• 충청수학회지
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• 제23권1호
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• pp.73-82
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• 2010

In an infinite-dimensional Hilbert space, the normal Mann iteration has only weak convergence, in general, even for nonexpansive mappings. The purpose of this paper is to modify the normal Mann iteration to have strong convergence for a closed quasi-$\phi$-nonexpansive mapping in the framework of Banach spaces.

• 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.

• 대한수학회지
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• 제57권3호
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• pp.617-640
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• 2020

In this paper, we introduce a parallel iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in a real reflexive Banach space. Moreover, we give some applications of the main theorem for solving some related problems. Finally, some numerical examples are developed to illustrate the behavior of the new algorithms with respect to existing algorithms.

• 대한수학회보
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• 제43권4호
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• pp.771-790
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• 2006

The iterative algorithms with errors for nonexpansive mappings are investigated in Banach spaces. Strong convergence theorems for these algorithms are obtained. Our results improve the corresponding results in [5, 13-15, 23, 27-29, 32] as well as those in [1, 16, 19, 26] in framework of a Hilbert space.

• 충청수학회지
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• 제23권4호
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• pp.801-812
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• 2010

In this paper, we consider the convergence of the shrinking projection method for quasi-$\phi$-nonexpansive mappings. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property.

• East Asian mathematical journal
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• 제24권1호
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• pp.35-43
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• 2008

Let E be a uniformly convex Banach space and K be a nonempty closed convex subset of E. Let ${\{T_i\}}^N_{i=1}$ be N nonexpansive self-mappings of K with $F\;=\;{\cap}^N_{i=1}F(T_i)\;{\neq}\;{\theta}$ (here $F(T_i)$ denotes the set of fixed points of $T_i$). Suppose that one of the mappings in ${\{T_i\}}^N_{i=1}$ is semi-compact. Let $\{{\alpha}_n\}\;{\subset}\;[{\delta},\;1-{\delta}]$ for some ${\delta}\;{\in}\;(0,\;1)$ and $\{{\beta}_n\}\;{\subset}\;[\tau,\;1]$ for some ${\tau}\;{\in}\;(0,\;1]$. For arbitrary $x_0\;{\in}\;K$, let the sequence {$x_n$} be defined iteratively by $\{{x_n\;=\;{\alpha}_nx_{n-1}\;+\;(1-{\alpha}_n)T_ny_n,\;\;\;\;\;\;\;\;\; \atop {y_n\;=\;{\beta}nx_{n-1}\;+\;(1-{\beta}_n)T_nx_n},\;{\forall}_n{\geq}1,}$, where $T_n\;=\;T_{n(modN)}$. Then {$x_n$} convergence strongly to a common fixed point of the mappings family ${\{T_i\}}^N_{i=1}$. The result presented in this paper generalized and improve the corresponding results of Chidume and Shahzad [C. E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62(2005), 1149-1156] even in the case of ${\beta}_n\;{\equiv}\;1$ or N=1 are also new.

• Kyungpook Mathematical Journal
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• 제47권1호
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• pp.81-89
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• 2007

In this paper, we introduce the concept of a system of asymptotically quasinonexpansive type mappings. Furthermore, we define a $k$-step iterative sequence approximating common fixed point for a system of asymptotically quasi-nonexpansive type mappings and study its stability in real Banach spaces.

• East Asian mathematical journal
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• 제28권3호
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• pp.265-280
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• 2012

In this paper, we introduce an iterative sequence by using a hybrid generalized $f$-projection algorithm for finding a common element of the set of fixed points of a relatively weak nonexpansive mapping an the set of solutions of a generalized variational inequality in a Banach space. Our results extend and improve the recent ones announced by Y. Liu [Strong convergence theorems for variational inequalities and relatively weak nonexpansive mappings, J. Glob. Optim. 46 (2010), 319-329], J. Fan, X. Liu and J. Li [Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear Analysis 70 (2009), 3997-4007], and many others.

• Nonlinear Functional Analysis and Applications
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• 제26권5호
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• pp.1045-1057
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• 2021

We present some fixed point theorems for ((𝛼₁, 𝛼₂, ⋯, 𝛼_n), p)-nonexpansive mappings in CAT(0) spaces. Moreover the properties of the fixed points set are studied. Many of them have been derived from new condition on these mappings, which makes the nonexpansive mapping T_𝛼 := 𝛼₁T ⊕ 𝛼₂T² ⊕ ⋯ ⊕ 𝛼_nTⁿ.

• Kyungpook Mathematical Journal
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• 제59권4호
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• pp.703-723
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• 2019

In this paper, we consider a common solution of three problems in real Hilbert spaces: the Ky Fan inequality problem, the variational inequality problem and the fixed point problem for demicontractive single-valued and quasi-nonexpansive multi-valued mappings. To find the solution we present a new iterative algorithm and prove a strong convergence theorem under mild conditions. Moreover, we provide a numerical example to illustrate the convergence behavior of the proposed iterative method.