Convergence Theorems of Three-Step Iterative Scheme for a Finite Family of Uniformly Quasi-Lipschitzian Mappings in Convex Metric Spaces

Abstract

We consider a new Noor-type iterative procedure with errors for approximating the common fixed point of a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved for such iterative sequences involving a finite family of uniformly quasi-Lipschitzian mappings. The results presented in this paper extend, improve and unify some main results in previous work.

1. Introduction and Preliminaries

Takahashi [1] introduced a notion of convex metric spaces and studied the fixed point theory for nonexpansive mappings in such setting. For the convex metric spaces, Kirk [2] and Goebel and Kirk [3] used the term "hyperbolic type space" when they studied the iteration processes for nonexpansive mappings in the abstract framework. For the Banach space, Petryshyn and Williamson [4] proved a sufficient and necessary condition for Picard iterative sequences and Mann iterative sequence to converge to fixed points for quasi-nonexpansive mappings. In 1997, Ghosh and Debnath [5] extended the results of [4] and gave the sufficient and necessary condition for Ishikawa iterative sequence to converge to fixed points for quasi-nonexpansive mappings. Liu [6–8] proved some sufficient and necessary conditions for Ishikawa iterative sequence and Ishikawa iterative sequence with errors to converge to fixed point for asymptotically quasi-nonexpansive mappings in Banach space and uniform convex Banach space. Tian [9] gave some sufficient and necessary conditions for an Ishikawa iteration sequence for an asymptotically quasi-nonexpansive mapping to converge to a fixed point in convex metric spaces. Very recently, Wang and Liu [10] gave some iteration sequence with errors to approximate a fixed point of two uniformly quasi-Lipschitzian mappings in convex metric spaces. The purpose of this paper is to give some sufficient and necessary conditions for a new Noor-type iterative sequence with errors to approximate a common fixed point for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. The results presented in this paper generalize, improve, and unify some main results of [1–14].

First of all, let us list some definitions and notations.

Let be a given self mapping of a nonempty convex subset of an arbitrary real normed space.The sequence defined by

(1.1)

is called the Noor iterative procedure with errors [11], where and are appropriate sequences in [] with and ,, and are bounded sequences in . If then (1.1) reduces to the Ishikawa iterative procedure with errors [15] defined as follows:

(1.2)

If then (1.2) reduces to the following Mann type iterative procedure with errors [15]:

(1.3)

Let be a metric space. A mapping is said to be asymptotically nonexpansive, if there exists a sequence [1,], such that

(1.4)

Let be the set of fixed points of in and , a mapping is said to be asymptotically quasi-nonexpansive, if there exists with such that

(1.5)

Moreover, is said to be uniformly quasi-Lipschitzian, if there exists such that

(1.6)

Remark 1.1.

If is nonempty, then it follows from the above definitions that an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, and an asymptotically quasi-nonexpansive mapping must be a uniformly quasi-Lipschitzian with . However, the inverse is not true in general.

Definition 1.2 (see [ 9]).

Let be a metric space, and let [0,1],,, be real sequences in [] with . A mapping is said to be a convex structure on if, for any and ,

(1.7)

If is a metric space with a convex structure , then is called a convex metric space. Let be a convex metric space, a nonempty subset of is said to be convex if

(1.8)

Definition 1.3.

Let be a convex metric space with a convex structure and be a finite family of uniformly quasi-Lipschitzian mappings with . Let ,,, and be nine sequences in [with

(1.9)

For a given define a sequence as follows:

(1.10)

where , is a Lipschitz continuous mapping with a Lipschitz constant and , are any given three sequences in Then is called the Noor-type iterative sequence with errors for a finite family of uniformly quasi-Lipschitzian mappings . If in (1.10), then the sequence defined by (1.10) can be written as follows:

(1.11)

If for all in (1.10), then for all and the sequence defined by (1.10) can be written as follows:

(1.12)

If and for all , then the sequence defined by (1.10) can be written as follows:

(1.13)

which is the Ishikawa type iterative sequence with errors considered in [9]. Further, if and for all , then for all and (1.10) reduces to the following Mann type iterative sequence with errors [9]:

(1.14)

In order to prove our main results, the following lemmas will be needed.

Lemma 1.4.

Let be a convex metric space, be a uniformly quasi-Lipschitzian mapping for such that . Then there exists a constant such that, for all

(1.15)

Proof.

In fact, for each , since is a uniformly quasi-Lipschitzian mapping, we have

(1.16)

where

(1.17)

This completes the Proof.

Lemma 1.5 (see [7]).

Let be three nonexpansive squences satisfying the following conditions:

(1.18)

Then

(1) exists;

(2)In addition, if , then .

Lemma 1.6.

Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of uniformly quasi-Lipschitzian mapping for such that and be a contractive mapping with a contractive constant Let be the iterative sequence with errors defined by (1.10) and , be three bounded sequences in . Let ,,, be sequences in [0,1] satisfying the following conditions:

(i);

(ii);

(iii)

Then the following conclusions hold:

(1)for all and ,

(1.19)

where, for alland

(1.20)

(2)there exists a constantsuch that

(1.21)

for all.

Proof.

• (1) It follows from (1.7),(1.10), and Lemma 1.4 that

  

  (1.22)

(1.23)

(1.24)

Substituting (1.23) into (1.22) and simplifying it, we have

(1.25)

Substituting (1.24) into (1.25) and simplifying it, we get

(1.26)

where

(1.27)

• (2) Since for all , it follows from (1.26) that, for and ,

  

  (1.28)

where

(1.29)

This completes the proof.

2. Main Results

Theorem 2.1.

Let be a complete convex metric space and be a nonempty closed convex subset of Let be a finite family of uniformly quasi-Lipschitzian mapping for such that and be a contractive mapping with a contractive constant . Let be the iterative sequence with errors defined by (1.10) and ,, be three bounded sequence in and ,,,,,,, and be nine sequences in [ 0,1] satisfying the following conditions:

(i),,

(ii),

(iii).

Then the sequence converges to a common fixed point if and only if , where

Proof.

The necessity is obvious. Now prove the sufficiency. In fact, from Lemma 1.6, we have

(2.1)

where . By conditions (i) and (ii), we know that

(2.2)

It follows from Lemma 1.5 that exists. Since , we have

(2.3)

Next prove that is a Cauchy sequence in . In fact, for any given , there exists a positive integer such that

(2.4)

From (2.4), there exist and positive integer such that

(2.5)

Thus Lemma 1.6 implies that, for any positive integers with ,

(2.6)

This shows that is a Cauchy sequence in a nonempty closed convex subset of a complete convex metric space . Without loss of generality, we can assume that Next prove that . In fact, for any given , there exists a positive integer such that for all ,

(2.7)

Again from (2.7), there exist and positive integer such that

(2.8)

Thus, for any , from (2.7) and (2.8), we have

(2.9)

By the arbitrariness of , we know that for all , that is, . This completes the Proof of Theorem 2.1.

Taking in Theorem 2.1, then we have the following theorem.

Theorem 2.2.

Let be a complete convex metric space and be a nonempty closed convex subset of Let be a finite family of uniformly quasi-Lipschitzian mapping for such that . Let be the iterative sequence with errors defined by (1.11) and ,, be three bounded sequence in , and ,,,,,,,and be nine sequence in [ 0,1] satisfying the conditions (i)–(iii) of Theorem 2.1. Then the sequence converges to a common fixed point if and only if

(2.10)

where

Taking in Theorem 2.1, then we have the following theorem.

Theorem 2.3.

Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of uniformly quasi-Lipschitzian mapping for such that and be a contractive mapping with a contractive constant . Let be the iterative sequence with errors defined by (1.12) and , be two bounded sequences in and ,,,,, be nine sequences in [ ] satisfying the conditions (ii) and (iii) of Theorem 2.1 and for all . Then the sequence converges to a common fixed point if and only if

(2.11)

where

Remark 2.4.

Theorems  2.1–2.3 generalize, improve, and unify some corresponding results in [1–14].

Similarly, we can obtain the following results.

Theorem 2.5.

Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of asymptotically quasi-nonexpansive mapping for such that and be a contractive mapping with a contractive constant . Let be the iterative sequence with errors defined by (1.10) and ,, be three bounded sequences in and ,,,,,, and be nine sequences in [] satisfying the conditions (i)–(iii) of Theorem 2.1. Then the sequence converges to a common fixed point if and only if

(2.12)

where

Proof.

From Remark 1.1, we know that each asymptotically quasi-nonexpansive mapping must be a uniformly quasi-Lipschitzian with

(2.13)

where is the sequence appeared in (1.5). Hence the conclusion of Theorem 2.5 can be obtained from Theorem 2.1 immediately. This completes the Proof.

Theorem 2.6.

Let be a complete convex metric space and be a nonempty closed convex subset of . Let be a finite family of asymptotically quasi-nonexpansive mapping for, such that . Let be the iterative sequence with errors defined by (1.11) and ,, be three bounded sequence in and ,,,,,,, and be nine sequence in [] satisfying the conditions (i)–(iii) of Theorem 2.1. Then the sequence converges to a common fixed point if and only if

(2.14)

where