Abstract
The purpose of this article is to introduce two iterative algorithms for finding a common fixed point of a semigroup of asymptotically nonexpansive mappings which is a unique solution of some variational inequality. We provide two algorithms, one implicit and another explicit, from which strong convergence theorems are obtained in a uniformly convex Banach space, which admits a weakly continuous duality mapping. The results in this article improve and extend the recent ones announced by Li et al. (Nonlinear Anal. 70:3065-3071, 2009), Zegeye et al. (Math. Comput. Model. 54:2077-2086, 2011) and many others.
MSC:47H05, 47H09, 47H20, 47J25.
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1 Introduction
Throughout this paper, we denote by ℕ and the set of all positive integers and all positive real numbers, respectively. Let X be a real Banach space. A mapping is said to be nonexpansive if
and T is asymptotically nonexpansive (see [1]) if there exists a sequence of positive real numbers with such that
We denote by the set of fixed points of T, i.e., .
Recall that a self-mapping is a contraction if there exists a constant such that
A one-parameter family of X into itself is said to be a strongly continuous semigroup of Lipschitzian mappings if the following conditions are satisfied:
-
(i)
for all ;
-
(ii)
for all ;
-
(iii)
for each the mapping from into X is continuous;
-
(iv)
for each , there exists a bounded measurable function such that
A strongly continuous semigroup of Lipschitzian mappings is called strongly continuous semigroup of nonexpansive mappings if for all and strongly continuous semigroup of asymptotically nonexpansive mappings if . Note that for asymptotically nonexpansive semigroup , we can always assume that the Lipschitzian constant is such that for each , is nonincreasing in t and ; otherwise, we replace for each with . is said to have a fixed point if there exists such that for all . We denote by the set of fixed points of , i.e., (for more details, see [2–4]).
A continuous operator of the semigroup is said to be uniformly asymptotically regular on X if for all and any bounded subset C of X, (see [5] for examples of uniformly asymptotically regular semigroups).
Recently, convergence theorems for common fixed points of a strongly continuous semigroup of nonexpansive mappings and their generalizations have been studied by numerous authors (see, e.g., [6–10]). Construction of fixed points of nonexpansive mappings (and of common fixed points of nonexpansive semigroups) is an important subject in the theory of nonexpansive mappings and finds application in a number of applied areas, in particular, in image recovery and signal processing (see, e.g., [11–16]). In the last ten years, the iterative methods for nonexpansive mappings have been applied to solve convex minimization problems; see, e.g., [17–19]. Let H be a real Hilbert space, whose inner product and norm are denoted by and , respectively. Let A be a strongly positive bounded linear operator on H; that is, there is a constant with the property
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:
where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H.
In 2003, Xu [19] proved that the sequence defined by the iterative method below, with the initial guess chosen arbitrarily,
converges strongly to the unique solution of the minimization problem (1.2) provided the sequence satisfies certain conditions. Using the viscosity approximation method, Moudafi [20] introduced the following iterative process for nonexpansive mappings (see [19] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial , we define the sequence recursively by
where is a sequence in . It is proved in [19, 20] that under certain appropriate conditions imposed on , the sequence generated by (1.4) strongly converges to a unique solution of the variational inequality
In 2006, Marino and Xu [21] combined the iterative method (1.3) with the viscosity approximation method (1.4) considering the following general iterative process:
where . They proved that the sequence generated by (1.6) converges strongly to a unique solution of the variational inequality
which is the optimality condition for the minimization problem
where h is a potential function for γf (i.e., for ).
On the other hand, Li et al.[22] considered the implicit and explicit viscosity iteration processes for a nonexpansive semigroup in a Hilbert space as follows:
where and are two sequences satisfying certain conditions. They proved the sequence defined by (1.8) and (1.9) converges strongly to , which solves the variational inequality (1.7). Under the framework of a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, Chen and Song [23] studied the strong convergence of the implicit and explicit viscosity iteration processes for a nonexpansive semigroup with as follows:
Very recently, Zegeye et al.[7] introduced the implicit and explicit iterative processes for a strongly continuous semigroup of asymptotically nonexpansive mappings in a reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm as follows:
They proved that defined by (1.12) and (1.13) converges strongly to a common fixed point of provided certain conditions are satisfied.
In this paper, motivated by the above results, we introduce two iterative algorithms for finding a common fixed point of a semigroup of asymptotically nonexpansive mappings which is a unique solution of some variational inequality. We establish the strong convergence results in a uniformly convex Banach space which admits a weakly continuous duality mapping. The results in this article improve and extend the recent ones announced by Li et al.[22], Zegeye et al.[7] and many others.
2 Preliminaries
Throughout this paper, we write (respectively ) to indicate that the sequence weakly (respectively weak∗) converges to x; as usual will symbolize strong convergence; also, a mapping I will denote the identity mapping. Let X be a real Banach space, be its dual space. Let . A Banach space X is said to be uniformly convex if, for each , there exists a such that for each , implies . It is know that a uniformly convex Banach space is reflexive and strictly convex (see also [24]). A Banach space is said to be smooth if the limit exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for .
Let be a continuous strictly increasing function such that and as . This function φ is called a gauge function. The duality mapping associated with a gauge function φ is defined by
where denotes the generalized duality paring. In particular, the duality mapping with the gauge function , denoted by J is referred to as the normalized duality mapping. Clearly, the relation holds for each (see [25]).
Browder [25] initiated the study of certain classes of nonlinear operators by means of the duality mapping . Following Browder [25], we say a Banach space X has a weakly continuous duality mapping if there exits a gauge function φ for which the duality mapping is single-valued and continuous from the weak topology to the weak∗ topology; that is, for each with , the sequence converges weakly∗ to . It is known that has a weakly continuous duality mapping with a gauge function for all . Set , , then , where ∂ denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function at is the set ).
In a Banach space X which admits a duality mapping with a gauge function φ, we say that an operator A is strongly positive (see [26]) if there exists a constant with the property
and
As special cases of (2.1), we have the following results.
-
(1)
If X is a smooth Banach space and for all (see [27]), then the inequality (2.1) reduces to
(2.3) -
(2)
If is a real Hilbert space, then the inequality (2.1) reduces to (1.1).
The first part of the next lemma is an immediate consequence of the subdifferential inequality and the proof of the second part can be found in [28].
Lemma 2.1 ([28])
Assume that a Banach space X has a weakly continuous duality mappingwith a gauge φ.
-
(i)
For all , the following inequality holds:
-
(ii)
Assume that a sequence in X converges weakly to a point . Then the following identity holds:
Lemma 2.2 ([26])
Assume that a Banach space X admits a duality mappingwith a gauge φ. Let A be a strongly positive linear bounded operator on X with a coefficientand. Then.
Definition 2.3 Let C be a closed convex subset of a real Banach space X. Let be a strongly continuous semigroup of asymptotically nonexpansive mappings from C into itself such that . Then is said to be almost uniformly asymptotically regular (in short a.u.a.r.) on C, if for all ,
Lemma 2.4 ([7])
Let C be a closed convex subset of a uniformly convex Banach space X andbe a strongly continuous semigroup of asymptotically nonexpansive mappings from C into itself with a sequencesuch that. Then for eachand,
Lemma 2.5 ([29])
Assume that is a sequence of nonnegative real numbers such that
whereis a sequence inandis a sequence in ℝ such that
-
(i)
;
-
(ii)
or .
Then.
3 Implicit iteration scheme
Theorem 3.1 Let X be a uniformly convex Banach space which admits a weakly continuous duality mappingwith a gauge φ such that φ is invariant on. Letbe a strongly continuous semigroup of asymptotically nonexpansive mappings from X into itself with a sequencesuch that. Letbe a contraction mapping with a constantandbe a strongly positive linear bounded operator with a constantsuch that. Letbe a sequence defined by
whereis a sequence inandis a positive real divergent sequence which satisfy the following conditions:
(C1) ;
(C2) .
Then the sequencedefined by (3.1) converges strongly to, whereis the unique solution of the variational inequality
Proof First, we show that defined by (3.1) is well defined. For all , let us define the mapping
Indeed, for all , we have
Since implies
for sufficiently large , that is,
Thus, by the Banach contraction mapping principle, there exits a unique fixed point , that is, defined by (3.1) is well defined.
Next, we show the uniqueness of a solution of the variational inequality (3.2). Suppose that are solutions of (3.2), then
and
Adding up (3.3) and (3.4), we obtain
which is a contradiction. We must have and the uniqueness is proved. Below, we use to denote the unique solution of the variational inequality (3.2).
Next, we show that is bounded. Take . Then from (3.1), we get that
It follows that
where . Thus, there exists such that
Hence, is bounded, so are and .
Next, we show that as . From (3.1), we note that
By the condition (C1), we obtain
For all , we note that
By Lemma 2.4 and (3.5), we obtain
Next, we show that . By reflexivity of X and boundedness of , there exists a weakly convergent subsequence of such that as . Since is weakly continuous, we have by Lemma 2.1 that
Let for all . It follows that
Since Φ is continuous and , it follows from (3.6) that
On the other hand, we note that
Combining (3.7) and (3.8), we obtain . The property of Φ implies that . In fact, since for all and , then we have
for all . Hence, .
Next, we show that is sequentially compact. Since , and is the gauge function, then for , and
By Lemma 2.1, we have
which implies that
where . Thus, there exists such that
In particular, we have
Since is single-valued and weakly continuous, it follows that as . The property of Φ implies that as .
Next, we show that solves the variational inequality (3.2). From (3.1), we derive that
For all , it follows from (3.10) that
Now, replacing n by in (3.11) and letting , we notice that
By the condition (C2), we obtain that
That is, is a solution of the variational inequality (3.2).
Finally, we show that converges strongly to . Suppose that there exists another subsequence as . We note that is the solution of the variational inequality (3.2). Hence, by uniqueness. In summary, we have shown that is sequentially compact and each cluster point of the sequence is equal to . Therefore, we conclude that as . This proof is complete. □
Remark 3.2 Theorem 3.1 extends and generalizes Theorem 3.5 of Zegeye et al.[7], Theorem 3.1 of Chen and Song [23] and Theorem 3.1 of [22], in the following respects:
-
(1)
Theorem 3.1 generalizes Theorem 3.5 of Zegeye et al. [7] to the viscosity iterative method in a different Banach space which admits a weakly continuous duality mapping.
-
(2)
Theorem 3.1 improves Theorem 3.5 of Zegeye et al. [7] in the sense that our theorem is applicable in a uniformly convex Banach space without the requirement that is almost uniformly asymptotically regular.
(3) Theorem 3.1 extends Theorem 3.1 of Chen and Song [23] from a class of strongly continuous semigroups of nonexpansive mappings to a more general class of strongly continuous semigroups of asymptotically nonexpansive mappings.
-
(4)
Theorem 3.1 includes Theorem 3.1 of Li et al. [22] as a special case.
If is a strongly continuous semigroup of nonexpansive mappings, we have and then Theorem 3.1 is reduced to the following results.
Corollary 3.3 Let X be a uniformly convex Banach space which admits a weakly continuous duality mappingwith a gauge φ such that φ is invariant on. Letbe a strongly continuous semigroup of nonexpansive mappings from X into itself such that. Letbe a contraction mapping with a constantandbe a strongly positive linear bounded operator with a constantsuch that. Letbe a sequence defined by
whereis a sequence insuch thatandis a positive real divergent sequence. Then the sequencedefined by (3.12) converges strongly to, whereis the unique solution of the variational inequality
Corollary 3.4 (Li et al. [[22], Theorem 3.1])
Let H be a real Hilbert space and C be a nonempty closed convex subset of X such that. Letbe a strongly continuous semigroup of nonexpansive mappings from C into itself such that. Letbe a contraction mapping with a constantandbe a strongly positive linear bounded operator with a constantsuch that. Letbe a sequence defined by
whereis a sequence insuch thatandis a positive real divergent sequence. Then the sequencedefined by (3.14) converges strongly to, whereis the unique solution of the variational inequality
4 Explicit iteration scheme
Theorem 4.1 Let X be a uniformly convex Banach space which admits a weakly continuous duality mappingwith a gauge φ such that φ is invariant on. Letbe a strongly continuous semigroup of asymptotically nonexpansive mappings from X into itself with a sequencesuch that. Letbe a contraction mapping with a constantandbe a strongly positive linear bounded operator with a constantsuch that. For given, letbe a sequence defined by
whereis a sequence inandis a positive real divergent sequence which satisfy the following conditions:
(C1) and;
(C2) .
Then the sequencedefined by (4.1) converges strongly to, whereis the unique solution of the variational inequality
Proof By the condition , we may assume, with no loss of generality, that for all . By Lemma 2.2, we have . First, we show that is bounded. Take and .
Since implies for sufficiently large . Then from (4.1), we get that
By induction, we have
Hence, is bounded, so are and .
Next, we show that as . From (4.1), we note that
By the condition (C1), we obtain
For all , we note that
By Lemma 2.4 and (4.3), we obtain and hence
Next, we show that
Let be a subsequence of such that
By reflexivity of X and boundedness of , there exists a weakly convergent subsequence of such that as . Since is weakly continuous, we have by Lemma 2.1 that
Let for all . It follows that
Since Φ is continuous and , it follows from (4.4) that
On the other hand, we note that
Combining (4.5) and (4.6), we obtain . The property of Φ implies that . In fact, since for all and , then we have
for all . Hence, . Since is single-valued and weakly continuous, we obtain that
Finally, we show that as . Now, from Lemma 2.1, we have
where . Put and
Then (4.8) reduces to formula
It follows from the conditions (C1), (C2) and (4.7) that and
Hence, by Lemma 2.5, we obtain that as . The property of Φ implies that as . This proof is complete. □
Applications 4.2 Let X be a uniformly convex Banach space which admits a weakly continuous duality mapping. Let be the space of all bounded linear operators on X. For , define of bounded linear operators by using the following exponential expression:
Then, clearly, the family satisfies the semigroup properties. Moreover, this family forms a one-parameter semigroup of self-mappings of X because exists for each .
Next, the following example shows that all conditions of Theorem 4.1 are satisfied.
Example 4.3 For instance, let , and . Then, clearly, the sequences , and satisfy our assumptions and the condition (C1) in Theorem 4.1. We show that the condition (C2) is achieved. Indeed, we have
Furthermore, if we take such that and (see, e.g., p.160 of [30]) then the sequence defined by (4.1) converges strongly to .
Remark 4.4 Theorem 4.1 extends and generalizes Theorem 3.5 of Zegeye et al.[7], Theorem 3.2 of Chen and Song [23] and Theorem 3.2 of Li et al.[22] in the following respects:
-
(1)
Theorem 4.1 generalizes Theorem 3.5 of Zegeye et al. [7] to the viscosity iterative method in a different Banach space which admits a weakly continuous duality mapping.
-
(2)
Theorem 4.1 improves Theorem 3.5 of Zegeye et al. [7] in the sense that our theorem is applicable in a uniformly convex Banach space without the requirement that is almost uniformly asymptotically regular.
(3) Theorem 4.1 extends Theorem 3.2 of Chen and Song [23] from a class of strongly continuous semigroups of nonexpansive mappings to a more general class of strongly continuous semigroups of asymptotically nonexpansive mappings.
-
(4)
Theorem 4.1 includes Theorem 3.2 of Li et al. [22] as a special case.
If is a strongly continuous semigroup of nonexpansive mappings, we have and then Theorem 4.1 is reduced to the following result.
Corollary 4.5 Let X be a uniformly convex Banach space which admits a weakly continuous duality mappingwith a gauge φ such that φ is invariant on. Letbe a strongly continuous semigroup of nonexpansive mappings from X into itself such that. Letbe a contraction mapping with a constantandbe a strongly positive linear bounded operator with a constantsuch that. For given, letbe a sequence defined by
whereis a sequence insuch thatand, andis a positive real divergent sequence. Then the sequencedefined by (4.9) converges strongly to, whereis the unique solution of the variational inequality
Corollary 4.6 (Li et al. [[22], Theorem 3.2])
Let H be a real Hilbert space and C be a nonempty closed convex subset of X such that. Letbe a strongly continuous semigroup of nonexpansive mappings from C into itself such that. Letbe a contraction mapping with a constantandbe a strongly positive linear bounded operator with a constantsuch that. For given, letbe a sequence defined by
whereis a sequence insuch thatand, andis a positive real divergent sequence. Then the sequencedefined by (4.11) converges strongly to, whereis the unique solution of the variational inequality
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The authors were supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No. 55000613).
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Sunthrayuth, P., Kumam, P. Fixed point solutions of variational inequalities for a semigroup of asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl 2012, 177 (2012). https://doi.org/10.1186/1687-1812-2012-177
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DOI: https://doi.org/10.1186/1687-1812-2012-177