Research Article | Open Access

# Convergence Theorem for a Family of Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

**Academic Editor:**Ram U. Verma

#### Abstract

Let be a real reflexive and strictly convex Banach space with a uniformly GΓ’teaux differentiable norm. Let be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of , with functions . Let and be a weakly contractive map. For some positive real numbers and satisfying , let be a -strongly accretive and -strictly pseudocontractive map. Let be an increasing sequence in with , and let and be sequences in satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality for all , is proved in a framework of a real Banach space.

#### 1. Introduction

Let be a real Banach space. We denote by the normalized duality map from to ( is the dual space of ), and it is defined by

A mapping is said to be contractive if , for , and some constant . It is said to be weakly contractive if there exists a nondecreasing function satisfying if and only if and , for all . It is known that the class of weakly contractive maps contain properly the class of contractive ones, see [1, 2]. A mapping is said to be nonexpansive if , for all and asymptotically nonexpansive if there exists a sequence with as and , for all . We denote by the set of fixed points of a map .

A mapping is said to be total asymptotically nonexpansive (see [3]) if there exist nonnegative real sequences and , with and as and strictly increasing and continuous functions with such that

*Remark 1.1. *If , the total asymptotically nonexpansive mapping coincides with generalized asymptotically nonexpansive mapping. In addition, for all , if , then generalized asymptotically nonexpansive mapping coincides with asymptotically nonexpansive mapping; if where , then generalized asymptotically nonexpansive mapping coincide with asymptotically nonexpansive mapping in the intermediate sense; if , and then we obtain from (1.2) the class of nonexpansive mapping.

A one-parameter family of generalized asymptotically nonexpansive semigroup is a family of self-mapping of such that (i) for ,(ii) for all and , (iii) for , (iv)there exist functions such that as , and

We will denote by the common fixed-point set of , that is,

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 , a mapping is said to be -strongly accretive if for any , there exists such that
and it is called -strictly pseudocontractive if

Let be a real Banach space, and let , and be positive real numbers satisfying and . Let be a -strongly accretive and -strictly pseudocontractive, then the following holds, see [4], for
that is, and are contractive mappings.

Let be a nonempty closed-convex subset of and a map. Then, a variational inequality problem with respect to and is found to be such that

Recently, convergence theorems for fixed points of nonexpansive mappings, common fixed points of family of nonexpansive mappings, nonexpansive semigroup, and their generalisation have been studied by numerous authors (see, e.g., [5β21]).

Acedo and Suzuki [22], 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.

Li et al. [23] proved strong convergence theorems for implicit viscosity schemes for common fixed points of family of generalized asymptotically nonexpansive semigroups in Banach spaces.

Let be a semigroup and 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 , 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 [24] studied the problem of approximating common fixed point of a family of nonexpansive semigroup and solution of some variational inequality problem in a real Hilbert space. They proved the following theorem.

Theorem 1.2 (Saeidi and Naseri [24]). * Let be a nonexpansive semigroup in 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 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 and . Let , and let 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 for all . Equivalently one has . *

More recently, as commented by Golkarmanesh and Naseri [25], Piri and Vaezi [4] gave a minor variation of Theorem 1.2 as follows.

Theorem 1.3 (Piri and Vaezi [4]). * 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 with constant , 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 and . Let be a sequence generated by ,
**
where , then, converges strongly to a common fixed point of the family which is the unique solution of the variational inequality for all . Equivalently one has . *

Very recently, Ali [26] continued the study of the problem in [4, 24] and proved a strong convergence theorem in a Banach space setting much more general than Hilbert space. He actually proved the following theorem.

Theorem 1.4 (Ali [26]). * Let be a real Banach space with local uniform Opial's property whose duality mapping is sequentially continuous. Let be a 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
*

*Remark 1.5. *It is well known that all spaces satisfy Opial's condition and possess a weakly sequentially continuous duality mapping. However, ββ spaces and consequently all Sobolev spaces do not satisfy either of the properties.

It is our purpose in this paper to prove a strong convergence theorem for approximating common fixed points of family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup in a real reflexive and strictly convex Banach space with a uniformly GΓ’teaux differentiable norm. Our theorem is applicable in spaces, (and consequently in sobolev spaces). Our theorem extends and improves some recent important results. For instance, our theorem presents a convergence of an explicit scheme that extends Theorem 1.4 to a more general setting of Banach spaces that includes spaces on one hand and for more general class of maps on the other hand.

#### 2. Preliminaries

Let denote the unit sphere of a real Banach space . is said to have a * GΓ’teaux differentiable * norm if the limit
exists for each ; is said to have a * uniformly GΓ’teaux differentiable * norm if for each , the limit is attained uniformly for . A Banach space is said to be strictly convex if for and .

Let be a nonempty, closed, convex, and bounded subset of a real Banach space , and let the diameter of be defined by . For each , let and denote the Chebyshev radius of relative to itself. The * normal structure coefficient * of (introduced in 1980 by Bynum [27], see also Lim [28] and the references contained therein) is defined by : is a closed convex and bounded subset of with . A space such that is said to have * uniform normal structures *. It is known that every space with a uniform normal structure is reflexive, and that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see, e.g., [29]).

Let be a real Banach space with uniformly GΓ’teaux differentiable norm, then the normalized duality mapping , defined by (1.1), is singled valued and uniformly continuous from the norm topology of to the topology of on each bounded subset of , see, for example [30].

*Definition 2.1. *Let be a continuous linear functional on , and let . We write instead of . The function is called a Banach limit when satisfies and for each .

For a Banach limit , it is known that for every . So if and as , we have .

We will make use of the following well-known result.

Lemma 2.2. *Let be a real-normed linear space. Then, the following inequality holds:
*

In the sequel, we shall also make use of the following lemmas.

Lemma 2.3 (Suzuki [31]). * Let and be bounded sequences in a real Banach space , and let be a sequence in with . Suppose that for all integer and . Then, .*

Lemma 2.4 (Shioji and Takahashi [32]). * Let be such that for all Banach limits . If , then . *

Lemma 2.5 (Xu [33]). *Let be a sequence of nonnegative real numbers satisfying the following relation:
**
where (i)ββ(ii)ββ (iii) and . Then, as .*

#### 3. Main Results

Theorem 3.1. * Let be a real reflexive and strictly convex Banach space with a uniformly GΓ’teaux differentiable norm, and let be uniformly asymptotically regular family of generalized asymptotically nonexpansive semigroup of , with functions and . Let be weakly contractive, and let be -strongly accretive and -strictly pseudocontractive with . Let and . Let and be sequences in and 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 solution of the variational inequality (3.3) in is at most one. Assume that are solutions of the variational inequality (3.3), then

Adding these two inequalities, we get

Therefore,
Since , we obtain that , and so the solution is unique in .

Now, let , since and as , then there exists such that and for all . 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 that
and by Lemma 2.3,

Thus,

Next, we show that , for all .

Since
we have

From as and (3.15), we obtain

Also,

Since and is uniformly asymptotically regular,
where is any bounded subset of containing . Since is continuous, we get that

This implies that

Next, we show that

Define a map by

Then, as , is continuous and convex, so as is reflexive, there exists such that . Hence, the set

Since , and is continuous for all , we have

Hence, .

Let . Since is a closed-convex set, there exists a unique such that

Since and ,

Therefore, . Since for all and , we have

Therefore, and so .

Let and . Then, it follows that , and using Lemma 2.2, we obtain that
which implies that

Moreover,

Since is norm-to-weak* uniformly continuous on bounded subsets of , we have that

Observe that from (3.14) and (3.15), we have

This implies that
and so we obtain by Lemma 2.4 that

Finally, we show that as . Since , if we denote by the value , then we clearly have . Let be large enough such that , for all , and let be a positive real number such that for all . Then, using the recursion formula (3.2) and for , we have

so that
where denotes .

Observeββthat and

Applying Lemma 2.5, we obtain as . This completes the proof.

The following corollaries follow from Theorem 3.1.

Corollary 3.2. *Let be a real uniformly convex and uniformly smooth Banach space, , and let , and be as in Theorem 3.1. Then, the sequence converges strongly to a common fixed point of the family which solves the variational inequality (3.3). *

Corollary 3.3. *Let be a real Hilbert space, and let ,, and be as in Theorem 3.1. Then, the sequence converges strongly to a common fixed point of the family which solves the variational inequality
*

Corollary 3.4. *Let be a family of nonexpansive semigroup of a real reflexive and strictly convex Banach space with a uniformly GΓ’teaux differentiable norm , and let , and be as in Theorem 3.1. Then, the sequence converges strongly to a common fixed point of the family which solves the variational inequality (3.3). *

#### Acknowledgment

The authors thank the anonymous referees for useful comments and observations, that helped in improving the presentation of this paper.

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