- Open Access
Strong convergence for total asymptotically pseudocontractive semigroups in Banach spaces
© Wang and Quan; licensee Springer 2012
Received: 23 August 2012
Accepted: 7 November 2012
Published: 28 November 2012
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.
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  proved Krasnoselskii’s fixed point theorem for general classes of maps,  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.  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.  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 
for each ;
for any and ;
For any , the mapping is continuous;
- (d)For all , there exists such that
- (e)For all , , there exists a sequence of positive real numbers with , , and there exists such that
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.
In order to prove the results of this paper, we shall need the following lemmas.
Lemma 2.1 
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 
for all , ;
If and , then ⇔ , ;
For , if and only if .
Qin 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.
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  has studied total asymptotically strict pseudocontractive semigroups. We now introduce the following semigroup mappings.
- (f)For all , , there exist sequences , with as and strictly increasing continuous functions with , for all , , there exists such that
- (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 .
When , , so we have , , i.e., , , so , . By the continuity of , we have . □
where , then the iterative sequence converges strongly to a common fixed point in C.
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 .
We prove that and are closed and convex for all .
We prove for each .
We prove that as .
Now, we prove as for all .
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 .
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 3.2 It is significant to remove the convexity of the duality mapping J and the total asymptotically pseudocontractive semigroup in Theorem 7 of this article.
All the authors contributed equally to the writing of the present article. And they also read and approved the final manuscript.
This work was supported by Scientific Research Fund Project of SiChuan Provincial Education Department (No. 11ZA172 and No. 12ZB345).
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