• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.325-336
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• 2008

In this paper, we prove the weak and strong convergence of the three-step iterative scheme with errors to a common fixed point for two asymptotically nonexpansive mappings in a uniformly convex Banach space under a condition weaker than compactness. Our theorems improve and generalize some previous results.

• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.525-536
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• 2007

In this paper, we prove the weak and strong convergence of the Noor iterative scheme to a common fixed point for two asymptotically nonexpansive mappings in a uniformly convex Banach space under a condition weaker than compactness. Our theorems improve and generalize some previous results.

• East Asian mathematical journal
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• 제24권1호
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• pp.81-95
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• 2008

Let E be a uniformly convex Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm, C a nonempty closed convex subset of E, and $T\;:\;C\;{\rightarrow}\;E$ a nonexpansive mapping satisfying the weak inwardness condition. Assume that every weakly compact convex subset of E has the fixed point property. For $f\;:\;C\;{\rightarrow}\;C$ a contraction and $t\;{\in}\;(0,\;1)$, let $x_t$ be a unique fixed point of a contraction $T_t\;:\;C\;{\rightarrow}\;E$, defined by $T_tx\;=\;tf(x)\;+\;(1\;-\;t)Tx$, $x\;{\in}\;C$. It is proved that if {$x_t$} is bounded, then $x_t$ converges to a fixed point of T, which is the unique solution of certain variational inequality. Moreover, the strong convergence of other implicit and explicit iterative schemes involving the sunny nonexpansive retraction is also given in a reflexive and strictly convex Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm.

• East Asian mathematical journal
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• 제27권1호
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• pp.35-49
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• 2011

In this paper, we study multi-step iterative algorithm with errors and give the necessary and sufficient condition to converge to com mon fixed points for a finite family of asymptotically quasi-nonexpansive type mappings in Banach spaces. Also we have proved a strong convergence theorem to converge to common fixed points for a finite family said mappings on a nonempty compact convex subset of a uniformly convex Banach spaces. Our results extend and improve the corresponding results of [2, 4, 7, 8, 9, 10, 12, 15, 20].

• Journal of applied mathematics & informatics
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• 제33권1_2호
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• pp.13-31
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• 2015

In this paper, we introduce a general iterative algorithm for finding a common element of the common fixed point set of an infinite family of ${\lambda}_i$-strict pseudo-contractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in q-uniformly smooth Banach spaces. Then, we analyze the strong convergence of the iterative sequence generated by the proposed iterative algorithm under mild conditions.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권4호
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• pp.293-308
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• 2004

Let K be a nonempty convex subset of an arbitrary Banach space X and $T\;:\;K\;{\rightarrow}\;K$ be a uniformly continuous strictly hemi-contractive operator with bounded range. We prove that certain Ishikawa iteration scheme with errors both converges strongly to a unique fixed point of T and is almost T-stable on K. We also establish similar convergence and almost stability results for strictly hemi-contractive operator $T\;:\;K\;{\rightarrow}\;K$, where K is a nonempty convex subset of arbitrary uniformly smooth Banach space X. The convergence results presented in this paper extend, improve and unify the corresponding results in Chang [1], Chang, Cho, Lee & Kang [2], Chidume [3, 4, 5, 6, 7, 8], Chidume & Osilike [9, 10, 11, 12], Liu [19], Schu [25], Tan & Xu [26], Xu [28], Zhou [29], Zhou & Jia [30] and others.

• Journal of applied mathematics & informatics
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• 제40권1_2호
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• pp.191-204
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• 2022

This paper introduces the statistical convergence concepts of double sequences of complex uncertain variables: statistical convergence almost surely(a.s.), statistical convergence in measure, statistical convergence in mean, statistical convergence in distribution and statistical convergence uniformly almost surely(u.a.s.).

• Korean Journal of Mathematics
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• 제16권2호
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• pp.215-231
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• 2008

Let E be a uniformly convex Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm, C a nonempty closed convex subset of E, and $T:C{\rightarrow}{\mathcal{K}}(E)$ a multivalued nonself-mapping such that $P_T$ is nonexpansive, where $P_T(x)=\{u_x{\in}Tx:{\parallel}x-u_x{\parallel}=d(x,Tx)\}$. For $f:C{\rightarrow}C$ a contraction and $t{\in}(0,1)$, let $x_t$ be a fixed point of a contraction $S_t:C{\rightarrow}{\mathcal{K}}(E)$, defined by $S_tx:=tP_T(x)+(1-t)f(x)$, $x{\in}C$. It is proved that if C is a nonexpansive retract of E and $\{x_t\}$ is bounded, then the strong ${\lim}_{t{\rightarrow}1}x_t$ exists and belongs to the fixed point set of T. Moreover, we study the strong convergence of $\{x_t\}$ with the weak inwardness condition on T in a reflexive Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm. Our results provide a partial answer to Jung's question.

• 대한수학회지
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• 제54권3호
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• pp.1031-1047
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• 2017

In this paper, we introduce two general iterative algorithms (one implicit algorithm and other explicit algorithm) for nonexpansive mappings in a reflexive Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm. Strong convergence theorems for the sequences generated by the proposed algorithms are established.

• 대한수학회논문집
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• 제29권1호
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• pp.83-95
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• 2014

In a uniformly convex Banach space, we introduce a iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings and utilize a new inequality to prove several convergence results for the iterative sequence. The results generalize and unify many important known results of relevant scholars.