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Coupled fixed point theorems for asymptotically nonexpansive mappings
Fixed Point Theory and Applications volume 2013, Article number: 68 (2013)
Abstract
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.
MSC:47H10, 47H09.
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 [1], while the authors in [2] introduced the convex approximation property of a space, proved that contractions satisfy an inequality analogue to the Zarantonello inequality (see [3]) and then studied the asymptotic behavior of contractions.
Given a nonempty subset D of a real linear normed space E, a self-mapping is said to be nonexpansive if the following inequality holds for all :
Many more general classes of mappings have been considered, including the class of asymptotically nonexpansive mappings introduced by Goebel and Kirk [4], defined by the relation
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. [5] 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.
Definition 1.1 Let D be a nonempty subset of a real normed linear space E. A mapping is said to be nonexpansive if
Definition 1.2 F is said to be asymptotically nonexpansive if there exists a sequence with such that
where the sequence is defined (see [7]) as follows:
Definition 1.3 F is said to be uniformly L-Lipschitzian (where L is a positive constant) if
When the equality is verified for , i.e., when
F is said to be Lipschitz with the constant L (or L-Lipschitzian).
Remark 1.4
-
1.
It is easy to see that if is a nonexpansive mapping, then F is an asymptotically nonexpansive mapping with a constant sequence .
-
2.
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 [6]) as follows:
(1.6)
In [5], Chang et al. defined demi-closed maps at the origin as follows.
Definition 1.5 [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 .
Lemma 1.7 Let E be a uniformly convex Banach space, C be a nonempty bounded closed convex subset of E. Then there exists a strictly increasing continuous convex function with such that for any Lipschitzian mapping with the Lipschitz constant , any finite many elements in and any finite many nonnegative numbers with , the following inequality holds:
Proof Let us prove by induction. For , (1.7) is trivial.
Let us prove for : Let δ be the modulus of uniform convexity of E and define by
It is well known (e.g., see [1, 9]) that d is strictly increasing, continuous, convex, satisfying and
for all such that and such that and .
Similarly,
Hence, if and , then from (1.9) and (1.10),
Putting (1.11) in (1.8), we have
i.e.,
Since is strictly increasing, and , where , we have
Let be defined as ; then
Hence,
Thus (1.7) is true for .
Now suppose that
Define a strictly increasing continuous convex function satisfying and
For example, if ∗ defines the inf-convolution, I the identity function on ℝ and the functions in the functions are extended to be +∞ on , then one could take (see [2])
Define also .
Fix and let . The case is trivial and therefore omitted. For the rest of the proof, we let the subscript i range through , while j ranges through . Put
We have that and after computation
The induction hypothesis (1.13) can also be written
We also have, by the triangle inequality,
From (1.12) we have
Similarly,
We also have
Similarly,
Put
Then, by (1.18) and (1.19),
Using (1.20) and (1.22) in (1.16) and also (1.21) and (1.22) in (1.17), we have
and
by summing,
Using (1.23) in (1.15) yields
Finally, when (1.22) and (1.24) are used in (1.14), we obtain
Therefore
To complete the proof, we show in the sequel that the dependence of on n can actually be omitted.
Since E is uniformly convex, E is B-convex (see [2]) and since the product of B-convex spaces is also B-convex (see [10]), is B-convex, hence has the convex approximation property (C.A.P.) (see [2]), i.e., for each , there exists a positive integer p such that
where is the open sphere centered at the origin and with r as radius, coM is the convex hall of M and
for every .
Put . Suppose satisfy
Consider . Thus, for each , there exist and such that
Now
Since
and
we have that
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
Let denote the projection of the first coordinate, i.e., for all . Note that if is a mapping, we have that is demi-closed at the origin if the existence of a sequence in such that
implies that
i.e.,
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., [4]), for all , there exist sequences and such that and , where , and , .
Since the sequences and converge strongly to 0 respectively, for any given and positive integer , there is an integer such that and
Since F is asymptotically nonexpansive,
and
Hence, for any ,
Similarly,
Since is asymptotically nonexpansive, is also asymptotically nonexpansive, hence is a Lipschitzian mapping with the Lipschitz constant . We have the following inequality:
By (2.1), we know that
By Lemma 1.7, (2.1) and (2.2), we have
Inequalities (2.4) and (2.5) into (2.3) yield
Taking the limit superior as in the above inequality and noting that is arbitrary, we have
Also, by the definition of the sequences and , we have that for all ,
Taking the limit superior in the above inequality, from (2.6) we have
Finally, the limit superior in the above inequality yields , which implies that as .
Similar computations yield that as . Hence,
Since F is continuous, we have
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.
Proof Let be fixed. Define the set as follows:
where is the open sphere in E of center x and radius r. D is bounded, so if (diameter of D), , hence . Let be the g.l.b. of . For each , define the set
The sets are nonempty and convex. Since E is reflexive, is nonempty. Now, for any and , there exists such that implies that .
Let us show that there exist such that the sequences and converge to x and y respectively. Suppose that for all such sequences do not converge. Then and a strictly increasing sequence of integers is such that
For ,
Hence
Let δ be the modulus of convexity of the space E. Assume and so that , and select n so that and so that .
If is sufficiently large, then implies that
and also
Now, by the uniform convexity of E,
Since , letting and , we have if
This contradicts the definition of , the g.l.b. of , since
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 [5] to nonexpansive maps defined in a product space. The proof of Theorem 2.2 follows the methodology in [4], 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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Olaoluwa, H., Olaleru, J.O. & Chang, SS. Coupled fixed point theorems for asymptotically nonexpansive mappings. Fixed Point Theory Appl 2013, 68 (2013). https://doi.org/10.1186/1687-1812-2013-68
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DOI: https://doi.org/10.1186/1687-1812-2013-68