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Strong convergence of a CQ method for k-strictly asymptotically pseudocontractive mappings
Fixed Point Theory and Applications volume 2012, Article number: 208 (2012)
Abstract
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.
1 Introduction
Let E be a real Banach space and be the dual of E. We denote the value of at by . The normalized duality mapping J from E to is defined by
for all . It is known that a Banach space E is smooth if and only if the normalized duality mapping J is single valued. Some properties of the duality mapping have been given in [1, 2].
Let C be a nonempty subset of E. The mapping is called nonexpansive if
for all . Also, T is called uniformly L-Lipschitz if there exists a constant such that
for all and each . The mapping is called k-strictly asymptotically pseudocontractive if there exist a sequence in with and a constant , and for any , there exists such that
for each . If I denotes the identity operator, then (1.1) can be written in the form
The class of k-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, j is the identity and it is shown [4] that (1.1) (and hence (1.2)) is equivalent to the inequality
which is the inequality considered by Qihou [3]. 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 [13] 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.
On the other hand, using the metric projection, Matsushita and Takahashi [17] introduced the following iterative algorithm for nonexpansive mappings: and
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.
In this paper, motivated by these facts, we introduce the following iterative algorithm for finding fixed points of a k-strictly asymptotically pseudocontractive mapping T in a uniformly convex and q-uniformly smooth Banach space: , and
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.
2 Preliminaries
The modulus of smoothness of a Banach space E is the function defined by
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 [1].
Let C be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E. Then for any , there exists a unique point such that
The mapping defined by is called the metric projection from E onto C. Let and . Then it is known that if and only if
In the sequel, we need the following results.
Proposition 2.1 (See [19])
Let C be a bounded closed convex subset of a uniformly convex Banach space E. Then there exists a strictly increasing convex continuous function with depending only on the diameter of C such that
holds for any nonexpansive mapping , any elements in C, and any numbers with .
Corollary 2.2 [[20], Corollary 1.2]
Under the same suppositions as in Proposition 2.1, there exists a strictly increasing convex continuous function with depending only on the diameter of C such that
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 [4]
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 [[21], Lemma 3.1]
Let E be a real q-uniformly smooth Banach space and C be a nonempty convex subset of E. Let be a k-strictly asymptotically pseudocontractive map, and let be a real sequence in . Define by for all . Then for all , we have
where L is the uniformly Lipschitzian constant of T and is the constant which appeared in [[21], Theorem 2.1].
Let and choose . Set for all in Lemma 2.4 and observe that . Thus,
for all and each .
Theorem 2.5 [[21], 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.
Proof It is easy to check that is closed and convex and for each . Moreover, and so . Suppose for . Then there exists a unique element such that . If , then it follows from (2.1) that
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.
Lemma 3.2 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 for any ,
Proof Fix and put . Since , we have . Since , there exist and with such that
and for all . It follows from Lemma 2.3 that T is uniformly L-Lipschitzian. Put , and . Thus,
for all . Define by
for all , where and is as in (2.2). It follows from (2.2) that is nonexpansive. Using (3.2) and the fact that , we have
for all . It follows from Corollary 2.2, (3.1), and (3.3) that
Since and , it follows from the last inequality that . Thus, and so . This completes the proof. □
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 .
Proof Put . Since and , we have that
for all . By Lemma 3.2, we have
Since is bounded, there exists such that . It follows from Theorem 2.5 (demiclosedness of T) that . From the weakly lower semicontinuity of norm and (3.4), we obtain
This together with the uniqueness of implies , and hence . Therefore, we obtain . Furthermore, we have that
Since E is uniformly convex, using the Kadec-Klee property, we have that . It follows that . This completes the proof. □
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Dehghan, H., Shahzad, N. Strong convergence of a CQ method for k-strictly asymptotically pseudocontractive mappings. Fixed Point Theory Appl 2012, 208 (2012). https://doi.org/10.1186/1687-1812-2012-208
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DOI: https://doi.org/10.1186/1687-1812-2012-208