto and Takahashi (2009), and many others. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.

A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities

In this paper, we introduce and study a new iterative scheme for finding the common element of the set of common fixed points of a sequence of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the general system of variational inequality for α andxa0μ-inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. This main theorem extends a recent result of Ceng etxa0al. (Math Meth Oper Res 67:375–390, 2008) and many others.

Implicit Iterations of Two Finite Families for Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to study the weak and strong convergence of an implicit iteration process to a common fixed point for two finite families of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Xu and Ori, Numer. Funct. Anal. Optim. 2001; 22:767–773, Zhou and Chang, Numer. Fund. Anal. Optim. 2002; 23:911–921, Chidume and Shahzad, Nonlinear. Anal. 2005; 62: 1149–1156.

An Extragradient Method and Proximal Point Algorithm for Inverse Strongly Monotone Operators and Maximal Monotone Operators in Banach Spaces

We introduce an iterative scheme for finding a common element of the solution set of a maximal monotone operator and the solution set of the variational inequality problem for an inverse strongly-monotone operator in a uniformly smooth and uniformly convex Banach space, and then we prove weak and strong convergence theorems by using the notion of generalized projection. The result presented in this paper extend and improve the corresponding results of Kamimura et al. (2004), and Iiduka and Takahashi (2008). Finally, we apply our convergence theorem to the convex minimization problem, the problem of finding a zero point of a maximal monotone operator and the complementary problem.

Existence Result of Generalized Vector Quasiequilibrium Problems in Locally -Convex Spaces

This paper deals with the generalized strong vector quasiequilibrium problems without convexity in locally -convex spaces. Using the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, the existence theorems for them are established. Moreover, we also discuss the closedness of strong solution set for the generalized strong vector quasiequilibrium problems.