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Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 142 (2012)
Abstract
The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive semigroups to prove strong convergence in Banach spaces. The main results presented in this paper extend and improve the corresponding results of many authors.
MSC:47H05, 47H09, 49M05.
1 Introduction
Throughout this article, we assume that E is a real Banach space with norm , is the dual space of E; is the duality pairing between E and ; C is a nonempty closed convex subset of E; and denote the natural number set and the set of nonnegative real numbers respectively. The mapping defined by
is called the normalized duality mapping. Let be a nonlinear mapping; denotes the set of fixed points of mapping T.
Alber et al.[1] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied the methods of approximation of fixed points. They are defined as follows.
Definition 1.1 Let be a mapping. T is said to be total asymptotically nonexpansive if there exist sequences , with as and a strictly increasing continuous function with such that holds for all and all .
T is said to be total asymptotically quasi-nonexpansive if , there exist sequences , with as and a strictly increasing continuous function with such that holds for all , and all .
Chidume and Ofoedu [2] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings and total asymptotically quasi-nonexpansive mappings in Banach spaces. Chidume et al.[3] gave a new iterative sequence and necessary and sufficient conditions for this sequence to converge to a common fixed point of finite total asymptotically nonexpansive mappings. Chang [4] established some new approximation theorems of common fixed points for a countable family of total asymptotically nonexpansive mappings in Banach spaces.
Recently, many researchers have focused on studying the convergence of iterative algorithms for quasi-ϕ-asymptotically nonexpansive (see [5–9]) and total quasi-ϕ-asymptotically nonexpansive (see [10–12]) mappings. Ye et al.[13] used a new hybrid projection algorithm to obtain strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces. Kim [14] used hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-ϕ-nonexpansive mappings to prove the strong convergence theorems. Saewan [15] used the shrinking projection method for solving generalized equilibrium problems and common fixed points for asymptotically quasi-ϕ-nonexpansive mappings.
A Banach space E is said to be strictly convex if for and ; it is also said to be uniformly convex if for any two sequences , in E such that and . Let be the unit sphere of E, then the Banach space E is said to be smooth provided exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for each . It is well known that if E is reflexive and smooth, then the duality mapping J is single valued. A Banach space E is said to have the Kadec-Klee property if a sequence of E satisfies that and , then . It is known that if E is uniformly convex, then E has the Kadec-Klee property.
In the sequel, we assume that E is a smooth, strictly convex and reflexive Banach space and C is a nonempty closed convex subset of E. We use to denote the Lyapunov functional defined by
It is obvious that
and
Following Alber [16], the generalized projection is defined by
The quasi-ϕ-asymptotically nonexpansive and total quasi-ϕ-asymptotically nonexpansive mappings are defined as follows.
Definition 1.2 A mapping is said to be quasi-ϕ-asymptotically nonexpansive, if , there exist sequences with as such that
holds for all , and all .
A mapping is said to be total quasi-ϕ-asymptotically nonexpansive, if , there exist sequences , with as and a strictly increasing continuous function with such that
holds for all , and all .
In recent years, many researchers have considered the convergence of asymptotically nonexpansive semigroups [17, 18]. The asymptotically nonexpansive semigroups are defined as follows.
Definition 1.3[17]
One-parameter family of mappings from C into itself is said to be an asymptotically nonexpansive semigroup on C, if the following conditions are satisfied:
-
(a)
for each ;
-
(b)
for any and ;
-
(c)
For any , the mapping is continuous;
-
(d)
There exist sequences with as such that
holds for all , .
We use to denote the common fixed point set of the semigroup T, i.e., .
Chang [19] used the modified Halpern-Mann type iteration algorithm for quasi-ϕ-asymptotically nonexpansive semigroups to prove the strong convergence in the Banach space. The quasi-ϕ-asymptotically nonexpansive semigroups are defined as follows.
Definition 1.4[19]
One-parameter family of mappings from C into itself is said to be a quasi-ϕ-asymptotically nonexpansive semigroup on C if the conditions (a), (b), (c) in Definition 1.3 and following condition (e) are satisfied:
-
(e)
For all , , , there exist sequences with as , such that
holds for all .
2 Preliminaries
This section contains some definitions and lemmas which will be used in the proofs of our main results in the following section.
Definition 2.1 One-parameter family of mappings from C into itself is said to be a total quasi-ϕ-asymptotically nonexpansive semigroup on C if conditions (a), (b), (c) in Definition 1.3 and following condition (f) are satisfied:
-
(f)
If , there exist sequences , with as and a strictly increasing continuous function with such that
holds for all , and all .
A total quasi-ϕ-asymptotically nonexpansive semigroup T is said to be uniformly Lipschitzian if there exists a bounded measurable function such that
The purpose of this article is to use the modified Halpern-Mann type iteration algorithm for total quasi-ϕ-nonexpansive asymptotically semigroups to prove the strong convergence in Banach spaces. The results presented in the article improve and extend the corresponding results of [5, 6, 9–12, 14, 15, 19] and many others.
In order to prove the results of this paper, we shall need the following lemmas:
Lemma 2.1 (See [16])
Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:
-
(i)
for all , ;
-
(ii)
If and , then , ;
-
(iii)
For , if and only if .
Lemma 2.2[19]
Let E be a uniformly convex and smooth Banach space and letandbe two sequences of E. Ifand eitheroris bounded, then.
Lemma 2.3[10]
Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and C be a nonempty closed convex subset of E. Letbe a closed and total quasi-ϕ-asymptotically nonexpansive mapping defined by Definition 1.2. If, then the fixed point setof T is a closed and convex subset of C.
3 Main results
Theorem 3.1 Let E be a real uniformly convex and uniformly smooth Banach space and C be a nonempty closed convex subset of E. Letbe a total quasi-ϕ-asymptotically nonexpansive semigroup from C into itself defined by Definition 2.1. Supposeis closed, uniformly L-Lipschitz and. Suppose there existssuch that. Letbe a sequence inandbe a sequence insatisfying the following conditions: , . Letbe a sequence generated by
where. Ifandis bounded in C, then the iterative sequenceconverges strongly to a common fixed pointin C.
Proof (I) We prove and all are closed and convex subsets in C.
It follows from Lemma 2.3 that , is a closed and convex subset of C. So is closed and convex in C. By the assumption we know that is closed and convex. We suppose that is closed and convex for some . By the definition of ϕ, we have that
This shows that is closed and convex.
-
(II)
We prove that .
In fact . Suppose that , . Let
It follows from (2) that for any , we have
and
Therefore, we have
Where . This shows that , so .
-
(III)
We prove that is a Cauchy sequence in C.
Since , from Lemma 2.1(ii), we have
Again since , , we have
It follows from Lemma 2.1(i) that for each , ,
Therefore, is bounded. By virtue of (1), is also bounded. Since and , we have . This implies that is nondecreasing. Hence, the limit exists. By the construction of , for any positive integer , we have and . This shows that
It follows from Lemma 2.2 that . Hence is a Cauchy sequence in C. Since C is complete, without loss of generality, we can assume that (some point in C). By the assumption, we have that
-
(IV)
Now we prove .
Since and , it follows from (3) and (4) that
Since , by Lemma 2.2, for each , we have
Since is bounded, and is a total quasi-ϕ-asymptotically nonexpansive semigroup with sequence , we have
This implies that is uniformly bounded. Since for each ,
This implies that , is also uniformly bounded. Since , from (3) we have
Since E is uniformly smooth, is uniformly continuous on each bounded subset of , it follows from (5) and (6) that
Since and J is uniformly continuous on each bounded subset of E, we have , and for each ,
By condition , we have that
Since J is uniformly continuous, this shows that uniformly for . Again by the assumptions that the semigroup is closed and uniformly L-Lipschitzian, we have
By uniformly for , and is a bounded and measurable function, and from (7) we have that
and
so we get
By virtue of the closeness of semigroup T, we have that , i.e., . By the arbitrariness of , we have .
-
(V)
Finally, we prove .
Let . Since and , we get , . This implies that
In view of the definition of , from (8), we have . Therefore, . This completes the proof of Theorem 3.1. □
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Acknowledgements
This work was supported by National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of Sichuan Provincial Education Department (No. 12ZB345 and No. 11ZA172).
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Quan, J., Chang, Ss. & Wang, X. Strong convergence for total quasi-ϕ-asymptotically nonexpansive semigroups in Banach spaces. Fixed Point Theory Appl 2012, 142 (2012). https://doi.org/10.1186/1687-1812-2012-142
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DOI: https://doi.org/10.1186/1687-1812-2012-142