Jong Kyu Kim

Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces

The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi--asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

Outer approximation algorithms for pseudomonotone equilibrium problems

We propose a new method for solving equilibrium problems on a convex subset C, where the underlying function is continuous and pseudomonotone which is called an outer approximation algorithm. The algorithm is to define new approximating subproblems on the convex domains Ck@?C,k=0,1,... , which forms a generalized iteration scheme for finding a global equilibrium point. Finally we present some numerical experiments to illustrate the behavior of the proposed algorithms.

An extragradient algorithm for solving bilevel pseudomonotone variational inequalities

We present an extragradient-type algorithm for solving bilevel pseudomonone variational inequalities. The proposed algorithm uses simple projection sequences. Under mild conditions, the convergence of the iteration sequences generated by the algorithm is obtained.

COMMON FIXED POINTS OF TWO NONEXPANSIVE MAPPINGS BY A MODIFIED FASTER ITERATION SCHEME

We introduce an iteration scheme for approximating common fixed points of two mappings. On one hand, it extends a scheme due to Agarwal et al. (2) to the case of two mappings while on the other hand, it is faster than both the Ishikawa type scheme and the one studied by Yao and Chen (18) for the purpose in some sense. Using this scheme, we prove some weak and strong convergence results for approximating common fixed points of two nonexpansive self mappings. We also outline the proofs of these results to the case of nonexpansive nonself mappings.

Regularization Inertial Proximal Point Algorithm for Monotone Hemicontinuous Mapping and Inverse Strongly Monotone Mappings in Hilbert Spaces

The purpose of this paper is to present a regularization variant of the inertial proximal point algorithm for finding a common element of the set of solutions for a variational inequality problem involving a hemicontinuous monotone mapping and for a finite family of -inverse strongly monotone mappings from a closed convex subset of a Hilbert space into .

Viscosity Approximation of Common Fixed Points for

We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007) and the pseudocontractive mapping in (Zegeye et al., 2007) to the pseudocontractive semigroup in Banach spaces under different conditions.

Implicit Iteration Process for Common Fixed Points of Strictly Asymptotically Pseudocontractive Mappings in Banach Spaces

In this paper, a new implicit iteration process with errors for finite families of strictly asymptotically pseudocontractive mappings and nonexpansive mappings is introduced. By using the iterative process, some strong convergence theorems to approximating a common fixed point of strictly asymptotically pseudocontractive mappings and nonexpansive mappings are proved. The results presented in the paper are new which extend and improve some recent results of Osilike et al. (2007), Liu (1996), Osilike (2004), Su and Li (2006), Gu (2007), Xu and Ori (2001).

Viscosity Approximation of Common Fixed Points for -Lipschitzian Semigroup of Pseudocontractive Mappings in Banach Spaces

We study the strong convergence of two kinds of viscosity iteration processes for approximating common fixed points of the pseudocontractive semigroup in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we get the strong convergence of the implicit viscosity iteration process for approximating common fixed points of the nonexpansive semigroup in Banach spaces satisfying some conditions. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in (Chen and He, 2007) and the pseudocontractive mapping in (Zegeye et al., 2007) to the pseudocontractive semigroup in Banach spaces under different conditions.

Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.

A viscosity hybrid steepest-descent method for a system of equilibrium and fixed point problems for an infinite family of strictly pseudo-contractive mappings

Based on a viscosity hybrid steepest-descent method, in this paper, we introduce an iterative scheme for finding a common element of a system of equilibrium and fixed point problems of an infinite family of strictly pseudo-contractive mappings which solves the variational inequality 〈(γf−μF)q,p−q〉≤0 for p∈⋂i=1∞F(Ti). Furthermore, we also prove the strong convergence theorems for the proposed iterative scheme and give a numerical example to support and illustrate our main theorem.MSC:46C05, 47D03, 47H05,47H09, 47H10, 47H20.