On Equivalence of Some Iterations Convergence for Quasi-Contraction Maps in Convex Metric Spaces

We show the equivalence of the convergence of Picard and Krasnoselskij, Mann, and Ishikawa iterations for the quasi-contraction mappings in convex metric spaces.

Let be a complete metric space and . Denote . A continuous mapping W is said to be a convex structure on [1] if for all with such that

If satisfies the conditions of convex structure, then is called convex metric space that is denoted as .

In the following part, we will consider a few iteration sequences in convex metric space . Suppose that is a self-map of .

Picard iteration is as follows:

Krasnoselskij iteration is as follows:

where .

Mann iteration is as follows:

where .

Ishikawa iteration is as follows:

where for all .

A mapping is called contractive if there exists such that

for all .

The map is called Kannan mapping [2] if there exists such that

for all .

A similar definition of mapping is due to the work Chatterjea [3] (that is called Chatterjea mapping), if there exists such that

for all .

Combining above three definitions, Zamfirescu [4] showed the following result.

Theorem 1.1.

Let be a complete metric space and a mapping for which there exist the real numbers and satisfying such that, for any pair , at least one of the following conditions holds:

(z1)  

(z2)  

(z3)  

Then has a unique fixed point, and the Picard iteration converges to fixed point. This class mapping is called Zamfirescu mapping.

In 1974, irić [5] introduced one of the most general contraction mappings and obtained that the unique fixed point can be approximated by Picard iteration. This mapping is called quasi-contractive if there exists such that

for any .

Clearly, every quasi-contraction mapping is the most general of above mappings.

Later on, in 1992, Xu [6] proved that Ishikawa iteration can also be used to approximate the fixed points of quasi-contraction mappings in real Banach spaces.

Theorem 1.2.

Let be any nonempty closed convex subset of a Banach space and a quasi-contraction mapping. Suppose that for all and . Then the Ishikawa iteration sequence defined by (1)–(3) converges strongly to the unique fixed point of .

In this paper, we will show the equivalence of the convergence of Picard and Krasnoselskij, Mann, and Ishikawa iterations for the quasi-contraction mappings in convex metric spaces.

Lemma 1.3.

Let be a nonnegative sequence which satisfies the following inequality

where , and as . Then as (see [7]).

Theorem 2.1.

Let be a convex metric space, a quasi-contraction mapping with . Suppose that are defined by the iterative processes (1.3) and (1.4), respectively. Then, the following two assertions are equivalent:

(i)Picard iteration (1.3) converges strongly to the unique fixed point ;

(ii)Krasnoselskij iteration (1.4) converges strongly to the unique fixed point .

Proof.

First, we show , that is, as as .

From (1.3), (1.4), and (1.1), we can get

Next, we consider . Using (1.10) with , to obtain

Set

Then is bounded. Without loss of generality, we let for each . Indeed, we will show this conclusion from the some following cases.

Case 1 .

Let for some . Then, from (1.10) and the above , we have

and it leads to a contradiction. Thus, . Similarity to or is also impossible.

Case 2 .

Let for some .

(i)If , then .

(ii)If , then, from (1.4) and (1.1)

that is, . By induction on , we can obtain .

(iii) If , from (1.4) and (1.1)

it implies that . By induction on , we can get .

Case 3 .

Let for some . Without loss of generality, we set . Then, from (1.4), (1.1)

it implies that , and by induction on , we may get , which is a contradiction.

Case 4 .

Let for some .

(i)If , then .

(ii)If , from (1.4), (1.1), then

it implies that and by induction on , then .

Case 5 .

Let for some .

(i)If , then .

(ii)If , then, from (1.3) and (1.10)

this is a contradiction.

Case 6 .

let for some .

(i)If , then .

(ii)If , then, from (1.4) and (1.10)

it implies that .

Case 7 .

Let for some .

(i)If , then .

(ii)If , then, from (1.3), (1.10)

it is a contradiction.

Case 8 .

let for some .

(i)If , then .

(ii)If , then, from (1.3) and (1.10)

which is a contradiction.

Set

where .

In view of the above cases, then , and we obtain that is bounded.

Indeed, suppose that for some . Then,

which implies that . Similarly, if or , we also obtain .

On the other hand, suppose that for some . Then,

which implies that . Similarly, if or , we also obtain . Therefore, from the above results, we obtain that , that is, is bounded.

For each , define

Then, using the same proof above, it can be shown that

If , and using (1.1) and (1.4), then

as . Since , hence as . Similarly, if or , , we may obtain the similar results. Therefore, from (2.1), we get

where

In (2.19), set . Then (2.19) is as follows:

By Lemma 1.3, we have as . From the inequality , we have .

Conversely, we will prove that . If , then is Picard iteration.

Theorem 2.2.

Let be as in Theorem 2.1. Suppose that are defined by the iterative processes (1.5) and (1.6), respectively, and are real sequences in such that . Then, the following two assertions are equivalent:

(i)Mann iteration (1.5) converges strongly to the unique fixed point ;

(ii)Ishikawa iteration (1.6) converges strongly to the unique fixed point .

Proof.

If the Ishikawa iteration (1.6) converges strongly to , then setting , in (1.6), we can get the convergence of Mann iteration (1.5). Conversely, we will show that . Letting , we want to prove .

From (1.5) and (1.6),

Using (1.10) with , to obtain

set

Applying the similar proof methods of Theorem 2.1, we obtain that is also bounded. The other proof is the same as that of Theorem 2.1 and is here omitted.

The authors are extremely grateful to Professor B. E. Rhoades of Indiana University for providing useful information and many help. They also thank the referees for their valuable comments and suggestions.

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