Strong convergence theorems for modifying Halpern iterations for a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping in reflexive Banach spaces
© Liu and Li; licensee Springer 2013
Received: 21 October 2012
Accepted: 26 February 2013
Published: 26 March 2013
In this paper, we discuss an iterative sequence for a totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping for modifying Halpern’s iterations and establish some strong convergence theorems under certain conditions. We utilize the theorems to study a modified Halpern iterative algorithm for a system of equilibrium problems. The results improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:6489-6497, 2012).
MSC:47J05, 47H09, 49J25.
Keywordsmulti-valued mapping quasi-ϕ-asymptotically nonexpansive total quasi-ϕ-asymptotically nonexpansive Halpern iterative sequence
for , where . The multi-valued mapping is called nonexpansive if for all . An element is called a fixed point of if . The set of fixed points of T is represented by .
exists. In this case, X is said to be smooth. The norm of the 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.
where denotes the generalized duality pairing.
X (, resp.) is uniformly convex if and only if (X, resp.) is uniformly smooth.
If X is smooth, then J is single-valued and norm-to-weak∗ continuous.
If X is reflexive, then J is onto.
If X is strictly convex, then for all .
If X has a Fréchet differentiable norm, then J is norm-to-norm continuous.
If X is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of X.
Each uniformly convex Banach space X has the Kadec-Klee property, i.e., for any sequence , if and , then .
for all and .
Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.
Remark 1.2 (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.
where is a real sequence in and denotes the metric projection from a Hilbert space H onto a closed convex subset K of H. It should be noted here that the iteration above works only in the Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings and quasi-ϕ-nonexpansive mappings have been introduced by Aoyama et al. , Chang et al. [9, 10], Chidume et al. , Matsushita et al. [12–14], Qin et al. , Song et al. , Wang et al.  and others.
Inspired by the work of Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of a multi-valued mapping and prove some strong convergence theorems. The results presented in the paper improve and extend the corresponding results in .
In the sequel, we denote the strong convergence and weak convergence of the sequence by and , respectively.
Lemma 2.1 (see )
- (a)if and only if .
- (b), .
If and , then if and only if , .
Lemma 2.2 (see )
Let X be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and let D be a nonempty closed convex subset of X. Let and be two sequences in D such that and , where ϕ is the function defined by (1.2), then .
Definition 2.1 A point is said to be an asymptotic fixed point of a multi-valued mapping if there exists a sequence such that and . Denote the set of all asymptotic fixed points of T by .
A multi-valued mapping is said to be relatively nonexpansive if , and , , , .
A multi-valued mapping is said to be closed if for any sequence with and , then .
Remark 2.1 If H is a real Hilbert space, then and is the metric projection of H onto D.
Next, we present an example of a relatively nonexpansive multi-valued mapping.
Example 2.1 (see )
Let X be a smooth, strictly convex and reflexive Banach space, let D be a nonempty closed and convex subset of X, and let 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 mapping, therefore, it is a closed quasi-ϕ-nonexpansive mapping; (3) .
A multi-valued mapping is said to be quasi-ϕ-nonexpansive if and , , , .
- (2)A multi-valued mapping is said to be quasi-ϕ-asymptotically nonexpansive if and there exists a real sequence , , such that(2.2)
Remark 2.2 From the definitions, it is obvious that a relatively nonexpansive multi-valued mapping is a quasi-ϕ-nonexpansive multi-valued mapping, and a quasi-ϕ-nonexpansive multi-valued mapping is a quasi-ϕ-asymptotically nonexpansive multi-valued mapping, and a quasi-ϕ-asymptotically nonexpansive multi-valued mapping is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping, but the converse is not true.
Lemma 2.3 Let X and D be as in Lemma 2.2. Let be a closed and totally quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences , and a strictly increasing continuous function with . If (as ) and , then is a closed and convex subset of D.
By Lemma 2.1(a), we obtain . Hence, . So, we have . This implies is closed.
By Lemma 2.2, we have . This implies that . Since T is closed, we have , i.e., . This completes the proof of Lemma 2.3. □
Definition 2.4 A mapping is said to be uniformly L-Lipschitz continuous if there exists a constant such that , where , , .
3 Main results
where , is the fixed point set of T, and is the generalized projection of X onto . If is nonempty and , then .
Proof (I) First, we prove that is a closed and convex subset in D.
Next, we prove that for all .
Now we prove that converges strongly to some point .
Now we prove that .
Finally, we prove that and so .
which yields that . Therefore, . The proof of Theorem 3.1 is completed. □
By Remark 2.2, the following corollaries are obtained.
where is the set of fixed points of T, and is the generalized projection of X onto , then converges strongly to .
Corollary 3.2 Let X and D be as in Theorem 3.1, and be a closed and uniformly L-Lipschitz continuous quasi-ϕ-nonexpansive multi-valued mapping. Let be a sequence of real numbers such that for all and satisfy . Let be the sequence generated by (3.14). Then converges strongly to .
where , is the fixed point set of T, and is the generalized projection of X onto , if is nonempty, then converges strongly to .
We utilize Corollary 3.2 to study a modified Halpern iterative algorithm for a system of equilibrium problems.
If , then converges strongly to , which is a common solution of the system of equilibrium problems for f.
Therefore the conclusion of Theorem 4.1 can be obtained from Corollary 3.2. □
The authors are very grateful to both reviewers for carefully reading this paper and for their comments.
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