Convergence of Iterative Sequences for Common Zero Points of a Family of -Accretive Mappings in Banach Spaces

Abstract

We introduce implicit and explicit viscosity iterative algorithms for a finite family of -accretive operators. Strong convergence theorems of the iterative algorithms are established in a reflexive Banach space which has a weakly continuous duality map.

1. Introduction

Let be a real Banach space, and let denote the normalized duality mapping from into given by

(1.1)

where denotes the dual space of and denotes the generalized duality pairing. In the sequel, we denote a single-valued normalized duality mapping by .

Let be a nonempty subset of . Recall that a mapping is said to be a contraction if there exists a constant such that

(1.2)

Recall that a mapping is said to be nonexpansive if

(1.3)

A point is a fixed  point of provided . Denote by the set of fixed points of , that is, . Given a real number and a contraction , we define a mapping

(1.4)

It is obviously that is a contraction on . In fact, for , we obtain

(1.5)

Let be the unique fixed point of , that is, is the unique solution of the fixed point equation

(1.6)

A special case has been considered by Browder [1] in a Hilbert space as follows. Fix and define a contraction on by

(1.7)

We use to denote the unique fixed point of , which yields that . In 1967, Browder [1] proved the following theorem.

Theorem B.

In a Hilbert space, as , converges strongly to a fixed point of , that is, closet to , that is, the nearest point projection of onto .

In [2], Moudafi proposed a viscosity approximation method which was considered by many authors [2–8]. If is a Hilbert space, is a nonexpansive mapping and is a contraction, he proved the following theorems.

Theorem M1.

The sequence generated by the following iterative scheme:

(1.8)

converges strongly to the unique solution of the variational inequality

(1.9)

where is a sequence of positive numbers tending to zero.

Theorem M2.

With and initial defined the sequence by

(1.10)

Suppose that , and and . Then, converges strongly to the unique solution of the unique solutions of the variational inequality

(1.11)

Recall that a (possibly multivalued) operator with domain and range in is accretive if for each and   , there exists a such that

(1.12)

An accretive operator is -accretive if for each . The set of zeros of is denoted by . Hence,

(1.13)

For each , we denote by the resolvent of , that is, . Note that if is -accretive, then is nonexpansive and , for all . We also denote by the Yosida approximation of , that is, . It is known that is a nonexpansive mapping from to .

Recently, Kim and Xu [9] and Xu [10] studied the sequence generated by the following iterative algorithm:

(1.14)

where is a real sequence and . They obtained the strong convergence of the iterative algorithm in the framework of uniformly smooth Banach spaces and reflexive Banach space, respectively. Xu [10] also studied the following iterative algorithm by viscosity approximation method

(1.15)

where is a real sequence , is a contractive mapping, and is a nonexpansive mapping with a fixed point. Strong convergence theorems of fixed points are obtained in a uniformly smooth Banach space; see [10] for more details.

Very recently, Zegeye and Shahzad [11] studied the common zero problem of a family of -accretive mappings. To be more precise, they proved the following result.

Theorem ZS.

Let be a strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm, a nonempty, closed, convex subset of , and    a family of -accretive mappings with . For any , let be generated by the algorithm

(1.16)

where is a real sequence which satisfies the following conditions: ; ; or and with for for and . If every nonempty, closed, bounded convex subset of has the fixed point property for a nonexpansive mapping, then converges strongly to a common solution of the equations for .

In this paper, motivated by the recent work announced in [3, 5, 9, 11–20], we consider the following implicit and explicit iterative algorithms by the viscosity approximation method for a finite family of -accretive operators . The algorithms are as following:

(1.17)

(1.18)

where with for , and is a real sequence in . It is proved that the sequence generated in the iterative algorithms (1.17) and (1.18) converges strongly to a common zero point of a finite family of -accretive mappings in reflexive Banach spaces, respectively.

2. Preliminaries

The norm of is said to be Gâteaux differentiable (and is said to be smooth) if

(2.1)

exists for each in its unit sphere . It is said to be uniformly Fréchet differentiable (and is said to be uniformly smooth) if the limit in (2.1) is attained uniformly for .

A Banach space is said to be strictly convex if, for , , such that ,

(2.2)

with , , and for some . In a strictly convex Banach space , we have that, if

(2.3)

for , , , where , then (see [21]).

Recall that a gauge is a continuous strictly increasing function such that and   as  . Associated to a gauge is the duality map defined by

(2.4)

Following Browder [22], we say that a Banach space has a weakly continuous duality map if there exists a gauge for which the duality map is single valued and weak-to-weak* sequentially continuous (i.e., if is a sequence in weakly convergent to a point , then the sequence converges weakly* to ). It is known that has a weakly continuous duality map for all with the gauge . In the case where for all , we write the associated duality map as and call it the (normalized) duality map. Set

(2.5)

then

(2.6)

where denotes the subdifferential in the sense of convex analysis. It also follows from (2.5) that is convex and .

In order to prove our main results, we also need the following lemmas.

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 [23].

Lemma 2.1.

Assume that has a weakly continuous duality map with the gauge .

(i)For all and , there holds the inequality

(2.7)

In particular, for and ,

(2.8)

(ii)For and for nonzero ,

(2.9)

Lemma 2.2 (see [24]).

Let be a Banach space satisfying a weakly continuous duality map, let be a nonempty, closed, convex subset of , and let be a nonexpansive mapping with a fixed point. Then, is demiclosed at zero, that is, if is a sequence in which converges weakly to and if the sequence converges strongly to zero, then .

Lemma 2.3 (see [11]).

Let be a nonempty, closed, convex subset of a strictly convex Banach space . Let , , be a family of -accretive mappings such that . Let be real numbers in such that and , where . Then, is nonexpansive and .

Lemma 2.4 (see [25]).

Let be a sequence of nonnegative real numbers satisfying the condition

(2.10)

where and such that

(i) and ,

(ii)either or .

Then converges to zero.

3. Main Results

Theorem 3.1.

Let be a strictly convex and reflexive Banach space which has a weakly continuous duality map with the gauge . Lek be a nonempty, closed, convex subset of and a contractive mapping with the coefficient . Let be a family of -accretive mappings with . Let , for each . For any , let be generated by the algorithm (1.17), where with for , and is a sequence in . If , then converges strongly to a common solution of the equations for , which solves the following variational inequality:

(3.1)

Proof.

From Lemma 2.3, we see that is a nonexpansive mapping and

(3.2)

Notice that is convex. From Lemma 2.1, for any fixed , we have

(3.3)

which in turn implies that

(3.4)

Note that (3.4) actually holds for all duality maps ; in particular, if we take the normalized duality (in which case, we have ), then we get

(3.5)

that is,

(3.6)

This implies that the sequence is bounded. Now assume that is a weak limit point of and a subsequence of converges weakly to . Then, by Lemma 2.2, we see that is a fixed point of . Hence, . In (3.4), replacing with and with , respectively, and taking the limit as , we obtain from the weak continuity of the duality map that

(3.7)

Hence, we have .

Next, we show that solves the variation inequality (3.1). For , we obtain

(3.8)

which implies that

(3.9)

Replacing with in (3.9) and passing through the limit as , we conclude that

(3.10)

It follows from Lemma 2.1 that is a positive-scalar multiple of . We, therefore, obtain that is a solution to (3.1).

Finally, we prove that the full sequence actually converges strongly to . It suffices to prove that the variational inequality (3.1) can have only one solution. This is an easy consequence of the contractivity of . Indeed, assume that both and are solutions to (3.1). Then, we see that

(3.11)

Adding them yields that

(3.12)

This implies that

(3.13)

which guarantees . So, (3.1) can have at most one solution. This completes the proof.

Next, we shall consider the explicit algorithm (1.18) which is rephrased below, the initial guess is arbitrary and

(3.14)

We need the strong convergence of the implicit algorithm (1.17) to prove the strong convergence of the explicit algorithm (3.14).

Theorem 3.2.

Let be a strictly convex and reflexive Banach space which has a weakly continuous duality map with the gauge . Lek be a nonempty, closed, convex subset of and a contractive mapping. Let be a family of -accretive mappings with . Let for each . For any , let be generated by the algorithm (1.18), where with for , , and is a sequence in which satisfies the following conditions: and . Assume also that

(i),

(ii) converges strongly to , where is the sequence generated by the implicity algorithm (1.17).

Then, converges strongly to , which solves the variational inequality (3.1).

Proof.

From Lemma 2.3, we obtain that is a nonexpansive mapping and

(3.15)

We observe that is bounded. Indeed, take and notice that

(3.16)

By simple inductions, we have

(3.17)

which gives that the sequence is bounded, so are and . From (1.17), we have

(3.18)

This implies that

(3.19)

which in turn implies that

(3.20)

It follows from that

(3.21)

From the assumption and the weak continuity of imply that,

(3.22)

Letting in (3.21), we obtain that

(3.23)

Finally, we show the sequence converges stongly to . Observe that

(3.24)

It follows from Lemma 2.1 that

(3.25)

which yields that

(3.26)

where is a appropriate constant such that . In view of Lemma 2.4, we can obtain the desired conclusion easily. This completes the proof.

As an application of Theorems 3.1 and 3.2, we have the following results for a single mapping.

Corollary 3.3.

Let be a reflexive Banach space which has a weakly continuous duality map with the gauge . Lek be a nonempty, closed, convex subset of and a contractive mapping with the coefficient . Let be a -accretive mapping with . Let . For any , let be generated by the following iterative algorithm:

(3.27)

Then, converges strongly to a solution of the equations .

Corollary 3.4.

Let be a reflexive Banach space which has a weakly continuous duality map with gauge . Let be a nonempty, closed, convex subset of and a contractive mapping. Let be a -accretive mappings with . Let . For any , let be generated by the following algorithm:

(3.28)

where is a sequence in which satisfies the following conditions: and . Also assume that

(i),

(ii) converges strongly to , where is the sequence generated by the implicity scheme (3.27) and .

Then, the sequence generated by the following iterative algorithm

(3.29)

converges strongly to a solution of the equation .