Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces
© Zhang and Xie; licensee Springer 2012
Received: 9 June 2012
Accepted: 13 September 2012
Published: 2 October 2012
Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. We obtain a weak convergence theorem of the explicit averaging cyclic algorithm for a finite family of asymptotically strictly pseudocontractive mappings of K under suitable control conditions, and elicit a necessary and sufficient condition that guarantees strong convergence of an explicit averaging cyclic process to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces. The results of this paper are interesting extensions of those known results.
Let E and be a real Banach space and the dual space of E, respectively. Let () denote the generalized duality mapping from E into given by for all , where denotes the generalized duality pairing between E and . In particular, is called the normalized duality mapping and it is usually denoted by J. If E is smooth or is strictly convex, then is single-valued. In the sequel, we will denote the single-valued generalized duality mapping by .
where , .
It is shown in  that the class of asymptotically κ-strictly pseudocontractive mappings and the class of κ-strictly pseudocontractive mappings are independent.
Let be N asymptotically strictly pseudocontractive self-mappings of K, and denote the common fixed points set of by , where . We consider the following explicit averaging cyclic algorithm.
where with , a positive integer and . The cyclic algorithm was first studied by Acedo and Xu  for the iterative approximation of common fixed points of a finite family of strictly pseudocontractive mappings in Hilbert spaces, and it is better than implicit iteration methods.
In  Xiaolong Qin et al. proved the following theorem in a Hilbert space.
Theorem QCKS Let K be a closed and convex subset of a Hilbert space H and be an integer. Let, for each , be an asymptotically -strictly pseudo-contractive mapping for some and a sequence such that . Let and . Assume that . For any , let be the sequence generated by the cyclic algorithm (5). Assume that the control sequence is chosen such that for all and a small enough constant . Then converges weakly to a common fixed point of the family .
Osilike and Shehu  extended the result of Theorem QCKS from a Hilbert space to 2-uniformly smooth Banach spaces which are also uniformly convex. They proved the following theorem.
where and is the constant appearing in the inequality (7) with . Let be the sequence generated by the cyclic algorithm (5). Then converges weakly to a common fixed point of the family .
where , , then the iterative sequence (5) converges weakly to a common fixed point of the family .
Furthermore, we elicit a necessary and sufficient condition that guarantees strong convergence of the iterative sequence (5) to a common fixed point of the family in q-uniformly smooth Banach spaces.
⇀ for weak convergence.
denotes the weak ω-limit set of .
E is uniformly smooth if and only if .
Theorem HKX ([, p.1130])
It is well known (see, for example, [, p.107]) that a q-uniformly smooth Banach space has a Fréchet differentiable norm.
Lemma 2.1 ([, p.1338])
Let E be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a nonempty, closed and convex subset of E and be an asymptotically κ-strictly pseudocontractive mapping with a nonempty fixed point set. Then is demiclosed at zero, that is, if whenever such that converges weakly to and converges strongly to 0, then .
Lemma 2.2 ([, p.80])
If and , then exists. If, in addition, has a subsequence which converges strongly to zero, then .
Lemma 2.3 ([, p.78])
Let E be a real Banach space, K be a nonempty subset of E and be an asymptotically κ-strictly pseudocontractive mapping. Then T is uniformly L-Lipschitzian.
exists for every ;
, for each ;
exists for all and for all .
Then the sequence converges weakly to a common fixed point of the family .
Hence, . Since , then exists. Since , for all . Set . We have , that is, . Hence, is a singleton, so that converges weakly to a common fixed point of the family . □
3 Main results
Theorem 3.1 Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. Let be an integer and . Let, for each , be an asymptotically -strictly pseudocontractive mapping for some with sequences such that , where , and . Let . Let satisfy the conditions (6) and be the sequence generated by the cyclic algorithm (5). Then converges weakly to a common fixed point of the family .
then (10) implies exists by Lemma 2.2 (and hence the sequence is bounded, that is, there exists a constant such that ).
Since , then (11) implies that . Thus .
By the choice of , we have , so it follows that . For the convenience of the following discussion, set , then .
Hence , this ensures that exists for all .
Now apply Lemma 2.4 to conclude that converges weakly to a common fixed point of the family . □
Thus , and it follows from Lemma 2.2 that exists.
we have .
Conversely, suppose , then the existence of implies that . Thus, for arbitrary , there exists a positive integer such that for any .
Thus , , and hence . □
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