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Relaxed Halpern-type iteration method for countable families of totally quasi-ϕ-asymptotically nonexpansive mappings
Journal of Inequalities and Applications volume 2013, Article number: 367 (2013)
Based on an original idea, namely, a specific way of choosing the indexes of involved mappings, we propose a relaxed Halpern-type iterative algorithm for approximating some common fixed point of a kind of nonlinear mappings and obtain a strong convergence theorem under suitable conditions. Since the involved mappings need no assumption of being uniformly totally quasi-ϕ-asymptotically nonexpansive, and there is no need to compute projections onto intersections of countably many closed and convex sets, the results improve those of the authors with related interest.
MSC:47H09, 47H10, 47J25.
Throughout this paper, we assume that E is a real Banach space with its dual , C is a nonempty closed convex subset of E, and is the normalized duality mapping defined by
In the sequel, we use to denote the set of fixed points of a mapping T.
Definition 1.1 
A 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
where denotes the Lyapunov functional defined by
It is obvious from the definition of ϕ that
Definition 1.2 
A countable family of mappings is said to be totally uniformly quasi-ϕ-asymptotically nonexpansive if and there exist nonnegative real sequences , with (as ) and a strictly increasing continuous function with such that(1.4)
A mapping is said to be uniformly L-Lipschitz continuous if there exists a constant such that(1.5)
In 2012, Chang et al.  used the following modified Halpern-type iteration algorithm for totally quasi-ϕ-asymptotically nonexpansive mappings to have the strong convergence under a limit condition only in the framework of Banach spaces.
where is a countable family of closed and uniformly totally quasi-ϕ-asymptotically nonexpansive mappings; and , is the generalized projection (see (2.1)) of E onto . Their results extended and improved the corresponding results of Qin et al. [2, 3], Wang et al. , Martinez-Yanes and Xu  and others.
However, it is obviously a very strong condition that the involved mappings are assumed to be uniformly totally quasi-ϕ-asymptotically nonexpansive. Additionally, the assumption conditions imposed on all () are not weak at all since each one is in fact an intersection of countably many sets. This fact makes the projection very hard to compute, and therefore the method proposed in their paper does not seem to be valuable in practice.
Inspired and motivated by the studies mentioned above, in this article, we introduce a relaxed iterative algorithm for approximating some common fixed point of a countable family of totally quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem.
Following Alber , the generalized projection is defined by
Lemma 2.1 
Let E be a smooth, strictly convex and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then the following conclusions hold:
for all and ;
If and , then , ;
For , if and only if .
Remark 2.2 The following basic properties for a Banach space E can be found in Cioranescu .
If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E;
If E is reflexive and strictly convex, then is norm-weak-continuous;
If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping is single-valued, one-to-one and onto;
A Banach space E is uniformly smooth if and only if is uniformly convex;
Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence , if and , then as .
Lemma 2.3 
Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let and be two sequences in C such that and , where ϕ is the function defined by (1.2), then .
Lemma 2.4 
Let E and C be the same as in Lemma 2.3. Let be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences , and a strictly increasing continuous function such that and . If , then the fixed point set of T is a closed and convex subset of C.
Lemma 2.5 
The unique solutions to the positive integer equation
where denotes the maximal integer that is not larger than x.
3 Main results
Theorem 3.1 Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty closed convex subset of E, and let , , be a countable family of closed and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , satisfying and (as and for each ) and a strictly increasing and continuous function satisfying condition (1.1) and each is uniformly -Lipschitz continuous. Let be a sequence in with . Let be the sequence generated by
where , is the generalized projection of E onto , and and are the solutions to the positive integer equation: (, ), that is, for each , there exist unique and such that
If is bounded and for each , then converges strongly to .
Proof We divide the proof into several steps.
F and () both are closed and convex subsets in C.
In fact, it follows from Lemma 2.4 that each is a closed and convex subset of C, so is F. In addition, with (=C) being closed and convex, we may assume that is closed and convex for some . In view of the definition of ϕ, we have that
where and . This shows that is closed and convex.
F is a subset of .
It is obvious that . Suppose that for some . Then, for any , we have
This implies that , and so .
In fact, since , from Lemma 2.1(2) we have , . Again since , we have , . It follows from Lemma 2.1(1) that for each and for each ,
which implies that is bounded, so is . Since for all , and , we have . This implies that is nondecreasing, hence the limit
Since E is reflexive, there exists a subsequence of such that as . 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 semi-continuous, we have
This implies that , and so as . Since , by virtue of the Kadec-Klee property of E, we obtain that
Since is convergent, this, together with , shows that . If there exists some subsequence of such that as , then from Lemma 2.1(1) we have that
that is, and so
is a member of F.
Set for each . For example, by Lemma 2.5 and the definition of , we have and . Then we have
Note that , i.e., as . It follows from (3.3) and (3.4) that
Since , it follows from (3.1), (3.3) and (3.5) that
as . Since as , it follows from (3.6) and Lemma 2.3 that
Note that whenever for each . Since is bounded, so is . In view of , hence from (3.1) we have that
In addition, implies that . Remark 2.2(ii) yields that, as ,
Again, since for each , as ,
this, together with (3.10) and the Kadec-Klee property of E, shows that
On the other hand, by the assumptions that for each , is uniformly -Lipschitz continuous, and noting that for all , we then have
From (3.11) and (), we have that and , i.e., . It then follows that, for each ,
In view of the closedness of , it follows from (3.11) that , namely, for each , and hence .
, and so as .
Put . Since and , we have , . Then
which implies that since , and hence . This completes the proof. □
Remark 3.2 Note that algorithm (3.1) just depends on the projection onto a single closed and convex set for each fixed n. An example  of how to compute such a projection is given as follows.
Dykstra’s algorithm Let be closed and convex subsets of . For any and , the sequences are defined by the following recursive formulae:
for with initial values and for . If , then converges to , where , .
We now give a nontrivial example of the calculation of common fixed points for specific mappings.
Example 3.3 Let with the standard norm and . Let be a sequence of nonexpansive mappings defined by . Consider the following iteration sequence generated by
where , and . Note that and for all since E is a Hilbert space. Moreover, it is not difficult to obtain that for all . Then (3.16) is reduced to
where is the solution to the positive integer equation: (, ). It is clear that is a sequence of closed and totally quasi-ϕ-asymptotically nonexpansive mappings with a common fixed point zero. It then can be shown by a similar way of Theorem 3.1 that converges strongly to zero. The numerical experiment outcome obtained by using MATLAB 220.127.116.119 shows that as , the computations of , , and are 0.025141746, 0.00078472044, 0.000025282198 and 0.00000082531714, respectively. This example illustrates the effectiveness of the introduced algorithm for countable families of totally quasi-ϕ-asymptotically nonexpansive mappings.
The so-called convex feasibility problem for a family of mappings is to find a point in the nonempty intersection , which exactly illustrates the importance of finding common fixed points of infinite families. The following example also clarifies the same thing.
Example 4.1 Let E be a smooth, strictly convex and reflexive Banach space, let C be a nonempty and closed convex subset of E, and let be a sequence of bifunctions satisfying the conditions: for each ,
(A2) is monotone, i.e., ;
(A4) the mapping is convex and lower semicontinuous.
A system of equilibrium problems for is to find an such that
whose set of common solutions is denoted by , where denotes the set of solutions to the equilibrium problem for (). It is shown in [, Theorem 4.3] that such a system of problems can be reduced to the approximation of some fixed point of a sequence of nonlinear mappings.
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This work is supported by the National Natural Science Foundation of China (Grant No. 11061037).
The author declares that they have no competing interests.
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Deng, WQ. Relaxed Halpern-type iteration method for countable families of totally quasi-ϕ-asymptotically nonexpansive mappings. J Inequal Appl 2013, 367 (2013). https://doi.org/10.1186/1029-242X-2013-367
- Halpern-type iteration
- totally quasi-ϕ-asymptotically nonexpansive mappings
- generalized projection