Common fixed point and solution of nonlinear functional equations
© Khan; licensee Springer. 2013
Received: 16 May 2013
Accepted: 7 October 2013
Published: 9 November 2013
Convergence of a new iterative scheme, containing Mann and Ishikawa iterative schemes, for asymptotically nonexpansive mappings on a 2-uniformly convex hyperbolic space is studied. As application, we find a solution of a system of certain nonlinear functional equations in uniformly convex Banach spaces.
The Banach contraction principle asserts that a contraction on a complete metric space has a unique fixed point and its proof hinges on ‘Picard iterations’. This principle is applicable to a variety of subjects such as integral equations, partial differential equations and engineering of image processing.
Many important nonlinear problems of mathematics reduce to nonlinear functional equations such as nonlinear integral equations and boundary value problems for nonlinear ordinary or partial differential equations which can be translated in terms of a fixed point equation for a given nonlinear mapping T on a Banach space or a metric space.
for an initial value .
We know that Picard iterates of nonexpansive mappings fail to converge even on a Banach space. Therefore, Mann  iterates were introduced to approximate fixed points of nonexpansive mappings. Mann iterates were not adequate for the approximation of fixed points of pseudocontractive mappings and this led to the introduction of Ishikawa iterates .
Let C be a nonempty subset of a metric space . A mapping T of C into itself is (i) asymptotically nonexpansive if there is a sequence with as and for all (when for each , it becomes nonexpansive); (ii) semi-continuous if for any bounded sequence in C satisfying , there exists a subsequence of such that ; (iii) completely continuous if every bounded sequence in C implies that has a convergent subsequence.
Nonexpansive mappings (the class of nonlinear mappings containing contractions as a subclass) remain a popular area of research in various fields. The iterative construction of fixed points of these mappings is a fascinating field of research. The fixed point problem for some nonlinear mappings has been studied on linear as well as nonlinear domains [4–8].
Numerous papers have appeared on the iterative construction of fixed points of asymptotically nonexpansive and asymptotically quasi-nonexpansive mappings in uniformly convex Banach spaces [3, 4, 7, 9–18].
Rhoades  established Mann and Ishikawa type convergence results as two separate results as follows.
Theorem 1 (, Theorem 2)
Let C be a nonempty bounded, closed and convex subset of a uniformly convex Banach space. Let T be a completely continuous asymptotically nonexpansive mapping on C with satisfying . Define to satisfy for all and . Then the Mann type iterative scheme in (1.4) converges to a fixed point of T.
Theorem 2 (, Theorem 3)
Let C be a nonempty bounded, closed and convex subset of a uniformly convex Banach space. Let T be a completely continuous asymptotically nonexpansive mapping on C with satisfying . Define , to satisfy for all and . Then the Ishikawa type iterative scheme in (1.3) converges to a fixed point of T.
An extension of a linear version (usually in Banach spaces) of a known result to metric fixed point theory has its own importance. As Mann and Ishikawa iterative schemes involve general convex combinations, we need some convex structure in a metric space to investigate their convergence on a nonlinear domain.
for all and (see also ).
A nonempty subset C of a convex metric space X is convex if for all and .
Normed spaces and their subsets are linear hyperbolic spaces while Hadamard manifolds , the Hilbert open unit ball equipped with the hyperbolic metric  and the spaces qualify for the criteria of nonlinear hyperbolic spaces [21, 23, 24, 29].
for any , and .
From now onwards we assume that X is a uniformly convex hyperbolic space with the property that for every , , there exists depending on s and ε such that for any .
where the infimum is taken over all such that , and .
It was shown in  that any space is 2-uniformly convex with .
Using the concept of a unique point in a metric segment , we express (1.2)-(1.4) in a convex hyperbolic space as follows:
In the sequel, the following results are needed.
Lemma 1 
for any .
Lemma 2 
If and , then exists.
Our purpose in this paper is to approximate a common fixed point of two asymptotically nonexpansive mappings through iterative scheme (1.6) in 2-uniformly convex hyperbolic spaces. This work provides a unified approach to convergence results for Mann and Ishikawa iterative schemes.
2 Convergence in 2-uniformly convex hyperbolic spaces
We set and .
Lemma 3 Let C be a nonempty convex subset of a hyperbolic space X, and let be asymptotically nonexpansive mappings with sequence such that . Then, for the sequence in (1.6), exists for all .
where . As , so by Lemma 2, exists. □
for all , and some .
Theorem 3 Let C be a nonempty closed and convex subset of a complete hyperbolic space X, and let be asymptotically nonexpansive mappings with sequence such that . Then in (1.6) converges to a point in F if and only if , where .
Proof We only prove sufficiency. Suppose that . It has been shown in the proof of Lemma 3 that . By the properties of inf, we have that and hence, by Lemma 2, exists. Therefore the hypothesis gives that . Next we show that is a Cauchy sequence. Let . Since , there exists such that . Hence there must exist such that .
This proves that is a Cauchy sequence. Since X is complete and C is its closed subset, therefore . Now gives that . As F is closed, so . □
Lemma 5 Let C be a nonempty convex subset of a 2-uniformly convex hyperbolic space X, and let be asymptotically nonexpansive mappings with sequence such that . Define and to satisfy for all . Then, for the sequence in (1.6), .
for some .
The following concept is needed to proceed further.
Using Condition (A) and Theorem 3, we prove a convergence theorem in complete 2-uniformly convex spaces as follows.
Theorem 4 Let C be a nonempty convex subset of a complete 2-uniformly convex hyperbolic space X. Let be asymptotically nonexpansive mappings with sequence such that and satisfy Condition (A). Define and to satisfy for . Then the sequence in (1.6) converges to a point in F.
Proof By Lemma 5, . Using Condition (A), we get that . Now Theorem 3 gives that converges to a point in F. □
Another convergence theorem is established in the following result under any of Conditions (ii)-(iii) without requiring the completeness of the space X.
Theorem 5 Let C be a nonempty convex subset of a 2-uniformly convex hyperbolic space X. Let be asymptotically nonexpansive mappings with sequence such that and either S or T is semi-compact. Define and to satisfy for all . Then the sequence in (1.6) converges to a point in F.
Proof Lemma 5 gives that . Suppose that T is semi-compact. Since exists, therefore is bounded. As and T is semi-compact, so there is a subsequence of such that , and hence and . Therefore implies that . That is, . As exists and , therefore . □
where for all .
Following the line of action of the proofs of Theorem 3 and Lemma 5, we can easily prove the following results.
Theorem 6 Let C be a nonempty closed and convex subset of a complete 2-uniformly convex hyperbolic space X, and let be a family of asymptotically quasi-nonexpansive self-mappings of C, i.e., for all and , . Suppose that , and for all i. Then the iterative sequence , defined by (2.6), converges to a common fixed point of the family of mappings if and only if .
Theorem 7 Let C be a nonempty closed and convex subset of a 2-uniformly convex hyperbolic space X, and let be a family of asymptotically nonexpansive mappings of C, i.e., for all and , where are sequences in with for each . Assume that and the sequence is in (2.6) with for some . If for some i, , is semi-compact, then converges to a point in .
Remark 1 (1) Theorem 5 extends (unifies) Theorem 2 of Khan and Takahashi  (Theorems 1-2) in the setting of 2-uniformly convex hyperbolic spaces.
(2) Theorem 6 establishes Theorem 1 by Qihou  together with its Corollaries 1 and 2, which are themselves extensions of the results of Ghosh and Debnath  and Petryshyn and Williamson , for two asymptotically nonexpansive mappings on a 2-uniformly convex hyperbolic space.
(3) All the results of this paper, in particular, hold in spaces.
for a family of asymptotically nonexpansive mappings on B.
The author is grateful to King Fahd University of Petroleum & Minerals for supporting the research project IN 121037.
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