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Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces with Applications
Fixed Point Theory and Applications volume 2011, Article number: 979586 (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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ2_HTML.gif)
Then, is a Picard operator.
Proof.
Let be such that
. Suppose first that
. Then, from (ii) we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ3_HTML.gif)
From (vi), by induction, we get, for each , that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ4_HTML.gif)
Since as
, for an arbitrary
we can choose
such that
, for each
. Since
for all
, we have for all
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ5_HTML.gif)
Now since (see (iii)) we have for any
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ6_HTML.gif)
By induction, for each , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ8_HTML.gif)
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ9_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ12_HTML.gif)
(iv)there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ13_HTML.gif)
Then the integral equation (4.1) has a unique solution in
.
Proof.
Let and the family of pseudonorms
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ14_HTML.gif)
Define now for
.
Then is family of gauges on
. Consider on
the partial order defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ16_HTML.gif)
First observe that from (ii) is increasing. Also, for each
with
and for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ17_HTML.gif)
Hence, for we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ18_HTML.gif)
From (iv) we have that . The conclusion follows now from Theorem 3.4.
Consider now the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ20_HTML.gif)
(iv)there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ21_HTML.gif)
Then the integral equation (4.9) has a unique solution in
.
Proof.
We consider the gauge space where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ22_HTML.gif)
and the operator defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ23_HTML.gif)
As before, consider on the partial order defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ25_HTML.gif)
Thus, for any , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F979586/MediaObjects/13663_2010_Article_1451_Equ26_HTML.gif)
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.
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Chifu, C., Petruşel, G. Fixed-Point Results for Generalized Contractions on Ordered Gauge Spaces with Applications. Fixed Point Theory Appl 2011, 979586 (2011). https://doi.org/10.1155/2011/979586
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DOI: https://doi.org/10.1155/2011/979586