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Modified block iterative procedure for solving the common solution of fixed point problems for two countable families of total quasi-ϕ-asymptotically nonexpansive mappings with applications
Fixed Point Theory and Applications volume 2012, Article number: 198 (2012)
Abstract
In this paper, we introduce a new iterative procedure which is constructed by the modified block hybrid projection method for solving a common solution of fixed point problems for two countable families of uniformly total quasi-ϕ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Finally, we apply the problem of a strong convergence theorem concerning maximal monotone operators in Banach spaces.
MSC:47H09, 47H10, 47H20, 47J20.
1 Introduction
Throughout this paper, we assume that E is a real Banach space, is the dual space of E. Let C be a nonempty, closed, and convex subset of E and be the pairing between E and . We denote the strong convergence and weak convergence of a sequence by and , respectively, and is the normalized duality mapping defined by
In the sequel, we use to denote the set of fixed points of a mapping T and use ℝ and to denote the set of all real numbers and the set of all nonnegative real numbers, respectively.
Definition 1.1 Let E be a Banach space.
-
(1)
E is said to be strictly convex if for all with .
-
(2)
E is said to be uniformly convex if for each , there exists such that for all with .
-
(3)
E is said to be smooth if the limit (1.2)
(1.2)
exists for each .
-
(4)
E is said to be uniformly smooth if the limit (1.2) is attained uniformly for all .
Remark 1.2 The basic properties below hold (see [1, 2]).
-
(1)
If E is a real uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E.
-
(2)
If E is a strictly convex reflexive Banach space, then is hemicontinuous, that is, is norm-to-weak∗-continuous.
-
(3)
If E is a smooth and strictly convex reflexive Banach space, then J is single-valued, one-to-one, and onto.
-
(4)
A Banach space E is uniformly smooth if and only if is uniformly convex.
-
(5)
Each uniformly convex Banach space E has the Kadec-Klee property; that is, for any sequence , if and , then .
-
(6)
A Banach space E is strictly convex if and only if J is strictly monotone; that is,
-
(7)
Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.
-
(8)
is uniformly convex, then J is uniformly norm-to-norm continuous on each bounded subset of E.
Let E be a smooth and strictly convex reflexive Banach space, and let C be a nonempty, closed, and convex subset of E. We assume that the Lyapunov functional is defined by [3, 4]
From the definition of ϕ, it is easy to see that
Let C be a nonempty, closed, and convex subset of E. For each , the generalized projection [3] is defined by
If C is a nonempty, closed, and convex subset of a smooth and strictly convex reflexive real Banach space E, then
-
(1)
for and , one has
-
(2)
, , .
-
(3)
if and only if , .
Remark 1.4 If E is a real Hilbert space H, then and (the metric projection of H onto C).
Definition 1.5 Let E be a smooth, strictly convex, and reflexive real Banach space, C be a nonempty, closed, and convex subset of E, be a mapping, and be the set of fixed points of T.
-
(1)
A point is said to be an asymptotic fixed point of T if there exists a sequence such that and . We denote the set of all asymptotic fixed points of T by .
-
(2)
A point is said to be a strong asymptotic fixed point of T if there exists a sequence such that and . We denote the set of all strong asymptotic fixed points of T by .
Definition 1.6 Let E be a smooth, strictly convex, and reflexive real Banach space, and let C be a nonempty, closed, and convex subset of E.
-
(1)
A mapping is said to be closed if for each with and , then .
-
(2)
A mapping is said to be relatively nonexpansive [5, 6] if , , and
-
(3)
A mapping is said to be weak relatively nonexpansive [7] if , , and
-
(4)
A mapping is said to be quasi-ϕ-nonexpansive (relatively quasi-nonexpansive) if and
-
(5)
A mapping is said to be quasi-ϕ-asymptotically nonexpansive (asymptotically relatively nonexpansive) if and there exists a sequence with such that
-
(6)
A mapping is said to be total quasi-ϕ-asymptotically nonexpansive if and there exists nonnegative real sequences and with , (as ), and a strictly increasing continuous function with such that
Remark 1.7 From Definition 1.6, it is easy to know that
-
(1)
every relatively nonexpansive mapping is closed;
-
(2)
every quasi-ϕ-asymptotically nonexpansive mapping is a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true;
-
(3)
every quasi-ϕ-nonexpansive mapping is a quasi-ϕ-asymptotically nonexpansive mapping with , but the converse is not true;
-
(4)
every weak relatively nonexpansive mapping is a quasi-ϕ-nonexpansive mapping because it does not require the condition , but the converse is not true;
-
(5)
every relatively nonexpansive mapping is a weak relatively nonexpansive mapping, but the converse is not true.
Regarding the iterative methods of nonlinear operator equations for relatively nonexpansive mappings, in 2005 Matsushita and Takahashi [5] and in 2008 Plubtieng and Ungchittrakool [6] proved the following result, respectively.
Theorem MT Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed and convex subset of E. Let be a relatively nonexpansive mapping, and let be a real sequence in with . Let be a sequence defined by
where J is the duality mapping on E. If , then the sequence converges strongly to , where is the generalized projection from C onto .
Theorem PU Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty, closed, and convex subset of E. Let be two relatively nonexpansive mappings with . Let be a sequence defined by
with the following restrictions:
-
(1)
and ;
-
(2)
, , and .
Then the sequence converges strongly to , where is the generalized projection from C onto Ω.
In 2010, Su et al. [7] introduced the concept of a countable family of weak relatively nonexpansive mappings and proved the following theorem which extends and improves Theorem MT and Theorem PU.
Theorem SXZ Let E be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty, closed, and convex subset of E. Let be two countable families of weak relatively nonexpansive mappings such that .
Let be a sequence defined by
with the conditions:
-
(1)
for some ;
-
(2)
such that for each ;
-
(3)
and .
Then the sequence converges strongly to , where is the generalized projection from C onto Ω.
In 2011, Chang et al. [8] proved some approximation theorems for common fixed points of countable families of total quasi-ϕ-asymptotically nonexpansive mappings which contain several kinds of mappings as their special cases in Banach spaces. Next, Chang et al. [9] modified the Halpern-type iteration algorithm for a total quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. Recently, Chang, Lee, and Chan [10] introduced a block hybrid projection algorithm for solving the convex feasibility problem and the generalized equilibrium problems for an infinite family of total quasi-ϕ-asymptotically nonexpansive mappings and they proved strong convergence theorems in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.
Theorem CLCY Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , , and a strictly increasing continuous function such that , , (as ), and . Let be a sequence generated by
where , is the generalized projection of E onto . Let , , and be sequences in satisfying the following conditions:
-
(1)
for each , ;
-
(2)
for all ;
-
(3)
for some .
If is a nonempty and bounded subset of C, then the sequence converges strongly to .
Theorem CLCZ Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , , and a strictly increasing continuous function such that , , (as ), and . Let be a sequence generated by
where , and is the generalized projection of E onto . If is a nonempty and bounded subset of C, then the sequence converges strongly to .
Theorem CLC Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , , and a strictly increasing continuous function such that , , (as ), and . Let () be a finite family of continuous and monotone mappings. Let () be a finite family of lower semi-continuous and convex functions, and let () be a finite family of bifunctions satisfying the conditions (A1)-(A4). Suppose that is a nonempty and bounded subset of C, where () is the set of the following generalized mixed quasi-equilibrium problems:
Let be a sequence generated by
where , , is the generalized projection of E onto . Let , be sequences in satisfying the following conditions:
-
(1)
for each , ;
-
(2)
for all .
Then the sequence converges strongly to .
In this paper, motivated and inspired by the previously mentioned results, we introduce a new iterative procedure by the modified block hybrid projection method for solving a common solution of fixed point problems for two countable families of uniformly total quasi-ϕ-asymptotically nonexpansive and uniformly Lipschitz continuous mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Then, we prove a strong convergence theorem of the iterative procedure generated by these conditions. The results obtained in this paper extend and improve several recent results in this area.
2 Preliminaries
Definition 2.1 Let C be a nonempty, closed, and convex subset of a real Banach space E.
-
(1)
A mapping is said to be nonexpansive if
-
(2)
A mapping is said to be uniformly L-Lipschitz continuous if there exists a constant such that
Definition 2.2 [11]
Let C be a nonempty, closed, and convex subset of a real Banach space E.
-
(1)
A countable family of mappings is said to be a uniformly quasi-ϕ-asymptotically nonexpansive mapping if and there exists a sequence with such that for each ,
-
(2)
A countable family of mappings is said to be a uniformly total quasi-ϕ-asymptotically nonexpansive mapping if and there exist nonnegative real sequences and with , (as ), and a strictly increasing continuous function with such that for each ,
Lemma 2.3 [4]
Let E be a uniformly smooth and strictly convex real Banach space, and let and be two sequences of E. If and either or is bounded, then .
Lemma 2.4 [8]
Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. Let and be two sequences in C and . If and , then .
Lemma 2.5 [8]
Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. Let be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences , , and a strictly increasing continuous function such that , (as ), and . If , then the fixed point set is a closed and convex subset of C.
Lemma 2.6 [11]
Let E be a uniformly convex Banach space, be a positive number, and be a closed ball of E. Then for any given sequence and for any given with , there exists a continuous, strictly increasing, and convex function g : with such that for any positive integers i, j with ,
3 Main results
In this section, we shall use the modified block hybrid projection method to study a common solution of fixed point problems for two countable families of closed and -Lipschitz continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces. For the purpose, we assume the following hypotheses.
(A1) Let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , and a strictly increasing function such that , (as ), , and , and for each ,
(A2) Let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , and a strictly increasing function such that , (as ), , and , and for each ,
Theorem 3.1 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, let C be a nonempty, closed, and convex subset of E, and let and satisfy the above conditions (A1)-(A2), respectively.
Suppose that is nonempty and bounded in C. Let , , and be sequences generated by
where and .
Let and be coefficient sequences in satisfying the following conditions:
-
(1)
and ;
-
(2)
, and , .
Then the sequence converges strongly to some point , where .
Proof We shall complete this proof of Theorem 3.1 in seven steps.
Step 1. We will show that Ω and are closed and convex for each .
In fact, it follows from Lemma 2.5 that and , for any , are closed and convex subsets of C. Therefore, Ω is closed and convex in C.
Clearly, is closed and convex. Suppose that is closed and convex for some .
By the assumption of ,
is equivalent to
So that .
Again, by the assumption of ,
is equivalent to
So that .
Hence, is closed and convex.
Step 2. We will show that is bounded and is a convergent sequence for all .
Indeed, it follows from (3.1) and Lemma 1.3(2) that
This implies that is bounded. By virtue of (1.3), the sequence is also bounded.
By the assumption of , we have , and .
This implies that and
Therefore, is a convergent sequence. Without loss of generality, we can assume that
Step 3. We will show that for all .
It is obvious that . Suppose that for some . For any given , from (3.1) and Lemma 2.6, we compute
It follows that
From (3.1) and Lemma 2.6, we compute
It follows that
By the assumptions of , , , and , and from (3.3) and (3.5), we obtain
and
So, we get . This implies that for all , and the sequence is well defined.
Step 4. We will show that there exists some point such that .
In fact, since is bounded and E is reflexive, then there exists a subsequence such that (some point in C).
Since is closed and convex and , it follows that is weakly closed and for each .
In view of , we have
Since the norm is weakly lower semi-continuous, we have
and so
This implies that , and so . Since , and by virtue of the Kadec-Klee property of E, we obtain
The sequence is convergent, and , which implies that . If there exists some subsequence such that , then from Lemma 1.3(2) we have
This implies that , and so
Step 5. We will show that and as .
Since , by the definition of , we have
Since exists, and we are taking in (3.10), then .
It follows from Lemma 2.3 that
By the definition of and , we get
From , , and , as , we obtain
Since , by virtue of Lemma 2.4, we get
It follows from (3.9) and (3.14) that
Since J is uniformly continuous on each bounded subset of E, then
Step 6. We will show that , where .
(6.1) First, we will show that .
For any and for any , it follows from (3.3), (3.7), (3.15), and (3.16) that
By the condition , , we obtain
It follows from the property of g that
Since and J is uniformly continuous on each bounded subset of E, it yields that .
Hence, from (3.17) we get
Since is norm-to-weak∗-continuous, we also have
Again, since for any ,
from (3.19) and the Kadec-Klee property of E, it follows that
On the other hand, by the assumption that for each , is uniformly -Lipschitz continuous, we get
It follows from (3.9), (3.11), and (3.20) that
and so
In view of (3.20) and the closeness of , it yields that for all . This implies that
(6.2) Next, we will show that .
For any and for any , it follows from (3.5), (3.8), (3.15), and (3.16) that
By the condition , , we obtain
It follows from the property of g that
Since and J is uniformly continuous on each bounded subset of E, it yields that .
Hence, from (3.22) we have
Since is norm-to-weak∗-continuous, we also have
Again, since for any ,
from (3.24) and the Kadec-Klee property of E, it follows that
On the other hand, by the assumption that for each , is uniformly -Lipschitz continuous, we have
From , and , as , we obtain
And it follows from (3.11) and (3.15) that
and so
In view of (3.25) and the closeness of , it yields that for all . This implies that
From (3.21) and (3.26), we can conclude that .
Step 7. Finally, we will show that .
Let . Since and , we have
This implies that
In view of the definition of , from (3.27), we have . Therefore, .
This completes the proof of Theorem 3.1. □
Theorem 3.2 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:
(B1) Let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a real sequence , .
(B2) Let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a real sequence , .
Suppose that is a nonempty and bounded in C. Let , , and be sequences generated by
where and .
Let and be coefficient sequences in satisfying the following conditions:
-
(1)
and ;
-
(2)
and , .
Then the sequence converges strongly to some point , where .
Proof Since , are countable families of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings, by virtue of Remark 1.7(2), , are countable families of closed and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative sequences , , and , , respectively, and a strictly increasing and continuous function , . Hence, and (as ). Therefore, all the conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 can be obtained from Theorem 3.1 immediately. □
Theorem 3.3 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:
(C1) Let be a countable family of closed and quasi-ϕ-nonexpansive mappings.
(C2) Let be a countable family of closed and quasi-ϕ-nonexpansive mappings.
Suppose that is nonempty and bounded in C. Let , , and be sequences generated by
and let and be coefficient sequences in satisfying the following conditions:
-
(1)
, and ;
-
(2)
, and , .
Then the sequence converges strongly to some point , where .
Proof Since , are countable families of closed and quasi-ϕ-nonexpansive mappings, by Remark 1.7(3), , are countable families of closed and quasi-ϕ-asymptotically nonexpansive mappings with nonnegative sequences and , respectively. Hence, and . Therefore, all the conditions in Theorem 3.2 as ‘Ω is bounded in C’ and ‘for each , , are uniformly -Lipschitz continuous’ are of no use here. Thus, all the conditions in Theorem 3.2 are satisfied. The conclusion of Theorem 3.3 can be obtained from Theorem 3.2 immediately. □
Theorem 3.4 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:
(D1) Let be a countable family of weak relatively nonexpansive mappings.
(D2) Let be a countable family of weak relatively nonexpansive mappings.
Suppose that is nonempty and bounded in C. Let , , and be sequences generated by
and let and be coefficient sequences in satisfying the following conditions:
-
(1)
and ;
-
(2)
and , .
Then the sequence converges strongly to some point , where .
Proof Since , are countable families of weak relatively nonexpansive mappings, from Remark 1.7(4), , are countable families of quasi-ϕ-nonexpansive mappings. Therefore, all the conditions in Theorem 3.3 are satisfied. The conclusion of Theorem 3.4 can be obtained from Theorem 3.3 immediately. □
Theorem 3.5 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:
(E1) Let be a countable family of relatively nonexpansive mappings.
(E2) Let be a countable family of relatively nonexpansive mappings.
Suppose that is nonempty and bounded in C. Let , , and be sequences generated by
and let and be coefficient sequences in satisfying the following conditions:
-
(1)
and ;
-
(2)
and , .
Then the sequence converges strongly to some point , where .
Proof Since , are countable families of relatively nonexpansive mappings, it follows from Remark 1.7(5) that , are countable families of weak relatively nonexpansive mappings. Therefore, all the conditions in Theorem 3.4 are satisfied. The conclusion of Theorem 3.5 can be obtained from Theorem 3.4 immediately. □
Remark 3.6 Theorems 3.1-3.5 generalize, improve, and extend the corresponding results in [5–8, 11–18], and [19] in the following aspects:
-
(a)
For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property. (Note that each uniformly convex Banach space must have the Kadec-Klee property.)
-
(b)
For the mappings, we extend the mappings from a nonexpansive mapping, a relatively nonexpansive mapping, a weakly relatively nonexpansive mapping, a quasi-ϕ-nonexpansive mapping or a quasi-ϕ-asymptotically nonexpansive mapping to a total quasi-ϕ-asymptotically nonexpansive mapping.
-
(c)
We extend a countable family of closed and uniformly -Lipschitz continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings to two countable families of closed and uniformly -Lipschitz continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings.
4 Deduced theorem
If we take in Theorem 3.1, then we obtain the following result.
Theorem 4.1 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:
(F1) Let be a closed, uniformly L-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences , , and a strictly increasing function such that , (as ), , and
(F2) Let be a closed, uniformly ℓ-Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences , , and a strictly increasing function such that , (as ), , and
Suppose that is nonempty and bounded in C. Let , , and be sequences generated by
where and .
Let and be coefficient sequences in satisfying the following conditions:
-
(1)
for some ;
-
(2)
for some .
Then the sequence converges strongly to some point , where .
If we set (identity mapping) for all in Theorem 3.1, then we obtain the following result.
Theorem 4.2 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:
(G1) Let be a countable family of closed, uniformly -Lipschitz continuous, and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences , , and a strictly increasing function such that , (as ), , and , and for each ,
(G2) Suppose that is nonempty and bounded in C. Let , , and be sequences generated by
where . Let and be coefficient sequences in satisfying the following conditions:
-
(1)
for some ;
-
(2)
;
-
(3)
, .
Then the sequence converges strongly to some point , where .
Remark 4.3 Theorem 4.2 contains the result of Chang et al. [8].
5 Applications
Now, we apply Theorem 3.5 to prove a strong convergence theorem concerning two maximal monotone operators in Banach spaces.
Let E be a smooth, strictly convex, and reflexive real Banach space, and let be a maximal monotone operator. For each , we can define a single value mapping by , and such a mapping is called the resolvent of A. It is easy to prove that for all . Using Theorem 3.5, we can obtain the following strong convergence theorem for maximal monotone operators.
Theorem 5.1 Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let C be a nonempty, closed, and convex subset of E. We assume the following:
(H1) Let A, B be two maximal monotone operators from E to , and let , be the resolvent of A and B, respectively, where .
(H2) Suppose that is nonempty, and let , , and be sequences generated by
where with , and let and be coefficient sequences in satisfying the following conditions:
-
(1)
, and ;
-
(2)
, and , .
Then the sequence converges strongly to some point , where .
Proof It is well known that for each , is a relatively nonexpansive mapping (see, for example, [5, 6, 19]). Therefore, for each and , we have
Again, by the same method, we can prove that the set of strong asymptotically fixed points
This implies that is a countable family of weak relatively nonexpansive mappings with the common fixed point set . By a similar way, we can prove that is a countable family of weak relatively nonexpansive mappings with the common fixed point set . Hence, the conclusion of Theorem 5.1 can be obtained from Theorem 3.5 immediately. □
Remark 5.2 Theorem 5.1 improves and extends Theorem 4.1 of Chang et al. [10] from one maximal monotone operator to two maximal monotone operators.
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Acknowledgements
The first author would like to thank the Bansomdejchaopraya Rajabhat University for financial support for the Ph.D. program at King Mongkut’s University of Technology Thonburi (KMUTT). The second author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand’s Office of the Higher Education Commission for financial support (Under NRU-CSEC Project No.55000613).
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Phuangphoo, P., Kumam, P. Modified block iterative procedure for solving the common solution of fixed point problems for two countable families of total quasi-ϕ-asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl 2012, 198 (2012). https://doi.org/10.1186/1687-1812-2012-198
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DOI: https://doi.org/10.1186/1687-1812-2012-198