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A new multi-step iteration for solving a fixed point problem of nonexpansive mappings
Fixed Point Theory and Applications volume 2013, Article number: 198 (2013)
Abstract
We introduce a new nonlinear mapping generated by a finite family of nonexpansive mappings. Weak and strong convergence theorems are also established in the setting of a Banach space.
MSC:47H09, 47H10.
1 Introduction
Let C be a nonempty, closed and convex subset of a real Banach space E. Let be a nonlinear mapping. The fixed point set of T is denoted by , that is, . Recall that a mapping T is said to be nonexpansive if for all , and a mapping is called a contraction if there exists a constant such that for all . We use to denote a class of contractions with constant α.
Fixed point problems are now arising in a wide range of applications such as optimization, physics, engineering, economics and applied sciences. Many related problems can be cast as the problem of finding fixed points for nonlinear mappings. The interdisciplinary nature of fixed point problems is evident through a vast literature which includes a large body of mathematical and algorithmic developments.
In the literature, several types of iterations have been constructed and proposed in order to get convergence results for nonexpansive mappings in various settings. One classical iteration process is defined as follows: and
where . This method was introduced in 1953 by Mann [1] and is known as the Mann iteration process. However, we note that it has only weak convergence in general; for instance, see [2].
In 1974, Ishikawa [3] proposed the following two-step iteration: and
where and are sequences in . This method is often called the Ishikawa iteration process.
Very recently, Agarwal et al. [4] introduced a new iteration process as follows: and
where and are sequences in . This method is called the S-iteration process. The weak convergence was studied in [5] for nonexpansive mappings. It was also shown in [5] that the convergence rate of the S-iteration process is faster than the Picard and Mann iteration processes for contractive mappings.
Firstly, motivated by Agarwal et al. [4], we have the aim to introduce and study a new mapping defined by the following definition.
Definition 1.1 Let C be a nonempty and convex subset of a real Banach space E. Let be a finite family of nonexpansive mappings of C into itself, and let be real numbers such that for all . Define the mapping as follows:
Such a mapping B is called the B-mapping generated by and . See [6–9] for the corresponding concept.
Secondly, using the definition above, we study weak convergence of the following Mann-type iteration process in a uniformly convex Banach space with a Fréchet differentiable norm or that satisfies Opial’s condition: and
where is a B-mapping generated by and (see Section 3).
Finally, we discuss strong convergence of the iteration scheme involving the modified viscosity approximation method [10] defined as follows: and
where , and are sequences in , and .
More references on earlier works promoting the theory of fixed points and common fixed points for nonexpansive mappings can be found in [11–21].
Throughout this paper, we use the notation:
-
⇀ for weak convergence and → for strong convergence.
-
denotes the weak ω-limit set of .
2 Preliminaries and lemmas
In this section, we begin by recalling some basic facts and lemmas which will be used in the sequel.
Let E be a real Banach space and let be the unit sphere of E. A Banach space E is said to be strictly convex if for any ,
It is also said to be uniformly convex if for each , there exists such that for any ,
It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a function called the modulus of convexity of E as follows:
Then E is uniformly convex if and only if for all . A Banach space E is said to be smooth if the limit
exists for all . The norm is said to be uniformly Gâteaux differentiable if for , the limit is attained uniformly for . It is said to be Fréchet differentiable if for , the limit is attained uniformly for . It is said to be uniformly smooth or uniformly Fréchet differentiable if the limit (2.1) is attained uniformly for . The normalized duality mapping is defined by
for all . It is known that E is smooth if and only if the duality mapping J is single valued, and that if E has a uniformly Gâteaux differentiable norm, J is uniformly norm-to-weak∗ continuous on each bounded subset of E. A Banach space E is said to satisfy Opial’s condition [22]. If and , then
Let . Then is demiclosed at 0 if for all sequence in C, and imply . It is known that if E is uniformly convex, C is nonempty closed and convex, and T is nonexpansive, then is demiclosed at 0 [23]. For more details, we refer the reader to [5, 24].
Lemma 2.1 [5]
Let E be a smooth Banach space. Then the following hold:
-
(i)
for all ;
-
(ii)
for all .
Lemma 2.2 [24]
In a strictly convex Banach space E, if
for all and , then .
Lemma 2.3 [25]
Let and be two sequences in a Banach space E such that
where satisfies the conditions .
If , then as .
Lemma 2.4 [26]
Let E be a uniformly convex Banach space with a Fréchet differentiable norm. Let C be a closed and convex subset of E, and let be a family of -Lipschitzian self-mappings on C such that and . For arbitrary , define for all . Then, for every , exists, in particular, for all and , .
Lemma 2.5 [15]
Let E be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm, let C be a nonempty closed convex subset of E, let be a continuous strongly pseudocontractive mapping with constant , and let be a continuous pseudocontractive mapping satisfying the weakly inward condition. If T has a fixed point in C, then the path defined by
converges strongly to a fixed point q of T as , which is a unique solution of the variational inequality
Remark 2.6 Lemma 2.5 holds if is a nonexpansive mapping and is a contraction.
The following lemma gives us a nice property of real sequences.
Lemma 2.7 [18]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(a)
;
-
(b)
or .
Then .
3 Weak convergence theorem
In this section, we give some properties concerning the B-mapping and then prove a weak convergence theorem for nonexpansive mappings.
Lemma 3.1 Let C be a nonempty, closed and convex subset of a strictly convex Banach space E. Let be a finite family of nonexpansive mappings of C into itself such that , and let be real numbers such that for all and . Let B be the B-mapping generated by and . Then the following hold:
-
(i)
;
-
(ii)
B is nonexpansive.
Proof (i) Since is trivial, it suffices to show that . To this end, let and . Then we have
This shows that
which turns out to be
Again by (3.1), we see that and thus
Using Lemma 2.2, we get that and hence . Again by (3.1), we have
which implies that
From (3.1) we see that . Since and ,
Using Lemma 2.2, we get that and hence .
By continuing this process, we can show that and for all . Finally, we obtain
which yields that since . Hence and thus .
(ii) The proof follows directly from (i). □
Lemma 3.2 Let C be a nonempty closed convex subset of a real Banach space E. Let be a finite family of nonexpansive mappings of C into itself such that , and let B be the B-mapping generated by and . Let be a real sequence in . For every , let be the B-mapping generated by and as follows:
If for all , then for all .
Proof Let and and be generated by and , and and , respectively. Then
Let and . Then
It follows that
Since as (), we thus complete the proof. □
Remark 3.3 It is easily seen that for all , is nonexpansive.
Lemma 3.4 Let C be a nonempty closed convex subset of a real Banach space E. Let be a finite family of nonexpansive mappings of C into itself such that . Let be a real sequence in . For every , let be the B-mapping generated by and .
If for all , then
for each bounded sequence in C.
Proof Let be a bounded sequence in C. For and for some , we have
Using the relation above, we can show that
Since for all , we obtain the desired result. □
Using the concept of B-mapping, we study weak convergence of the sequence generated by Mann-type iteration process (1.2).
Theorem 3.5 Let E be a uniformly convex Banach space having a Fréchet differentiable norm or that satisfies Opial’s condition. Let C be a nonempty, closed and convex subset of E. Let be a finite family of nonexpansive mappings of C into itself such that . Let be a real sequence in such that (). For every , let be the B-mapping generated by and . Let be a sequence in satisfying . Let be generated by and
Then converges weakly to .
Proof Let . Then for all and hence
It follows that is nonincreasing; consequently, exists. Assume . Since E is uniformly convex, it follows (see, for example, [27]) that
which implies that
Since exists and , by the continuity of , we have . Since (), let the mapping be generated by and . Then, by Lemma 3.2, we have for all . So we have
Since B is nonexpansive and E is uniformly convex, by the demiclosedness principle, . Moreover, by Lemma 3.1(i).
We next show that is a singleton. Indeed, suppose that . Define by
Then is nonexpansive and . Using Lemma 2.4, we have exists. Suppose that and are subsequences of such that and . Then
This shows that .
Assume that E satisfies Opial’s condition. Let and and be subsequences of such that and . If , then
which is a contradiction. It follows that . Therefore as . This completes the proof. □
4 Strong convergence theorem
In this section, we prove a strong convergence theorem for a finite family of nonexpansive mappings in Banach spaces.
Theorem 4.1 Let E be a strictly convex and reflexive Banach space having a uniformly Gâteaux differentiable norm. Let C be a nonempty, closed and convex subset of E. Let be a finite family of nonexpansive mappings of C into itself such that . Let be a real sequence in such that (). For every , let be the B-mapping generated by and . Let , and be sequences in which satisfy the conditions:
-
(C1)
;
-
(C2)
and ;
-
(C3)
.
Let and define the sequence by and
Then converges strongly to , where q is also the unique solution of the variational inequality
Proof We divide the proof into the following steps.
Step 1. We show that is bounded. Let . Then for all and hence, by the nonexpansiveness of , we have
By induction, we can conclude that is bounded. So are and .
Step 2. We show that . To this end, we define . From (1.3) we have
for some . It turns out that
From conditions (C2), (C3) and Lemma 3.4, we have
Lemma 2.3 yields that and hence
Step 3. We show that . Indeed, noting that
we have, by (C2) and (C3),
Let be the B-mapping generated by and . So, by Lemma 3.2, we have for all . It also follows that
For , we define a contraction as follows:
Then there exists a unique path such that
From Lemma 2.5, we know that as , where . Lemma 3.1(i) also yields that . Moreover, q is the unique solution of variational inequality (4.1).
Step 4. We show that . We see that
It follows, by Lemma 2.1(ii) that
which gives
So we have
for some . Since E has a uniformly Gâteaux differentiable norm, J is norm-to-weak∗ uniformly continuous on bounded subsets of E. So we have
and
as . On the other hand, we have
Since and are interchangeable, using (4.2)-(4.5), we obtain
Step 5. We show that as . In fact, we have
which implies that
Put and . So it is easy to check that is a sequence in such that and . Hence, by Lemma 2.7, we conclude that as . This completes the proof. □
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Acknowledgements
The author wishes to thank editor/referees for valuable suggestions and Professor Suthep Suantai for the guidance. This research was supported by the Thailand Research Fund, the Commission on Higher Education, and University of Phayao under Grant MRG5580016.
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Cholamjiak, P. A new multi-step iteration for solving a fixed point problem of nonexpansive mappings. Fixed Point Theory Appl 2013, 198 (2013). https://doi.org/10.1186/1687-1812-2013-198
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DOI: https://doi.org/10.1186/1687-1812-2013-198