Abstract
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.
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1 Introduction
Assume that X is a real Banach space with the dual , D is a nonempty closed convex subset of X. We also denote by J the normalized duality mapping from X to which is defined by
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 .
The norm of a Banach space X is said to be Gâteaux differentiable if for each , the limit
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.
Next, we assume that X is a smooth, strictly convex and reflexive Banach space and D is a nonempty closed convex subset of X. In the sequel, we always use to denote the Lyapunov functional defined by
It is obvious from the definition of the function ϕ that
and
for all and .
Following Alber [1], 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.
In the sequel, we denote the strong convergence and weak convergence of the sequence by and , respectively.
Lemma 1.1 (see [1])
Let X be a smooth, strictly convex and reflexive Banach space and D be a nonempty closed convex subset of X. Then the following conclusions hold:
-
(a)
if and only if ;
-
(b)
, ;
-
(c)
if and , then if and only if , .
Remark 1.1 (see [2])
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 [2])
If H is a real Hilbert space, then , and is the metric projection of H onto D.
Definition 1.1 Let be a nonexpansive retraction.
-
(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 [2])
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 .
Let , and define a mapping as follows:
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 [2])
Let D be a unit ball in a real Hilbert space , and let be a nonself mapping defined by
where is a sequence in such that .
It is proved in Goebal and Kirk [3] that
-
(i)
, ;
-
(ii)
, , .
Let , , , then . Letting (), () and be a nonnegative real sequence with , then from (i) and (ii) we have
Since D is a unit ball in a real Hilbert space , it follows from Remark 1.2 that , . The above inequality can be written as
Again, since and , this implies that . From above inequality, we get 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 [4])
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.
Proof Let be a sequence in such that . Since T is a totally quasi-ϕ-asymptotically nonexpansive nonself mapping, we have
for all . Therefore,
By Lemma 1.2, we obtain . So, we have . This implies is closed.
Let and , and put . We prove that . Indeed, in view of the definition of ϕ, let be a sequence generated by , , , we have
Since
Substituting (1.10) into (1.11) and simplifying it, we have
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. □
Definition 1.2 (1) (see [5]) A countable family of nonself mappings is said to be uniformly quasi-ϕ-asymptotically nonexpansive if , and there exist nonnegative real sequences , , such that for each ,
-
(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)
(3) (see [5]) A nonself mapping is said to be uniformly L-Lipschitz continuous if there exists a constant such that
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. [16], Kiziltunc et al. [10], Yildirim et al. [11], Yang et al. [22], Wang [18, 19], Pathak et al. [14], Thianwan [17], Qin et al. [15], Hao et al. [9], Guo et al. [7], Nilsrakoo et al. [13] and others.
2 Main results
Theorem 2.1 Let X be a real uniformly smooth and uniformly convex Banach space, D be a nonempty closed convex subset of X. Let be a family of uniformly totally quasi-ϕ-asymptotically nonexpansive nonself mappings with sequences , , with (as ), and a strictly increasing continuous function with such that for each , is uniformly -Lipschitz continuous. Let be a sequence in and be a sequence in satisfying the following conditions:
-
(i)
;
-
(ii)
.
Let be a sequence generated by
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.
In fact, by Lemma 1.3 for each , is closed and convex in D. Therefore, ℱ is a closed and convex subset in D. By the assumption that is closed and convex, suppose that is closed and convex for some . In view of the definition of ϕ, 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 that .
Let . Hence for any , by (1.5), we have
and
Therefore, we have
where . This shows that and so . The conclusion is proved.
-
(III)
Now, we prove that converges strongly to some point .
Since , from Lemma 1.1(c), we have
Again since , we have
It follows from Lemma 1.1(b) that for each and for each ,
Therefore, is bounded, and so is . Since and , we have . This implies that is nondecreasing. Hence, exists.
By the construction of , for any , we have and . This shows that
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).
By the assumption, it is easy to see that
-
(IV)
Now, we prove that .
Since , from (2.1) and (2.6), we have
Since , it follows from (2.7) and Lemma 1.2 that
Since is bounded and is a family of uniformly total quasi-ϕ-asymptotically nonexpansive nonself mappings, we have
This implies that is uniformly bounded.
Since
this implies that is also uniformly bounded.
In view of , from (2.1), we have that
for each .
Since is uniformly continuous on each bounded subset of , it follows from (2.8) and (2.9) that
for each . Since J is uniformly continuous on each bounded subset of X, we have
By condition (ii), we have that
Since J is uniformly continuous, this shows that
for each . Again, by the assumption that is uniformly -Lipschitz continuous for each , thus we have
for each .
We get . Since and , we have .
In view of the continuity of , it yields that . Since , it implies that . By the arbitrariness of , we have .
-
(V)
Finally, we prove that and so .
Let . Since and , we have . This implies that
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.
Let be a sequence generated by
where , , is the generalized projection of X onto . If ℱ is nonempty, then converges strongly to .
3 Application
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.
Theorem 3.1 Let H be a real Hilbert space, D be a nonempty closed and convex subset of H. , be the same as in Theorem 2.1. Let be a countable family of bifunctions satisfying conditions (A1)-(A4) as given in Example 1.1. Let be the family of mappings defined by (1.9), i.e.,
Let be the sequence generated by
If , then converges strongly to , which is a common solution of the system of equilibrium problems for f.
Proof In Example 1.1, we have pointed out that , is nonempty and convex for all , is a countable family of quasi-ϕ-nonexpansive nonself mappings. Since is nonempty, so is a countable family of quasi-ϕ-nonexpansive mappings and for all , is a uniformly 1-Lipschitzian mapping. Hence, (3.1) can be rewritten as follows:
Therefore, the conclusion of Theorem 3.1 can be obtained from Corollary 2.1. □
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Yi, L. Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications. J Inequal Appl 2012, 268 (2012). https://doi.org/10.1186/1029-242X-2012-268
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DOI: https://doi.org/10.1186/1029-242X-2012-268