### Convergence and stability results for a class of asymptotically quasi-nonexpansive mappings in the intermediate sense

#### Abstract

In this paper, we study the notion of a class of asymptotically

quasi-nonexpansive mappings in the intermediate sense and define a k-step iterative sequence to approximating common fixed point for a class of above mentioned mappings. Also, we study its stability in real Banach spaces. The results presented in this paper are improvement and generalization of several known

corresponding results in the existing literature (see, e.g.,

[3], [4], [9], [12]-[14], [17], [20]-[26]).

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R.P. Agarwal, N.J. Huang and Y.J. Cho,

Stability of iterate procedures with errors for nonlinear

equations of $phi$-strongly accretive type operators, Numer. Funct. Anal. Optimiz. textbf{22} (2001), 471-485.

R.P. Agarwal, Y.J. Cho, J. Li and N.J. Huang,

Stability of iterate procedures with errors approximating

common fixed point for a couple of quasi-contractive mappings in a $q$-uniformly smooth Banach spaces, J. Math. Anal. Appl. textbf{272} (2002), 435-447.

S.S. Chang, On the approximating problem of

fixed points for asymptotically nonexpansive mappings, Indian J. Pure and Appl., textbf{32(9)} (2001), 1-11.

M.K. Ghosh, L. Debnath, Convergence of

Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. textbf{207} (1997), 96-103.

K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. textbf{35} (1972), 171-174.

A.M. Harder, Fixed point theory and stability

results for fixed point iteration procedures, Ph.D. Thesis,

University of Missouru-Rolla, (1987).

A.M. Harder and T.L. Hicks, A stable

iteration procedure for nonexpansive mappings, Math. Japon.

textbf{33} (1988), 687-692.

A.M. Harder and T.L. Hicks, Stability

results for fixed point iteration procedures}, Math. Japon.

textbf{33} (1988), 693-706.

Z.Y. Huang, Mann and Ishikawa iterations with

errors for asymptotically nonexpansive mappings, Comput. Math. Appl. textbf{37} (1999), 1-7.

G.E. Kim and T.H. Kim, Mann and Ishikawa iterations

with errors for non-Lipschitzian mappings in Banach spaces,

Comput. Math. Appl. 42(2001), 1565-1570.

W.A. Kirk, Fixed point theorems for

non-lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. textbf{17} (1974), 339-346.

Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. textbf{259} (2001), 1-7.

Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J.

Math. Anal. Appl. textbf{259} (2001), 18-24.

Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mappings with error member of

uniformly convex Banach spaces, J. Math. Anal. Appl. textbf{266} (2002), 468-471.

M.O. Osilike, Stable iteration procedures for

strong pseudocontractions and nonlinear equations of the accretive type, J. Math. Anal. Appl. textbf{204} (1996), 677-692.

M.O. Osilike, Stability of the Mann and Ishikawa iteration procedures for $phi$-strong pseudocontractions

and nonlinear equations of the $phi$-strongly accretive type}, J. Math. Anal. Appl. textbf{227} (1998), 319-334.

W.V. Petryshyn, T.E. Williamson, Strong and weak convergence of the sequence of successive approximations

for quasi-nonexpansive mappings, J. Math. Anal. Appl. textbf{43} (1973), 459-497.

B.E. Rhoades, Fixed point theorems and

stability results for fixed point iteration procedures, Indian J.

Pure and Appl. textbf{21} (1990), 1-9.

B.E. Rhoades, Fixed point theorems and

stability results for fixed point iteration procedures II, Indian

J. Pure and Appl. textbf{24} (1993), 691-703.

D.R. Sahu and J.S. Jung, Fixed point iteration processes for non-Lipschitzian mappings of asymptotically quasi-nonexpansive type, Int. J. Math. Math. Sci. textbf{33} (2003), 2075-2081.

J. Schu, Iterative construction of fixed

points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. textbf{158} (1991), 407-413.

N. Shahzad and A. Udomene, Approximating

common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, Vol.2006, Article ID 18909, Pages 1-10.

K.K. Tan and H.K. Xu, Approximating fixed points of

nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. textbf{178} (1993), 301-308.

K.K. Tan and H.K. Xu, Fixed point iteration processes for

asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. textbf{122} (1994), 733-739.

Y. Yao and Y.-C. Liou, New iterative scheme

for asymptotically quasi-nonexpansive mappings, J. Inequal. Appl. textbf{2010}, Article ID 934692, 9 pages,

doi:10.1155/2010/934692.

L.C. Zeng, A note on approximating fixed

points of nonexpansive mapping by the Ishikawa iterative process, J. Math. Anal. Appl. textbf{226} (1998), 245-250.

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