Hybrid Extragradient-Type Methods for Finding a Common Solution of an Equilibrium Problem and a Family of Strict Pseudo-Contraction Mappings

This paper proposes a new hybrid variant of extragradient methods for finding a common solution of an equilibrium problem and a family of strict pseudo-contraction mappings. We present an algorithmic scheme that combine the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this algorithm is modified by projecting on a suitable convex set to get a better convergence property. The convergence of two these algorithms are investigated under certain assumptions.

A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM,FIXED POINT PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS,AND QUASI-VARIATIONAL INCLUSION PROBLEM

In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Wakahashi and Toyoda [9], Takahashi and Takahashi [I0] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [II], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12].

APPROXIMATION OF FIXED POINTS AND VARIATIONAL SOLUTIONS FOR PSEUDO-CONTRACTIVE MAPPINGS IN BANACH SPACES

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map- pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con- vex optimization problems, and split feasibility problems. Our result extends many recent important results.

SOME CONVERGENCE PROBLEM OF ITERATIVE SEQUENCESFOR ACCRETIVE AND PSEUDO-CONTRACTIVETYPE MAPPING IN BANACH SPACES

Some necessary and sufficient conditions for convergence of Ishikawa Mann and steepest descent iterative sequence for accretive and pseudo-contractive type mapping in Banach spaces were obtained. The results improve, extend and include some recent results.

CQ Method for Strictly Pseudo-Contractive Mapping

This paper obtains a strong convergence theorem for k-strictly pseudo-contractive mapping under the framework of Hilbert spaces using CQ method. Due to the fact that non-expansive mapping is only O-strictly pseudo-contractive, the main result obtained in this paper extends the corresponding main result of Nakajo-Takahashi from non-expansive mapping to k-strictly pseudo-contractive one, where k∈ [0,1）.

NECESSARY AND SUFFICIENT CONDITION OF THE STRONG CONVERGENCE FOR TWO FINITE FAMILIES OF UNIFORMLY L-LIPSCHITZIAN MAPPINGS

The purpose of this paper is to study necessary and su？cient condition for the strong convergence of a new parallel iterative algorithm with errors for two finite families of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in this paper improve and extend the recent ones announced by [2–7].

Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semi-groups in Banach spaces

The purpose of this paper is to study the weak convergence problems of the implicity iteration process for Lipschitzian pseudocontractive semi-groups in the general Banach spaces. The results presented in this paper extend and improve the corresponding results of some people.

ON THE EXISTENCE OF SOLUTIONS FOR EQUATIONS WITH ACCRETIVE MAPPINGS IN PROBABILISTIC NORMED SPACES

The purpose of this paper is to introduce the concept of accretive mapping in probabilistic normed space (PN space, in short) and to study the existence problem of solutions for equations with accretive mappings in PN space and some existence theorems are shown.

ITERATIVE PROCESS FOR CERTAIN NONLINEAR MAPPINGS WITH LIPSCHITZ CONDITION

Using the new analysis techniques, the problem of iterative approximation of solutions of the equation for Lipschitz phi-strongly accretive operators and of fixed points for Lipschitz phi-strongly pseudo-contractive mappings are discussed. The main results of this paper improve and extend the corresponding results obtained by Chang, Chidume, Deng, Ding, Tan-Xu and Osilike.

A Modified Averaging Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of k-Strictly Asymptotically Pseudocontractive Mappings

The composite implicit iteration process introduced by Su and Li [J. Math. Anal. Appl. 320 (2006) 882-891] is modified. A strong convergence theorem for approximation of common fixed points of finite family of k-strictly asymptotically pseudo-contractive mappings is proved in Banach spaces using the modified iteration process.

Three-step Iterations with Errors for Nonlinear Strongly Accretive Operator Equations

In this paper, we suggest and analyse a three-step iterative scheme with errors for solving nonlinear strongly accretive operator equation Tx = f without the Lipshitz condition. The results presented in this paper improve and extend current results in the more general setting.