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Fixed Point Theory and Applications

, 2010:458086 | Cite as

On Mappings with Contractive Iterate at a Point in Generalized Metric Spaces

Open Access
Research Article

Abstract

Using the setting of generalized metric space, the so-called G-metric space, fixed point theorems for mappings with a contractive and a generalized contractive iterate at a point are proved. These results generalize some comparable results in the literature. A common fixed point result is also proved.

Keywords

Positive Integer Continuous Mapping Differential Geometry Symmetric Space Arbitrary Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Sehgal in [1] proved fixed point theorem for mappings with a contractive iterate at a point and therefore generalized a well-known Banach theorem.

Theorem 1.1.

Let Open image in new window be a complete metric space and let Open image in new window be a continuous mapping with property that for every Open image in new window there exists Open image in new window so that for every Open image in new window

Then Open image in new window has a unique fixed point Open image in new window in Open image in new window and Open image in new window , for each Open image in new window .

Guseman [2] extended Sehgal's result by removing the condition of continuity of Open image in new window and weakening (1.1) to hold on some subset Open image in new window of Open image in new window such that Open image in new window , where, for some Open image in new window , Open image in new window contains the closure of the iterates of Open image in new window . Further extensions appear in [3, 4]. Our aim in this study is to show that these results are valid in more general class of spaces.

In 1963, S. Gähler introduced the notion of 2-metric spaces but different authors proved that there is no relation between these two function and there is no easy relationship between results obtained in the two settings. Because of that, Dhage [5] introduced a new concept of the measure of nearness between three or more object. But topological structure of so called Open image in new window -metric spaces was incorrect. Finally, Mustafa and Sims [6] introduced correct definition of generalized metric space as follows.

Definition 1.2 (see [6]).

Let Open image in new window be a nonempty set, and let Open image in new window be a function satisfying the following properties:

Open image in new window if Open image in new window ;

Open image in new window ; for all Open image in new window , with Open image in new window ;

Open image in new window , for all Open image in new window , with Open image in new window ;

Open image in new window , (symmetry in all three variables);

Open image in new window , for all Open image in new window .

Then the function Open image in new window is called a generalized metric, or, more specifically, a Open image in new window -metric on Open image in new window , and the pair Open image in new window is called a Open image in new window -metric space.

Clearly these properties are satisfied when Open image in new window is the perimeter of triangle with vertices at Open image in new window , Open image in new window , and Open image in new window , moreover taking Open image in new window in the interior of the triangle shows that Open image in new window is the best possible.

Example 1.3.

Let Open image in new window be an ordinary metric apace, then Open image in new window can define Open image in new window -metrics on Open image in new window by

Open image in new window ,

Open image in new window .

Example 1.4 (see [6]).

and extend Open image in new window to Open image in new window by using the symmetry in the variables. Then it is clear the Open image in new window is a Open image in new window -metric space.

Definition 1.5 (see [6]).

Let Open image in new window be a Open image in new window -metric space, and let Open image in new window be sequence of points of Open image in new window , a point Open image in new window is said to be the limit of the sequence Open image in new window , if Open image in new window , and one says that the sequence Open image in new window is Open image in new window -convergent to Open image in new window

Thus, if Open image in new window in a Open image in new window -metric space Open image in new window , then for any Open image in new window , there exists Open image in new window such that Open image in new window , for all Open image in new window .

Definition 1.6 (see [6]).

Let Open image in new window be a Open image in new window -metric space, a sequence Open image in new window is called Open image in new window -Cauchy if for every Open image in new window , there is Open image in new window such that Open image in new window , for all Open image in new window ; that is, if Open image in new window as Open image in new window .

A Open image in new window -metric space Open image in new window is said to be Open image in new window -complete (or complete Open image in new window -metric) if every Open image in new window -Cauchy sequence in Open image in new window is Open image in new window -convergent in Open image in new window .

Proposition 1.7 (see [6]).

Let Open image in new window be a Open image in new window -metric space, then the function Open image in new window is jointly continuous in all three of its variables.

Definition 1.8 (see [6]).

A Open image in new window -metric space Open image in new window is called symmetric Open image in new window -metric space if Open image in new window , for all Open image in new window .

Proposition 1.9 (see [6]).

Note that if Open image in new window is a symmetric Open image in new window -metric space, then
However, if Open image in new window is nonsymmetric, then by the Open image in new window -metric properties it follows that

and that in general these inequalities cannot be improved.

Proposition 1.10 (see [6]).

A Open image in new window -metric space Open image in new window is Open image in new window -complete if and only if Open image in new window is a complete metric space.

In recent years a lot of interesting papers were published with fixed point results in Open image in new window -metric spaces, see [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. This paper is our contribution to the fixed point theory in Open image in new window -metric spaces.

2. Fixed Point Results

for all Open image in new window , we write Open image in new window .

Theorem 2.1.

Let Open image in new window or Open image in new window . Let Open image in new window , with Open image in new window . If there exists Open image in new window such that for Open image in new window , Open image in new window , then Open image in new window is the unique fixed point of Open image in new window in Open image in new window . Moreover, Open image in new window , Open image in new window , for any Open image in new window for Open image in new window , and for Open image in new window if Open image in new window .

Proof.

If Open image in new window is a symmetric space than Open image in new window and (2.1) becomes
and (2.2) becomes
thus the result follows from Theorem 12 in [3] and it is valid for any Open image in new window . Suppose now that Open image in new window is nonsymmetric space. Then by inequality (1.5) we have that (2.1) becomes
and (2.2) becomes

Since Open image in new window need not be less then 1 we can use metric fixed point results only for Open image in new window . On the other side, using the concept of Open image in new window -metric space, we are going to prove the result, if the first case for any Open image in new window , and in the second one for Open image in new window . This means that our results are real generalization in the case of nonsymmetric Open image in new window -metric spaces.

Let Open image in new window . Uniqueness follows from (2.1), since for Open image in new window , it follows that Open image in new window . Now Open image in new window implies that Open image in new window .

so Open image in new window , Open image in new window .

If Open image in new window , uniqueness follows from (2.2) since for Open image in new window , it follows that Open image in new window and further Open image in new window . Now for any Open image in new window
which is a contradiction, and therefore

Therefore, Open image in new window , where Open image in new window . For Open image in new window , Open image in new window , Open image in new window .

For Open image in new window the set Open image in new window is called the orbit for Open image in new window .

Theorem 2.2.

Let Open image in new window be a complete Open image in new window -metric space and let Open image in new window be a mapping. Suppose that for some Open image in new window the orbit Open image in new window is complete, and that: for some Open image in new window and each Open image in new window there is an integer Open image in new window such that

for all Open image in new window .

If inequality in (2.14) holds for all Open image in new window , then Open image in new window and Open image in new window , Open image in new window .

Moreover, if Open image in new window , then Open image in new window is the fixed point of Open image in new window .

Proof.

If Open image in new window is a symmetric Open image in new window -metric space the statement easily follows from Guseman fixed point result [2]. Let Open image in new window be nonsymmetric Open image in new window -metric space. Then by inequality (1.5)
Thus, one can use the fixed point result in metric space only for Open image in new window . But here, using the concept of Open image in new window -metric, we prove the result for any Open image in new window . At first let us show that

so Open image in new window is Cauchy sequence and there exists Open image in new window , for some Open image in new window , and inequality (2.15) is proved.

If we suppose that inequality in (2.14) is satisfied for all Open image in new window , then, for all Open image in new window ,

so Open image in new window .

On the other hand,

implies that Open image in new window .

Since Open image in new window is continuous it means that

Hence Open image in new window .

Now, let us suppose that Open image in new window . Since Open image in new window by Theorem 2.1   Open image in new window is the fixed point of Open image in new window in Open image in new window and Open image in new window .

For Open image in new window , in inequality (2.14) independently on Open image in new window , we are going to simplify the proof and to relax the condition in (2.14).

Corollary 2.3.

If (2.24) holds, for all Open image in new window or Open image in new window is orbitally continuous at Open image in new window , then Open image in new window is a fixed point of Open image in new window .

Proof.

If Open image in new window is a symmetric space than Open image in new window so (2.24) becomes

and result follows from Theorem 2 in [19].

and there exists Open image in new window . If (2.24) holds for all Open image in new window , then by Theorem 2.2, since Open image in new window , it follows that Open image in new window .

The fact that Open image in new window is orbitally continuous at Open image in new window , and that Open image in new window , implies that Open image in new window , and therefore Open image in new window .

Remark 2.4.

Let us note that this result is very close to Theorem 2.1 in [8].

Remark 2.5.

In the statements above Open image in new window does not have to be continuous.

The next theorems are generalizations of Ćirić fixed point results in [4].

Theorem 2.6.

Let Open image in new window be a complete metric space and Open image in new window a mapping. Suppose that for each Open image in new window there exists a positive integer Open image in new window such that

holds for some Open image in new window and all Open image in new window . Then Open image in new window has a unique fixed point Open image in new window . Moreover, for every Open image in new window , Open image in new window .

Proof.

If Open image in new window is a symmetric space then Open image in new window and inequality (2.29) becomes

for all Open image in new window . Then the result follows from Theorem 2.1 in [4] and it is true for all Open image in new window .

Now suppose that Open image in new window is nonsymmetric space. Then, by definition of the metric Open image in new window and inequality (1.5) we have

But Open image in new window need not to be less than 1, so we will prove the statement by using Open image in new window -metric.

First, let us prove prove that
Clearly (2.32) is true for Open image in new window . Suppose that Open image in new window , and that (2.32) is true for Open image in new window and let us prove it for Open image in new window . Let Open image in new window . Now

Thus by induction we obtain (2.32).

Let us prove that Open image in new window is a Cauchy sequence. Let Open image in new window , Open image in new window , Open image in new window , and we define inductively a sequence of integers and a sequence of points Open image in new window in Open image in new window as follows: Open image in new window , and Open image in new window , Open image in new window . Evidently, Open image in new window is a subsequence of the orbit Open image in new window . Using this sequence we will prove that Open image in new window is a Cauchy sequence.

Let Open image in new window be any fixed member of Open image in new window and let Open image in new window and Open image in new window be any two members of the orbit which follow after Open image in new window . Then Open image in new window and Open image in new window for some Open image in new window and Open image in new window , respectively. Now, using (2.29) we get
Repeating this argument Open image in new window times we get
Since Open image in new window , it follows that Open image in new window is a Cauchy sequence. Let Open image in new window . We show that Open image in new window is a fixed point of Open image in new window . First, let us prove that Open image in new window , where Open image in new window . For Open image in new window , we now have
and on letting Open image in new window tend to infinity it follows that

For Open image in new window we have Open image in new window .

To show that Open image in new window is a fixed point of Open image in new window , let us suppose that Open image in new window and let Open image in new window . Then

Since Open image in new window , it follows that Open image in new window , which implies that Open image in new window is a fixed point of Open image in new window .

Let us suppose that for some Open image in new window , Open image in new window . Then,

implies that Open image in new window and thus Open image in new window is the unique fixed point in Open image in new window .

If we suppose that Open image in new window is continuous, then we may prove the following theorem.

Theorem 2.7.

Let Open image in new window be a complete Open image in new window -metric space and let Open image in new window be a continuous mapping which satisfies the condition: for each Open image in new window there is a positive integer Open image in new window such that

for some Open image in new window and all Open image in new window . Then Open image in new window has a unique fixed point Open image in new window and Open image in new window , for every Open image in new window .

Proof.

Let Open image in new window be an arbitrary point in Open image in new window . Then, as in the proof of Theorem 2.6, the orbit Open image in new window is bounded and is a Cauchy sequence in the complete Open image in new window -metric space Open image in new window and so it has a limit Open image in new window in Open image in new window . Since by the hypothesis Open image in new window is continuous,

Therefore, Open image in new window is a fixed point of Open image in new window . By the same argument as in the proof of Theorem 2.6, it follows that Open image in new window is a unique fixed point of Open image in new window .

Remark 2.8.

The condition that Open image in new window is a continuous mapping can be relaxed by the following condition: Open image in new window is continuous at a point Open image in new window .

3. A Common Fixed Point Result

Now, we are going to prove Hadžić [20] fixed point theorem in 2-metric space, in a manner of Open image in new window -metric spaces.

Theorem 3.1.

Let Open image in new window be a complete Open image in new window -metric space, Open image in new window and Open image in new window one to one continuous mappings, Open image in new window continuous mapping commutative with Open image in new window and Open image in new window . Suppose that there exists a point Open image in new window such that Open image in new window is complete and that the following conditions are satisfied:

Then there exists one and only one element Open image in new window such that

(e.g., there exists a unique common fixed point for Open image in new window , and Open image in new window )

Proof.

Since Open image in new window starting with Open image in new window we can define the sequence Open image in new window such that
We are going to prove that Open image in new window is Cauchy sequence
Thus we proved that Open image in new window is a Cauchy sequence, so there exists Open image in new window such that

It obvious that Open image in new window .

At first we will prove that Open image in new window

so Open image in new window .

Now, since that Open image in new window and Open image in new window are continuous we have that Open image in new window so Open image in new window .

Further, let us prove that Open image in new window .
implies that
and Open image in new window . Similarly one can see that Open image in new window , so we prove that
If we suppose that Open image in new window is some other common fixed point for Open image in new window , and Open image in new window then we have that

which is contradiction!

So, the common fixed point for Open image in new window , and Open image in new window is unique, and proof is completed.

Remark 3.2.

For Open image in new window condition (2.14) is satisfied but the Theorem 2.2 is not just a consequence of Theorem 3.1 since in Theorem 2.2 we do not suppose that Open image in new window is continuous.

Notes

Acknowledgments

The authors are thankful to professor B. E. Rhoades, for his advice which helped in improving the results. This work was supported by grants approved by the Ministry of Science and Technological Development, Republic of Serbia, for the first author by Grant no. 144016, and for the second author by Grant no. 144025.

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Copyright information

© Gajić and and Z. Lozanov-Crvenković. 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.

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsUniversity of Novi SadNovi SadSerbia

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