Convergence theorems for some multi-valued generalized nonexpansive mappings
© Chang et al.; licensee Springer. 2014
Received: 7 September 2013
Accepted: 24 January 2014
Published: 11 February 2014
In this paper, we propose an algorithms for finding a common fixed point of an infinite family of multi-valued generalized nonexpansive mappings in uniformly convex Banach spaces. Under suitable conditions some strong and weak convergence theorems for such mappings are proved. The results presented in the paper improve and extend the corresponding results of Suzuki (J. Math. Anal. Appl. 340:1088-1095, 2008), Eslamian and Abkar (Math. Comput. Model. 54:105-111, 2011), Abbas et al. (Appl. Math. Lett. 24:97-102, 2011) and others.
MSC:47J05, 47H09, 49J25.
The study of fixed points for multi-valued contractions and nonexpansive mappings using the Hausdorff metric was initiated by Markin  and Nadler . Since then the metric fixed point theory of multi-valued mappings has been rapidly developed. The theory of multi-valued mappings has been applied to control theory, convex optimization, differential equations, and economics. Different iterative processes have been used to approximate fixed points of multi-valued nonexpansive mappings [3–7]. Recently Abbas et al.  introduced an one-step iterative process to approximate a common fixed point of two multi-valued nonexpansive mappings in uniformly convex Banach spaces.
On the other hand, in 2008 Suzuki  introduced a class of mappings satisfying the condition (C) which is weaker than nonexpansive mappings (sometimes, such a mapping is called a generalized nonexpansive mapping). He then proved some fixed point and convergence theorems for such mappings. Very recently, Eslamian and Abkar [10, 11] generalized it to multi-valued case, and they proved some fixed point results in uniformly convex Banach spaces.
The aim of this paper is to introduce an iterative process for approximating a common fixed point of an infinite family of multi-valued mappings satisfying the condition (C). Under suitable conditions some weak and strong convergence theorems for such iterative process are proved in uniformly convex Banach spaces.
Throughout this paper, we assume that X is a real Banach space, K is a nonempty subset of X. We denote by the set of all positive integers. We denote by ‘’ and ‘’ the strong and weak convergence of , respectively.
Remark 2.1 It is well-known that weakly compact convex subsets of a Banach space and closed convex subsets of a uniformly convex Banach space are proximinal.
- (i)contraction, if there exists a constant such that for any
- (ii)nonexpansive, if for all
- (iii)quasi-nonexpansive, if and
If T is nonexpansive, then T satisfies the condition (C).
If T satisfies the condition (C) and , then T is a quasi-nonexpansive mapping.
Proof The conclusion (1) is obvious. Now we prove the conclusion (2).
We mention that there exist single-valued and multi-valued mappings satisfying the condition (C) which are not nonexpansive, for example:
Example 1 
Then T is a single-valued mapping satisfying condition (C), but T is not nonexpansive.
Example 2 
then it is easy to prove that T is a multi-valued mapping satisfying condition (C), but T is not nonexpansive.
This completes the proof of Lemma 2.5. □
Lemma 2.6 
Lemma 2.7 Let X be a strictly convex Banach space, K be a nonempty closed and convex subset of X and be a multi-valued mapping satisfying the condition (C). If is nonempty, then it is a closed and convex subset of K.
This implies that . Since Tp is closed, we have , i.e., and so is closed.
from (2.4), we have and from (2.5), we have . These implies that . Therefore and , i.e., . This completes the proof of Lemma 2.7. □
Lemma 2.8 (Demi-closed principle)
Let X be a uniformly convex Banach space satisfying the Opial condition, K be a nonempty closed and convex subset of X. Let be a multi-valued mapping with convex-values and satisfying the condition (C). Let be a sequence in K such that , and let , then , i.e., is demi-closed at zero.
By virtue of the Opial condition, we have , . And so .
This completes the proof of Lemma 2.8. □
3 Weak convergence theorems
We are now in a position to give the following theorem.
, for each ;
for each , ;
and , and ,
then the sequence converges weakly to some point .
This shows that is decreasing and bounded below. Hence the limit exists for each . And so and both are bounded.
Finally we prove that (some point in ℱ).
In fact, since is bounded, there exists a subsequence such that . By Lemma 2.8, is demi-closed at zero. Hence from (3.8), . By the arbitrariness of , we have .
This is a contradiction. Therefore and .
This completes the proof of Theorem 3.1. □
The following theorem can be obtained from Theorem 3.1 immediately.
where is the sequence as given in Theorem 3.1. If , then the sequence converges weakly to some point .
4 Some strong convergence theorems
Proof The necessity of condition (4.1) is obvious.
Next we prove the sufficiency of condition (4.1).
This implies that , for all . Therefore and .
This completes the proof of Theorem 4.1. □
The following theorem can be obtained from Theorem 4.1 immediately.
- (iv)there exists an increasing function with , such that for some(4.3)
then the sequence converges strongly to some point .
Proof As in the proof of Theorem 3.1, for each , . Especially we have . Hence from (4.3) we obtain . The conclusion of Theorem 4.2 can be obtained from Theorem 4.1 immediately.
We now intend to remove the condition that for each and each .
where . □
We have the following.
, for each ;
for each , ;
, and, for each , the mapping satisfies the condition (C);
- (iv)there exists an increasing function with for all such that for some
then the sequence converges strongly to some point .
Since (as ), it follows that for . Hence and converges strongly to q. Since exists, we conclude that converges strongly to q.
This completes the proof of Theorem 4.3. □
The convex feasibility problem (CFP) was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and in medical image reconstruction . Recently, it has been found that the CFP can also be used in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning.
Let X be a real Banach space, K be a nonempty closed and convex subset of X and be a countable family of subset of K. The ‘so called’ convex feasibility problem for the family of subsets is to find a point .
In this section, we shall utilize Theorem 3.2 to study the convex feasibility problem for an infinite family of single-valued mappings satisfying the condition (C). We have the following result.
where is the sequence as given in Theorem 3.2. If , then there exists a point which is a solution of the convex feasibility problem for the family of subsets , and the sequence defined by (5.1) converges weakly to .
The authors would like to express their thanks to the Reviewers and the Editors for their helpful suggestions and advices. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).
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