Weak and strong convergence of a scheme with errors for two nonexpansive mappings
Introduction
Let C be a nonempty convex subset of a normed space E and be two mappings. Xu [11] introduced the following iterative schemes known as Mann iterative scheme with errors and Ishikawa iterative scheme with errors:
(1) The sequence defined by where are sequences in such that and is a bounded sequence in C, is known as Mann iterative scheme with errors. This scheme reduces to Mann iterative scheme if .
(2) The sequence defined by where are sequences in satisfying and are bounded sequences in , is known as Ishikawa iterative scheme with errors. This scheme becomes Ishikawa iterative schemes if . Chidume and Moore [2] studied the above schemes in 1999.
A generalization of Mann and Ishikawa iterative schemes was given by Das and Debata [3] and Takahashi and Tamura [9]. This scheme dealt with two mappings:
We further generalize this scheme to the one with errors as follows.
(3) The sequence , in this case, is defined bywhere are sequences in with , , and are bounded sequences in .
Nonexpansive mappings since their introduction have been extensively studied by many authors in different frames of work. One is the convergence of iteration schemes constructed through nonexpansive mappings. In this paper, we study the iterative scheme given in (1.1) for weak and strong convergence for a pair of nonexpansive mappings in a uniformly convex Banach space. To proceed in this direction, we first recall the following definitions.
A Banach space E is said to satisfy Opial's condition [6] if for any sequence in E, implies that for all with .
A mapping is called demiclosed with respect to if for each sequence in C and each , and imply that and .
Next we state the following useful lemmas. Lemma 1 Schu [7] Suppose that E is a uniformly convex Banach space and for all positive integers . Also suppose that and are two sequences of such that , and hold for some . Then . Lemma 2 Tan and Xu [10] Let be two nonnegative sequences satisfying If then exists. Lemma 3 Browder [1] Let E be a uniformly convex Banach space satisfying Opial’s condition and let C be a nonempty closed convex subset of E. Let T be a nonexpansive mapping of C into itself. Then is demiclosed with respect to zero.
Section snippets
Convergence of the iteration scheme
In this section, we shall prove the weak and strong convergence of the iteration scheme (1.1) to a common fixed point of the nonexpansive mappings S and . Let denote the set of all fixed points of . We first prove the following lemmas. Lemma 4 Let E be a normed space and C its nonempty bounded convex subset. Let be nonexpansive mappings. Let be the sequence as defined in (1.1) with , . If , then exists for all . Proof Since C is
References (10)
- et al.
Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process
J. Math. Anal. Appl.
(1993) Ishikawa and Mann Iteration process with errors for nonlinear strongly accretive operator equations
J. Math. Anal. Appl.
(1998)- F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proceedings of the Symposium...
- et al.
Fixed points iteration for pseudocontractive maps
Proc. Amer. Math. Soc.
(1999) - et al.
Fixed points of quasi-nonexpansive mappings
Indian J. Pure Appl. Math.
(1986)
Cited by (116)
A new iterative method for approximating common fixed points of two non-self mappings in a CAT(0) space
2023, Rendiconti del Circolo Matematico di PalermoIterative approximation of common fixed points of two nonself asymptotically nonexpansive mappings in CAT(0) spaces with numerical examples
2023, Mathematical Methods in the Applied SciencesCOMMON FIXED POINT THEOREMS OF TWO FINITE FAMILIES OF ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS IN HYPERBOLIC SPACES
2023, Journal of Nonlinear Functional AnalysisOn fixed point results for some generalized nonexpansive mappings
2023, AIMS Mathematics