Open Access

Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces with Applications

Fixed Point Theory and Applications20112011:979586

https://doi.org/10.1155/2011/979586

Received: 6 December 2010

Accepted: 31 December 2010

Published: 19 January 2011

Abstract

The purpose of this paper is to present some fixed-point results for single-valued -contractions on ordered and complete gauge space. Our theorems generalize and extend some recent results in the literature. As an application, existence results for some integral equations on the positive real axis are given.

1. Introduction

Throughout this paper will denote a nonempty set endowed with a separating gauge structure , where is a directed set (see [1] for definitions). Let and . We also denote by the set of all real numbers and by .

A sequence of elements in is said to be Cauchy if for every and , there is an with for all and . The sequence is called convergent if there exists an such that for every and , there is an with , for all .

A gauge space is called complete if any Cauchy sequence is convergent. A subset of is said to be closed if it contains the limit of any convergent sequence of its elements. See also Dugundji [1] for other definitions and details.

If is an operator, then is called fixed point for if and only if . The set denotes the fixed-point set of .

On the other hand, Ran and Reurings [2] proved the following Banach-Caccioppoli type principle in ordered metric spaces.

Theorem 1.1 (Ran and Reurings [2]).

Let be a partially ordered set such that every pair has a lower and an upper bound. Let be a metric on such that the metric space is complete. Let be a continuous and monotone (i.e., either decreasing or increasing) operator. Suppose that the following two assertions hold:

(1)there exists  such that , for each with ;

(2)there exists such that or .

Then has an unique fixed point , that is, , and for each the sequence of successive approximations of starting from converges to .

Since then, several authors considered the problem of existence (and uniqueness) of a fixed point for contraction-type operators on partially ordered sets.

In 2005, Nieto and Rodrguez-López proved a modified variant of Theorem 1.1, by removing the continuity of . The case of decreasing operators is treated in Nieto and Rodrguez-López [3], where some interesting applications to ordinary differential equations with periodic boundary conditions are also given. Nieto, Pouso, and Rodrguez-López, in a very recent paper, improve some results given by Petruşel and Rus in [4] in the setting of abstract -spaces in the sense of Fréchet, see, for example, [5, Theorems  3.3 and 3.5]. Another fixed-point result of this type was given by O'Regan and Petruşel in [6] for the case of -contractions in ordered complete metric spaces.

The aim of this paper is to present some fixed-point theorems for -contractions on ordered and complete gauge space. As an application, existence results for some integral equations on the positive real axis are given. Our theorems generalize the above-mentioned theorems as well as some other ones in the recent literature (see; Ran and Reurings [2], Nieto and Rodrguez-López [3, 7], Nieto et al. [5], Petruşel and Rus [4], Agarwal et al. [8], O'Regan and Petruşel [6], etc.).

2. Preliminaries

Let be a nonempty set and be an operator. Then, , , denote the iterate operators of . Let be a nonempty set and let . Let a subset of and an operator. By definition the triple is called an -space (Fréchet [9]) if the following conditions are satisfied.

(i)If , for all , then and .

(ii)If and , then for all subsequences, , of we have that and .

By definition, an element of is a convergent sequence, is the limit of this sequence and we also write .

In what follow we denote an -space by .

In this setting, if , then an operator is called orbitally -continuous (see [5]) if [ and , as and for any ] imply [ , as ]. In particular, if , then is called orbitally continuous.

Let be a partially ordered set, that is, is a nonempty set and ≤ is a reflexive, transitive, and antisymmetric relation on . Denote
(2.1)

Also, if , with then by we will denote the ordered segment joining and , that is, . In the same setting, consider . Then, is the lower fixed-point set of , while is the upper fixed-point set of . Also, if and , then the cartesian product of and is denoted by , and it is defined in the following way: , .

Definition 2.1.

Let be a nonempty set. By definition is an ordered -space if and only if

(i) is an -space;

(ii) is a partially ordered set;

(iii) , and , for each .

If is a gauge space, then the convergence structure is given by the family of gauges . Hence, is an ordered -space, and it will be called an ordered gauge space, see also [10, 11].

Recall that is said to be a comparison function if it is increasing and , as . As a consequence, we also have , for each , and is right continuous at 0. For example, (where ), and , are comparison functions.

Recall now the following important abstract concept.

Definition 2.2 (Rus [12]).

Let be an -space. An operator is, by definition, a Picard operator if

(i) ;

(ii) as , for all .

Several classical results in fixed-point theory can be easily transcribed in terms of the Picard operators, see [4, 13, 14]. In Rus [12] the basic theory of Picard operators is presented.

3. Fixed-Point Results

Our first main result is the following existence, uniqueness, and approximation fixed-point theorem.

Theorem 3.1.

Let be an ordered complete gauge space and be an operator. Suppose that

(i)for each with there exists such that and ;

(ii) ;

(iii)if and , then ;

(iv)there exists such that ;

(v) is orbitally continuous;

(vi)there exists a comparison function such that, for each one has
(3.1)

Then, is a Picard operator.

Proof.

Let be such that . Suppose first that . Then, from (ii) we obtain
(3.2)
From (vi), by induction, we get, for each , that
(3.3)
Since as , for an arbitrary we can choose such that , for each . Since for all , we have for all that
(3.4)
Now since (see (iii)) we have for any that
(3.5)
By induction, for each , we have
(3.6)

Hence is a Cauchy sequence in . From the completeness of the gauge space we have , as .

Let be arbitrarily chosen. Then;

(1)If then and thus, for each , we have , for each . Letting we obtain that .

(2)If then, by (i), there exists such that and . From the second relation, as before, we get, for each , that , for each and hence , as . Then, using the first relation we infer that, for each , we have , for each . Letting again , we conclude .

By the orbital continuity of we get that . Thus . If we have for some , then from above, we must have , so .

If , then plays the role of .

Remark 3.2.

Equivalent representation of condition (iv) are as follows.

(iv)'There exists such that or

(iv)" .

Remark 3.3.

Condition (ii) can be replaced by each of the following assertions:

(ii)' is increasing,

(ii)" is decreasing.

However, it is easy to see that assertion (ii) in Theorem 3.1. is more general.

As a consequence of Theorem 3.1, we have the following result very useful for applications.

Theorem 3.4.

Let be an ordered complete gauge space and be an operator. One supposes that

(i)for each with there exists such that and ;

(ii) is increasing;

(iii)there exists such that ;

(iv)

(a) is orbitally continuous or

(b)if an increasing sequence converges to in , then for all ;

(v)there exists a comparison function such that
(3.7)

(vi)if and , then

Then is a Picard operator.

Proof.

Since is increasing and we immediately have . Hence from (v) we obtain , for each . By a similar approach as in the proof of Theorem 3.1 we obtain
(3.8)

Hence is a Cauchy sequence in . From the completeness of the gauge space we have that as .

By the orbital continuity of the operator we get that . If (iv)(b) takes place, then, since , given any there exists such that for each we have . On the other hand, for each , since , we have, for each that
(3.9)

Thus .

The uniqueness of the fixed point follows by contradiction. Suppose there exists , with . There are two possible cases.

(a)If , then we have as , which is a contradiction. Hence .

(b)If then there exists such that and . The monotonicity condition implies that and are comparable as well as and . Hence , as , which is again a contradiction. Thus .

4. Applications

We will apply the above result to nonlinear integral equations on the real axis
(4.1)

Theorem 4.1.

Consider (4.1). Suppose that

(i) and are continuous;

(ii) is increasing for each ;

(iii)there exists a comparison function , with for each and any , such that
(4.2)
(iv)there exists such that
(4.3)

Then the integral equation (4.1) has a unique solution in .

Proof.

Let and the family of pseudonorms
(4.4)

Define now for .

Then is family of gauges on . Consider on the partial order defined by
(4.5)

Then is an ordered and complete gauge space. Moreover, for any increasing sequence in converging to some we have , for any . Also, for every there exists which is comparable to and .

Define , by the formula
(4.6)
First observe that from (ii) is increasing. Also, for each with and for , we have
(4.7)
Hence, for we obtain
(4.8)

From (iv) we have that . The conclusion follows now from Theorem 3.4.

Consider now the following equation:
(4.9)

Theorem 4.2.

Consider (4.9). Suppose that

(i) and are continuous;

(ii) is increasing for each ;

(iii)there exists a comparison function , with for each and any , such that
(4.10)
(iv)there exists such that
(4.11)

Then the integral equation (4.9) has a unique solution in .

Proof.

We consider the gauge space where
(4.12)
and the operator defined by
(4.13)
As before, consider on the partial order defined by
(4.14)

Then is an ordered and complete gauge space. Moreover, for any increasing sequence in converging to a certain we have , for any . Also, for every there exists which is comparable to and . Notice that (ii) implies that is increasing.

From condition (iii), for with , we have
(4.15)
Thus, for any , we obtain
(4.16)

As before, from (iv) we have that . The conclusion follows again by Theorem 3.4.

Remark 4.3.

It is worth mentioning that it could be of interest to extend the above technique for other metrical fixed-point theorems, see [15, 16], and so forth.

Authors’ Affiliations

(1)
Faculty of Business, Babeş-Bolyai University

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Copyright

© Cristian Chifu and Gabriela Petruşel. 2011

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.