Open Access

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

Fixed Point Theory and Applications20102010:252871

DOI: 10.1155/2010/252871

Received: 23 July 2010

Accepted: 9 September 2010

Published: 19 September 2010

Abstract

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

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq1_HTML.gif be a complete metric space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq2_HTML.gif . Denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq3_HTML.gif . A continuous mapping W https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq4_HTML.gif is said to be a convex structure on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq5_HTML.gif [1] if for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq6_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq7_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ2_HTML.gif
(1.2)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq8_HTML.gif satisfies the conditions of convex structure, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq9_HTML.gif is called convex metric space that is denoted as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq10_HTML.gif .

In the following part, we will consider a few iteration sequences in convex metric space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq11_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq12_HTML.gif is a self-map of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq13_HTML.gif .

Picard iteration is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ3_HTML.gif
(1.3)
Krasnoselskij iteration is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq14_HTML.gif .

Mann iteration is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq15_HTML.gif .

Ishikawa iteration is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq16_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq17_HTML.gif .

A mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq18_HTML.gif is called contractive if there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq19_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ7_HTML.gif
(1.7)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq20_HTML.gif .

The map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq21_HTML.gif is called Kannan mapping [2] if there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq22_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ8_HTML.gif
(1.8)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq23_HTML.gif .

A similar definition of mapping is due to the work Chatterjea [3] (that is called Chatterjea mapping), if there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq24_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ9_HTML.gif
(1.9)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq25_HTML.gif .

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

Theorem 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq26_HTML.gif be a complete metric space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq27_HTML.gif a mapping for which there exist the real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq29_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq30_HTML.gif such that, for any pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq31_HTML.gif , at least one of the following conditions holds:

(z1)   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq32_HTML.gif

(z2)   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq33_HTML.gif

(z3)   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq34_HTML.gif

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq35_HTML.gif has a unique fixed point, and the Picard iteration converges to fixed point. This class mapping is called Zamfirescu mapping.

In 1974, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq36_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq37_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ10_HTML.gif
(1.10)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq38_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq39_HTML.gif be any nonempty closed convex subset of a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq40_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq41_HTML.gif a quasi-contraction mapping. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq42_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq44_HTML.gif . Then the Ishikawa iteration sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq45_HTML.gif defined by (1)–(3) converges strongly to the unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq46_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq47_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq48_HTML.gif be a nonnegative sequence which satisfies the following inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ11_HTML.gif
(1.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq49_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq50_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq51_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq52_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq53_HTML.gif (see [7]).

2. Results for Quasi-Contraction Mappings

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq54_HTML.gif be a convex metric space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq55_HTML.gif a quasi-contraction mapping with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq56_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq57_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq58_HTML.gif ;

(ii)Krasnoselskij iteration (1.4) converges strongly to the unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq59_HTML.gif .

Proof.

First, we show https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq60_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq61_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq62_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq63_HTML.gif .

From (1.3), (1.4), and (1.1), we can get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ12_HTML.gif
(2.1)
Next, we consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq64_HTML.gif . Using (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq65_HTML.gif , to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ13_HTML.gif
(2.2)
Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ14_HTML.gif
(2.3)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq66_HTML.gif is bounded. Without loss of generality, we let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq67_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq68_HTML.gif . Indeed, we will show this conclusion from the some following cases.

Case 1 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq69_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq70_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq71_HTML.gif . Then, from (1.10) and the above https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq72_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ15_HTML.gif
(2.4)

and it leads to a contradiction. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq73_HTML.gif . Similarity to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq74_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq75_HTML.gif is also impossible.

Case 2 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq76_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq77_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq78_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq79_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq80_HTML.gif .

(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq81_HTML.gif , then, from (1.4) and (1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ16_HTML.gif
(2.5)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq82_HTML.gif . By induction on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq83_HTML.gif , we can obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq84_HTML.gif .
  1. (iii)
    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq85_HTML.gif , from (1.4) and (1.1)
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ17_HTML.gif
    (2.6)
     

it implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq86_HTML.gif . By induction on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq87_HTML.gif , we can get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq88_HTML.gif .

Case 3 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq89_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq90_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq91_HTML.gif . Without loss of generality, we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq92_HTML.gif . Then, from (1.4), (1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ18_HTML.gif
(2.7)

it implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq93_HTML.gif , and by induction on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq94_HTML.gif , we may get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq95_HTML.gif , which is a contradiction.

Case 4 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq96_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq97_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq98_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq99_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq100_HTML.gif .

(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq101_HTML.gif , from (1.4), (1.1), then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ19_HTML.gif
(2.8)

it implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq102_HTML.gif and by induction on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq103_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq104_HTML.gif .

Case 5 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq105_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq106_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq107_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq108_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq109_HTML.gif .

(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq110_HTML.gif , then, from (1.3) and (1.10)

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ20_HTML.gif
(2.9)

this is a contradiction.

Case 6 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq111_HTML.gif .

let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq112_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq113_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq114_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq115_HTML.gif .

(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq116_HTML.gif , then, from (1.4) and (1.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ21_HTML.gif
(2.10)

it implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq117_HTML.gif .

Case 7 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq118_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq119_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq120_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq121_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq122_HTML.gif .

(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq123_HTML.gif , then, from (1.3), (1.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ22_HTML.gif
(2.11)

it is a contradiction.

Case 8 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq124_HTML.gif .

let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq125_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq126_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq127_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq128_HTML.gif .

(ii)If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq129_HTML.gif , then, from (1.3) and (1.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ23_HTML.gif
(2.12)

which is a contradiction.

Set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ24_HTML.gif
(2.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq130_HTML.gif .

In view of the above cases, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq131_HTML.gif , and we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq132_HTML.gif is bounded.

Indeed, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq133_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq134_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ25_HTML.gif
(2.14)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq135_HTML.gif . Similarly, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq136_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq137_HTML.gif , we also obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq138_HTML.gif .

On the other hand, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq139_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq140_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ26_HTML.gif
(2.15)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq141_HTML.gif . Similarly, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq142_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq143_HTML.gif , we also obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq144_HTML.gif . Therefore, from the above results, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq145_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq146_HTML.gif is bounded.

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq147_HTML.gif , define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ27_HTML.gif
(2.16)
Then, using the same proof above, it can be shown that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ28_HTML.gif
(2.17)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq148_HTML.gif , and using (1.1) and (1.4), then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ29_HTML.gif
(2.18)
as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq149_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq150_HTML.gif , hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq151_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq152_HTML.gif . Similarly, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq153_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq155_HTML.gif , we may obtain the similar results. Therefore, from (2.1), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ30_HTML.gif
(2.19)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq156_HTML.gif

In (2.19), set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq157_HTML.gif . Then (2.19) is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ31_HTML.gif
(2.20)

By Lemma 1.3, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq158_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq159_HTML.gif . From the inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq160_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq161_HTML.gif .

Conversely, we will prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq162_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq163_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq164_HTML.gif is Picard iteration.

Theorem 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq165_HTML.gif be as in Theorem 2.1. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq166_HTML.gif are defined by the iterative processes (1.5) and (1.6), respectively, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq167_HTML.gif are real sequences in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq168_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq169_HTML.gif . Then, the following two assertions are equivalent:

(i)Mann iteration (1.5) converges strongly to the unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq170_HTML.gif ;

(ii)Ishikawa iteration (1.6) converges strongly to the unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq171_HTML.gif .

Proof.

If the Ishikawa iteration (1.6) converges strongly to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq172_HTML.gif , then setting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq173_HTML.gif , in (1.6), we can get the convergence of Mann iteration (1.5). Conversely, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq174_HTML.gif . Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq175_HTML.gif , we want to prove https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq176_HTML.gif .

From (1.5) and (1.6),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ32_HTML.gif
(2.21)
Using (1.10) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq177_HTML.gif , to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ33_HTML.gif
(2.22)
set
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_Equ34_HTML.gif
(2.23)

Applying the similar proof methods of Theorem 2.1, we obtain that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F252871/MediaObjects/13663_2010_Article_1237_IEq178_HTML.gif is also bounded. The other proof is the same as that of Theorem 2.1 and is here omitted.

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Shijiazhuang Railway University
(2)
Department of Mathematics, Indiana University

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Copyright

© Zhiqun Xue et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.