Abstract
We introduce a general composite iterative scheme for nonexpansive semigroups in Banach spaces. We establish some strong convergence theorems of the general iteration scheme under different control conditions. The results presented in this paper improve and extend the corresponding results of Marino and Xu (2006), and others, from Hilbert spaces to Banach spaces.
1. Introduction
Let be a real Banach space, and let be a nonempty closed convex subset of . A mapping of into itself is said to be nonexpansive if for each . We denote the set of fixed points of . We know that is nonempty if is bounded, for more detail see [1]. A one-parameter family from of into itself is said to be a nonexpansive semigroup on if it satisfies the following conditions: (i) for all ;(ii) for all ;(iii)for each the mapping is continuous;(iv) for all and .
We denote by the set of all common fixed points of , that is . We know that is nonempty if is bounded, see [2]. Recall that a self-mapping is a contraction if there exists a constant such that for each . As in [3], we use the notation to denote the collection of all contraction on , that is, . Note that each has a unique fixed point in . Iterative methods for nonexpansive mappings have recently been applied to solve minimization problems; see, for example, [4–10] and references therein.
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 : where is the fixed point set of a nonexpansive mapping on and is a given point in . Assume is strongly positive; that is there is a constant such that for all .
In 2003, Xu [7] proved that the sequence generated by converges strongly to the unique solution of the minimization problem (1.1) provided the sequence satisfies certain conditions. On the other hand, Moudafi [11] introduced the viscosity approximation method and proved that if is a real Hilbert space, the sequence generated by the following algorithm: where is a contraction mapping with constant and satisfies certain conditions, converges strongly to a fixed point of in which is the unique solution of the following variational inequality:
Recently, Marino and Xu [12] combined the iterative method (1.2) with the viscosity approximation method (1.3) considering the following general iterative process: They proved the sequence , generated by (1.5), converges strongly to the unique solution of the variational inequality which is the optimality condition for the minimization problem where is the fixed point set of a nonexpansive mapping and is a potential function for (i.e., for all ). Xu [3] studied further the viscosity approximation method for nonexpansive semigroup in uniformly smooth Banach spaces. This result extends Theorem 2.2 of Moudafi [11] to a Banach space. Kim and Xu [13] studied the sequence generated by the algorithm and proved strong convergence of scheme (1.8) in the framework of a uniformly smooth Banach space. Later, Yao et al. [14] introduced a new iteration process by combining the modified Mann iteration [13] and the viscosity method introduced by Moudafi [11]. Let be a closed convex subset of a Banach space, a nonexpansive mapping such that ; and . Define in the following way: where and are two sequences in . They proved, under different control conditions on the sequences and , that converge strongly to a fixed point of .
In 2008, Sahu and O’Regan [15] studied several strong convergence theorems for a family of nonexpansive or pseudocontractive nonself-mappings in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm. Recently, Li and Gu [16] studied the sequence generated by the algorithm in Banach spaces, as follows: and they proved the sequence defined by (1.10) that, converges strongly to the unique solution of the variational inequality: Very recently, Kumam and Wattanawitoon [5] introduced the following new composite explicit iterative schemes defined by given and for the approximation of common fixed point of a one parameter nonexpansive semigroup in a real Hilbert space under some appropriate control conditions. They proved strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate conditions.
Question 1. Can the theorem of Marino and Xu [12] be extend from a Hilbert space to a general Banach space, such as uniformly smooth Banach space?
Question 2. Can we extend the iterative method of algorithm (1.10) to a general iterative process?
The purpose of this paper is to give affirmative answer to these questions mentioned above. In this paper, inspired and motivated by the iterative sequence (1.5) given by Marino and Xu [12] and (1.10) given by Li and Gu [16], we introduce a composite iterative algorithm in a Banach space as follows: where is a contraction mapping, is a nonexpansive semigroup and is a strongly positive linear bounded operator, and prove, under certain different control conditions on the sequences , , and , that defined by (1.13) converges strongly to a common fixed point, which solves some variational inequality in Banach spaces. The results presented in this paper extend the corresponding results announced by Marino and Xu [12] and some others from Hilbert spaces to Banach spaces.
2. Preliminaries
Throughout this paper, let be a real Banach space, be a closed convex subset of . Let be a normalized duality mapping by where denotes the dual space of and denotes the generalized duality paring. In the following, the notation and denote the weak and strong convergence, respectively. Also, a mapping denotes the identity mapping.
The norm of a Banach space is said to be Gâteaux differentiable if the limit exists for each on the unit sphere of . In this case is smooth. Recall that the Banach space is said to be smooth if duality mapping is single valued. In a smooth Banach space, we always assume that is strongly positive (see [17]), that is, a constant with the property
Moreover, if for each in the limit (2.2) is uniformly attained for , we say that the norm is uniformly Gâteaux differentiable. The norm of is said to be Frêchet differentiable, if for each , the limit (2.2) is attained uniformly for . The norm of is said to be uniformly Frêchet differentiable (or is said to be uniformly smooth), the limit (2.2) is attained uniformly for . A Banach space is said to be strictly convex if , implies ; uniformly convex if for all , where is modulus of convexity of defined by A uniformly convex Banach space is reflexive and strictly convex (see [18, Theorems 4.1.6 and 4.1.2]) and every uniformly smooth Banach space is a reflexive Banach with uniformly Gâteaux differentiable norm (see [18, Theorems 4.3.7 and 4.1.6]) (also see [19]).
Now, we present the concept of a uniformly asymptotically regular semigroup (see [20–22]). Let be a nonempty closed convex subset of a Banach space , is a continuous operator semigroup on . Then is said to uniformly asymptotically regular (in short, u.a.r.) on if for all and any bounded subset of ,
Lemma 2.1 (Chen and Song [23]). Let be a nonempty closed convex subset of a uniformly Banach space , a bounded closed convex subset of . If we denote a nonexpansive semigroup on such that . For all , the set , then
It is easy to check that the set defined by Lemma 2.1 is a u.a.r. nonexpansive semigroup on (see [24] for more detail).
Lemma 2.2 (Cai and Hu [17]). Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and . Then .
Lemma 2.3 (Suzuki [25]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .
If a Banach space admits a sequentially continuous duality mapping from weak topology to weak star topology, then by Lemma 1 of [26], we have that duality mapping is a single value. In this case, the duality mapping is said to be a weakly sequentially continuous duality mapping, that is, for each with , we have (see [26–28] for more details).
A Banach space is said to be satisfying Opial's condition if for any sequence for all implies
By Theorem 1 in [26], it is well known that if admits a weakly sequentially continuous duality mapping, then satisfies Opial's condition and is smooth.
Lemma 2.4 (Demiclosed principle (Jung [27])). Let be a nonempty closed convex subset of a reflexive Banach space which satisfies Opial's condition, and suppose is nonexpansive. Then the mapping is demiclosed at zero, that is, and implies .
Lemma 2.5 (Liu [29]). Let be a real Banach space and be the normalized duality mapping. Then, for any , we have for all with .
Lemma 2.6 (Aoyama et al. [30]). Let be a sequence of nonnegative real numbers, a sequence of with , a sequence of nonnegative real number with , and a sequence of real numbers with . Suppose that for all . Then .
3. Main Results
We need following the lemma to prove our theorem.
Lemma 3.1. Let be a nonempty bounded closed convex subset of a reflexive, smooth Banach space with admits a weakly sequentially continuous duality mapping from to with and let be a nonexpansive semigroup on such that . Let be a contraction mapping with a coefficient , be a strongly positive linear bounded operator with a coefficient such that and be a net of positive real sequence such that . Suppose that is a u.a.r. nonexpansive semigroup. Then the sequence define by converges strongly to the common fixed point , where solves the variational inequality
Proof. Observe that for . By Lemma 2.2, we have .
Firstly, we show that defined by (3.1) is well define. Define the mapping provided by , for each . Then, for each that
This show that is a contraction mapping. Thus, by Banach's contraction principle guarantees that has a unique fixed point , that is, defined by (3.1) is well define.
Next, we show the uniqueness of a solution of the variational inequality (3.2). Supposing satisfy the inequality (3.2), we have
Adding up both equations of (3.4), we get that
Since this implies that , which is a contradiction. Hence and the uniqueness is proved.
Next, we show that is bounded. Indeed, for any , we have
It follows that . Hence is bounded.
Next, we show that as . We observe that
for all . On the other hand, we note that
for all . By assuming that and be a u.a.r. nonexpansive semigroup, then for all , we get
From (3.7)–(3.9), letting , we get
for all . Assume is such that as . Put and . We will show that contains a subsequence converges strongly to , where . Since is bounded sequence and Banach space is reflexive, there exists a subsequence of which converges weakly to as . Again since Banach space has a weakly sequentially continuous duality mapping satisfying Opial's condition. It follows by Lemma 2.4 and noting (3.10), we have . For each , we note that
Thus, we have
It follows that
Hence,
In particular, we have
Since is bounded and the duality mapping is single-valued and weakly sequentially continuous from into , it follows (3.15), we have as .
Next, we show that solves the variational inequality (3.2). Since
we derive that
Notice that
It follows that, for any ,
Now, replacing and with and , respectively in (3.19), and letting , noting (3.8), we obtain . That is, is a solution of variational inequality (3.2). By uniqueness, as , we have shown that each cluster point of the sequence is equal to . Then, we conclude that as . This completes the proof.
Now, we prove the following theorem which is the main result of this paper.
Theorem 3.2. Let be a nonempty bounded closed convex subset of a uniformly smooth Banach space which admit a weakly sequentially continuous duality mapping from into with and be a nonexpansive semigroup on such that . Let be a contraction mapping with a coefficient and be a strongly positive bounded linear operator with coefficient such that . Let , , be the sequences in and be a positive real divergent sequence such that for all . Assume the following control conditions are hold: and ; , and ; for all .Then the sequence defined by (1.13) converges strongly to the common fixed point , where is the unique solution in of the variational inequality (3.2).
Proof. First, we show that is bounded. By the control condition , we may assume, without loss of generality, that for all . Since is a linear bounded operator on , by (2.3), we have . Observe that
that is to say is positive. It follows that
For any , we compute
By induction, we get
for . Hence is bounded, so are , and .
Next, we show that . From definition of , observing that
We note that
It follows that
Now, we consider the first term on the right side of (3.2), we have
Substituting (3.27) into (3.26), we get
Similarly, from definition of , observing that
We note that
It follows that
Substituting (3.28) into (3.31), we get
where is an appropriate constant such that
Putting , observing control conditions and , we have
Hence, by Lemma 2.6 to (3.32), we get that
Next, we show that
Observe that
It follows that
Observing control condition and noting (3.10), we have
Moreover, we note that
It follows that
Noting (3.19), hence
On the other hand, we note that
Therefor, by (3.26) and control condition imply that , we have
Next, we show that
For each , let be a unique point of such that . By Lemma 3.1, we have
where .
On the other hand, observe that is a strongly positive linear bounded operator, it follows from (2.3), we have
Combining (3.47) with (3.46), we have
It follows that
Now, taking limit superior as firstly, and then as in (3.49) (using (3.44)), we have
where is a constant such that for all and . Now, taking limit superior as in (3.50). Hence, we get
Moreover, we note that
Taking limit superior as in (3.52), we have
By Lemma 3.1, as . Since is a uniformly smooth Banach space, imply that is norm-to-norm uniformly continuous on bounded subset of (see, e.g., [18, Lemma 1]), we obtain
Therefore, from (3.53), we have
Finally, we show that . By Lemma 2.5, we have
It follows that
Put and . The above reduces to formula . Observing control condition and noting (3.55), it is easily seen that and . By Lemma 2.6, we conclude that . This completes the proof.
Corollary 3.3. Let be a nonempty bounded closed convex subset of a real Hilbert space with and be a nonexpansive semigroup on such that . Let be a contraction mapping with a coefficient and be a strongly positive bounded linear operator with coefficient such that . Let , , be the sequences in and be a positive real divergent sequence such that for all . Assume that the following control conditions that hold: and ; , , and ; for all .Then the sequence defined by (1.13) converges strongly to the common fixed point , where is the unique solution in of the variational inequality (1.6).
Proof. From Theorem 3.2, if is a real Hilbert space, we get the desired conclusion easily.
Corollary 3.4. Let be a nonempty bounded closed convex subset of a uniformly smooth Banach space which admit a weakly sequentially continuous duality mapping from into and be a nonexpansive semigroup mapping on such that . Let be a contraction mapping with a coefficient . Let , , be the sequences in and be a positive real divergent sequence such that for all . Assume that the following control conditions hold: and ; , , and ; for all .Then the sequence defined by converges strongly to the fixed point , where is the unique solution in of the variational inequality (1.11).
Proof. Taking and , then (1.13) is reduced to (3.58).
Corollary 3.5 (Marino and Xu [12]). Let be a real Hilbert space. Let be a nonexpansive mapping on such that , be a contraction mapping with a coefficient and be a strongly positive bounded linear operator with a coefficient such that . Assume that the following control conditions hold: and ; . Then the sequence defined by converges strongly to the fixed point , where is the unique solution in of the variational inequality (1.6).
Proof. If is a real Hilbert space, we get the desired conclusion easily. Taking , , and , then (1.13) is reduced to (3.59).
Now, we prove the following theorem, under certain different control conditions on the sequence , and .
Theorem 3.6. Let be a nonempty bounded closed convex subset of a uniformly smooth Banach space which admit a weakly sequentially continuous duality mapping from into with and be a nonexpansive semigroup on such that . Let be a contraction mapping with a coefficient and be a strongly positive bounded linear operator with coefficient such that . Let , , be the sequences in and be a positive real divergent sequence such that for all and for all . Assume that the following control conditions are hold: and ; ; . Suppose that is a u.a.r. nonexpansive semigroup on . Then the sequence defined by (1.13) converges strongly to the common fixed point , where is the unique solution in of the variational inequality (3.2).
Proof. By using the same argument and techniques as those of Theorem 3.2, we note that is bounded, and so are the set , and . We will show that .
From Theorem 3.2, we note that
Define . That is . Now, we compute .
Then, we note that
one has
Substituting (3.60) into (3.62), we have
By assuming that , it follows that
Since is a u.a.r. nonexpansive semigroup and , then for all , we have
Moreover, since , are bounded, observing control conditions , , , and noting (3.65) then, inequality (3.64) implies that
It follows by Lemma 2.5 that
Observing that
and hence,
By the same proof in Theorem 3.2, we note that
Hence . Moreover, by Theorem 3.2, we note that
Next, we show that . We note that
Since is a u.a.r. nonexpansive semigroup, and noting (3.65), we have
for all . By using the same argument and techniques as those of Theorem 3.2, we have converges strongly to a common fixed point . This completes the proof.
Corollary 3.7. Let be a nonempty bounded closed convex subset of a real Hilbert space with and let be a nonexpansive semigroup on such that . Let be a contraction mapping with a coefficient and be a strongly positive bounded linear operator with coefficient such that . Let , , be the sequences in and be a positive real divergent sequence such that for all and for all . Assume that the following control conditions are hold: and ; ; . Suppose that is a u.a.r. nonexpansive semigroup on . Then the sequence defined by (1.13) converges strongly to the fixed point , where is the unique solution in of the variational inequality (1.6).
Proof. From Theorem 3.6, if is a real Hilbert space, we get the desired conclusion easily.
If we taking and , then (1.13) is reduced to (3.74). So, we obtain the following corollary.
Corollary 3.8. Let be a nonempty bounded closed convex subset of a uniformly smooth Banach space which admit a weakly sequentially continuous duality mapping from into and let be a nonexpansive semigroup on such that . Let be a contraction mapping with a coefficient and , , be the sequences in and be a positive real divergent sequence such that for all and for all . Assume that the following control conditions hold: and ; ; . Suppose that is a u.a.r. nonexpansive semigroup on . Then the sequence defined by converges strongly to the fixed point , where is the unique solution in of the variational inequality (1.11).
Acknowledgment
The authors would like to thank The National Research Council of Thailand (NRCT) and the Faculty of Science KMUTT for financial support.