Convergence Of Implicit Iteration Process Research Articles

Convergence of an implicit iteration process with errors for two asymptotically nonexpansive mappings

AbstractThe purpose of this paper is to introduce an implicit iterative process with errors for approximating common fixed point of two finite families of asymptotically nonexpansive mappings in the framework of Banach space. The results presented in this paper extend and generalize the corresponding results of Qin et al. [Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings, Appl. Math. Comp. 210 (2009), 542–550], Thakur [Weak and strong convergence of composite implicit iteration process, Appl. Math. Comp. 190 (2007), 965–973] and some others.

Weak and strong convergence of implicit iterative process for a finite family of asymptotically TJ mappings

In this paper we study the weak and strong convergence of implicit iteration process to a common xed point for a nite family of asymptotically TJ mappings in Hilbert spaces. This paper is motivated by [Lin, Lai-Jiu; Chuang, Chih-Sheng; Yu, Zenn-Tsun Fixed point theorems for some new nonlinear mappings in Hilbert spaces, Fixed Point Theory Appl. (2011), 2011:51, 16 pp.].

Errata to Convergence of implicit iterative process for finite family of I-nonexpansive mappings

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Convergence of implicit iterative process for a finite family of I-nonexpansive mappings

AbstractWe prove that an implicit iterative process converges weakly and strongly to a common fixed point of a finite family of

On the convergence theorems of implicit iteration process for a finite family of [formula omitted]-asymptotically nonexpansive mappings

In this paper, we consider the weak and strong convergence of implicit iteration process to a common fixed point of I -asymptotically nonexpansive mappings. The main results extend to a finite family of I -asymptotically nonexpansive mappings in a Banach space.

On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings

The purpose of this work is to study the sufficient and necessary conditions and sufficient conditions on the strong convergence of the implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings in real uniformly convex Banach spaces. The results presented in this work give an affirmative answer to the open question raised by Xu and Ori [H.K. Xu, M.G. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773] and improve and generalize the corresponding results of Chang et al. [S.S. Chang, K.K. Tan, H.W.J. Lee, Chi Kin Chan, 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], Sun [Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003) 351–358], Zhou and Chang [Y.Y. Zhou, S.S. Chang, Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim. 23 (2002) 911–921].

Implicit iterative process for common fixed point of a finite family of asymptotically nonexpansive mappings

The purpose of this paper is to study the weak and strong convergence of implicit iteration process to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of [1,2,4–9,11–15].

Convergence of the new composite implicit iteration process with random errors

In this paper, strong convergence theorems for approximation of common fixed points of a finite family of asymptotically demicontractive mappings are proved in Banach spaces using the new composite implicit iteration scheme with errors. Our results of this paper improve and extend the corresponding results of Chen, Song, Zhou [R.D. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Gu [F. Gu, The new composite implicit iterative process with errors for common fixed points of a finite family of strictly pseudocontractive mappings, J. Math. Anal. Appl. 329 (2007) 766–776] and Yang and Hu [L.P. Yang, G. Hu, Convergence of implicit iteration process with random errors, Acta Math. Sinica (Chin. Ser.) 51 (1) (2008) 11–22].

On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces

The purpose of this paper is to introduce the concept of non-self asymptotically quasi-nonexpansive-type mappings and to construct a iterative sequence to converge to a common fixed point for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces. The results presented in this paper improve and extend the corresponding results in Alber, Chidume and Zegeye [Ya.I. Alber, C.E. Chidume, H. Zegeye, Approximating of total asymptotically nonexpansive mappings, Fixed Point Theory and Applications (2006) 1–20. Article ID10673], Ghosh and Debnath [M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 207 (1997) 96–103], Liu [Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive type mappings, Journal of Mathematical Analysis and Applications 259 (2001) 1–37; Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications 259 (2001) 18–24; Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space, Journal of Mathematical Analysis and Applications 266 (2002) 468–471], Petryshyn [W.V. Petryshyn, T.E. Williamson Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 43 (1973) 459–497], Quan and Chang [J. Quan, S.S. Chang, X.J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications V (2006) 1–38. Article ID 70830], Shahzad and Udomene [N. Shahzad, A. Udomene, Approximating common fixed point of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2006) 1–10. Article ID 18909] Xu [B.L. Xu, M.A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 267 (2002) 444–453], Zhang [S.S. Zhang, Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Mathematicae Applicatae Sinica 24 (2001) 236–241] and Zhou and Chang [Y.Y. Zhou, S.S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numerical Functional Analysis and Optimization 23 (2002) 911–921].

A new composite implicit iterative process for a finite family of nonexpansive mappings in Banach spaces

AbstractThe purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang (2002).

Strong and Weak Convergence of Implicit Iterative Process with Errors for Asymptotically Nonexpansive Mappings

Strong and Weak Convergence of Implicit Iterative Process with Errors for Asymptotically Nonexpansive Mappings

On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings

The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of [H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159; B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961; P.L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Paris, Ser. A 284 (1977), 1357–1359; S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292; Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003) 351–358; R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491; H.K. Xu, M.G. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optimiz. 22 (2001) 767–773; Y.Y. Zhou, S.S. Chang, Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optimiz. 23 (2002) 911–921].

CONVERGENCE OF IMPLICIT ITERATION PROCESS FOR A FINITE FAMILY OF ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN BANACH SPACES

ABSTRACT The purpose of this paper is to study the weak and strong convergence of implicit iteration process to a common fixed point for a finite family of asymptotically nonexpansive mappings and nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Refs. [1], 5-7, 10-11.