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Strong convergence for total asymptotically pseudocontractive semigroups in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 216 (2012)
Abstract
In this article, the demiclosedness principle for total asymptotically pseudocontractions in Banach spaces is established. The strong convergence to a common fixed point of total asymptotically pseudocontractive semigroups in Banach spaces is established based on the demiclosedness principle, the generalized projective operator, and the hybrid method. The main results presented in this article 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 the norm , is the dual space of E; is the duality pairing between E and ; C is a nonempty closed convex subset of E; â„• 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 the mapping T. We use ‘→’ to stand for strong convergence and ‘⇀’ for weak convergence.
As is well known, fixed theory has always been kept a watchful eye on in nonlinear analysis, such as [1] proved Krasnoselskii’s fixed point theorem for general classes of maps, [2] established strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in a Banach space. Common fixed points of semigroups have also been discussed; see [3–5]. The construction of fixed points of pseudocontractive mappings (asymptotically pseudocontractive mappings), and of common fixed points of pseudocontractive semigroups (asymptotically pseudocontractive semigroups) is an important problem in the theory of pseudocontractive mappings. Iterative approximation of fixed points or common fixed points for asymptotically pseudocontractive mappings, pseudocontractive semigroups, asymptotically pseudocontractive semigroups in Hilbert or Banach spaces has been studied extensively by many authors; see for example [4, 6–12]. Alber et al. [13] introduced a more general class of asymptotically nonexpansive mappings called total asymptotically nonexpansive mappings and studied the methods of approximation of fixed points. Qin et al. [14] have introduced total asymptotically pseudocontractive mappings and proved a weak convergence theorem of fixed points for total asymptotically pseudocontractive mappings in Hilbert spaces.
The pseudocontractive semigroup and asymptotically pseudocontractive semigroup are defined as follows.
Definition 1.1 [4]
One-parameter family of mappings from C into itself is said to be a pseudocontractive semigroup on C, if the following conditions are satisfied:
-
(a)
for each ;
-
(b)
for any and ;
-
(c)
For any , the mapping is continuous;
-
(d)
For all , there exists such that
holds.
Definition 1.2 One-parameter family of mappings from C into itself is said to be an asymptotically pseudocontractive semigroup on C, if the conditions (a), (b), (c) in Definition 1.1 and the following condition (e) are satisfied:
-
(e)
For all , , there exists a sequence of positive real numbers with , , and there exists such that
holds.
The purpose of this article is to introduce the concept of a total asymptotically pseudocontractive semigroup and to use the hybrid method for total asymptotically pseudocontractive semigroups to get the strong convergence in Banach spaces. The results presented in the article improve and extend the corresponding results of many authors.
2 Preliminaries
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 the following conditions hold, :
-
1.
;
-
2.
;
-
3.
.
Following Alber [15], the generalized projection is defined by
In order to prove the results of this paper, we shall need the following lemmas.
Lemma 2.1 [16]
Let E be a uniformly convex and smooth Banach space and let and be two sequences of E. If and either or is bounded, then .
Lemma 2.2 [15]
Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conditions hold:
-
(i)
for all , ;
-
(ii)
If and , then ⇔ , ;
-
(iii)
For , if and only if .
Qin[14] introduced the class of total asymptotically pseudocontractive mappings in Hilbert spaces and established a weak convergence theorem of fixed points. Now, we give the definition of total asymptotically pseudocontractive mappings in a Banach space.
Definition 2.1 The mapping T from C into itself is said to be total asymptotically pseudocontractive on C if, for all , , there exist sequences , with as and strictly increasing continuous functions with , and there exists such that
holds.
Remark 2.1 If , then reduces to , so the total asymptotically pseudocontractive mappings include asymptotically pseudocontractive mappings. If , for all , then total asymptotically pseudocontractive mappings coincide with the class of pseudocontractive mappings.
Yang [17] has studied total asymptotically strict pseudocontractive semigroups. We now introduce the following semigroup mappings.
Definition 2.2 (1) One-parameter family of mappings from C into itself is said to be a total asymptotically pseudocontractive semigroup on C if the conditions (a), (b), (c) in Definition 1.1 and the following condition (f) are satisfied:
-
(f)
For all , , there exist sequences , with as and strictly increasing continuous functions with , for all , , there exists such that
holds.
-
(2)
A total asymptotically pseudocontractive semigroup T is said to be uniformly Lipschitzian if there exists a bounded measurable function such that
In this article, we let .
3 Main results
Theorem 3.1 (Demiclosedness principle)
Let E be a reflexive smooth Banach space with a weakly sequential continuous duality mapping J and C be a nonempty bounded and closed convex subset of E. Let be a uniformly L-Lipschitzian and total asymptotically pseudocontractive semigroup from C into itself defined by Definition 2.2. Suppose there exists such that . Then is demiclosed at zero, where I is the identical mapping.
Proof Assume that , with and as . We want to prove and . Since C is a closed convex subset of E, C is weakly closed. So, . In the following, we prove .
Now, we choose , and let for . Because is uniformly L-Lipschitzian, we have
Since is total asymptotically pseudocontractive, we have
By assumption , and as , J is a weakly sequential continuous duality mapping, we have
By the L-Lipschitz of and the definition of , we have
Thus,
which implies that
When , , so we have , , i.e., , , so , . By the continuity of , we have . □
Theorem 3.2 Let E be a real uniformly convex and uniformly smooth Banach space with a weakly sequential continuous convex duality mapping J and C be a nonempty closed convex subset of E. Let be a total asymptotically pseudocontractive semigroup of mappings from C into itself defined by Definition 2.2. Suppose there exists such that . Suppose are closed, convex, uniformly L-Lipschitz and . Let be a sequence in , where . Let be a sequence generated by
where , then the iterative sequence converges strongly to a common fixed point in C.
Proof We divide the proof into six steps.
-
(I)
is well defined for every . Since is uniform L-Lipschitzian continuous and convex, so is closed and convex. Moreover, is nonempty, therefore, is well defined for every .
-
(II)
We prove that and are closed and convex for all .
From the definitions of and , it is obvious that is closed and convex and is closed for each . is convex for each , which follows from the convexity of J.
-
(III)
We prove for each .
We first show . Let , . By (7), and the uniform L-Lipschitz continuity of and the total asymptotical pseudocontractiveness of , we have
This implies that
This shows that for all . This proves that for all . Next, we prove for all . By induction, for , we have . Assume that . Since is the projection of x onto , by Lemma 2.2, we have
for any . This with the assumption , and the definition of leads to , .
-
(IV)
We prove that as .
From (7) and Lemma 2.2, we have , this shows, for any and , we get
for all . So, we have and is bounded. So, by Lemma 2.2, we have
as . By Lemma 2.1, we get as .
-
(V)
Now, we prove as for all .
It follows from as , , is bounded, is uniformly bounded, J is a weakly sequential continuous duality mapping, and that
So, as . In addition,
So, as .
-
(VI)
Finally, we prove as .
Let be a subsequence of such that , then by Theorem 3.1, we have . We let . For any , and , so we get .
On the other hand, from the weak lower semicontinuity of the norm, we have
From the definition of , we obtain and hence . So, we have . Using the Kadec-Klee property of E, we obtain that converges strongly to . Since is an arbitrary weakly convergent sequence of , we can conclude that converges strongly to . This completes the proof of Theorem 3.2. □
Remark 3.1 Theorem 7 extends the main results of Zhou [8, 9], improves the results of Yang [17] and many others.
Remark 3.2 It is significant to remove the convexity of the duality mapping J and the total asymptotically pseudocontractive semigroup in Theorem 7 of this article.
Authors’ information
All the authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.
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Acknowledgements
This work was supported by Scientific Research Fund Project of SiChuan Provincial Education Department (No. 11ZA172 and No. 12ZB345).
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Wang, X., Quan, J. Strong convergence for total asymptotically pseudocontractive semigroups in Banach spaces. Fixed Point Theory Appl 2012, 216 (2012). https://doi.org/10.1186/1687-1812-2012-216
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DOI: https://doi.org/10.1186/1687-1812-2012-216