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The modified Mann type iterative algorithm for a countable family of totally quasi-ϕ-asymptotically nonexpansive mappings by the hybrid generalized f-projection method
Fixed Point Theory and Applications volume 2013, Article number: 63 (2013)
Abstract
The purpose of this article is to introduce the modified Mann type iterative sequence, using a new technique, by the hybrid generalized f-projection operator for a countable family of totally quasi-ϕ-asymptotically nonexpansive mappings in a uniform smooth and strictly convex Banach space with the Kadec-Klee property. Then we prove that the modified Mann type iterative scheme converges strongly to a common element of the sets of fixed points of the given mappings. Our result extends and improves the results of Li et al. (Comput. Math. Appl. 60:1322-1331, 2010), Takahashi et al. (J. Math. Anal. Appl. 341:276-286, 2008) and many other authors.
MSC:47H05, 47H09, 47H10.
1 Introduction
Let E be a real Banach space and C be a nonempty closed and convex subset of E. A mapping is said to be totally asymptotically nonexpansive [1] if there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that
A point is a fixed point of T provided . Denote by the fixed point set of T, that is, . A point is called an asymptotic fixed point of T [2] if C contains a sequence which converges weakly to p such that . The asymptotic fixed point set of T is denoted by .
Let be a dual space of the Banach space E. We recall that for all and , we denote the value of at x by . Then the normalized duality mapping is defined by
If E is a Hilbert space, then , where I is the identity mapping. Next, consider the functional defined by
where J is the normalized duality mapping and denotes the duality pairing of E and .
If E is a Hilbert space, then . It is obvious from the definition of ϕ that
T is said to be relatively nonexpansive [3, 4] if and
T is said to be relatively asymptotically nonexpansive [5, 6] if and there exists a sequence with as such that
T is said to be ϕ-nonexpansive [7, 8] if
T is said to be quasi-ϕ-nonexpansive [7, 8] if and
T is said to be asymptotically ϕ-nonexpansive [8] if there exists a sequence with as such that
T is said to be quasi-ϕ-asymptotically nonexpansive [8] if and there exists a sequence with as such that
T 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
Remark 1.1 (1) Every relatively nonexpansive mapping implies a relatively quasi-nonexpansive mapping, a quasi-ϕ-nonexpansive mapping implies a quasi-ϕ-asymptotically nonexpansive mapping and a quasi-ϕ-asymptotically nonexpansive mapping implies a totally quasi-ϕ-asymptotically nonexpansive mapping, but the converses are not true.
(2) A relatively quasi-nonexpansive mapping is sometimes called hemi-relatively nonexpansive mapping. The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings (see [4, 9–13]), which requires the strong restriction .
(3) For other examples of relatively quasi-nonexpansive mappings such as the generalized projections and others, see [[7], Examples 2.3 and 2.4].
On the other hand, Alber [14] introduced that the generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional , that is, , where is a solution of the minimization problem
In 2006, Wu and Huang [15] introduced a new generalized f-projection operator in Banach spaces. They extended the definition of generalized projection operators introduced by Abler [16] and proved the properties of the generalized f-projection operator.
Now, we recall the concept of the generalized f-projection operator. Let be a functional defined by
where , , ρ is a positive number and is proper, convex and lower semicontinuous. From the definition of G, Wu and Huang [15] studied the following properties:
-
(1)
is convex and continuous with respect to ϖ when y is fixed;
-
(2)
is convex and lower semicontinuous with respect to y when ϖ is fixed.
Definition 1.2 Let E be a real Banach space with the dual space and C be a nonempty closed and convex subset of E. We say that is a generalized f-projection operator if
In 1953, Mann [17] introduced the following iteration process, which is now well known as Mann’s iteration:
where the initial guess element is arbitrary and is a sequence in . Mann’s iteration has been extensively investigated for nonexpansive mappings and some mappings. In an infinite-dimensional Hilbert space, Mann’s iteration can conclude only weak convergence (see [18, 19]). Bauschke and Combettes [20] introduced a modified Mann iteration method (1.5) in a Hilbert space and proved, under appropriate conditions, some strong convergence.
Recently, Takahashi et al. [21] studied the strong convergence theorem by the new hybrid method for a family of nonexpansive mappings in Hilbert spaces: , , and
where for all and is a sequence of nonexpansive mappings of C into itself such that . They proved that if satisfies some appropriate conditions, then converges strongly to .
The ideas to generalize the process (1.5) from Hilbert spaces to Banach spaces have recently been made. Especially, Matsushita and Takahashi [11] proposed the following hybrid iteration method with the generalized projection for a relatively nonexpansive mapping T in a Banach space E:
They proved that converges strongly to a point . Many authors studied methods for approximating fixed points of a countable family of (relatively quasi-) nonexpansive mappings (see [22–26]).
In 2008, Alber et al. [27] proved a new strong convergence result of the regularized successive approximation method for a total asymptotically nonexpansive mapping in a Hilbert spaces. In 2010, Li et al. [28] introduced the following hybrid iterative scheme for approximation fixed points of a relatively nonexpansive mapping using the generalized f-projection operator in a uniformly smooth real Banach space which is also uniformly convex: and
They proved strong convergence theorems for finding an element in the fixed point set of T.
One question is raised naturally as follows:
Are the results of Alber et al. [27], Li et al. [28]and Takahashi et al. [21]true in the framework of strictly convex Banach spaces for totally quasi-ϕ-asymptotically nonexpansive mappings?
Motivated and inspired by the works mentioned above, in this article we aim to introduce a new hybrid projection algorithm of the generalized f-projection operator for a countable family of totally quasi-ϕ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Our result extends and improves the results of Li et al. [28], Takahashi et al. [21] and many other authors.
2 Preliminaries
A Banach space E with the norm is called strictly convex if for all with and . Let be a unit sphere of E. A Banach space E is called smooth if the limit
exists for each . It is also called uniformly smooth if the limit exists uniformly for all . The modulus of smoothness of E is the function defined by
The modulus of convexity of E (see [29]) is the function defined by
In this paper, we denote the strong convergence and weak convergence of a sequence by and , respectively.
Remark 2.1 The basic properties of E, , J and are as follows (see [30]):
-
(1)
If E is an arbitrary Banach space, then J is monotone and bounded;
-
(2)
If E is strictly convex, then J is strictly monotone;
-
(3)
If E is smooth, then J is single-valued and semi-continuous;
-
(4)
If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E;
-
(5)
If E is reflexive smooth and strictly convex, then the normalized duality mapping J is single-valued, one-to-one and onto;
-
(6)
If E is a reflexive strictly convex and smooth Banach space and J is the duality mapping from E into , then is also single-valued, bijective and is also the duality mapping from into E and thus and ;
-
(7)
If E is uniformly smooth, then E is smooth and reflexive;
-
(8)
E is uniformly smooth if and only if is uniformly convex;
-
(9)
If E is a reflexive and strictly convex Banach space, then is norm-weak∗-continuous.
Remark 2.2 If E is a reflexive, strictly convex and smooth Banach space, then if and only if . It is sufficient to show that if then . From (1.1) we have . This implies that . From the definition of J, one has . Therefore, we have (see [30–32] for more details).
Recall that a Banach space E has the Kadec-Klee property [30, 31, 33] if, for any sequence and with and , as . It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.
The generalized projection [14] from E into C is defined by
The existence and uniqueness of the operator follows from the properties of the functional and the strict monotonicity of the mapping J (see, for example, [14, 30, 31, 34, 35]). If E is a Hilbert space, then and becomes the metric projection . If C is a nonempty closed and convex subset of a Hilbert space H, then is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces.
We also need the following lemmas for the proof of our main results.
Let T be a nonlinear mapping, T is said to be uniformly asymptotically regular on C if
A mapping T from C into itself is said to be closed if, for any sequence such that and , we have .
Lemma 2.3 (Chang et al. [36])
Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with the sequences and of nonnegative real numbers with , as and a strictly increasing continuous function with . If , then the fixed point set is a closed convex subset of C.
Lemma 2.4 (Wu and Hung [15])
Let E be a real reflexive Banach space with the dual space and C be a nonempty closed and convex subset of E. The following statements hold:
-
(1)
is a nonempty, closed and convex subset of C for all ;
-
(2)
If E is smooth, then for all , if and only if
-
(3)
If E is strictly convex and is positive homogeneous (i.e., for all such that , where ), then is a single-valued mapping.
In the following lemma, Fan et al.[37] showed that Lemma 2.1(iii) in [37] can be removed.
Lemma 2.5 (Fan et al. [37])
Let E be a real reflexive Banach space with its dual space and C be a nonempty closed and convex subset of E. If E is strictly convex, then is single-valued.
Note that J is a single-valued mapping when E is a smooth Banach space. There exists a unique element such that , where . This substitution in (1.4) gives the following:
Now, we consider the second generalized f-projection operator in Banach spaces (see [28]).
Definition 2.6 Let E be a real smooth Banach space and C be a nonempty, closed and convex subset of E. We say that is the generalized f-projection operator if
Lemma 2.7 (Deimling [38])
Let E be a Banach space and be a lower semicontinuous convex function. Then there exist and such that
Lemma 2.8 (Li et al. [28])
Let E be a reflexive smooth Banach space and C be a nonempty, closed and convex subset of E. The following statements hold:
-
(1)
is nonempty, closed and convex subset of C for all ;
-
(2)
For all , if and only if
-
(3)
If E is strictly convex, then is a single-valued mapping.
Lemma 2.9 (Li et al. [28])
Let E be a real reflexive smooth Banach space and C be a nonempty closed and convex subset of E. If for all , then
Remark 2.10 Let E be a uniformly convex and uniformly smooth Banach space and for all . Then Lemma 2.9 reduces to the property of the generalized projection operator considered by Alber [14].
If for all and , then the definition of totally quasi-ϕ-asymptotically nonexpansive T is equivalent to the following:
If and there exist nonnegative real sequences , with , as and a strictly increasing continuous function with such that
3 Main results
Theorem 3.1 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let be a countable family of closed and uniformly totally quasi-ϕ-asymptotically nonexpansive mappings with the sequences , of nonnegative real numbers with , as and a strictly increasing continuous function with . Let be a convex and lower semicontinuous function with such that for all and . Assume that is uniformly asymptotically regular for all and . For an initial point , let for each and and define the sequence by
where is a sequence in and . If , then converges strongly to a point .
Proof We split the proof into four steps.
Step 1. We first show that is closed and convex for all . From the definition, for all is closed and convex. Suppose that is closed and convex for some . For any , we know that is equivalent to the following:
Therefore, is closed and convex. Hence is closed and convex for all .
Step 2. We show, by induction, that for all . It is obvious that . Suppose that for some . Let . Since is a totally quasi-ϕ asymptotically nonexpansive mapping, for each , we have
This shows that , which implies that . Hence for all .
Step 3. We show that as . Since is a convex and lower semicontinuous function, from Lemma 2.7, it follows that there exist and such that for all . Since , it follows that
For any , since , we have
This implies that is bounded and so are and . From the fact that and , it follows from Lemma 2.9 that
This implies that is nondecreasing. Hence we know that exists. Taking , we obtain
Since is bounded, E is reflexive and is closed and convex for all , we can assume that . From the fact that and , we get
Since f is convex and lower semicontinuous, we have
By (3.6) and (3.7), we get
That is, , which implies that and so, by virtue of the Kadec-Klee property of E, it follows that
We also have
Since is bounded (we denote ), it follows that
From (3.8) and (3.9), we have . Since J is uniformly norm-to-norm continuous, it follows that
Since and by the definition of , it follows that
that is, we get
From (3.5) and (3.10), it follows that for each ,
Also, from (1.2), it follows that for each ,
Since J is uniformly norm-to-norm continuous, it follows that for each ,
That is, bounded in for all . Since E is reflexive and is also reflexive, we can assume that for all . Since E is reflexive, we see that . Hence there exists such that . It follows that for each ,
Taking on the both sides of the equality above and the property of weak lower semicontinuity of the norm , it follows that
That is, , which implies that . It follows that for each , . From (3.14) and the Kadec-Klee property of , we have as for all . Since is norm-weak∗-continuous, that is, , it follows from (3.13) and the Kadec-Klee property of E that
From (3.9), (3.15) and the triangle inequality, we have
Since J is uniformly norm-to-norm continuous, we obtain
From the definition of , it follows that
and so
Since , it follows from (3.11) and (3.17) that, for each ,
Since is uniformly norm-to-norm continuous, for each , we obtain
By using the triangle inequality, for each , we have
From (3.21) and as , it follows that for each ,
For each , we have
Since is uniformly asymptotically regular for all , it follows from (3.22) that
That is, as . From as and the closedness of , we have for all . We see that for all , which implies that .
Step 4. We show that . Since ℱ is a closed and convex set, it follows from Lemma 2.8 that is single-valued, which is denoted by v. By the definition and , we also have
By the definition of G and f, we know that, for any , is convex and lower semicontinuous with respect to ξ and so
From the definition of , since , we conclude that and as . This completes the proof. □
Setting and in Theorem 3.1, we have the following.
Corollary 3.2 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let be a countable family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings. Let be a convex and lower semicontinuous function with such that for all and . Assume that is uniformly asymptotically regular for all and . For an initial point , let , and define the sequence by
where is a sequence in . If , then converges strongly to a point .
Setting and in Theorem 3.1, we have the following.
Corollary 3.3 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let T be a closed totally quasi-ϕ-asymptotically nonexpansive mapping with the sequences , of nonnegative real numbers with , as and a strictly increasing continuous function with . Let be a convex and lower semicontinuous function with such that for all and . Assume that T is a uniformly asymptotically regular and . For an initial point , let and define the sequence by
where is a sequence in and . If , then converges strongly to a point .
Taking for all , we have and . Thus, from Theorem 3.1, we obtain the following.
Corollary 3.4 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let be a countable family of closed and uniformly totally quasi-ϕ-asymptotically nonexpansive mappings with the sequences , of nonnegative real numbers with , as and a strictly increasing continuous function with . Assume that is uniformly asymptotically regular for all and . For an initial point , let , and define the sequence by
where is a sequence in and . If , then converges strongly to a point .
Corollary 3.5 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let be a countable family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings. Assume that is uniformly asymptotically regular for all and . For an initial point , let , and define the sequence by
where is a sequence in . If , then converges strongly to a point .
Corollary 3.6 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let T be a closed totally quasi-ϕ-asymptotically nonexpansive mapping with the sequences , of nonnegative real numbers with , as and a strictly increasing continuous function with . Assume that T is uniformly asymptotically regular and . For an initial point , let and define the sequence by
where is a sequence in and . If , then converges strongly to a point .
Remark 3.7 (1) Corollary 3.3 extends and improves the results of Li et al. [28] from a relatively nonexpansive mapping to a totally quasi-ϕ-asymptotically nonexpansive mapping.
(2) Corollary 3.6 extends and generalizes the result of Takahashi et al. [21] from a Hilbert space to a Banach space and from a nonexpansive mapping to a totally quasi-ϕ asymptotically nonexpansive mapping.
(3) In the case of spaces, we extend Banach spaces from a uniformly smooth and uniformly convex Banach to a uniformly smooth and strictly convex Banach with the Kadec-Klee property, which can be found in the literature works by many authors (see [12, 21, 22, 28]).
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Acknowledgements
This research was supported by Thaksin University. Moreover, the forth author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170).
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Saewan, S., Kanjanasamranwong, P., Kumam, P. et al. The modified Mann type iterative algorithm for a countable family of totally quasi-ϕ-asymptotically nonexpansive mappings by the hybrid generalized f-projection method. Fixed Point Theory Appl 2013, 63 (2013). https://doi.org/10.1186/1687-1812-2013-63
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DOI: https://doi.org/10.1186/1687-1812-2013-63