Coupled fixed point theorems for asymptotically nonexpansive mappings
© Olaoluwa et al.; licensee Springer 2013
Received: 21 September 2012
Accepted: 1 February 2013
Published: 25 March 2013
We introduce the theory of asymptotical nonexpansiveness of mappings defined in the algebraic product and with values in the space E. We then prove the existence of coupled fixed points of such mappings when E is a uniformly convex Banach space. This paper is an extension of some recent results in the literature.
1 Introduction and preliminaries
In the past years, many researchers have proved various results on the theory of nonexpansive mappings (or contractions). The mean ergodic theorem for contractions in uniformly convex Banach spaces was proved in , while the authors in  introduced the convex approximation property of a space, proved that contractions satisfy an inequality analogue to the Zarantonello inequality (see ) and then studied the asymptotic behavior of contractions.
where the sequence converges to 1 as . They proved that a self asymptotically nonexpansive map of a nonempty closed convex bounded subset of a real uniformly convex Banach space has a fixed point. Then Chang et al.  established some convergence theorems for this class of mappings without the assumption of boundedness of the subset D.
Recently, the concept of coupled fixed points was introduced and developed by some authors (see [6–8]). A coupled fixed point of a map is defined as an element such that and . One could be interested in extending nonexpansiveness to maps defined on a product space (the algebraic product) and study the existence of their coupled fixed points. This is the main purpose of this paper.
We now introduce the definitions of nonexpansive maps, asymptotically nonexpansive maps, Lipschitzian and uniformly Lipschitzian maps defined in product spaces.
F is said to be Lipschitz with the constant L (or L-Lipschitzian).
It is easy to see that if is a nonexpansive mapping, then F is an asymptotically nonexpansive mapping with a constant sequence .
If is an asymptotically nonexpansive mapping with a sequence such that , then it must be uniformly L-Lipschitzian with .
- 3.The sequence can be written as the sequence defined (see ) as follows:(1.6)
In , Chang et al. defined demi-closed maps at the origin as follows.
Definition 1.5 
Let E be a real Banach space and D be a closed subset of E. A mapping is said to be demi-closed at the origin if, for any sequence in D, the conditions weakly and strongly imply .
We extend this definition to maps defined in as follows.
Definition 1.6 Let E be a real Banach space and D be a closed subset of E. A mapping is said to be demi-closed at the origin if, for any sequence in , the conditions , weakly and , strongly imply .
Proof Let us prove by induction. For , (1.7) is trivial.
Thus (1.7) is true for .
Define also .
To complete the proof, we show in the sequel that the dependence of on n can actually be omitted.
for every .
whenever . One can easily construct f such that , i.e., such that , which guarantees (1.7) with f independent of n. □
2 Existence of coupled fixed points
i.e., the existence of a coupled fixed point of F.
Now we prove our main theorem.
Theorem 2.1 Let D be a nonempty closed convex subset of a uniformly convex Banach space E and be an asymptotically nonexpansive map with the sequence as defined in (1.2). Then satisfies the demi-closedness at the origin property. In other terms, if any sequence in is such that weakly, weakly, strongly and strongly, then F has a coupled fixed point .
Proof Since converges weakly to , and are bounded in D. Therefore, there exists such that , where is the ball of E of radius r centered in 0. Hence C is a nonempty bounded closed convex subset in D.
Next, we prove that as , and , where and .
Since and converge weakly to and respectively, by Mazur’s theorem (see, e.g., ), for all , there exist sequences and such that and , where , and , .
Finally, the limit superior in the above inequality yields , which implies that as .
Hence is a coupled fixed point of F, which completes the proof. □
The conclusion of demi-closedness in the previous theorem states that if sequences and in D are such that , weakly and , strongly, then is a coupled fixed point of F. A more direct conclusion about the existence of a coupled fixed point of F can be obtained by adding the property boundedness of the subset D. This fact is expressed in the following theorem.
Theorem 2.2 Let D be a nonempty convex closed and bounded subset of a uniformly convex Banach space E. Then any asymptotically nonexpansive mapping has a coupled fixed point.
The sets are nonempty and convex. Since E is reflexive, is nonempty. Now, for any and , there exists such that implies that .
Let δ be the modulus of convexity of the space E. Assume and so that , and select n so that and so that .
Hence, and so and . The fact that implies that the sequences and are Cauchy, hence and . This completes the proof. □
Since nonexpansive maps are asymptotically nonexpansive, we have the following corollary.
Corollary 2.3 Let D be a nonempty convex closed and bounded subset of a uniformly convex Banach space. Then any nonexpansive mapping has a coupled fixed point.
Remark 2.4 Theorem 2.1 is an extension of Theorem 1 of  to nonexpansive maps defined in a product space. The proof of Theorem 2.2 follows the methodology in , extending the result therein to product spaces. Our results are, to the best of our knowledge, first of their kind in the theory of nonexpansiveness in product spaces dealing with the existence of a coupled fixed point.
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