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Strong Convergence Theorems for Modifying Halpern Iterations for Quasi--Asymptotically Nonexpansive Multivalued Mapping in Banach Spaces with Applications
An iterative sequence for quasi--asymptotically nonexpansive multivalued mapping for modifying Halpern's iterations is introduced. Under suitable conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in the work by Chang et al. 2011.
Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. Let be a nonempty closed subset of a real Banach space . A mapping is said to be nonexpansive, if , for all . Let and denote the family of nonempty subsets and nonempty closed bounded subsets of , respectively. The Hausdorff metric on is defined by for , , where . The multivalued mapping is called nonexpansive, if , for all . An element is called a fixed point of , if . The set of fixed points of is represented by .
Let be a real Banach space with dual . We denote by the normalized duality mapping from to which is defined by where denotes the generalized duality pairing.
A Banach space 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 .
The norm of Banach space is said to be Gâteaux differentiable, if for each , the limit exists, where . In this case, is said to be smooth. The norm of Banach space is said to be Fréchet differentiable, if for each , the limit (1.3) is attained uniformly, for , and the norm is uniformly Fréchet differentiable if the limit (1.3) is attained uniformly for . In this case, is said to be uniformly smooth.
Remark 1.1. The following basic properties for Banach space X and for the normalized duality mapping can be found in Cioranescu . (1) (, resp.) is uniformly convex if and only if (, resp.) is uniformly smooth.(2)If is smooth, then is single-valued and norm-to-weak* continuous.(3)If is reflexive, then is onto.(4)If is strictly convex, then , for all .(5)If has a Fréchet differentiable norm, then is norm-to-norm continuous.(6)If is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of .(7)Each uniformly convex Banach space has the Kadec-Klee property, that is, for any sequence , if and , then .(8)If is a reflexive and strictly convex Banach space with a strictly convex dual and is the normalized duality mapping in , then , and .
Next, we assume that is a smooth, strictly convex, and reflexive Banach space and is a nonempty, closed and convex subset of . In the sequel, we always use to denote the Lyapunov functional defined by It is obvious from the definition of the function that for all and .
Following Alber , the generalized projection is defined by Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.
Example 1.2 (see ). Let be the generalized projection from a smooth, reflexive and strictly convex Banach space onto a nonempty closed convex subset of , then is a closed and quasi--nonexpansive from onto .
In 1953, Mann  introduced the following iterative sequence : where the initial guess is arbitrary, and is a real sequence in . It is known that under appropriate settings the sequence converges weakly to a fixed point of . However, even in a Hilbert space, Mann iteration may fail to converge strongly . Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern  proposed the following so-called Halpern iteration: where , are arbitrary given and is a real sequence in . Another approach was proposed by Nakajo and Takahashi . They generated a sequence as follows: where is a real sequence in and denotes the metric projection from a Hilbert space onto a closed and convex subset of . It should be noted here that the iteration previous works only in Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings are introduced (see [8–12]).
Inspired by Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of multivalued mapping .
In the sequel, we denote the strong convergence and weak convergence of the sequence by and , respectively.
Lemma 2.1 (see ). Let be a smooth, strictly convex, and reflexive Banach space, and let be a nonempty closed and convex subset of . Then the following conclusions hold (a) if and only if , for all ; (b), for all , for all ; (c)if and , then , for all .
Remark 2.2. If is a real Hilbert space, then and is the metric projection of onto .
Definition 2.3. A point is said to be an asymptotic fixed point of , if there exists a sequence such that and . Denote the set of all asymptotic fixed points of by .
Definition 2.4. (1) A multivalued mapping is said to be relatively nonexpansive, if , , and , for all , , .
(2) A multivalued mapping is said to be closed, if for any sequence with and , then .
Next, we present an example of relatively nonexpansive multivalued mapping.
Example 2.5 (see ). Let be a smooth, strictly convex, and reflexive Banach space, let be a nonempty closed and convex subset of , and let be a bifunction satisfying the conditions: (A1) , for all ; (A2) , for all ; (A3) , for each ; (A4) the function is convex and lower semicontinuous, for each . The “so-called” equilibrium problem for is to find a such that , for all . The set of its solutions is denoted by .
Let , and define mapping as follows: then (1) is single-valued, and so ; (2) is a relatively nonexpansive mapping, therefore it is a closed and quasi--nonexpansive mapping; (3) .
Definition 2.6. (1) A multivalued mapping is said to be quasi--nonexpansive, if , and , for all , , .
(2) A multivalued mapping is said to be quasi--asymptotically nonexpansive, if , and there exists a real sequence , such that
(3) A multivalued mapping is said to be totally quasi--asymptotically nonexpansive, if , and there exist nonnegative real sequences , with , (as ) and a strictly increasing continuous function with such that
Remark 2.7. From the definitions, it is obvious that a relatively nonexpansive multivalued mapping is a quasi--nonexpansive multivalued mapping, and a quasi--nonexpansive multivalued mapping is a quasi--asymptotically nonexpansive multivalued mapping, but the converse is not true.
Lemma 2.8. Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let be a nonempty closed and convex subset of . Let and be two sequences in such that and , where is the function defined by (1.4), then .
Proof. For , we have . This implies that and so . Since is uniformly smooth, is reflexive and , therefore, there exist a subsequence and a point such that . Because the norm is weakly lower semi continuous, we have By Lemma 2.1(a), we have . Hence we have . Since and has the Kadec-Klee property, we have . By Remark 1.1, it follows that . Since , by using the Kadec-Klee property of , we get . If there exists another subsequence such that , then we have This implies that . So . The conclusion of Lemma 2.8 is proved.
Lemma 2.9. Let and be as in Lemma 2.8. Let be a closed and quasi--asymptotically nonexpansive multivalued mapping with nonnegative real sequences , if , then the fixed point set of is a closed and convex subset of .
Proof. Let be a sequence in , such that . Since is quasi--asymptotically nonexpansive multivalued mapping, we have
for all and for all . Therefore,
By Lemma 2.1, we obtain , Hence, . So, we have . This implies that is closed.
Let , and , and put . we prove that . Indeed, in view of the definition of , let , we have Since Substituting (2.8) into (2.9) and simplifying it, we have Hence, we have . This implies that . Since is closed, we have , that is, . This completes the proof of Lemma 2.9.
Definition 2.10. A mapping is said to be uniformly -Lipschitz continuous, if there exists a constant such that , where , , .
3. Main Results
Theorem 3.1. Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, let be a nonempty, closed and convex subset of , and let be a closed and uniformly -Lipschitz continuous quasi--asymptotically nonexpansive multivalued mapping with nonnegative real sequences and satisfying condition (2.2). Let be a sequence in . If is the sequence generated by where , is the fixed point set of , and is the generalized projection of onto . If is nonempty, then converges strongly to .
Proof. (I) First, we prove that are closed and convex subsets in . By the assumption that is closed and convex. Suppose that is closed and convex for some . In view of the definition of , we have
This shows that is closed and convex. The conclusions are proved.
(II) Next, we prove that , for all . In fact, it is obvious that . Suppose , for some . Hence, for any , by (1.6), we have This shows that and so .
(III) Now, we prove that converges strongly to some point . In fact, since , from Lemma 2.1(c), we have Again since , we have It follows from Lemma 2.1(b) that for each and for each , Therefore, is bounded, and so is . Since and , we have . This implies that is nondecreasing. Hence exists. Since is reflexive, there exists a subsequence such that (some point in ). Since is closed and convex and . This implies that is weakly closed and for each . In view of , we have Since the norm is weakly lower semicontinuous, we have and so This shows that , and we have . Since , by the virtue of Kadec-Klee property of , we obtain that . Since is convergent, this together with shows that . If there exists some subsequence such that , then from Lemma 2.1, we have that is, and hence By the way, from (3.11), it is easy to see that
(IV) Now, we prove that . In fact, since , from (3.1), (3.11), and (3.12), we have Since , it follows from (3.13) and Lemma 2.8 that Since is bounded and is quasi--asymptotically nonexpansive multivalued mapping, is bounded. In view of . Hence from (3.1), we have that Since , this implies . From Remark 1.1, it yields that Again since this together with (3.16) and the Kadec-Klee-property of shows that On the other hand, by the assumptions that is -Lipschitz continuous, thus we have From (3.18) and , we have that . In view of the closeness of , it yields that , this implies that .
(V) Finally, we prove that and so . Let . Since , we have . This implies that which yields that . Therefore, . This completes the proof of Theorem 3.1.
By Remark 2.7, the following corollaries are obtained.
Corollary 3.2. Let and be as in Theorem 3.1, and let be a closed and uniformly -Lipschitz continuous a relatively nonexpansive multivalued mapping. Let in with . Let be the sequence generated by where is the set of fixed points of , and is the generalized projection of onto , then converges strongly to .
Corollary 3.3. Let and be as in Theorem 3.1, and let be a closed and uniformly -Lipschitz continuous quasi--nonexpansive multivalued mapping. Let be a sequence of real numbers such that for all , and satisfying: . Let be the sequence generated by (3.21). Then, converges strongly to .
We utilize Corollary 3.3 to study a modified Halpern iterative algorithm for a system of equilibrium problems.
Theorem 4.1. Let , , and be the same as in Theorem 3.1. Let be a bifunction satisfying conditions (A1)–(A4) as given in Example 2.5. Let be a mapping defined by (2.1), that is, Let be the sequence generated by If , then converges strongly to which is a common solution of the system of equilibrium problems for .
Proof. In Example 2.5, we have pointed out that , , and is a closed and quasi--nonexpansive mapping. Hence (4.2) can be rewritten as follows: Therefore the conclusion of Theorem 4.1 can be obtained from Corollary 3.3.
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