1. Introduction

Let be a real Banach space with the dual space and be a normalized duality mapping defined by

(1.1)

where denotes the generalized duality pairing. It is well known that (see, e.g., [1, pages 107–113])

(i) is single-valued if is strictly convex;

(ii) is uniformly smooth if and only if is single-valued and uniformly continuous on any bounded subset of .

Let be a nonempty closed convex subset of . A mapping is said to be

  1. (i)

    nonexpansive if

    (1.2)

(ii)-Lipschitzian if there exists a constant such that

(1.3)

(iii)-strongly pseudocontractive if there exist a constant and such that

(1.4)

(iv)pseudocontractive if there exists such that

(1.5)

It is easy to see that the pseudocontractive mapping is more general than the nonexpansive mapping.

A pseudocontractive semigroup is a family,

(1.6)

of self-mappings on such that

(1) for all ;

(2) for all and ;

(3) is pseudocontractive for each ;

(4)for each , the mapping from into is continuous.

If the mapping in condition (3) is replaced by

is nonexpansive for each ;

then is said to be a nonexpansive semigroup on .

We denote by the common fixed points set of pseudocontractive semigroup , that is,

(1.7)

In the sequel, we always assume that .

In recent decades, many authors studied the convergence of iterative algorithms for nonexpansive mappings, nonexpansive semigroup, and pseudocontractive mapping in Banach spaces (see, e.g., [2–14]). Let be a nonexpansive semigroup from into itself and be a contractive mapping. It follows from Banach's fixed theorem that the following implicit viscosity iteration process is well defined:

(1.8)

where and . Some authors studied the convergence of iteration process (1.8) for nonexpansive mappings in certain Banach spaces (see [5, 10]). Recently, Xu [11] studied the following implicit iteration process: for any ,

(1.9)

where , , and obtained the convergence theorem as follows.

Theorem 1 (see [11]).

Let be a uniformly convex Banach space having a weakly continuous duality map with gauge , a nonempty closed convex subset of and

(1.10)

a nonexpansive semigroup on such that . If

(1.11)

then generated by (1.9) converges strongly to a member of .

Xu [11] also proposed the following problem.

Problem 1 (see [11]).

We do not know if Theorem X holds in a uniformly convex and uniformly smooth Banach (e.g., for).

This problem has been solved by Li and Huang [15] and Suzuki [8], respectively.

Moudafi's viscosity approximation method has been recently studied by many authors (see, e.g., [2, 3, 5, 10, 13, 15–17] and the references therein). Chen and He [3] studied the convergence of (1.8) constructed from a nonexpansive semigroup and a contractive mapping in a reflective Banach space with a weakly sequentially continuous duality mapping. Zegeye et al. [13] studied the convergence of (1.8) constructed from a pseudocontractive mapping and a contractive mapping.

On the other hand, many authors (see [2, 3, 5, 13]) studied the following explicit viscosity iteration process: for any given ,

(1.12)

where , and . Chen and He [3] studied the convergence of (1.12) constructed from a nonexpansive semigroup and obtained some convergence results.

An interesting work is to extend some results involving nonexpansive mapping, nonexpansive semigroup, and pseudocontractive mapping to the semigroup of pseudocontractive mappings. Li and Huang [15] generalized some corresponding results to pseudocontractive semigroup in Banach spaces. Some further study concerned with approximating common fixed points of the semigroup of pseudocontractive mappings in Banach spaces, we refer to Li and Huang [16].

Motivated by the works mentioned above, in this paper, we study the convergence of implicit viscosity iteration process (1.8) constructed from the pseudocontractive semigroup and -strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differential norms. As special cases, we obtain the convergence of the implicit iteration process for approximating the common fixed point of the nonexpansive semigroup in certain Banach spaces. We also study the convergence of the explicit viscosity iteration process (1.12) constructed from the pseudocontractive semigroup and -strongly pseudocontractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differential norms. The results presented in this paper extend and generalize some results concerned with the nonexpansive semigroup in [3] and the pseudocontractive mapping in [13] to the pseudocontractive semigroup in Banach spaces under different conditions.

2. Preliminaries

A real Banach space is said to have a weakly continuous duality mapping if is single-valued and weak-to- sequentially continuous (i.e., if each is a sequence in weakly convergent to , then converges to ). Obviously, if has a weakly continuous duality mapping, then is norm-to- sequentially continuous. It is well known that posses duality mapping which is weakly continuous (see, e.g., [11]).

Let be the Banach space of all bounded real-valued sequences. A Banach limit (see [1]) is a linear continuous functional on such that

(2.1)

for each . If is a Banach limit, then it follows from [1, Theorem 1.4.4] that

(2.2)

for each .

A mapping with domain and range in is said to be demiclosed at a point if whenever is a sequence in which converges weakly to and converges strongly to , then .

For the sake of convenience, we restate the following lemmas that will be used.

Lemma 2.1 (see [18]).

Let be a Banach space, be a nonempty closed convex subset of , and be a strongly pseudocontractive and continuous mapping. Then has a unique fixed point in .

Lemma 2.2 (see [19]).

Let be a Banach space and be the normalized duality mapping. Then for any and ,

(2.3)

Lemma 2.3 (see [12]).

Let . Then a real Banach space is uniformly convex if and only if there exists a continuous and strictly increasing convex function with such that

(2.4)

for all , , where .

Lemma 2.4 (see [9]).

Let be a sequence of nonnegative real numbers such that

(2.5)

where , , is fixed, , and . Then .

3. Main Results

We first discuss the convergence of implicit viscosity iteration process (1.8) constructed from a pseudocontractive semigroup .

Theorem 3.1.

Let be a nonempty closed convex subset of a real Banach space . Let be an -Lipschitzian semigroup of pseudocontractive mappings and be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that for any bounded subset ,

(3.1)

Then the sequence generated by (1.8) is well defined. Moreover, if

(3.2)

then for any .

Proof.

Let

(3.3)

Since

(3.4)

we know that is strongly pseudocontractive and strongly continuous. It follows from Lemma 2.1 that has a unique fixed point (say) , that is, generated by (1.8) is well defined.

Taking , we have

(3.5)

and so . This means is bounded. By the Lipschitzian conditions of and , it follows that and are bounded. Therefore,

(3.6)

For any given ,

(3.7)

where is the integral part of . Since and is continuous for any , it follows from (3.1) that

(3.8)

This completes the proof.

Theorem 3.2.

Let be a uniformly convex Banach space with the uniformly Gâteaux differential norm and be a nonempty closed convex subset of . Let be an -Lipschitzian semigroup of pseudocontractive mappings satisfying (3.1) and let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that is a sequence generated by (1.8) and

(1);

(2), where with for all .

Then converges strongly to a common fixed point of that is the unique solution in to the following variational inequality:

(3.9)

Proof.

From Theorem 3.1, we know that is bounded and . It is easy to see that is a nonempty bounded closed convex subset of (see, e.g., [10]).

Now, we show that there exists a common fixed point of in . For any and , it follows from that

(3.10)

and so

(3.11)

Next, we prove that is a singleton. In fact, since is uniformly convex, by Lemma 2.3 that there exists a continuous and strictly increasing convex function with such that, for any and ,

(3.12)

Taking Banach limit on the above inequality, it follows that

(3.13)

This implies and so is a singleton. Therefore, (3.11) implies that there exists such that .

For any , from (1.8), we have

(3.14)

Since , it follows from (3.14) that

(3.15)

Furthermore, for any , by Lemma 2.2, we have

(3.16)

For any , since has a uniformly Gâteaux differential norm, we know that is norm-to- uniformly continuous on any bounded subset of (see, e.g., [1, pages 107–113]) and so there exists sufficient small such that

(3.17)

This implies that

(3.18)

By the arbitrariness of , it follows that

(3.19)

Adding inequalities (3.15) and (3.19), we have

(3.20)

This implies that there exists subsequence which converges strongly to . From the proof of (3.20), we know that for any subsequence and so there exists subsequence of which converges strongly to . If there exists another subsequence which converges strongly to , then it follows from Theorem 3.1 that . From (3.14), we have

(3.21)

Thus

(3.22)

This implies that and so . Therefore, converges strongly to . From (3.14) and the deduction above, we know that is also the unique solution to the variational inequlity

(3.23)

This completes the proof.

Remark 3.3.

  1. (1)

    Theorem 3.2 extends and generalizes Theorem 3.1 of [3] from nonexpansive semigroup to Lipschitzian pseudocontractive semigroup in Banach spaces with different conditions; (2) If is a pseudocontractive mapping, then condition (3.1) is trivial.

If is a nonexpansive semigroup, then is an -Lipschitzian semigroup of pseudocontractive mappings, condition of Theorem 3.2 holds trivially. From Theorem 3.2, we have the following result.

Corollary 3.4.

Let be a uniformly convex Banach space with the uniformly Gâteaux differential norm and be a nonempty closed convex subset of . Let be a nonexpansive semigroup satisfying (3.1) and let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that is a sequence generated by (1.8). If

(3.24)

then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Theorem 3.5.

Let be a uniformly smooth Banach space and be a nonempty closed convex subset of . Let be a nonexpansive semigroup satisfying (3.1) and let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that is a sequence generated by (1.8). If

(3.25)

then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Proof.

For the nonexpansive semigroup , condition of Theorem 3.2 is trivial and so formula (3.11) holds. Since uniformly smooth Banach space has the fixed point property for nonexpansive mapping (see, e.g., [10]), has a fixed point . The rest proof is similar to the proof of Theorem 3.2 and so we omit it. This completes the proof.

Theorem 3.6.

Let be a real Hilbert space and be a nonempty closed convex subset of . Let be an -Lipschitzian semigroup of pseudocontractive mappings satisfying (3.1) and let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that is a sequence generated by (1.8). If

(3.26)

then converges strongly to a common fixed point of that is the unique solution in to the following variational inequality:

(3.27)

Proof.

From the proof of Theorem 3.1, we know that is bounded and so there exists subsequence which converges weakly to some point . By Theorem 3.1, we have

(3.28)

It follows from [20, Theorem 3.18b] that is demiclosed at zero for each , where is an identity mapping. This implies that .

In addition, from (1.8), we have

(3.29)

and so

(3.30)

This implies that converges strongly to . Similar to the proof of Theorem 3.2, it is easy to show that converges strongly to that is also the unique solution to VI (3.27). This completes the proof.

Now we turn to discuss the convergence of explicit viscosity iteration process (1.12) for approximating the common fixed point of the pseudocontractive semigroup .

Theorem 3.7.

Let be a nonempty closed convex subset of a real Banach space . Let be an -Lipschitzian semigroup of pseudocontractive mappings with such that (3.1) holds. Let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and the following conditions hold:

(i), , for all ;

(ii), ,   ;

  1. (iii)

    there exists some constant such that

    (3.31)

(iv)The following equation holds:

(3.32)

Then for any .

Proof.

Let denote the sequence defined as in (1.8) with . By virtue of condition (ii) and Theorem 3.1, we know that is well defined and for any . From (1.8), we have

(3.33)
(3.34)

To obtain the assertion of Theorem 3.7, we first give a serial of estimations: using (3.33), we get

(3.35)

which implies that

(3.36)

where . From the proof of Theorem 3.1, we know that is bounded. Therefore, there exists a constant such that

(3.37)

By using (1.12) and (3.33), we have

(3.38)

It follows from (1.12) and (3.34) that

(3.39)

By virtue of (1.12), (3.34), and Lemma 2.2, we have

(3.40)

Since , then by condition (iii). Thus for sufficient large , we know

(3.41)

Consequently, by condition (iv) we can have

(3.42)

Squaring on both sides of (3.42) and using (3.37), we get

(3.43)

Setting and , then it follows from conditions (i)–(iv) that

(3.44)

By Lemma 2.4, we know that , which implies that

(3.45)

Consequently, since by (3.37), we have

(3.46)

Now we prove that for any . Since

(3.47)

by Theorem 3.1 and (3.46) we know that for any ,

(3.48)

This completes the proof.

Remark 3.8.

An example for the conditions (i)–(iii) of Theorem 3.7 is given by

(3.49)

for all , where is an any given positive real number. It is easy to see that the conditions with regard to and in Theorem 3.7 hold. If the mapping is Lipschitz continuous for any , then condition (iv) in Theorem 3.7 also holds.

Theorem 3.9.

Let be a uniformly convex Banach space with the uniformly Gâteaux differential norm and be a nonempty closed convex subset of . Let be an -Lipschitzian semigroup of pseudocontractive mappings with such that (3.1) holds. Let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Assume further that condition (2) of Theorem 3.2 holds, where is generated by (1.8) with . Then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Proof.

By Theorem3.2, we know that converges strongly to a fixed point of that is the unique solution in to VI (3.9), where is generated by (1.8) with . It follows from (3.46) that . This completes the proof.

Remark 3.10.

  1. (1)

    Theorem 3.9 extends Theorem 4.1 of [13] from Lipschitzian pseudocontractive mapping to Lipschitzian pseudocontractive semigroup in Banach spaces under different conditions; (2) Theorem 3.9 also extends Theorem 3.2 of [3] from nonexpansive semigroup to Lipschitzian pseudocontractive semigroup in Banach spaces under different conditions.

If is a nonexpansive semigroup, then is an -Lipschitzian semigroup of pseudocontractive mappings, condition of Theorem 3.2 holds trivially. Therefore, Theorem 3.9 gives the following result.

Corollary 3.11.

Let be a uniformly convex Banach space with the uniformly Gâteaux differential norm and be a nonempty closed convex subset of . Let be a nonexpansive semigroup satisfying (3.1) and be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Then converges strongly to a common fixed point of that is the unique solution in F(T) to VI (3.9).

Theorem 3.12.

Let be a uniformly smooth Banach space and be a nonempty closed convex subset of . Let be a nonexpansive semigroup satisfying (3.1) and let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Then converges strongly to a common fixed point of that is the unique solution in to VI (3.9).

Proof.

Let denote the sequence defined as in (1.8) with . By Theorem 3.5, we know that converges strongly to a fixed point of that is the unique solution in to VI (3.9). It follows from (3.46) that . This completes the proof.

Theorem 3.13.

Let be a real Hilbert space and be a nonempty closed convex subset of . Let be an -Lipschitzian semigroup of pseudocontractive mappings with such that (3.1) holds. Let be an -Lipschitzian -strongly pseudocontractive mapping. Suppose that the sequence is generated by (1.12) and conditions (i)–(iv) of Theorem 3.7 hold. Then converges strongly to a common fixed point of that is the unique solution in to VI (3.27).

Proof.

Let denote the sequence defined as in (1.8) with . By Theorem 3.6, we know that converges strongly to a fixed point of that is the unique solution in to VI (3.27). It follows from (3.46) that . This completes the proof.