A new multi-step iteration for solving a fixed point problem of nonexpansive mappings
© Cholamjiak; licensee Springer 2013
Received: 16 April 2013
Accepted: 5 July 2013
Published: 22 July 2013
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.
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.
where and are sequences in . This method is often called the Ishikawa iteration process.
where and are sequences in . This method is called the S-iteration process. The weak convergence was studied in  for nonexpansive mappings. It was also shown in  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. , we have the aim to introduce and study a new mapping defined by the following definition.
where is a B-mapping generated by and (see Section 3).
where , and are sequences in , and .
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 . 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 . For more details, we refer the reader to [5, 24].
Lemma 2.1 
for all ;
for all .
Lemma 2.2 
for all and , then .
Lemma 2.3 
where satisfies the conditions .
If , then as .
Lemma 2.4 
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 
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 
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.
B is nonexpansive.
Using Lemma 2.2, we get that and hence .
which yields that since . Hence and thus .
(ii) The proof follows directly from (i). □
If for all , then for all .
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 .
for each bounded sequence in C.
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).
Then converges weakly to .
Since B is nonexpansive and E is uniformly convex, by the demiclosedness principle, . Moreover, by Lemma 3.1(i).
This shows that .
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.
Proof We divide the proof into the following steps.
By induction, we can conclude that is bounded. So are and .
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).
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. □
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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