##### Title: Convergence Theorems of an Implicit Iterates with Errors for Non-Lipschitzian Asymptotically Quasi-Nonexpansive Type Mappings

##### Pages: 79-99

##### Cite as:

G. S. Saluja, Convergence Theorems of an Implicit Iterates with Errors for Non-Lipschitzian Asymptotically Quasi-Nonexpansive Type Mappings, Int. J. Anal. Appl., 1 (1) (2013), 79-99.#### Abstract

The aim of this paper is to study an implicit iterative process with errors for two finite families of non-Lipschitzian asymptotically quasi-nonexpansive type mappings in the framework of real Banach spaces. In this paper, we have obtained a necessary and sufficient condition to converge to common fixed points for proposed scheme and mappings and also obtained strong convergence theorems by using semi-compactness and *Condition (B’)*.

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#### References

- C.E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal., TMA, 62 (2005), no. 6, 1149-1156.
- J.B. Diaz, F.T. Metcalf, On the structure of the set of subsequential limit points of successive approximations, Bull. Amer. Math. Soc. 73 (1967), 516-519.
- H. Fukhar-ud-din, S.H. Khan, Convergence of two-step iterative scheme with errors for two asymptotically nonexpansive mappings, Int. J. Math. Math. Sci. (2004), no. 37-40, 1965-1971.
- H. Fukhar-ud-din, A.R. Khan, Convergence of implicit iterates with errors for mappings with unbounded domain in Banach spaces, Int. J. Math. Math. Sci. 10 (2005), 1643-1653.
- M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasinonexpansive mappings, J. Math. Anal. Appl. 207 (1997), 96-103.
- K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.
- S.H. Khan, W. Takahashi, Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn. 53 (2001), no. 1, 143-148.
- W.A. Kirk, Fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17 (1974), 339-346.
- L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), no. 1, 114-125.
- Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 259 (2001), 1-7.
- Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl. 259 (2001), 18-24.
- W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
- W.V. Petryshyn, T.E. Williamson, Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973), 459-497.
- X. Qin, S.M. Kang, R.P. Agarwal, On the convergence of an implicit iterative process for generalized asymptotically quasi-nonexpansive mappings, Fixed Point Theory Appl. (2010), Article ID 714860, 19pp. (doi:10.1155/2010/74860)
- D.R. Sahu, J.S. Jung, Fixed point iteration processes for non-Lipschitzian mappings of asymptotically quasi-nonexpansive type, Int. J. Math. Math. Sci. 33 (2003), 2075-2081.
- J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Astra. Math. Soc. 43 (1991), no. 1, 153-159.
- N. Shahzad, A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2006), Article ID 18909, 10pp.
- T. Shimizu, W. Takahashi, Strong convergence theorem for asymptotically nonexpansive mappings, Nonlinear Anal. 26 (1996), no. 2, 265-272.
- Z. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003), no. 1, 351-358.
- K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.
- R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992), 486-491.
- H.K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127-1138.
- Y. Xu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998), no. 1, 91-101.
- H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001), no. 5-6, 767-773.