- Open Access
Strong convergence of a CQ method for k-strictly asymptotically pseudocontractive mappings
© Dehghan and Shahzad; licensee Springer 2012
Received: 6 February 2012
Accepted: 26 October 2012
Published: 22 November 2012
Let E be a real q-uniformly smooth Banach space, which is also uniformly convex (for example, or spaces, ), and C be a nonempty bounded closed convex subset of E. Let be a k-strictly asymptotically pseudocontractive map with a nonempty fixed point set. A hybrid algorithm is constructed to approximate fixed points of such maps. Furthermore, strong convergence of the proposed algorithm is established.
which is the inequality considered by Qihou . In the same paper, the author proved strong convergence of the modified Mann iteration processes for k-strictly asymptotically pseudocontractive mappings in Hilbert spaces. The modified Mann iteration scheme was introduced by Schu [5, 6] and has been used by several authors (see, for example, [7–12]). In  Osilike extended Qihou’s result from Hilbert spaces to much more general real q-uniformly smooth Banach spaces, .
The classes of nonexpansive and asymptotically nonexpansive mappings are important classes of mappings because they have applications to solutions of differential equations which have been studied by several authors (see, e.g., [14–16] and references contained therein). It would be of interest to study the class of k-strictly asymptotically pseudocontractive mappings in view of the fact that it is closely related to the above two classes.
where denotes the convex closure of the set D, J is the normalized duality mapping, is a sequence in with , and is the metric projection from E onto . Then, they proved that generated by (1.3) converges strongly to a fixed point of the mapping T.
where denotes the convex closure of the set D, J is the normalized duality mapping, is a sequence in with , and is the metric projection from E onto .
The purpose of this paper is to establish a strong convergence theorem of the iterative algorithm (1.4) for k-strictly asymptotically pseudocontractive mappings in a uniformly convex and q-uniformly smooth Banach space.
E is uniformly smooth if and only if . Let . The Banach space E is said to be q-uniformly smooth if there exists a constant such that . Hilbert spaces, (or ) spaces, , and the Sobolev spaces, , , are q-uniformly smooth.
When is a sequence in E, we denote strong convergence of to by and weak convergence by . The Banach space E is said to have the Kadec-Klee property if for every sequence in E, and imply that . Every uniformly convex Banach space has the Kadec-Klee property .
In the sequel, we need the following results.
Proposition 2.1 (See )
holds for any nonexpansive mapping , any elements in C, and any numbers with .
Corollary 2.2 [, Corollary 1.2]
holds for any nonexpansive mapping , any elements in C, and any numbers with . (Note that γ does not depend on T.)
In order to utilize Corollary 2.2 for k-strictly asymptotically pseudocontractive mappings, we need the following lemmas.
Lemma 2.3 
Let E be a real Banach space, C be a nonempty subset of E, and be a k-strictly asymptotically pseudocontractive mapping. Then T is uniformly L-Lipschitzian.
Lemma 2.4 [, Lemma 3.1]
where L is the uniformly Lipschitzian constant of T and is the constant which appeared in [, Theorem 2.1].
for all and each .
Theorem 2.5 [, Theorem 3.1]
Let E be a real q-uniformly smooth Banach space which is also uniformly convex. Let C be a nonempty closed convex subset of E and be a k-strictly asymptotically pseudocontractive mapping with a nonempty fixed point set. Then is demiclosed at zero, i.e., if and , then , where is the set of all fixed points of T.
3 Strong convergence theorem
In this section, we study the iterative algorithm (1.4) for finding fixed points of k-strictly asymptotically pseudocontractive mappings in a uniformly convex and q-uniformly smooth Banach space. We first prove that the sequence generated by (1.4) is well defined. Then, we prove that converges strongly to , where is the metric projection from E onto .
Lemma 3.1 Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E, and let be a mapping. If , then the sequence generated by (1.4) is well defined.
which implies . Therefore, . By the mathematical induction, we obtain that for all . Therefore, is well defined. □
In order to prove our main result, the following lemma is needed.
Theorem 3.3 Let C be a nonempty bounded closed convex subset of a real q-uniformly smooth and uniformly convex Banach space E. Let be a k-strictly asymptotically pseudocontractive mapping with such that . Let be the sequence generated by (1.4). Then converges strongly to the element of , where is the metric projection from E onto .
Since E is uniformly convex, using the Kadec-Klee property, we have that . It follows that . This completes the proof. □
The authors are grateful to the reviewers for their useful comments.
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