Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications
© Yi; licensee Springer 2012
Received: 21 May 2012
Accepted: 5 November 2012
Published: 22 November 2012
In this paper, we introduce a class of totally quasi-ϕ-asymptotically nonexpansive nonself mappings and study the strong convergence under a limit condition only in the framework of Banach spaces. Meanwhile, our results are applied to study the approximation problem of a solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:7864-7870, 2012).
MSC:47J05, 47H09, 49J25.
where denotes the generalized duality pairing.
Let D be a nonempty closed subset of a real Banach space X. A mapping is said to be nonexpansive if for all . An element is called a fixed point of if . The set of fixed points of T is represented by .
A Banach space X is said to be strictly convex if for all with and . A Banach space is said to be uniformly convex if for any two sequences with and .
exists, where . In this case, X is said to be smooth. The norm of a Banach space X is said to be Fréchet differentiable if for each , the limit (1.1) is attained uniformly for and the norm is uniformly Fréchet differentiable if the limit (1.1) is attained uniformly for . In this case, X is said to be uniformly smooth.
A subset D of X is said to be a retract of X if there exists a continuous mapping such that for all . It is well known that every nonempty closed convex subset of a uniformly convex Banach space X is a retract of X. A mapping is said to be a retraction if . It follows that if a mapping P is a retraction, then for all y in the range of P. A mapping is said to be a nonexpansive retraction if it is nonexpansive and it is a retraction from X to D.
for all and .
Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.
In the sequel, we denote the strong convergence and weak convergence of the sequence by and , respectively.
Lemma 1.1 (see )
if and only if ;
if and , then if and only if , .
Remark 1.1 (see )
Let be the generalized projection from a smooth, reflexive and strictly convex Banach space X onto a nonempty closed convex subset D of X. Then is a closed and quasi-ϕ-nonexpansive from X onto D.
Remark 1.2 (see )
If H is a real Hilbert space, then , and is the metric projection of H onto D.
- (1)A nonself mapping is said to be quasi-ϕ-nonexpansive if , and(1.7)
- (2)A nonself mapping is said to be quasi-ϕ-asymptotically nonexpansive if , and there exists a real sequence , (as ), such that(1.8)
- (3)A nonself mapping is said to be totally quasi-ϕ-asymptotically nonexpansive if , and there exist nonnegative real sequences , with (as ) and a strictly increasing continuous function with such that(1.9)
Remark 1.3 From the definitions, it is obvious that a quasi-ϕ-nonexpansive nonself mapping is a quasi-ϕ-asymptotically nonexpansive nonself mapping, and a quasi-ϕ-asymptotically nonexpansive nonself mapping is a totally quasi-ϕ-asymptotically nonexpansive nonself mapping, but the converse is not true.
Next, we present an example of a quasi-ϕ-nonexpansive nonself mapping.
Example 1.1 (see )
Let H be a real Hilbert space, D be a nonempty closed and convex subset of H and be a bifunction satisfying the conditions: (A1) , ; (A2) , ; (A3) for each , ; (A4) for each given , the function is convex and lower semicontinuous. The so-called equilibrium problem for f is to find an such that , . The set of its solutions is denoted by .
then (1) is single-valued, and so ; (2) is a relatively nonexpansive nonself mapping, therefore it is a closed quasi-ϕ-nonexpansive nonself mapping; (3) and is a nonempty and closed convex subset of D; (4) is nonexpansive. Since is nonempty, and so it is a quasi-ϕ-nonexpansive nonself mapping from D to H, where , .
Now, we give an example of a totally quasi-ϕ-asymptotically nonexpansive nonself mapping.
Example 1.2 (see )
where is a sequence in such that .
, , .
where P is the nonexpansive retraction. This shows that the mapping T defined as above is a totally quasi-ϕ-asymptotically nonexpansive nonself mapping.
Lemma 1.2 (see )
Let X be a uniformly convex and smooth Banach space, and let and be two sequences of X such that and are bounded; if , then .
Lemma 1.3 Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty closed convex subset of X. Let be a totally quasi-ϕ-asymptotically nonexpansive nonself mapping with , then is a closed and convex subset of D.
By Lemma 1.2, we obtain . So, we have . This implies is closed.
Hence, we have . This implies that . Since TP is closed and , we have . Since , and so , i.e., . This implies is convex. This completes the proof of Lemma 1.3. □
- (2)A countable family of nonself mappings is said to be uniformly totally quasi-ϕ-asymptotically nonexpansive if , and there exist nonnegative real sequences , with (as ) and a strictly increasing continuous function with such that for each ,(1.14)
Considering the strong and weak convergence of asymptotically nonexpansive self or nonself mappings, relatively nonexpansive, quasi-ϕ-nonexpansive and quasi-ϕ-asymptotically nonexpansive self or nonself mappings have been considered extensively by several authors in the setting of Hilbert or Banach spaces (see [4–19]).
The purpose of this paper is to modify the Halpern and Mann-type iteration algorithm for a family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings to have the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. [5, 6, 20, 21], Su et al. , Kiziltunc et al. , Yildirim et al. , Yang et al. , Wang [18, 19], Pathak et al. , Thianwan , Qin et al. , Hao et al. , Guo et al. , Nilsrakoo et al.  and others.
2 Main results
where , , is the generalized projection of X onto . If ℱ is nonempty, then converges strongly to .
Proof (I) First, we prove that ℱ and are closed and convex subsets in D.
Next, we prove that for all .
In fact, it is obvious that . Suppose that .
Now, we prove that converges strongly to some point .
Therefore, is bounded, and so is . Since and , we have . This implies that is nondecreasing. Hence, exists.
It follows from Lemma 1.2 that . Hence, is a Cauchy sequence in D. Since D is complete, without loss of generality, we can assume that (some point in D).
Now, we prove that .
This implies that is uniformly bounded.
this implies that is also uniformly bounded.
for each .
for each .
We get . Since and , we have .
Finally, we prove that and so .
which yields that . Therefore, . The proof of Theorem 3.1 is completed. □
By Remark 1.3, the following corollary is obtained.
Corollary 2.1 Let X, D, , be the same as in Theorem 2.1. Let be a family of uniformly quasi-ϕ-asymptotically nonexpansive nonself mappings with the sequence , , such that for each , is uniformly -Lipschitz continuous.
where , , is the generalized projection of X onto . If ℱ is nonempty, then converges strongly to .
In this section we utilize Corollary 2.1 to study a modified Halpern iterative algorithm for a system of equilibrium problems. We have the following result.
If , then converges strongly to , which is a common solution of the system of equilibrium problems for f.
Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □
The author is very grateful to both reviewers for careful reading of this paper and for their comments.
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