Jung Im Kang

Local stability of the Pexiderized Cauchy and Jensen's equations in fuzzy spaces

Lex X be a normed space and Y be a Banach fuzzy space. Let D = {(x, y) ∈ X × X : ||x|| + ||y|| ≥ d} where d > 0. We prove that the Pexiderized Jensen functional equation is stable in the fuzzy norm for functions defined on D and taking values in Y. We consider also the Pexiderized Cauchy functional equation.2000 Mathematics Subject Classification: 39B22; 39B82; 46S10.

On iterations methods for zeros of accretive operators in Banach spaces

In this paper, three iterations are designed to approach zeros of set-valued accretive operators in Banach spaces. The first one is the continuous Picard type iteration involving the resolvent, the second one is the approximate Picard type iteration involving the resolvent and the third one is the Halpern type iteration involving the resolvent. Some strong convergence theorems for three iterations are proved.

Approximation of fixed points for nonexpansive semigroups in Hilbert spaces

In this paper, we propose two new algorithms for finding a common fixed point of a nonexpansive semigroup in Hilbert spaces and prove some strong convergence theorems for nonexpansive semigroups. Our results improve and generalize the corresponding results given by Shimizu and Takahashi (J. Math. Anal. Appl. 211:71-83, 1997), Shioji and Takahashi (Nonlinear Anal. TMA 34:87-99, 1998), Lau et al. (Nonlinear Anal. TMA 67:1211-1225, 2007) and many others.MSC:47H05, 47H10, 47H17.

CONVERGENCE THEOREMS FOR UNIFORMLY QUASI-LIPSCHITZIAN MAPPINGS

We prove some convergence theorems of the modified Ishikawa iterative sequence with errors for uniformly quasi-Lipschitzian mappings in metric spaces. Our results generalize and improve the corresponding results of Petryshyn and Williamson, Ghosh and Debnath, Liu, and many others.

Composite schemes for variational inequalities over equilibrium problems and variational inclusions

AbstractLet C be a nonempty closed convex subset of a Hilbert space H, and let T:H→H be a nonlinear mapping. It is well known that the following classical variational inequality has been applied in many areas of applied mathematics, modern physical sciences, computerized tomography and many others. Find a point x∗∈C such that A〈Tx∗,x−x∗〉≥0,∀x∈C. In this paper, we consider the following variational inequality. Find a point x∗∈C such that B〈(F−γf)x∗,x−x∗〉≥0,∀x∈C, and, for solutions of the variational inequality (B) with the feasibility set C, which is the intersection of the set of solutions of an equilibrium problem and the set of a solutions of a variational inclusion, construct the two composite schemes, that is, the implicit and explicit schemes to converge strongly to the unique solution of the variational inequality (B).Recently, many authors introduced some kinds of algorithms for solving the variational inequality problems, but, in fact, our two schemes are more simple for finding solutions of the variational inequality (B) than others.MSC:49J30, 47H10, 47H17, 49M05.

Generalized nonlinear implicit quasivariational inclusions

We introduce and study a new class of generalized nonlinear implicit quasivariational inclusions involving relaxed Lipschitzian mappings. We prove the existence of solution for the generalized nonlinear implicit quasivariational inclusions and construct some new stable perturbed iterative algorithms with errors. We also give an application to a class of generalized nonlinear implicit variational inequalities.

CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS

In this paper, we will prove the following: Let D be a nonempty subset of a normed linear space X and T : D ! X be a nonexpansive mapping. Let fxng be a sequence in D and ftng, fsng be real sequences such that (i) 0 • tnt < 1 and P 1=1 tn = 1, (ii) (a) 0 • sn • 1; sn ! 0 as n ! 1 and P 1=1 tnsn < 1 or (b) sn = s for all n ‚ 1 and s 2 (0;1), (iii) xn+1 = (1itn)xn+tnT(snTxn+(1isn)xn) for all n ‚ 1. Then, if the sequence fxng is bounded, then