• Journal of the Korean Mathematical Society
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• v.38 no.6
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• pp.1275-1284
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• 2001

In this paper, we introduce an iteration generated by countable nonexpansive mappings and prove a weak convergence theorem which is connected with the feasibility problem. This result is used to solve the problem of finding a solution of the countable convex inequality system and the problem of finding a common fixed point for a commuting countable family of nonexpansive mappings.

• Communications of the Korean Mathematical Society
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• v.28 no.2
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• pp.335-351
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• 2013

In this paper, we introduce a new iterative scheme for finding a common element of the fixed points set of a countable family of uniformly Lipschitzian generalized asymptotically quasi-nonexpansive mappings and the solutions set of equilibrium problems. Some strong convergence theorems of the proposed iterative scheme are established by using the concept of W-mappings of a countable family of uniformly Lipschitzian generalized asymptotically quasi-nonexpansive mappings.

• Journal of applied mathematics & informatics
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• v.35 no.3_4
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• pp.313-321
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• 2017

In this note we establish a new non-convex hybrid iteration algorithm corresponding to Khan iterative process [4] and prove strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Moreover, the main results are applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. The results presented in this article are interesting extensions of some current results.

• Kyungpook Mathematical Journal
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• v.54 no.3
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• pp.375-385
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• 2014

We investigate strong convergence of Halpern's iteration for a countable family of strongly relatively nonexpansive mappings in the framework of uniformly convex and uniformly smooth Banach spaces. Our results extend those announced by many authors.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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• pp.1393-1404
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• 2008

Let K be a nonempty closed convex subset of a Banach space E. Suppose $\{T_{n}\}$ (n = 1,2,...) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ${\cap}_{n=1}^{\infty}$ F$\(T_n){\neq}{\phi}$. For $x_0{\in}K$, define $x_{n+1}={\lambda}_{n+1}x_{n}+(1-{\lambda}_{n+1})T_{n+1}x_{n},n{\geq}0$. If ${\lambda}_n{\subset}[0,1]$ satisfies $lim_{n{\rightarrow}{\infty}}{\lambda}_n=0$, we proved that $\{x_n\}$ weakly converges to some $z{\in}F\;as\;n{\rightarrow}{\infty}$ in the framework of reflexive Banach space E which satisfies the Opial's condition or has $Fr{\acute{e}}chet$ differentiable norm or its dual $E^*$ has the Kadec-Klee property. We also obtain that $\{x_n\}$ strongly converges to some $z{\in}F$ in Banach space E if K is a compact subset of E or there exists one map $T{\in}\{T_{n};n=1,2,...\}$ satisfy some compact conditions such as T is semi compact or satisfy Condition A or $lim_{n{\rightarrow}{\infty}}d(x_{n},F(T))=0$ and so on.