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Strong convergence theorems for a finite family of total asymptotically strict pseudocontractive semigroups in Banach spaces
Fixed Point Theory and Applications volume 2013, Article number: 178 (2013)
Abstract
The purpose of this paper is to introduce the concepts of total asymptotically strictly pseudocontractive semigroup, asymptotically strictly pseudocontractive semigroup etc., and to prove some strong convergence theorems of the explicit iteration process for these kinds of semigroups in arbitrary Banach spaces. The results presented in the paper extend and improve some recent results announced in the current literature.
MSC:47H20, 47H10.
1 Introduction and preliminaries
Let E be a real Banach space, be the dual space of E. In the sequel we assume that C is a nonempty closed convex subset of E, is the set of nonnegative real numbers and is the normalized duality mapping defined by
Let be a mapping, we denote by the set of fixed points of T. We also use ‘→’ to stand for strong convergence and ‘⇀’ for weak convergence.
We first recall some definitions.
A one-parameter family of self-mappings of C is said to be a nonexpansive semigroup if the following conditions are satisfied:
-
(i)
, for any and ;
-
(ii)
for each ;
-
(iii)
for each , is continuous;
-
(iv)
for any , is a nonexpansive mapping on C, that is, for any ,
(1.2)
If the family satisfies conditions (i)-(iii), then it is:
-
(a)
Pseudocontractive semigroup if for any , there exists such that
(1.3) -
(b)
Uniformly Lipschitzian semigroup if there exists a bounded measurable function such that, for any and ,
(1.4)
In this case, we also say that â„‘ is a uniformly -Lipschitzian semigroup.
-
(c)
Strictly pseudocontractive semigroup if there exists a bounded function and for any given , there exists such that
(1.5)
for any .
It is easy to see that such a semigroup is -Lipschitzian and pseudocontractive semigroup.
-
(d)
Demicontractive semigroup if , and there exists a bounded function for any , and , there exists such that
(1.6)
In this paper, we introduce the following semigroups.
Definition 1.1 A one-parameter family of self-mappings of C satisfies conditions (i)-(iii), then it is:
-
(e)
Total asymptotically strictly pseudocontractive semigroup if there exist a bounded function and sequences and with and as . For any given , there exists such that
(1.7)
where is a continuous and strictly increasing function with .
In this case, we also say that â„‘ is a -total asymptotically strict pseudocontractive semigroup.
-
(f)
Asymptotically strictly pseudocontractive semigroup if there exist a bounded function and a sequence with as ; for any given , there exists such that
(1.8)
for any .
In this case, we also say that â„‘ is a -asymptotically strict pseudocontractive semigroup.
-
(g)
Asymptotically demicontractive semigroup if and there exist a bounded function and a sequence with as ; for any , and , there exists such that
(1.9)
for any .
In this case, we also say that â„‘ is a -asymptotically demicontractive semigroup.
Remark 1.2 If and , then a total asymptotically strict pseudocontractive semigroup is an asymptotically strict pseudocontractive semigroup. Every asymptotically strict pseudocontractive semigroup with is an asymptotically demicontractive semigroup. If , , an asymptotically strict pseudocontractive semigroup is a strict pseudocontractive semigroup, an asymptotically demicontractive semigroup is a demicontractive semigroup.
It is easy to see that condition (1.7) is equivalent to the following condition: for any , and , there exists such that
The convergence problem of implicit and explicit iterative sequences for nonexpansive semigroups to a common fixed point has been considered by some authors in various spaces; see, for example, [1–11].
In 1998, Shioji-Takahashi [1] introduced and studied a Halpern-type scheme for a common fixed point of a family of asymptotically nonexpansive semigroups in the framework of a real Hilbert space.
In 2003, Suzuki [2] proved that the implicit scheme defined by
converges strongly to a common fixed point of the family of nonexpansive semigroups in a real Hilbert space. Xu [3] extended the result of Suzuki to a more general real uniformly convex Banach space having a weakly sequentially continuous duality mapping.
In 2005, Aleyner and Reich [4] proved the strong convergence of an explicit Halpern-type scheme defined by
to a common fixed point of the family of nonexpansive semigroups in a reflexive Banach space with a uniformly Gâteaux differentiable norm.
Recently, Chang et al. [11] introduced the following explicit iteration process:
for the Lipschitzian and demicontractive semigroup in general Banach spaces. Under appropriate assumptions imposed upon the parameter sequences and , they proved that the sequence defined by (1.13) converges strongly to some point in .
Inspired and motivated by the above works of Shioji-Takahashi [1], Suzuki [2], Xu [3], Aleyner-Reich [4] and Chang [11], in this paper we aim to study the strong convergence to a common fixed point for a finite family of uniformly Lipschitzian and total asymptotically strict pseudocontractive semigroups , in general Banach spaces. The results presented in the paper extend and improve some recent results given in [4, 5, 7, 9].
The following lemmas will be needed in proving our main results.
Lemma 1.3 Let , and be sequences of nonnegative real numbers satisfying
where is some nonnegative integer. If and , then the limit exists.
Lemma 1.4 [12]
Let E be any real Banach space, let be the dual space of E and let be the normalized duality mapping. Then, for any , we have
2 Main results
Let E be a real Banach space, and let C be a nonempty, closed convex subset of E. For some fixed , let be a uniformly -Lipschitzian with a bounded measurable function and -total asymptotically strictly pseudocontractive semigroup with a bounded function , sequences with and as , such that
Then, for , and ,
where is a continuous and strictly increasing function with , and
Consider a family of uniformly -Lipschitzian and -total asymptotically strict pseudocontractive semigroups of C and let
For , , and any ,
and
Now, we are ready to give our main results.
Theorem 2.1 Let E be a real Banach space, and let C be a nonempty, closed convex subset of E. Let be a finite family of uniformly -Lipschitzian and -total asymptotically strictly pseudocontractive semigroups of C, , , , , , , , , L, λ, F, ϕ, and be the same as above. In addition, there exist positive constants M and such that for all . Let be the sequence generated by
where , , is a sequence in and is an increasing sequence in . If the following conditions are satisfied:
-
(1)
, , , .
-
(2)
Assume for any and for any bounded subset ,
(2.8) -
(3)
There exists a compact subset G of E such that for some .
Then the sequence converges strongly to some element in F.
Proof The proof of Theorem 2.1 is divided into four steps.
Step 1. First we prove that exists for all .
For any , by (1.4) we have
It follows from (2.7) and (2.9) that
and
By (1.10), for the point and p, there exists such that
Since Ï• is an increasing function, it results that if and if . In either case, we can obtain that
Thus, by Lemma 1.4, (2.10)-(2.13), we have
By condition (1), it follows from Lemma 1.3 that the limit exists and so the sequence is bounded in C.
Step 2. Now we prove that
In fact, it follows from (2.14) that
where . Hence, for some ,
Letting , we have
By condition (1), we obtain
which implies
Since exists for all and , using (2.11), we have
This implies that
It follows from (2.20) and (2.21) that
For any , we have
from (2.8) and (2.23), conclusion (2.15) is proved.
Step 3. Now we prove that
For each , let
Since is uniformly -Lipschitzian continuous, it follows from (2.1) and (2.4) that
from (2.15) and (2.21), we have
This implies that
where , .
For , we have
from (2.22) and (2.27), we obtain
It follows from (2.28) that , . This completes the proof.
Step 4. Finally, we prove the sequence converges strongly to some element in F.
By (2.25), we have , . If for some compact subset G of E and some , then there exists a subsequence of and such that
Hence, it follows from (2.29) that
Now, for any , since , there exists a subsequence of such that . Using (2.30) and the fact that is Lipschitzian, we get . Since is arbitrarily chosen, we have .
Since as and the limit exists, this implies that as . This completes the proof. □
The following theorem can be obtained from Theorem 2.1 immediately.
Theorem 2.2 Let E be a real Banach space, and let C be a nonempty, closed convex subset of E. Let be a finite family of uniformly -Lipschitzian and -asymptotically strictly pseudocontractive semigroups of C, , , , , , L, λ, F be as in Theorem 2.1. . Let be the sequence defined by (2.7), is a sequence in and let be an increasing sequence in . If the following conditions are satisfied:
-
(1)
, , .
-
(2)
Assume for any and for any bounded subset ,
-
(3)
There exists a compact subset G of E such that for some .
Then the sequence converges strongly to some element in F.
Proof Taking , , in Theorem 2.1, since all conditions in Theorem 2.1 are satisfied. It follows from Theorem 2.1 that the sequence as .
This completes the proof of Theorem 2.2. □
Remark 2.3 Theorems 2.1 and 2.2 extend and improve the corresponding results of Chang et al. [11], Shioji and Takahashi [1], Suzuki [2], Xu [3], Aleyner and Reich [4] and others.
Open problem It may be interesting to post the following open problem: Can Theorem 2.1 be generalized to a finite family of semigroups of mappings S which are representation, so commutative or left reversible (see Liu and Zhang [13] and related references there), which is not necessarily the positive real number with addition?
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Acknowledgements
The authors would like to express their thanks to the editor and the referees for their helpful comments and suggestions. This work was supported by the Natural Science Foundation of Sichuan Province (No. 08ZA008).
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Yang, L., Chang, Ss. & Zhao, F.H. Strong convergence theorems for a finite family of total asymptotically strict pseudocontractive semigroups in Banach spaces. Fixed Point Theory Appl 2013, 178 (2013). https://doi.org/10.1186/1687-1812-2013-178
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DOI: https://doi.org/10.1186/1687-1812-2013-178