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Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Mappings
Fixed Point Theory and Applications volume 2009, Article number: 402602 (2009)
Abstract
We propose a relaxed composite implicit iteration process for finding approximate common fixed points of a finite family of strictly pseudocontractive mappings in Banach spaces. Several convergence results for this process are established.
1. Introduction and Preliminaries
Let be a real Banach space, and let be its dual space. Denote by the normalized duality mapping from into defined by
where is the generalized duality pairing between and . If is smooth, then is single valued and continuous from the norm topology of to the weak* topology of .
A mapping with domain and range in is called -strictly pseudocontractive in the terminology of Browder and Petryshyn [1], if there exists a constant such that
for all and all . Without loss of generality, we may assume . If denotes the identity operator, then (1.2) can be written in the form
for all and all . In (1.2) and (1.3), the positive number is said to be a strictly pseudocontractive constant.
The class of strictly pseudocontractive mappings has been studied by several authors (see, e.g., [1–10]). It is shown in [4] that a strictly pseudocontractive map is -Lipschitzian (i.e., for some ). Indeed, it follows immediately from (1.3) that
and hence where . It is clear that in Hilbert spaces the important class of nonexpansive mappings (mappings for which ) is a subclass of the class of strictly pseudocontractive maps.
Let be a nonempty convex subset of , and let be a finite family of nonexpansive self-maps of . In [11], Xu and Ori introduced the following implicit iteration process; for any initial and , the sequence is generated as follows:
The scheme is expressed in a compact form as
where . Moreover, they proved the following convergence theorem in a Hilbert space.
Theorem 1.1 ([11]).
Let be a Hilbert space, and let be a nonempty closed convex subset of . Let be nonexpansive self-maps of such that where . Let , and let be a sequence in , such that . Then the sequence defined implicitly by (1.6) converges weakly to a common fixed point of the mappings .
Subsequently, Osilike [12] extended their results from nonexpansive mappings to strictly pseudocontractive mappings and derived the following convergence theorems in Hilbert and Banach spaces.
Theorem 1.2 ([12]).
Let be a real Hilbert space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where . Let , and let be a sequence in such that . Then the sequence defined by (1.6) converges weakly to a common fixed point of the mappings .
Theorem 1.3 ([12]).
Let be a real Banach space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where , and let be a real sequence satisfying the conditions:
(i)
(ii);
(iii).
Let , and let be defined by (1.6). Then
(i) exists for all ;
(ii).
Let be a nonempty closed convex subset of a real Banach space . Very recently, Su and Li [13] introduced a new implicit iteration process for strictly pseudocontractive self-maps of :
that is,
where and . First, they established the following convergence theorem.
Theorem 1.4 ([13, Theorem ]).
Let be a real Banach space, and let be a nonempty closed convex subset of . Let be strictly pseudocontractive self-maps of such that , where , and let be two real sequences satisfying the conditions:
(i);
(ii);
(iii);
(iv), where is common Lipschitz constant of .
For , let be defined by (1.8). Then
(i) exists for all ;
(ii).
Second, they derived the following result by using Theorem 1.4.
Theorem 1.5 ([13, Theorem ]).
Let be a nonempty closed convex subset of a real Banach space , let be a semicompact strictly pseudocontractive self-map of such that , where , and let be a real sequence satisfying the conditions:
(i);
(ii).
Then for , the sequence defined by Mann iterative process,
converges strongly to a fixed point of .
On the other hand, Zeng and Yao [14] introduced a new implicit iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive self-maps of a real Hilbert space and established some convergence theorems for this implicit iteration scheme. To be more specific, let be a finite family of nonexpansive self-maps of , and let be a mapping such that for some constants is a -Lipschitz and -strongly monotone mapping. Let and and take a fixed number . The authors proposed the following implicit iteration process with perturbed mapping .
For an arbitrary initial point , the sequence is generated as follows:
The scheme is expressed in a compact form as
It is clear that if , then the implicit iteration scheme (1.11) with perturbed mapping reduces to the implicit iteration process (1.6).
Theorem 1.6 ([14, Theorem ]).
Let be a real Hilbert space, and let be a mapping such that for some constants ; is -Lipschitz and -strongly monotone. Let be nonexpansive self-maps of such that . Let , let , , and let satisfying the conditions: and , for some . Then the sequence defined by (1.11) converges weakly to a common fixed point of the mappings .
The above Theorem 1.6 extends Theorem 1.1 from the implicit iteration process (1.6) to the implicit iteration scheme (1.11) with perturbed mapping.
Let be a real Banach space, and let be a nonempty convex subset of . Recall that a mapping is said to be -strongly accretive if there exists a constant such that
for all and all .
Proposition 1.7.
Let be a real Banach space, and let be a mapping:
(i)if is -strictly pseudocontractive, then is a Lipschitz mapping with constant .
(ii)if is both -strictly pseudocontractive and -strongly accretive with , then is nonexpansive.
Proof.
It is easy to see that statement (i) immediately follows from the definition of strict pseudocontraction. Now utilizing the definitions of strict pseudocontraction and strong accretivity, we obtain
Since ,
and hence is nonexpansive.
Let be a real Banach space, and let be a nonempty convex subset of such that . Let be strictly pseudocontractive self-maps of , and let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . In this paper we introduce a general implicit iteration process as follows:
where , and . In particular, whenever , it is easy to see that (1.15) reduces to (1.8).
Let denote common Lipschitz constant of strictly pseudocontractive self-maps of . Since is a nonempty convex subset of such that , for each , the operator
maps into itself.
Utilizing Proposition 1.7, we have
for all . Thus, is strongly pseudocontractive, if for each . Since is also Lipschitz mapping, it follows from [12, 15, 16] that has a unique fixed point , that is, for each
Therefore, if , then the composite implicit iteration process (1.15) with perturbed mapping can be employed for the approximation of common fixed points of strictly pseudocontractive self-maps of .
The purpose of this paper is to investigate the problem of approximating common fixed points of strictly pseudocontractive mappings of Browder-Petryshyn in an arbitrary real Banach space by this general implicit iteration process (1.15). To this end, we need the following lemma and definition.
Lemma 1.8 ([8]).
Let , and be sequences of nonnegative real numbers satisfying the inequality
If
then exists.
The following definition can be found, for example, in [13].
Definition 1.9.
Let be a closed subset of a real Banach space , and let be a mapping. is said to be semicompact if, for any bounded sequence in such that , there must exist a subsequence such that .
2. Main Results
We are now in a position to prove our main results in this paper.
Theorem 2.1.
Let be a real Banach space, and let be a nonempty closed convex subset of such that . Let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . Let be strictly pseudocontractive self-maps of such that , where , and let , and be three real sequences in satisfying the conditions:
(i);
(ii);
(iii);
(iv);
(v), where is the common Lipschitz constant of .
For , let be defined by
where , then
(i) exists for all ;
(ii).
Proof.
First, since each strictly pseudocontractive mapping is a Lipschitz mapping, there exists a constant such that
It is now well known (see, e.g., [15]) that
for all and all . Take arbitrarily. Then it follows from (2.1) that
Utilizing (2.3), we obtain
Since each , is strictly pseudocontractive, there exists such that
Thus, utilizing Proposition 1.7(ii) we know from (2.5) that
From (2.1), we also have that
Since is a Lipschitz mapping with constant , we have
Substituting (2.8) and (2.9) into (2.7), we deduce that
and hence
Setting
we conclude from (2.11) that
Thus
Since
and , we get
Setting , it follows from condition (ii) that and so there must exist a natural number such that for all ,
Therefore, it follows from (2.14) that
In order to consider the second term on the right-hand side of (2.18), we will prove that is bounded. Indeed, utilizing (2.8) and (2.9) and simplifying these inequalities, we have
and hence
This implies that
Now, we consider the second term on the right-hand side of (2.21). Since , we have
Since , there exists a natural number such that for all ,
Again, it follows from condition that
Therefore, it follows from (2.21) that
According to conditions (ii)–(iv), we can readily see that
Thus, in terms of Lemma 1.8 we deduce that exists, and hence is bounded.
Now, we consider the second term on the right-hand side of (2.18). Since is bounded, and , there exists a constant and a natural number such that for all ,
Thus, it follows from (2.18) that
Since is bounded, there exists a constant such that . It follows from (2.28) that
and hence
Utilizing conditions (ii)–(iv), we know from (2.30) that
Since , we have
This completes the proof of Theorem 2.1.
The iterative scheme (1.15) becomes the explicit version as follows, whenever :
In the case when , (2.33) is the Mann iteration process as follows:
The conclusion of Theorem 2.1 remains valid for the iteration processes (2.33) and (2.34). Furthermore, we have the following result.
Theorem 2.2.
Let be a real Banach space, and let be a nonempty closed convex subset of such that . Let be a perturbed mapping which is both -strongly accretive and -strictly pseudocontractive with . Let be a semicompact strictly pseudocontractive self-map of such that , where , and let and be two real sequences in satisfying the conditions:
(i);
(ii);
(iii).
Then Mann iteration process (2.34) converges strongly to a fixed point of .
Proof.
Since
there exists a subsequence of such that
By the semicompactness of , there must exist a subsequence of such that
It follows from (2.36) that , and hence . Since exists, we have
This completes the proof of Theorem 2.2.
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Acknowledgments
The first author was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Shanghai Leading Academic Discipline Project (S30405) and Innovation Program of Shanghai Municipal Education Commission (09ZZ133). The third author was partially supported by the Grant NSF 97-2115-M-110-001
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Ceng, L.C., Shyu, D.S. & Yao, J.C. Relaxed Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of Strictly Pseudocontractive Mappings. Fixed Point Theory Appl 2009, 402602 (2009). https://doi.org/10.1155/2009/402602
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DOI: https://doi.org/10.1155/2009/402602