Strong Convergence Theorems of Viscosity Iterative Methods for a Countable Family of Strict Pseudo-contractions in Banach Spaces

Let be a real Banach space and a nonempty closed convex subset of . A mapping is called -contraction if there exists a constant such that for all , . We use to denote the collection of all contractions on . That is, . A mapping is said to be -strictly pseudo-contractive mapping (see, e.g., [1]) if there exists a constant , such that

(1.1)

for all , . Note that the class of -strict pseudo-contractions strictly includes the class of nonexpansive mappings which are mapping on such that , for all , . That is, is nonexpansive if and only if is a 0-strict pseudo-contraction. A mapping is said to be -strictly pseudo-contractive mapping with respect to if, for all , , there exists a constant such that

(1.2)

A countable family of mapping is called a family of uniformly -strict pseudo-contractions with respect to , if there exists a constant such that

(1.3)

We denote by the set of fixed points of , that is, .

In order to find a fixed point of nonexpansive mapping , Halpern [2] was the first to introduce the following iteration scheme which was referred to as Halpern iteration in a Hilbert space: , , ,

(1.4)

He pointed out that the control conditions (C1) and (C2) are necessary for the convergence of the iteration scheme (1.4) to a fixed point of . Furthermore, the modified version of Halpern iteration was investigated widely by many mathematicians. Recently, for the sequence of nonexpansive mappings with some special conditions, Aoyama et al. [3] introduced a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings satisfying some conditions. Let and

(1.5)

for all where is a nonempty closed convex subset of a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and is a sequence in . They proved that defined by (1.5) converges strongly to a common fixed point of Very recently, Song and Zheng [4] also studied the strong convergence theorem of Halpern iteration (1.5) for a countable family of nonexpansive mappings satisfying some conditions in either a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm or a reflexive Banach space with a weakly continuous duality mapping. Other investigations of approximating common fixed points for a countable family of nonexpansive mappings can be found in [3, 5–10] and many results not cited here.

On the other hand, in the last twenty years or so, there are many papers in the literature dealing with the iteration approximating fixed points of Lipschitz strongly pseudo-contractive mappings by using the Mann and Ishikawa iteration process. Results which had been known only for Hilbert spaces and Lipschitz mappings have been extended to more general Banach spaces and a more general class of mappings (see, e.g., [1, 11–13] and the references therein).

In 2007, Marino and Xu [12] proved that the Mann iterative sequence converges weakly to a fixed point of -strict pseudo-contractions in Hilbert spaces, which extend Reich's theorem [14, Theorem ] from nonexpansive mappings to -strict pseudo-contractions in Hilbert spaces.

Recently, Zhou [13] obtained some weak and strong convergence theorems for -strict pseudo-contractions in Hilbert spaces by using Mann iteration and modified Ishikawa iteration which extend Marino and Xu's convergence theorems [12].

More recently, Hu and Wang [11] obtained that the Mann iterative sequence converges weakly to a fixed point of -strict pseudo-contractions with respect to in -uniformly convex Banach spaces. To be more precise, they obtained the following theorem.

Theorem HW

Let be a real -uniformly convex Banach space which satisfies one of the following:

(i) has a Fréchet differentiable norm;

(ii) satisfies Opial's property.

Let a nonempty closed convex subset of . Let be a -strict pseudo-contractions with respect to , and . Assume that a real sequence in satisfy the following conditions:

(1.6)

Then Mann iterative sequence defined by

(1.7)

converges weakly to a fixed point of .

Very recently, Hu [15] obtained strong convergence theorems on a mixed iteration scheme by the viscosity approximation methods for -strict pseudo-contractions in -uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. To be more precise, Hu [15] obtained the following theorem.

Theorem H.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ,

(ii) and ,

(iii) , where .

Let be the sequence generated by the following:

(1.8)

Then the sequence converges strongly to a fixed point of .

In this paper, motivated by Hu and Wang [11], Hu [15], Aoyama et al. [3] and Song and Zheng [4], we introduce a viscosity iterative approximation method for finding a common fixed point of a countable family of strictly pseudo-contractions which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in either -uniformly convex Banach space which admits a weakly continuous duality mapping or -uniformly convex Banach space with uniformly Gâteaux differentiable norm. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The results presented in this paper improve and extend the corresponding results announced by Hu and Wang [11], Hu [15], Aoyama et al. [3] Song and Zheng [4], and many others.

Throughout this paper, let be a real Banach space and its dual space. We write (resp., ) to indicate that the sequence weakly (resp., weak*) converges to ; as usual will symbolize strong convergence. Let denote the unit sphere of a Banach space . A Banach space is said to have

(i)a Gâteaux differentiable norm (we also say that is smooth), if the limit

(2.1)

exists for each , ,

(ii)a uniformly Gâteaux differentiable norm, if for each in , the limit (2.1) is uniformly attained for ,

(iii)a Fréchet differentiable norm, if for each , the limit (2.1) is attained uniformly for ,

(iv)a uniformly Fréchet differentiable norm (we also say that is uniformly smooth), if the limit (2.1) is attained uniformly for .

The modulus of convexity of is the function defined by

(2.2)

is uniformly convex if and only if, for all such that . is said to be -uniformly convex, if there exists a constant such that .

The following facts are well known which can be found in [16, 17]:

(i)the normalized duality mapping in a Banach space with a uniformly Gâteaux differentiable norm is single-valued and strong-weak* uniformly continuous on any bounded subset of ;

(ii)each uniformly convex Banach space is reflexive and strictly convex and has fixed point property for nonexpansive self-mappings;

(iii)every uniformly smooth Banach space is a reflexive Banach space with a uniformly Gâteaux differentiable norm and has fixed point property for nonexpansive self-mappings.

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1 (see [11]).

Let be a real -uniformly convex Banach space and a nonempty closed convex subset of . let be a -strict pseudo-contraction with respect to , and a real sequence in . If is defined by , , then for all , , the inequality holds

(2.3)

where is a constant in [18, Theorem ]. In addition, if , , and , then , for all , .

Lemma 2.2 (see [19, 20]).

Let be a nonempty closed convex subset of a Banach space which has uniformly Gâteaux differentiable norm, a nonexpansive mapping with and a -contraction. Assume that every nonempty closed convex bounded subset of has the fixed points property for nonexpansive mappings. Then there exists a continuous path: , satisfying , which converges to a fixed point of as .

Lemma 2.3 (see [21]).

Let and be bounded sequences in Banach space such that

(2.4)

where is a sequence in such that . Assume

(2.5)

Then .

Definition 2.4 (see [3]).

Let be a family of mappings from a subset of a Banach space into with . We say that satisfies the AKTT-condition if for each bounded subset of ,

(2.6)

Remark 2.5.

The example of the sequence of mappings satisfying AKTT-condition is supported by Lemma 4.1.

Lemma 2.6 (see [3, Lemma ]).

Suppose that satisfies AKTT-condition. Then, for each , converses strongly to a point in . Moreover, let the mapping be defined by

(2.7)

Then for each bounded subset of ,

Lemma 2.7 (see [22]).

Assume that is a sequence of nonnegative real numbers such that

(2.8)

where is a sequence in and is a sequence such that

(a)

(b) or

Then

By a gauge function we mean a continuous strictly increasing function such that and as . Let be the dual space of . The duality mapping associated to a gauge function is defined by

(2.9)

In particular, the duality mapping with the gauge function , denoted by , is referred to as the normalized duality mapping. Clearly, there holds the relation for all (see [23]). Browder [23] initiated the study of certain classes of nonlinear operators by means of the duality mapping . Following Browder [23], we say that a Banach space has a if there exists a gauge for which the duality mapping is single-valued and continuous from the weak topology to the weak* topology, that is, for any with , the sequence converges weakly* to . It is known that has a weakly continuous duality mapping with a gauge function for all . Set

(2.10)

then

(2.11)

where denotes the subdifferential in the sense of convex analysis (recall that the subdifferential of the convex function at is the set .

The following lemma is an immediate consequence of the subdifferential inequality. 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 [24].

Lemma 2.8 (see [24]).

Assume that a Banach space has a weakly continuous duality mapping with gauge .

(i)For all , , the following inequality holds:

(2.12)

In particular, in a smooth Banach space , for all , ,

(2.13)

(ii)Assume that a sequence in converges weakly to a point . Then the following identity holds:

(2.14)

For a nonexpansive mapping, and , defines a contraction mapping. Thus, by the Banach contraction mapping principle, there exists a unique fixed point satisfying

(3.1)

For simplicity we will write for provided no confusion occurs. Next, we will prove the following lemma.

Lemma 3.1.

Let be a reflexive Banach space which admits a weakly continuous duality mapping with gauge . Let be a nonempty closed convex subset of , a nonexpansive mapping with and . Then the net defined by (3.1) converges strongly as to a fixed point of which solves the variational inequality:

(3.2)

Proof.

We first show that the uniqueness of a solution of the variational inequality (3.2). Suppose both and are solutions to (3.2), then

(3.3)

Adding (3.3), we obtain

(3.4)

Noticing that for any , ,

(3.5)

From (3.4), we conclude that . This implies that and the uniqueness is proved. Below we use to denote the unique solution of (3.2). Next, we will prove that is bounded. Take a ; then we have

(3.6)

It follows that

(3.7)

Hence is bounded, so are and . The definition of implies that

(3.8)

If follows from reflexivity of and the boundedness of sequence that there exists which is a subsequence of converging weakly to as . Since is weakly sequentially continuous, we have by Lemma 2.8 that

(3.9)

Let

(3.10)

It follows that

(3.11)

Since

(3.12)

we obtain

(3.13)

On the other hand, however,

(3.14)

It follows from (3.13) and (3.14) that

(3.15)

This implies that . Next we show that as . In fact, since and is a gauge function, then for , and

(3.16)

Following Lemma 2.8, we have

(3.17)

This implies that

(3.18)

Now observing that implies , we conclude from the last inequality that

(3.19)

Hence as . Next we prove that solves the variational inequality (3.2). For any , we observe that

(3.20)

Since

(3.21)

we can derive that

(3.22)

Thus

(3.23)

Noticing that

(3.24)

Now replacing in (3.23) with and letting , we have

(3.25)

So, is a solution of the variational inequality (3.2), and hence by the uniqueness. In a summary, we have shown that each cluster point of (at ) equals . Therefore, as . This completes the proof.

Theorem 3.2.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let be a family of uniformly -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii) , where .

Let be the sequence generated by the following:

(3.26)

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all and suppose that . Then the sequence converges strongly to which solves the variational inequality:

(3.27)

Proof.

Rewrite the iterative sequence (3.26) as follows:

(3.28)

where , and , is the identity mapping. By Lemma 2.1, is nonexpansive such that for all . Taking any , from (3.28), it implies that

(3.29)

Therefore, the sequence is bounded, and so are the sequences , . Since and , we know that is bounded. We note that for any bounded subset of ,

(3.30)

From and satisfing AKTT-condition, we obtain that

(3.31)

that is, the sequence satisfies AKTT-condition. Applying Lemma 2.6, we can take the mapping defined by

(3.32)

Moreover, we have is nonexpansive and

(3.33)

It is easy to see that . Hence The iterative sequence (3.28) can be expressed as follows:

(3.34)

where

(3.35)

We estimate from (3.35)

(3.36)

Hence

(3.37)

Since , and , we have from (3.37) that

(3.38)

Hence, by Lemma 2.3, we obtain

(3.39)

From (3.35), we get

(3.40)

and so it follows from (3.39) and (3.40) that

(3.41)

It follows from Lemma 2.6 and (3.41), we have

(3.42)

Since is a nonexpansive mapping, we have from Lemma 3.1 that the net generated by

(3.43)

converges strongly to , as . Next, we prove that

(3.44)

Let be a subsequence of such that

(3.45)

If follows from reflexivity of and the boundedness of sequence that there exists which is a subsequence of converging weakly to as . Since is weakly continuous, we have by Lemma 2.8 that

(3.46)

Let

(3.47)

It follows that

(3.48)

From (3.42), we obtain

(3.49)

On the other hand, however,

(3.50)

It follows from (3.49) and (3.50) that

(3.51)

This implies that , that is, . Since the duality map is single-valued and weakly continuous, we get that

(3.52)

as required. Finally, we show that as .

(3.53)

It follows that from condition (i) and (3.44) that

(3.54)

Apply Lemma 2.7 to (3.53) to conclude as ; that is, as . This completes the proof.

If is a family of nonexpansive mappings, then we obtain the following results.

Corollary 3.3.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let be a family of nonexpansive mappings such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii) .

Let be the sequence generated by the following:

(3.55)

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all and suppose that . Then the sequence converges strongly which solves the variational inequality:

(3.56)

Corollary 3.4.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let be a -strict pseudo-contraction with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii) , where .

Let be the sequence generated by the following

(3.57)

Then the sequence converges strongly to which solves the following variational inequality:

(3.58)

Theorem 3.5.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a family of uniformly -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii) , where .

Let be the sequence generated by the following:

(3.59)

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all and suppose that . Then the sequence converges strongly to a common fixed point of .

Proof.

It follows from the same argumentation as Theorem 3.2 that is bounded and , where is a nonexpansive mapping defined by (3.32). From Lemma 2.2 that the net generated by converges strongly to , as . Obviously,

(3.60)

In view of Lemma 2.8, we calculate

(3.61)

and therefore

(3.62)

Since , and are bounded and , we obtain

(3.63)

where . We also know that

(3.64)

From the fact that , as , is bounded and the duality mapping is norm-to-weak uniformly continuous on bounded subset of , it follows that as ,

(3.65)

Combining (3.63), (3.64) and two results mentioned above, we get

(3.66)

From (3.28) and Lemma 2.8, we get

(3.67)

Hence applying in Lemma 2.7 to (3.67), we conclude that .

Corollary 3.6.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a family of nonexpansive mappings such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii)

Let be the sequence generated by the following:

(3.68)

Suppose that satisfies the AKTT-condition. Let be a mapping of into itself defined by for all and suppose that . Then the sequence converges strongly to a common fixed point of .

Corollary 3.7.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of which has the fixed point property for nonexpansive mappings. Let be a -strict pseudo-contractions with respect to , and . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii) , where .

Let be the sequence generated by the following:

(3.69)

Then the sequence converges strongly to a common fixed point of .

We consider the problem of finding a zero of an accretive operator. An operator is said to be accretive if for each and , there exists such that An accretive operator is said to satisfy the range condition if for all where is the domain of is the identity mapping on , is the range of and is the closure of . If is an accretive operator which satisfies the range condition, then we can define, for each a mapping by which is called the resolvent of . We know that is nonexpansive and for all We also know the following [25]: For each , and , it holds that

(4.1)

By the proof of Theorem in [3], we have the following lemma.

Lemma 4.1.

Let be a Banach space and a nonempty closed convex subset of Let be an accretive operator such that and Suppose that is a sequence of such that and Then

(i)The sequence satisfies the AKTT-condition.

(ii) for all and where as

By Corollary 3.3, we obtain the following result.

Theorem 4.2.

Let be a real -uniformly convex Banach space with a weakly continuous duality mapping , and a nonempty closed convex subset of . Let is an -accretive operator in such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii) ;

(iv) is a sequence of such that and

Let be the sequence generated by the following:

(4.2)

Then the sequence converges strongly which solves the following variational inequality:

(4.3)

By Corollary 3.6, we obtain the following result.

Theorem 4.3.

Let be a real -uniformly convex Banach space with uniformly Gâteaux differentiable norm, and a nonempty closed convex subset of . Let is an -accretive operator in such that . Let be a -contraction with . Assume that real sequences , and in satisfy the following conditions:

(i) for all ;

(ii) and ;

(iii) ;

(iv) is a sequence of such that and

Let be the sequence generated by the following:

(4.4)

Then the sequence converges strongly in .