1. Introduction and Preliminaries

Throughout this paper, we assume that is a real Banach space, is the dual space of , is a nonempty closed convex subset of , and is the normalized duality mapping defined by

(1.1)

where denotes the duality pairing between and . The single-valued normalized duality mapping is denoted by .

Definition 1.1.

Let be a mapping. Therefore, the following are given.

() is said to be uniformly -Lipschitzian if there exists such that, for any ,

(1.2)

() is said to be asymptotically nonexpansive if there exists a sequence with such that, for any given ,

(1.3)

() is said to be asymptotically pseudocontractive if there exists a sequence with such that, for any , there exists as follows:

(1.4)

Remark 1.2.

() It is easy to see that if is an asymptotically nonexpansive mapping, then is a uniformly -Lipschitzian mapping, where . And every asymptotically nonexpansive mapping is asymptotically pseudocontractive, but the inverse is not true, in general.

() The concept of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [7], while the concept of asymptotically pseudocontractive mappings was introduced by Schu [4] who proved the following theorem.

Theorem 1.3 (see Schu [4]).

Let be a Hilbert space, be a nonempty bounded closed convex subset of , and let be a completely continuous, uniformly -Lipschitzian and asymptotically pseudocontractive mapping with a sequence satisfying the following conditions:

(i) as ,

(ii), where .

Suppose further that and are two sequences in such that for all , where and are some positive number. For any , let be the iterative sequence defined by

(1.5)

Then converges strongly to a fixed point of in .

In [1], the first author extended Theorem to a real uniformly smooth Banach space and proved the following theorem.

Theorem 1.4 (see Chang [1]).

Let be a uniformly smooth Banach space, be a nonempty bounded closed convex subset of , and be an asymptotically pseudocontractive mapping with a sequence with , and let , where is the set of fixed points of in . Let be a sequence in satisfying the following conditions:

(i),

(ii).

For any , let be the iterative sequence defined by

(1.6)

If there exists a strict increasing function with such that

(1.7)

where is some fixed point of in , then as .

Very recently, in [3] Ofoedu proved the following theorem.

Theorem 1.5 (see Ofoedu [3]).

Let be a real Banach space, let be a nonempty closed convex subset of , and let be a uniformly -Lipschitzian asymptotically pseudocontractive mapping with a sequence , such that , where is the set of fixed points of in . Let be a sequence in satisfying the following conditions:

(i),

(ii),

(iii).

For any , let be the iterative sequence defined by

(1.8)

If there exists a strict increasing function with such that

(1.9)

then converges strongly to .

Remark 1.6.

It should be pointed out that although Theorem 1.5 extends Theorem 1.4 from a real uniformly smooth Banach space to an arbitrary real Banach space, it removes the boundedness condition imposed on .

In [8], Xu and Ori introduced the following implicit iteration process for a finite family of nonexpansive mappings (here ), with as a real sequence in (0, 1), and an initial point :

(1.10)

where (here the mod  function takes values in ). Xu and Ori proved the weak convergence of this process to a common fixed point of the finite family defined in a Hilbert space.

Chidume and Shahzad [9] and Zhou and Chang [10] studied the weak and strong convergences of this implicit process to a common fixed point for a finite family of nonexpansive mappings, respectively.

Recently, Feng Gu [11] introduced a composite implicit iteration process with errors for a finite family of strictly pseudocontractive mappings as follows:

(1.11)

where , , , , , are four real sequences in satisfying and for all , and are two bounded sequences in , and is a given point in . Feng Gu proved the strong convergence of this process to a common fixed point for a finite family of strictly pseudocontractive mappings in a real Banach space.

Inspired and motivated by the abovesaid facts, we introduced a two-step implicit iteration process with errors for a finite family of -Lipschitzian mappings as follows:

(1.12)

where , , , and , are four real sequences in satisfying and for all , and are two bounded sequences in , and is a given point in .

Observe that if is a nonempty closed convex subset of and be uniformly -Lipschitzian mappings. If , where , then for given , and , the mapping defined by

(1.13)

is a contractive mapping. In fact, the following are observed

(1.14)

Since for all , hence is a contractive mapping. By Banach contractive mapping principle, there exists a unique fixed point such that

(1.15)

Therefore, if , then the iterative sequence (1.12) can be employed for the approximation of common fixed points for a finite family of uniformly -Lipschitzian mappings.

Especially, if and are two sequences in satisfying for all , is a bounded sequence in , and is a given point in , then the sequence defined by

(1.16)

is called the one-step implicit iterative sequence with errors for a finite family of operators .

The purpose of this paper is, by using a simple and quite different method, to study the convergence of implicit iterative sequence defined by (1.12) and (1.16) to a common fixed point for a finite family of -Lipschitzian mappings instead of the assumption that is a uniformly -Lipschitzian and asymptotically pseudocontractive mapping in a Banach space. Our results extend and improve some recent results in [1–6]. Even in the case of , for all or are also new.

For the main results, the following lemmas are given.

Lemma 1.7 (see Petryshyn [12]).

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

(1.17)

Lemma 1.8 (see Moore and Nnoli [13]).

Let be a sequence of nonnegative real numbers and be a real sequence satisfying the following conditions:

(1.18)

If there exists a strictly increasing function such that

(1.19)

where is some nonnegative integer and is a sequence of nonnegative number such that , then as .

Lemma 1.9.

Let and be two nonnegative real sequences satisfying the following condition:

(1.20)

where is a sequence in (0, 1) with . If , then exists.

2. Main Results

In this section, we shall prove our main theorems in this paper.

Theorem 2.1.

Let be a real Banach space, be a nonempty closed convex subset of , be uniformly -Lipschitzian mappings with , where is the set of fixed points of in , and let be a point in . Let be a sequence with . Let , , , and be four sequences in satisfying the following conditions: , , for all . Let and be two bounded sequences in , and let be the iterative sequence with errors defined by (1.12), then the following conditions are satisfied:

(i),

(ii),

(iii),

(iv),

(v),

(vi),

(vii), for all , where .

If there exists a strict increasing function with such that

(2.1)

for all and , then converges strongly to .

Proof.

The proof is divided into two steps.

  1. (i)

    First, we prove that the sequence defined by (1.12) is bounded.

In fact, it follows from (1.12) and Lemma 1.7 that

(2.2)

where

(2.3)

Note that

(2.4)

where . By the conditions (iii) and (v), the following are given:

(2.5)

Substituting (2.4) into (2.2), we have

(2.6)

and hence

(2.7)

where

(2.8)

Since , , , and as , there exists a positive integer such that for all . Therefore, it follows from (2.7) that

(2.9)

and so

(2.10)

By the conditions (ii), (iv)(vi), and (2.5), the following are considered:

(2.11)

It follows from Lemma 1.9 that the limit exists. Therefore, the sequence is bounded. Without loss of generality, we can assume that , where is a positive constant.

  1. (ii)

    Now, we consider (2.9) and prove that .

Taking , , and

(2.12)

then (2.9) can be written as

(2.13)

By the conditions (i)(vi), we know that all the conditions in Lemma 1.8 are satisfied. Therefore, it follows that

(2.14)

that is, as . This completes the proof of Theorem 2.1.

Remark 2.2.

() Theorem 2.1 extends and improves the corresponding results in Chang [1], Cho et al. [2], Ofoedu [3], Schu [4], and Zeng [5, 6].

() The method given by the proof of Theorem 2.1 is quite different from the method given in Ofoedu [3].

() Theorem 2.1 extends and improves Theorem of Ofoedu [3]; it abolishes the assumption that is an asymptotically pseudocontractive mapping.

The following theorem can be obtained from Theorem 2.1 immediately.

Theorem 2.3.

Let be a real Banach space, let be a nonempty closed convex subset of , let be uniformly -Lipschitzian mappings with , where is the set of fixed points of in , and let be a point in . Let be a sequence with . Let and be two sequences in satisfying the following conditions: , for all . Let be a bounded sequence in , and let be the iterative sequence with errors defined by (1.16), then the following conditions are satisfied:

(i),

(ii),

(iii),

(iv),

(v),for all , where .

If there exists a strict increasing function with such that

(2.15)

for all and , then converges strongly to .

Proof.

Taking in Theorem 2.1, then the conclusion of Theorem 2.3 can be obtained from Theorem 2.1 immediately. This completes the proof of Theorem 2.3.

Theorem 2.4.

Let be a real Banach space, let be a nonempty closed convex subset of , let be a uniformly -Lipschitzian mappings with , where is the set of fixed points of in , and let be a point in . Let be a sequence with . Let and be two sequences in satisfying the following condition: , , and let be a bounded sequence in satisfying the following conditions:

(i),

(ii),

(iii),

(iv).

(v), .

For any , let be the iterative sequence defined by

(2.16)

If there exists a strict increasing function with such that

(2.17)

for all and , then converges strongly to .

Proof.

Taking in Theorem 2.3, then the conclusion of Theorem 2.4 can be obtained from Theorem 2.3 immediately. This completes the proof of Theorem 2.4.

Remark 2.5.

In Theorem 2.4 without the assumption that is an asymptotically pseudocontractive mapping, Theorem 2.4 extends and improves Theorem of Ofoedu [3].