Fixed point approximation of asymptotically nonexpansive mappings in hyperbolic spaces
© Fukhar-ud-din and Kalsoom; licensee Springer. 2014
Received: 31 October 2013
Accepted: 27 February 2014
Published: 17 March 2014
Convergence theorems are established in a hyperbolic space for the modified Noor iterations with errors of asymptotically nonexpansive mappings. The obtained results extend and improve the several known results in Banach spaces and spaces simultaneously.
Nonexpansive mappings are Lipschitzian with Lipschitz constant equal to 1. The class of nonexpansive mappings enjoys the fixed point property and even the approximate fixed point property in the general setting of metric spaces. The importance of this class lies in its powerful applications in initial value problems of the differential equations, game-theoretic model, image recovery and minimax problems. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk  as an important generalization of the class of nonexpansive mappings. Therefore, it is natural to extend powerful results for nonexpansive mappings to the class of asymptotically nonexpansive mappings. Iterative construction of fixed points of various nonlinear mappings emerged as the most powerful tool for solving such nonlinear problems. Approximation of fixed points of asymptotically nonexpansive mappings has been studied extensively by many authors; see for example [2–13] and the references cited therein.
In 1989, Glowinski and Le Tallec  used a three-step iterative process to find approximate solutions of elastoviscoplasticity problem, liquid crystal theory and eigenvalue computation. They observed that the three-step iterative process gives better numerical computations than two-step and one-step iterative processes. In 1998, Haubruge et al.  studied convergence analysis of a three-step iterative process of Glowinski and Le Tallec  and applied this process to obtain new splitting type iterations for solving variational inequalities, separable convex programming and minimization of a sum of convex functions. They also proved that the three-step iterative process leads to highly parallel iterations under certain conditions. Thus we conclude that the three-step iterative process plays an important and significant role in solving various numerical problems which arise in pure and applied sciences.
In 2000, Noor  introduced a three-step iterative process and studied the approximate solutions of variational inclusion in Hilbert spaces. In 2002, Xu and Noor  presented a three-step iterative process to approximate fixed points of asymptotically nonexpansive mappings in a Banach space. Cho et al.  extended Xu and Noor’s iterative process to a three-step iterative process with errors in Banach spaces and used it to approximate fixed points of asymptotically nonexpansive mappings. In 2005, Suantai  proposed and analyzed the modified three-step Noor iterative process. This process was further studied for different kinds of mappings by Khan and Hussain  and Khan  for example. Nammanee et al.  extended this process to the one with errors as follows.
where , , , , , , and are sequences in ; , and are bounded sequences in C.
By different choices of parameters , , , , , , , to be zero, one can see that one-step iterations of Mann , two-step iterations of Ishikawa , three-step iterations of Xu and Noor , three-step iterations with errors of Cho et al.  and modified three-step iterations of Suantai  all are the special cases of iteration process (1).
Most of phenomena in nature are nonlinear. Therefore, mathematicians and scientists are always in pursuit of finding methods to solve nonlinear real world problems. So translating a linear version of known problems into its equivalent nonlinear version has a great importance.
Keeping in mind the occurrence of such phenomena, we translate modified three-step Noor iterations with errors in a nonlinear domain, namely, hyperbolic spaces and study their convergence analysis in a new setup.
for all and (see also ); the space is convex  if only (a) is satisfied. A subset C of the hyperbolic 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].
Throughout the paper, a hyperbolic space will simply be denoted by X. A hyperbolic space X is uniformly convex (UC)  if for any , and , there exists such that , whenever and .
A mapping such that for a given and (as in the definition of UC) is known as a modulus of uniform convexity. We call η monotone if it decreases with respect to r (for a fixed ε).
it becomes nonexpansive if for all . It was shown in  that an asymptotically nonexpansive mapping on a nonempty, bounded, closed and convex subset of a (UC) hyperbolic space has a fixed point.
We translate (1) in a hyperbolic space as follows.
where , , , , , , , are sequences in and , and are bounded sequences in C and , and .
- (i)modified Noor iterations (with ):(4)
- (ii)Noor iterations with errors (with ):(5)
- (iii)Noor iterations (with ):(6)
- (iv)Ishikawa iterations (with ):(7)
- (v)Mann iterations (with ):(8)
The purpose of this paper is to establish convergence results of iteration process (3) for asymptotically nonexpansive mappings on a nonlinear domain ((UC) hyperbolic spaces) which includes both (UC) Banach spaces and spaces. Therefore, our results extend and improve the corresponding ones proved by Suantai , Xu and Noor  and others in a (UC) Banach space and are also valid in spaces, simultaneously.
In the sequel, we need the following lemmas.
Lemma 1.1 ()
Let , and be sequences of non-negative real numbers such that and . If , , then exists.
Lemma 1.2 ()
Let X be a (UC) hyperbolic space with monotone modulus of uniform convexity η. Let and be a sequence in for some . If and are sequences in X such that , and for some , then .
2 Main results
The following lemma is crucial for proving the convergence results.
Lemma 2.1 Let X be a (UC) hyperbolic space with monotone modulus of uniform convexity η, and let C be a nonempty, bounded, closed and convex subset of X. Let T be an asymptotically nonexpansive self-mapping on C with a sequence such that . For a given , compute , and as in (3) satisfying , , and .
If q is a fixed point of T, then exists.
- (ii)If , then
- (iii)If and , then
- (iv)If and , then
where , and .
Since , , and , it follows from Lemma 1.1 that exists.
(ii) Since C is bounded, there exists such that .
If , then there exist such that for all . We have shown in part (i) that exists, therefore (say).
If , then there exist such that for all . Similarly, gives that there exist such that for all .
If and , then there exist such that and for all .
Next we show that and .
, and .
Then , and in (3) converge to the same fixed point of T.
Since T is completely continuous and is bounded, there exists a subsequence of such that converges. Therefore from (24), converges. Let . By the continuity of T and (24), we have that , so q is a fixed point of T. By Lemma 2.1(i), exists. But . Thus . Further the inequalities and give that and , respectively.
For , Theorem 2.2 reduces to the following.
Then , and in (4) converge to the same fixed point of T.
For in Theorem 2.2, we obtain the following result.
Then , and in (6) converge to the same fixed point of T.
For in Theorem 2.2, we can obtain the Ishikawa-type convergence result.
Then and in (7) converge to the same fixed point of T.
For , Theorem 2.2 reduces to the Mann-type convergence result.
Corollary 2.6 Let C be a nonempty bounded, closed and convex subset of a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let T be a completely continuous asymptotically nonexpansive self-map of C with satisfying and . Let be real sequences in satisfying . Then in (8) converge to a fixed point of T.
As a direct consequence of Theorem 2.2, we formulate the following result in spaces.
, and .
where is the geodesic path between x and y in X. Then , and converge to the same fixed point of T.
Proof Any space is a (UC) hyperbolic space (take ), therefore conclusion follows from Theorem 2.2. □
Theorem 2.2 itself is a nonlinear version of Theorem 2.3 in .
Corollary 2.3 extends and generalizes Theorem 2.3 in .
Corollary 2.4 extends and generalizes Theorem 2.1 in .
Corollary 2.5 extends and generalizes Theorem 3 in .
(2) Our results also hold in a space.
The first author owes a lot to Professor Wataru Takahashi from whom he received his doctorate at Tokyo Institute of Technology, Tokyo, Japan. He is extremely indebted to Professor Takahashi and wishes him a long healthy active life. The first author is also grateful to King Fahd University of Petroleum and Minerals for supporting research project IN121055. The second author Amna Kalsoom gratefully acknowledges Higher Education Commission (HEC) of Pakistan for financial support during this research.
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