Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces

Abstract

Using -strongly accretive and -strictly pseudocontractive mapping, we introduce a general iterative method for finding a common fixed point of a semigroup of non-expansive mappings in a Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded real-valued functions of the semigroup. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality.

1. Introduction

Let be a real Hilbert space. A mapping of into itself is called non-expansive if , for all . By , we denote the set of fixed points of (i.e., ).

Mann [1] introduced an iteration procedure for approximation of fixed points of a non-expansive mapping on a Hilbert space as follows. Let and

(1.1)

where is a sequence in . See also [2].

On the other hand, Moudafi [3] introduced the viscosity approximation method for fixed point of non-expansive mappings (see [4] for further developments in both Hilbert and Banach spaces). Let be a contraction on a Hilbert space (i.e., for all and ). Starting with an arbitrary initial , define a sequence recursively by

(1.2)

where is sequence in . It is proved in [3, 4] that, under appropriate condition imposed on , the sequence generated by (1.2) converges strongly to the unique solution in of the variational inequality:

(1.3)

Assume that is strongly positive, that is, there is a constant with the property

(1.4)

In [4] (see also [5]), it is proved that the sequence defined by the iterative method below, with the initial guess chosen arbitrarily,

(1.5)

converges strongly to the unique solution of the minimization problem

(1.6)

provided that the sequence satisfies certain conditions. Marino and Xu [6] combined the iterative (1.5) with the viscosity approximation method (1.2) and considered the following general iterative methods:

(1.7)

where . They proved that if is a sequence in satisfying the following conditions:

either or ,

then, the sequence generated by (1.7) converges strongly, as , to the unique solution of the variational inequality:

(1.8)

which is the optimality condition for minimization problem

(1.9)

where is a potential function for (i.e., , for all ).

Let be the topological dual of a Banach space . The value of at will be denoted by or . With each , we associate the set

(1.10)

Using the Hahn-Banach theorem, it is immediately clear that for each . The multivalued mapping from into is said to be the (normalized) duality mapping. A Banach space is said to be smooth if the duality mapping is single valued. As it is well known, the duality mapping is the identity when is a Hilbert space; see [7].

Let and be two positive real numbers such that . Recall that a mapping with domain and range in is called -strongly accretive if, for each , there exists such that

(1.11)

Recall also that a mapping is called -strictly pseudo-contractive if, for each , there exists such that

(1.12)

It is easy to see that (1.12) can be rewritten as

(1.13)

see [8].

In this paper, motivated and inspired by Atsushiba and Takahashi [9], Lau et al. [10], Marino and Xu [6] and Xu [4, 11], we introduce the iterative below, with the initial guess chosen arbitrarily,

(1.14)

where is -strongly accretive and -strictly pseudo-contractive with , is a contraction on a Hilbert space with coefficient , is a positive real number such that , and is a non-expansive semigroup on such that the set of common fixed point of is nonempty, is a subspace of such that and the mapping is an element of for each , and is a sequence of means on . Our purpose in this paper is to introduce this general iterative algorithm for approximating a common fixed points of semigroups of non-expansive mappings which solves some variational inequality. We will prove that if is left regular and is a sequence in satisfying the conditions and , then converges strongly to , which solves the variational inequality:

(1.15)

Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of non-expansive mappings are also presented. It is worth mentioning that we obtain our result without assuming condition .

2. Preliminaries

Let be a semigroup and let be the space of all bounded real-valued functions defined on with supremum norm. For and , we define elements and in by

(2.1)

Let be a subspace of containing , and let be its dual. An element in is said to be a mean on if . We often write instead of for and . Let be left invariant (resp., right invariant), that is, (resp., ) for each . A mean on is said to be left invariant (right invariant) if (resp. ) for each and . is said to be left (resp., right) amenable if has a left (resp., right) invariant mean. is amenable if is both left and right amenable. As it is well known, is amenable when is a commutative semigroup; see [12]. A net of means on is said to be left regular if

(2.2)

for each , where is the adjoint operator of .

Let be a nonempty closed and convex subset of a reflexive Banach space . A family of mapping from into itself is said to be a non-expansive semigroup on if is non-expansive and for each . We denote by the set of common fixed points of , that is,

(2.3)

The open ball of radius centered at is denoted by . For subset of , by , we denote the closed convex hull of . Weak convergence is denoted by , and strong convergence is denoted by .

Lemma 2.1 (see [12, 13]).

Let f be a function of semigroup into a reflexive Banach space such that the weak closure of is weakly compact, and let be a subspace of containing all functions with . Then, for any , there exists a unique element in such that

(2.4)

for all . Moreover, if is a mean on then

(2.5)

One can write by

Lemma 2.2 (see [13]).

Let be a closed convex subset of a Hilbert space , a semigroup from into such that , the mapping an element of for each and , and a mean on . If one writes instead of , then the following holds.

(i) is non-expansive mapping from into .

(ii) for each .

(iii) for each .

(iv)If is left invariant, then is a non-expansive retraction from onto .

Let be a nonempty subset of a normed space , and let . An element is said to be the best approximation to if

(2.6)

where . The number is called the distance from to or the error in approximating by . The (possibly empty) set of all best approximation from to is denoted by

(2.7)

This defines a mapping from into and is called metric (the nearest point) projection onto .

Lemma 2.3 (see [7]).

Let be a nonempty convex subset of a smooth Banach space and let and . Then, the following is equivalent.

(i) is the best approximation to .

(ii) is a solution of the variational inequality

(2.8)

Let be a nonempty subset of a Banach space and a mapping. Then is said to be demiclosed at if, for any sequence in , the following implication holds:

(2.9)

Lemma 2.4 (see [14]).

Let be a nonempty closed convex subset of a Hilbert space and suppose that is non-expansive. Then, the mapping is demiclosed at zero.

The following lemma is well known.

Lemma 2.5.

Let be a real Hilbert space. Then, for all

(i)

(ii)

Lemma 2.6 (see [11]).

Let be a sequence of nonnegative real numbers such that

(2.10)

where and are sequences of real numbers satisfying the following conditions:

(i)

(ii)either or

Then,

The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.

Lemma 2.7.

Let be a real smooth Banach space and a mapping.

(i)If is -strongly accretive and -strictly pseudo-contractive with , then, is contractive with constant .

(ii)If is -strongly accretive and -strictly pseudo-contractive with , then, for any fixed number , is contractive with constant .

Proof.

• (i) From (1.11) and (1.13), we obtain

  

  (2.11)

  Because , we have

  

  (2.12)

  and, therefore, is contractive with constant.

• (ii) Because is contractive with constant, for each fixed number , we have

  

  (2.13)

This shows that is contractive with constant .

Throughout this paper, will denote a -strongly accretive and -strictly pseudo-contractive mapping with , and is a contraction with coefficient on a Hilbert space . We will also always use to mean a number in .

3. Strong Convergence Theorem

The following is our main result.

Theorem 3.1.

Let be a non-expansive 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 left regular sequence of means on , and let be a sequence in such that and . Let and be generated by the iteration algorithm (1.14). Then, converges strongly, as , to , which is a unique solution of the variational inequality (1.15). Equivalently, one has

(3.1)

Proof.

First, we claim that is bounded. Let ; by Lemmas 2.2 and 2.7 we have

(3.2)

By induction,

(3.3)

Therefore, is bounded and so is .

Set . We remark that is -invariant bounded closed convex set and . Now we claim that

(3.4)

Let . By [15, Theorem ], there exists such that

(3.5)

Also by [15, Corollary ], there exists a natural number such that

(3.6)

for all and . Let . Since is strongly left regular, there exists such that for and . Then we have

(3.7)

By Lemma 2.2 we have

(3.8)

It follows from (3.5), (3.6), (3.7), and (3.8) that

(3.9)

for all and . Therefore,

(3.10)

Since is arbitrary, we get (3.4). In this stage, we will show that

(3.11)

Let and . Then, there exists , which satisfies (3.5). Take

(3.12)

From and (3.4) there exists such that and , for all . By Lemma 2.7, we have

(3.13)

for all . Therefore, we have

(3.14)

for all . This shows that

(3.15)

Since is arbitrary, we get (3.11).

Let . Then is a contraction of into itself. In fact, we see that

(3.16)

and hence is a contraction due to

Therefore, by Banach contraction principal, has a unique fixed point . Then using Lemma 2.3, is the unique solution of the variational inequality

(3.17)

We show that

(3.18)

Indeed, we can choose a subsequence of such that

(3.19)

Because is bounded, we may assume that . In terms of Lemma 2.4 and (3.11), we conclude that . Therefore,

(3.20)

Finally, we prove that as . By Lemmas 2.5 and 2.7 we have

(3.21)

On the other hand

(3.22)

Since and are bounded, we can take a constant such that

(3.23)

So from the above, we reach the following:

(3.24)

Substituting (3.24) in (3.21), we obtain

(3.25)

It follows that

(3.26)

where

(3.27)

Since is bounded and , by (3.18), we get

(3.28)

Consequently, applying Lemma 2.6, to (3.26), we conclude that .

Corollary 3.2.

Let , , , and be as in Theorem 3.1. Suppose that a strongly positive bounded linear operator on with coefficient and . Let be defined by the iterative algorithm

(3.29)

Then, converges strongly, as , to , which is a unique solution of the variational inequality

(3.30)

Proof.

Because is strongly positive bounded linear operator on with coefficient , we have

(3.31)

Therefore, is -strongly accretive. On the other hand,

(3.32)

Since is strongly positive if and only if is strongly positive, we may assume, with no loss of generality, that , so that

(3.33)

This shows that is -strictly pseudo-contractive. Now apply Theorem 3.1 to conclude the result.

Corollary 3.3.

Let , , and be as in Theorem 3.1. Suppose and define a sequence by the iterative algorithm

(3.34)

Then, converges strongly, as , to a , which is a unique solution of the variational inequality

(3.35)

Proof.

It is sufficient to take and in Theorem 3.1.

4. Some Application

Corollary 4.1.

Let and be non-expansive mappings on a Hilbert space with such that . Let be a sequence in satisfying conditions and . Let , and define a sequence by the iterative algorithm:

(4.1)

Then, converges strongly, as , to which solves the variational inequality:

(4.2)

Proof.

Let for each . Then is a semigroup of non-expansive mappings on . Now, for each and , we define Then, is regular sequence of means [16]. Next, for each and , we have

(4.3)

Therefore, applying Theorem 3.1, the result follows.

Corollary 4.2.

Let be a strongly continuous semigroup of non-expansive mappings on a Hilbert space such that . Let be a sequence in satisfying conditions and . Let and . Let be a sequence defined by the iterative algorithm:

(4.4)

where is an increasing sequence in such that and . Then, converges strongly, as , to , which solves the variational inequality

(4.5)

Proof.

For , we define for each , where denotes the space of all real-valued bounded continuous functions on with supremum norm. Then, is regular sequence of means [16]. Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.

Corollary 4.3.

Let be a strongly continuous semigroup of non-expansive mappings on a Hilbert space such that . Let be a sequence in satisfying conditions and . Let and . Let be a sequence defined by the iterative algorithm

(4.6)

where is an decreasing sequence in such that . Then converges strongly, as , to , which solves the variational inequality

(4.7)

Proof.

For , we define for each . Then is regular sequence of means [16]. Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.

Corollary 4.4.

Let be a non-expansive mapping on a Hilbert space such that . Let be a sequence in satisfying conditions and and let be a strongly regular matrix. Let and . Let be a sequence defined by the iterative algorithm

(4.8)

Then, converges strongly, as , to which solves the variational inequality

(4.9)

Proof.

For each , we define

(4.10)

for each . Since is a strongly regular matrix, for each , we have , as ; see [17]. Then, it is easy to see that is regular sequence of means. Furthermore, for each , we have Now, apply Theorem 3.1 to conclude the result.