Open Access

Remarks on Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings

Fixed Point Theory and Applications20102010:315398

DOI: 10.1155/2010/315398

Received: 20 March 2010

Accepted: 4 May 2010

Published: 6 June 2010

Abstract

We discuss the newly introduced concept of cone metric spaces. We also discuss the fixed point existence results of contractive mappings defined on such metric spaces. In particular, we show that most of the new results are merely copies of the classical ones.

1. Introduction

Cone metric spaces were introduced in [1]. A similar notion was also considered by Rzepecki in [2]. After carefully defining convergence and completeness in cone metric spaces, the authors proved some fixed point theorems of contractive mappings. Recently, more fixed point results in cone metric spaces appeared in [38]. Topological questions in cone metric spaces were studied in [6] where it was proved that every cone metric space is first countable topological space. Hence, continuity is equivalent to sequential continuity and compactness is equivalent to sequential compactness. It is worth mentioning the pioneering work of Quilliot [9] who introduced the concept of generalized metric spaces. His approach was very successful and used by many (see references in [10]). It is our belief that cone metric spaces are a special case of generalized metric spaces. In this work, we introduce a metric type structure in cone metric spaces and show that classical proofs do carry almost identically in these metric spaces. This approach suggest that any extension of known fixed point result to cone metric spaces is redundant. Moreover the underlying Banach space and the associated cone subset are not necessary.

For more on metric fixed point theory, the reader may consult the book [11].

2. Basic Definitions and Results

First let us start by making some basic definitions.

Definition 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq1_HTML.gif be a real Banach space with norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq3_HTML.gif a subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq4_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq5_HTML.gif is called a cone if and only if

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq6_HTML.gif is closed, nonempty, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq7_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq8_HTML.gif is the zero vector in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq9_HTML.gif ;

(2)if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq10_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq11_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq12_HTML.gif ;

(3)if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq13_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq14_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq15_HTML.gif .

Given a cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq16_HTML.gif in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq17_HTML.gif , we define a partial ordering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq18_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq19_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ1_HTML.gif
(2.1)
We also write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq20_HTML.gif whenever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq22_HTML.gif , while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq23_HTML.gif will stand for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq24_HTML.gif (where Int( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq25_HTML.gif ) designate the interior of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq26_HTML.gif ). The cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq27_HTML.gif is called normal if there is a number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq28_HTML.gif , such that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq29_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ2_HTML.gif
(2.2)

The least positive number satisfying this inequality is called the normal constant of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq30_HTML.gif . The cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq31_HTML.gif is called regular if every increasing sequence which is bounded from above is convergent. Equivalently the cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq32_HTML.gif is called regular if every decreasing sequence which is bounded from below is convergent. Regular cones are normal and there exist normal cones which are not regular.

Throughout the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq33_HTML.gif and the cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq34_HTML.gif will be omitted.

Definition 2.2.

A cone metric space is an ordered pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq35_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq36_HTML.gif is any set and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq37_HTML.gif is a mapping satisfying

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq38_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq39_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq40_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq41_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq42_HTML.gif ;

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq43_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq44_HTML.gif ;

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq45_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq46_HTML.gif .

Convergence is defined as follows.

Definition 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq47_HTML.gif be a cone metric space, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq48_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq50_HTML.gif . If for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq51_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq52_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq53_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq55_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq56_HTML.gif is said to be convergent. We will say https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq57_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq58_HTML.gif and write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq59_HTML.gif .

It is easy to show that the limit of a convergent sequence is unique. Cauchy sequences and completeness are defined by

Definition 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq60_HTML.gif be a cone metric space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq61_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq62_HTML.gif . If for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq63_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq64_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq65_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq67_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq68_HTML.gif is called Cauchy sequence. If every Cauchy sequence is convergent in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq69_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq70_HTML.gif is called a complete cone metric space.

The basic properties of convergent and Cauchy sequences may be found at [1]. In fact the properties and their proofs are identical to the classical metric ones. Since this work concerns the fixed point property of mappings, we will need the following property.

Definition 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq71_HTML.gif be a cone metric space. A mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq72_HTML.gif is called Lipschitzian if there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq73_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ3_HTML.gif
(2.3)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq74_HTML.gif . The smallest constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq75_HTML.gif which satisfies the above inequality is called the Lipschitz constant of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq76_HTML.gif , denoted https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq77_HTML.gif . In particular https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq78_HTML.gif is a contraction if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq79_HTML.gif .

As we mentioned earlier cone metric spaces have a metric type structure. Indeed we have the following result.

Theorem 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq80_HTML.gif be a metric cone over the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq81_HTML.gif with the cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq82_HTML.gif which is normal with the normal constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq83_HTML.gif . The mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq84_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq85_HTML.gif satisfies the following properties:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq86_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq87_HTML.gif ;

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq88_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq89_HTML.gif ;

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq90_HTML.gif , for any points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq91_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq92_HTML.gif .

Proof.

The proofs of (1) and (2) are easy and left to the reader. In order to prove (3), let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq93_HTML.gif be any points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq94_HTML.gif . Using the triangle inequality satisfied by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq95_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ4_HTML.gif
(2.4)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq96_HTML.gif is normal with constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq97_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ5_HTML.gif
(2.5)
which implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ6_HTML.gif
(2.6)

This completes the proof of the theorem.

Note that the property (3) is discouraging since it does not give the classical triangle inequality satisfied by a distance. But there are many examples where the triangle inequality fails (see, e.g., [12]).

The above result suggest the following definition.

Definition 2.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq98_HTML.gif be a set. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq99_HTML.gif be a function which satisfies

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq100_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq101_HTML.gif ;

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq102_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq103_HTML.gif ;

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq104_HTML.gif , for any points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq106_HTML.gif , for some constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq107_HTML.gif .

The pair https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq108_HTML.gif is called a metric type space.

Similarly we define convergence and completeness in metric type spaces.

Definition 2.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq109_HTML.gif be a metric type space.

(1)The sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq110_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq111_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq112_HTML.gif .

(2)The sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq113_HTML.gif is Cauchy if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq114_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq115_HTML.gif is complete if and only if any Cauchy sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq116_HTML.gif is convergent.

3. Some Fixed Point Results

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq117_HTML.gif be a map. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq118_HTML.gif is called Lipschitzian if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq119_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ7_HTML.gif
(3.1)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq120_HTML.gif . The smallest constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq121_HTML.gif will be denoted https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq122_HTML.gif .

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq123_HTML.gif be a complete metric type space. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq124_HTML.gif be a map such https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq125_HTML.gif is Lipschitzian for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq126_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq127_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq128_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq129_HTML.gif . Moreover for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq130_HTML.gif , the orbit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq131_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq132_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq133_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq134_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ8_HTML.gif
(3.2)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ9_HTML.gif
(3.3)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq135_HTML.gif is convergent, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq136_HTML.gif . This forces https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq137_HTML.gif to be a Cauchy sequence. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq138_HTML.gif is complete, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq139_HTML.gif converges to some point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq140_HTML.gif . First let us show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq141_HTML.gif is a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq142_HTML.gif . Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ10_HTML.gif
(3.4)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ11_HTML.gif
(3.5)
If we let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq143_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq144_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq145_HTML.gif . Next we show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq146_HTML.gif has at most one fixed point. Indeed let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq148_HTML.gif be two fixed points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq149_HTML.gif . Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ12_HTML.gif
(3.6)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq150_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq151_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq152_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq153_HTML.gif . Therefore we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq154_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq155_HTML.gif , which completes the proof of the theorem.

The condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq156_HTML.gif is needed because of the condition (3) satisfied by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq157_HTML.gif . In fact a more natural condition should be

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq159_HTML.gif , for any points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq160_HTML.gif , for some constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq161_HTML.gif .

An example of such https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq162_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq163_HTML.gif is given below.

Example 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq164_HTML.gif be the set of Lebesgue measurable functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq165_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ13_HTML.gif
(3.7)
Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq166_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ14_HTML.gif
(3.8)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq167_HTML.gif satisfies the following properties:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq169_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq170_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq172_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq173_HTML.gif ;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq175_HTML.gif , for any points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq176_HTML.gif .

In the next result we consider the case of metric type spaces https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq177_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq178_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq179_HTML.gif . Recall that a subset https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq180_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq181_HTML.gif is said to be bounded whenever https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq182_HTML.gif .

Theorem 3.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq183_HTML.gif be a complete metric type space, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq184_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq185_HTML.gif instead of (3). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq186_HTML.gif be a map such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq187_HTML.gif is Lipschitzian for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq189_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq190_HTML.gif has a unique fixed point if and only if there exists a bounded orbit. Moreover if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq191_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq192_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq193_HTML.gif , the orbit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq194_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq195_HTML.gif .

Proof.

Clearly if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq196_HTML.gif has a fixed point, then its orbit is bounded. Conversely let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq197_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq198_HTML.gif is bounded, that is, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq199_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq200_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq201_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq202_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ15_HTML.gif
(3.9)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq203_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq204_HTML.gif is a Cauchy sequence. Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq205_HTML.gif converges to some point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq206_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq207_HTML.gif is complete. The remaining part of the proof follows the same as in the previous theorem.

The connection between the above results and the main theorems of [1] are given in the following corollary.

Corollary 3.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq208_HTML.gif be a metric cone over the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq209_HTML.gif with the cone https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq210_HTML.gif which is normal with the normal constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq211_HTML.gif . Consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq212_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq213_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq214_HTML.gif be a contraction with constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq215_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_Equ16_HTML.gif
(3.10)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq217_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq218_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq219_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq220_HTML.gif is convergent, which implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq221_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq222_HTML.gif , and any orbit converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq223_HTML.gif .

From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq224_HTML.gif in the above Corollary, we easily see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq225_HTML.gif -convergence and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq226_HTML.gif -convergence are identical.

Remark 3.5.

In [1] the authors gave an example of a map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq227_HTML.gif which is contraction for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq228_HTML.gif but not for the euclidian distance. From the above corollary, we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq229_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq230_HTML.gif may not be less than 1, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq231_HTML.gif may not be a contraction for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq232_HTML.gif . This is why the above theorems were stated in terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq233_HTML.gif .

Using the ideas described above one can prove fixed point results for mappings which contracts orbits and obtain similar results as Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F315398/MediaObjects/13663_2010_Article_1256_IEq234_HTML.gif for example in [1].

Authors’ Affiliations

(1)
Department of Mathematical Science, The University of Texas at El Paso
(2)
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals

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Copyright

© Mohamed A. Khamsi. 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.