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Chen, Rudong,Miao, Qian The Youngnam Mathematical Society Korea 2008 East Asian mathematical journal Vol.24 No.1
The aim of this paper is to prove convergence of implicit iteration process to a common fixed point for a finite family of strong successive $\Phi$-pseudocontractive mappings. The results presented in this paper extend and improve the corresponding results of S. S. Chang [On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 313(2006), 273-283], M. O. Osilike[Implicit iteration process for common fixed points of a finite finite family of strictly pseudocontractive maps, Appl. Math. Comput. 189(2) (2007), 1058-1065].
Rudong Chen,Qian Miao 영남수학회 2008 East Asian mathematical journal Vol.24 No.1
The aim of this paper is to prove convergence of implicit iteration process to a common fixed point for a finite family of strong successive Φ-pseudocontractive mappings. The results presented in this paper extend and improve the corresponding results of S. S. Chang [On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 313 (2006), 273–283], M. O. Osilike [Implicit iteration process for common fixed points of a finite finite family of strictly pseudocontractive maps, Appl. Math. Comput. 189(2) (2007), 1058–1065].
Yisheng Song,Rudong Chen 대한수학회 2008 대한수학회지 Vol.45 No.5
Let K be a nonempty closed convex subset of a Banach space E. Suppose {Tn} (n = 1, 2, . . .) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ∩∞n=1 F(Tn) ≠0. For x0 ∈ K, define xn+1 = λn+1xn + (1 − λn+1)Tn+1xn, n ≥ 0. If λn ⊂ [0, 1] satisfies limn→∞ λn = 0, we proved that {xn} weakly converges to some z ∈ F as n → ∞ in the framework of reflexive Banach space E which satisfies the Opial’s condition or has Fr´echet differentiable norm or its dual E* has the Kadec-Klee property. We also obtain that {xn} strongly converges to some z ∈ F in Banach space E if K is a compact subset of E or there exists one map T ∈ {Tn; n = 1, 2, . . .} satisfy some compact conditions such as T is semicompact or satisfy Condition A or limn→∞ d(xn, F(T)) = 0 and so on. Let K be a nonempty closed convex subset of a Banach space E. Suppose {Tn} (n = 1, 2, . . .) is a uniformly asymptotically regular sequence of nonexpansive mappings from K to K such that ∩∞n=1 F(Tn) ≠0. For x0 ∈ K, define xn+1 = λn+1xn + (1 − λn+1)Tn+1xn, n ≥ 0. If λn ⊂ [0, 1] satisfies limn→∞ λn = 0, we proved that {xn} weakly converges to some z ∈ F as n → ∞ in the framework of reflexive Banach space E which satisfies the Opial’s condition or has Fr´echet differentiable norm or its dual E* has the Kadec-Klee property. We also obtain that {xn} strongly converges to some z ∈ F in Banach space E if K is a compact subset of E or there exists one map T ∈ {Tn; n = 1, 2, . . .} satisfy some compact conditions such as T is semicompact or satisfy Condition A or limn→∞ d(xn, F(T)) = 0 and so on.
Song, Yisheng,Chen, Rudong Korean Mathematical Society 2008 대한수학회지 Vol.45 No.5
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.
STRONG CONVERGENCE OF A NEW ITERATIVE ALGORITHM FOR AVERAGED MAPPINGS IN HILBERT SPACES
Yonghong Yao,Haiyun Zhou,Rudong Chen 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.3
Let H be a real Hilbert space. Let T : H → H be an averaged mapping with F(T ) ≠ 0. Let {αn} be a real numbers in (0, 1). For given x0 ∈ H, let the sequence {xn} be generated iteratively by [수식]Assume that the following control conditions hold:[수식]Then {xn} converges strongly to a fixed point of T.
STRONG CONVERGENCE OF A NEW ITERATIVE ALGORITHM FOR AVERAGED MAPPINGS IN HILBERT SPACES
Yao, Yonghong,Zhou, Haiyun,Chen, Rudong The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.3
Let H be a real Hilbert space. Let T : $H\;{\rightarrow}\;H$ be an averaged mapping with $F(T)\;{\neq}\;{\emptyset}$. Let {$\alpha_n$} be a real numbers in (0, 1). For given $x_0\;{\in}\;H$, let the sequence {$x_n$} be generated iteratively by $x_{n+1}\;=\;(1\;-\;{\alpha}_n)Tx_n$, $n\;{\geq}\;0$. Assume that the following control conditions hold: (i) $lim_{n{\rightarrow}{\infty}}\;{\alpha}_n\;=\;0$; (ii) $\sum^{\infty}_{n=0}\;{\alpha}_n\;=\;{\infty}$. Then {$x_n$} converges strongly to a fixed point of T.