International Scholarly Research Notices

1. Introduction

Let be a real Banach space and let be the dual space of . A mapping is called a gauge function if it is strictly increasing, continuous and . Let be a gauge function, a generalized duality mapping with respect to is defined by, , where denotes the duality pairing between element of and that of . If , then is simply called the normalized duality mapping and is denoted by . For any , an element of is denoted by .

The modulus of convexity of is the function defined by and is called uniformly convex if . A Banach space is said to satisfy Opial’s condition [1] if, for any sequence in , as implies that All Hilbert spaces and spaces, satisfy Opial’s condition. However , do not satisfy this condition; see, for example, [2]. The space is said to have weakly (sequentially) continuous duality map if there exists a gauge function such that is singled valued and (sequentially) continuous from with weak topology to with topology. It is known that every Banach space with weakly sequentially continuous duality mapping satisfies Opial’s condition (see [3]). Every space, () has a weakly sequentially continuous duality map.

The space is said to have uniform Opial’s condition [4] if for each , there exists an such that for each with and each sequence satisfying as , and .

is said to satisfy the local uniform Opial’s condition [5] if, for any weak null sequence in with and any , there exists such that for all with . Observe that uniform Opial’s condition implies local uniform Opial’s condition which in turn implies Opial’s condition.

A self-mapping is said to be contraction if , where is a fixed constant. It is said to be weakly contractive if there exists a nondecreasing function satisfying if and only if and . It is known that the class of weakly contractive maps contain properly the class of contractive ones; see [6, 7]. The map is said to be nonexpansive if and asymptotically nonexpansive if there exists a sequence with such that and . The set of fixed point of is defined as .

A one parameter family of self-mapping of is called nonexpansive semigroup if the following conditions are satisfied:(i); (ii); (iii)for each , the mapping is continuous;(iv)for and , .

The family is said to be asymptotically nonexpansive semigroup if conditions (i)–(iii) are satisfied and, in addition, there exists a function satisfying and .

The family is said to be asymptotically regular if for all and . It is said to be uniformly asymptotically regular if, for any and for any bounded subset of , For some positive real numbers and , the mapping is said to be -strongly accretive if for any , there exists such that and it is called strictly pseudocontractive if Let be a nonempty closed convex subset of and be a map. Then, a variational inequality problem with respect to and is find such that The problem of solving a variational inequality of the form (1.10) has been intensively studied by numerous authors due to its various applications in several physical problems, such as in operations research, economics, and engineering design; see, for example, [8–10] and the references therein. Iterative methods for approximating fixed points of nonexpansive mappings, nonexpansive semigroups, and their generalizations which solves some variational inequalities problems have been studied by a number of authors (see, e.g., [11–17] and the references therein).

A typical problem is to minimize a quadratic function over the set of the fixed points of some nonexpansive mapping in a real Hilbert space : Here, is the fixed point set of a nonexpansive mapping of is a point in , and is some bounded, linear, and strongly positive operator on , where a map is said to be strongly positive if there exists a constant such that For a strongly positive bonded linear operator and any , we have This implies that is -strongly accretive (or in particular -strongly monotone). On the other hand, by simple calculation, the following relation also holds: This implies that is strictly pseudocontractive.

Let be a real Hilbert space. In 2003, Xu [18] proved that the sequence defined by chosen arbitrarily, converges strongly to the unique solution of the minimization problem (1.11) provided that the sequence satisfies certain control conditions.

In 2000, Moudafi [12] introduced the viscosity approximation method for nonexpansive mappings. Let be a contraction on . Starting with an arbitrary initial point , define a sequence recursively by where is a sequence in . It was proved in [12] that, under certain appropriate conditions impose on , the sequence generated by (1.16) converges strongly to the unique solution of the variational inequality: For a strongly positive linear bounded map on with coefficient , Marino and Xu [11] combined the iterative method (1.15) with the viscosity approximation method (1.16) and studied the following general iterative method: They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.18) converges strongly to the unique solution of the variational inequality: which is also the optimality condition for the minimization problem where is a potential function for .

Yao et al. [19] proved that the iterative scheme defined by where and are sequences in satisfying some control conditions, converges to a fixed point of a nonexpansive mapping which solves the variational inequality (1.19).

Acedo and Suzuki [20], recently, proved the strong convergence of the Browder’s implicit scheme, , to a common fixed point of a uniformly asymptotically regular family of nonexpansive semigroup in the framework of a real Hilbert space.

Let be a semigroup and let be the subspace of all bounded real valued functions defined on with supremum norm. For each , the left translator operator on is defined by for each and . Let be a subspace of containing 1 and let be its topological dual. An element of is said to be a mean on if . Let be invariant; that is, for each . A mean on is said to be left invariant if for each and .

Recently, Saeidi and Naseri [14] studied the problem of approximating common fixed point of a family of nonexpansive semigroup and solution of some variational inequality problem and proved the following theorem.

Theorem 1.1 (Saeidi and Naseri [14]). Let be a nonexpansive semigroup on a real Hilbert space such that . Let be a left invariant subspace of B(S) such that , and the function is an element of for each . Let be a contraction with constant and let be strongly positive map with constant . Let be a left regular sequence of means on and let be a sequence in such that (i) and (ii) . Let and be a sequence generated by Then, converges strongly to a common fixed point of the family which is the unique solution of the variational inequality . Equivalently one has .

More recently, as commented by Golkarmanesh and Naseri [21], Piri and Vaezi [13] gave a minor variation of Theorem 1.1 as follows.

Theorem 1.2 (Piri and Vaezi [13]). Let be a nonexpansive semigroup on a real Hilbert space such that . Let be a left invariant subspace of such that , and the function is an element of for each . Let be a contraction and let be -strongly accretive and strictly pseudocontractive with . Let be a left regular sequence of means on and let be a sequence in such that (i) and (ii).Let be generated by : where . Then, converges strongly to a common fixed point of the family which is the unique solution of the variational inequality . Equivalently one has .

Motivated by these results, it is our purpose in this paper to continue the study of this problem and prove new strong convergence theorem for common fixed point of family of asymptotically nonexpansive semigroup and solution of some variational inequality problem in the framework of a real Banach space much more general than Hilbert. Our theorem, proved for more general classes of maps, is applicable in spaces, .

2. Preliminaries

In the sequel, we will make use of the following lemmas.

Lemma 2.1. Let be a real normed linear space. Then, the following inequality holds:

Lemma 2.2 (Suzuki [22]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all integers and . Then, .

Lemma 2.3 (Kim et al. [23]). Let be a real Banach space satisfying the local uniform Opial’s condition and a nonempty weakly compact convex subset of . If is asymptotically nonexpansive semigroup on , then is demiclosed at zero.

Lemma 2.4 (Acedo and Suzuki, [20]). Let be a set of a separated topological vector space . Let be a family of mappings on such that for all . Assume that is asymptotically regular, then holds for all .

Lemma 2.5 (Xu [24]). Let be a sequence of nonnegative real numbers satisfying the following relation: where, (i) , (ii) , (iii) . Then, as .

Let be a real Banach space and , and positive real numbers satisfying and . Let be a -strongly accretive and strictly pseudocontractive then, as shown in [13], and are strict contractions. In fact, for , which implies Also, for,

3. Main Results

Theorem 3.1. Let be a real Banach space with local uniform Opial’s property whose duality mapping is sequentially continuous. Let be uniformly asymptotically regular family of asymptotically nonexpansive semigroup of , with function and . Let be weakly contractive and let be -strongly accretive and strictly pseudocontractive with . Let and . Let and be sequences in and let be an increasing sequence in satisfying the following conditions: Define a sequence by , Then, the sequence converges strongly to a common fixed point of the family which solves the variational inequality:

Proof. We start by showing that the solution of the variational inequality (3.3) in is unique. Assume are solutions of the variational inequality (3.3), then Adding these two relations, we get Therefore, Since , we obtain that and so the solution is unique in .
Now, let , since as there exists such that . Hence, for , we have the following: so that By induction, we have Thus, is bounded and so are , and . Observe that so that From this, we obtain that which implies and by Lemma 2.2, Thus, as .
Next, we show that and .
Since we have From as , we obtain .
Now, for any , we have Using this and the uniform asymptotic regularity of , we get We also have This implies that Let be a subsequence of such that and assume without loss of generality that . By Lemma 2.3, is demiclosed at zero, so and, by Lemma 2.4, .
Since the duality map of is weakly sequentially continuous, we obtain We now conclude by showing that as . Since , if we denote by the value , then we clearly have that . Let be large enough such that , and let be a positive real number such that . Then, using the recursion formula (3.2) and for , we have Observe that and Applying Lemma 2.5, we obtain as . This completes the proof.