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Fixed Point and Best Proximity Theorems under Two Classes of Integral-Type Contractive Conditions in Uniform Metric Spaces
Fixed Point Theory and Applications volume 2010, Article number: 510974 (2010)
Abstract
This paper investigates the existence of fixed points and best proximity points of p-cyclic self-maps on a set of subsets of a certain uniform space under integral-type contractive conditions. The parallel properties of the associated restricted composed maps from any of the subsets to itself are also investigated. The subsets of the uniform space are not assumed to intersect.
1. Introduction
Fixed point theory is of an intrinsic theoretical interest but also a useful tool in a wide class of practical problems. There is an exhaustive variety of results concerning fixed point theory in Banach spaces and metric spaces involving different types of contractive conditions including those associated with the so-called Kannan maps and with Meir-Keeler contractions (see, e.g., [1–6]). There is also a rich background literature concerning the use of contractive conditions in integral form using altering distances, Lebesgue integrable test functions, and comparison functions, [7–9]. Also, the so-called reasonable expansive mappings have been investigated in [10], and conditions for the existence of fixed points have been given. It has been used, for instance, for the study of the Lyapunov stability of delay-free dynamic systems and also for that of dynamic systems subject to delays and then described by functional differential equations (see, for instance, [11, 12]) concerning a related fixed point background for those systems and [12–15] concerning some related background for stability. On the other hand, it has also been useful for investigating the stability of hybrid systems consisting of coupled continuous-time and discrete-time or digital dynamic subsystems [16]. This paper considers -cyclic self-maps in a uniform space (
, where
is a nonempty set equipped with a nonempty family
of subsets of
satisfying certain uniformity properties. The family
is called the uniform structure of
, and its elements are called entourages, neighbourhoods, or surroundings. The uniform space
is assumed to be endowed with an
-distance or and
-distance. The existence of fixed points and best proximity points in restricted
-cyclic self-maps
) of
, [8], subject to the constraint
for each pair of adjacent subsets
of
,  
, under the cyclic condition
;
, is investigated separately under two groups of integral-type contractive conditions. One of such groups involves a positive integrand test function while the other combines a positive integrand with a comparison function. Some properties of the composed restricted self-maps on each of the subsets are also investigated. The subsets of the uniform structure do not necessarily intersect. If the sets do not intersect, then it is proven that
if
is a metric space endowed with a distance map
, some
, and
is said to be a best proximity point. Also, it follows that
for some
. If the self-map
of
is nonexpanding, then
, for all
, [8].
2. Basic Results about
-Distances,
-Distances, and
-Closeness
Define the nonempty family of subsets of
of the form
with
. Note by construction that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ1_HTML.gif)
The following definitions of -closeness and an
and
-distances are used throughout the paper.
Definition 2.1 , (see [7, 9]).
If and
and
, then
and
are said to be
-close. A sequence
is a Cauchy sequence for
if for any
, there exists
such that
and
are
-close for
.
Definition 2.2.
A function is said to be an
-distance if
(1);
(2)for each ,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ2_HTML.gif)
Definition 2.2 generalizes slightly that of [7] by admitting to depend on
since it is being used on distinct sets
. Note that
is symmetrical, that is,
then
so that
and
are
-close under Definition 2.2.
Definition 2.3 (see [7]).
A function is said to be an
-distance if
(1)it is an -distance;
(2).
Assertion 1.
Assume that any is symmetrical, that is,
. Then,
is an
-distance if and only if
and
are
-close for all
provided that
for some
and some
.
Proof.
It follows from the symmetry of all and Definition 2.3 by a simple contradiction argument. Take a pair
from Definition 2.3 since
is an
-distance fulfilling
for some
and some
. Such a pair always exists for any
. Since
is symmetrical, then
. Since
if and only if
then
and
are
-close.
Assertion 2.
-
(1)
and
are symmetrical.
-
(2)
If
is of the form
then
is symmetrical.
-
(3)
If
is nonempty and
with
, then
is not symmetrical.
If, in-addition, then there are no
in
being
-close.
Proof.
-
(1)
is symmetrical.
is symmetrical. Assertion 2(1) has been proven.
-
(2)
for some
is symmetrical. Assertion 2 (2) is proven.
-
(3)
Proceed by contradiction:
symmetrical
what contradicts
.
Assertion 2 states that some, but not all, nonempty subsets of
are symmetrical. For instance, if
, then
is not symmetrical since there are
such that
are not in
; that is, there are pairs
,
which are not
-close. If furthermore the sets
are disjoint, then there is no pair in
being
-close (Assertion 2(3)). Note that under symmetry of
, the second property of an
-distance can be rewritten in an equivalent form by replacing
with
being
-close. The subsequent result states that, contrarily to results in former studies related to
and
-distances [7, 9], the second property guaranteeing an
-distance necessarily involves
-values exceeding distances between the various subsets
.
Lemma 2.4.
Assume that is an
-distance and
for some
. If
, then
for some
and some
.
Proof.
Assume that , so that
and
, and
, for some
. The following cases can occur.
(1)If , and since
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ3_HTML.gif)
which leads to the contradiction .
(2)If , and since
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ4_HTML.gif)
which leads to the same contradiction as in Case (1).
(3)If and if
, the above contradiction of cases (1) and (2), is also obtained by replacing
.
(4)If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ5_HTML.gif)
which leads to the same contradiction as that of case (1).
The following lemma is a direct consequence of Lemma 2.4:
Lemma 2.5.
Assume that is an
-distance and
for
. If
then
for some
and some
.
3. Main Results about Fixed Points and Best Proximity Points
Consider -cyclic self-maps
subject to
;  
. The objective is to first investigate if each of them has a fixed point and then if they have a common fixed point through contraction conditions on Lebesgue integrals and use of comparison functions. Without loss of generality, we discuss the fixed points of self-maps
of
. Consider a Lebesgue-integrable map
which satisfies
,  
such that
, for all 
.
Define also the composed self-map as
from the self-map
whose restrictions to
,
, are defined via the restriction
;
by
for each
;
. Note that the domain of the self-map
of
is
while that of
is
. The paper investigates, under two types of integral-type contractive conditions of self-maps
of
, the existence of fixed points of such a self-map in
, provided that the intersection is nonempty. In that case, the fixed points coincide with those of the self-map
. It also investigated the existence of best proximity points between adjacent and nonadjacent subsets
; 
for the case that
. In such a case, the best proximity points at each pair of adjacent subsets
;
are also fixed points of the composed self-maps
from each subset
to itself;
even under weaker contractive integral-type conditions. A key basic result used in the mathematical proofs is that the distance between any pair of (adjacent or nonadjacent) subsets is identical for nonexpansive contractions.
It is first assumed that the integral-type contractive Condition 1 below holds.
Condition 1.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ6_HTML.gif)
where are sequences of nonnegative real numbers subject to
for all
, for all
. The self-map
of
is said to be reasonably nonexpansive through this paper if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ7_HTML.gif)
for some nonnegative real constants and
. In particular,
is reasonably nonexpansive if
is nonexpansive. The following result follows from Condition 1.
Theorem 3.1.
The following properties hold under Condition 1 for any -distance
:
(i)The restricted self-maps ; for all 
satisfying (3.1) are all nonexpansive, and so it is the self-map
;
(ii); for all 
;
(iii); for all 
, with
and
being monotone increasing with
;
(iv)If that is,
, then there is a fixed point
of the self-map
of
and of its restrictions to
and
defined through the natural set inclusions
. Also,
for the self-map
.
Proof.
Consider some -distance
. Note that for each
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ8_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ9_HTML.gif)
If, in particular, ;
, then
if
with any
and
; for all 
, and if
then
are
-close. It is first proven that the self-map
of
satisfying (3.1) is nonexpansive. Proceed by contradiction by assuming that it is expansive. Then, one gets the following by defining a real sequence
with general term
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ10_HTML.gif)
for some which is a contradiction, and the self-map
(and then the self-map
) of
is nonexpansive and property (i) holds. Now, its is proven by contradiction that
. Assume that there exist
satisfying
such that
. Then there are best proximity points
and some
such that, since
and the self-map
of
is nonexpanding, one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ11_HTML.gif)
for some with the last inequality being strict unless
, what is a contradiction if
. Now, assume that
, then the best proximity point
since
and
, that is,
. This is a contradiction to the assumption
. Then,
and
if and only if
. Since the self-map
of
restricted to
is nonexpansive, then the self-map
of
restricted to
is reasonably nonexpansive. It also follows by contradiction that
. Assume that
. Then, the following contradiction follows from (3.1):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ12_HTML.gif)
for some ,
, unless
(and then
;
) provided that
. Such a
always exists since
;
. Then,
, and Property (ii) follows.
Note that (3.1) yields directly via recursion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ13_HTML.gif)
Note that . Define
with
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ14_HTML.gif)
Note also that the cardinal (or discrete measure) of is
(i.e., infinity numerable), since otherwise,
for some
(contrarily to one of the given hypothesis) and
. Since
and
, so that
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ15_HTML.gif)
Then, since the distance between any two adjacent sets is a real constant
, one gets the following from (3.8), and (3.10):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ16_HTML.gif)
where since
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ17_HTML.gif)
Note that is monotone increasing with it argument
and that
. Property (iii) has been proven. If
, then
so that there is a fixed point
of
which is also a fixed point of its extensions
and
since
,
and
. It turns out that
is also a fixed point of
, [17]. Property (iv) has been proven.
Note that the proved boundedness property of the -distance
also relies on the fact that this is a distance between best proximity points in adjacent sets. It is well known that a distance map in a metric space has always a uniform equivalent distance which is finite. The following two concluding results from (3.11) are direct since
; for all
.
Corollary 3.2.
Assume that and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ18_HTML.gif)
Then, there is a set of card
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ19_HTML.gif)
If  , then the points of the set
satisfy
so that
is a set of best proximity points in
of the self-maps
and
of
. Each
is a fixed point of
, a best proximity point of
and satisfies
and
so that
;  for all
.
Corollary 3.3.
If so that
;
, then Corollary 3.2 still holds with the set
consisting only of a set of identical points
in
such that
,  
, and
;
.
Since an -distance is also an
-distance, the following conclusion is direct from Theorem 3.1 and Corollary 3.2.
Corollary 3.4.
Theorem 3.1 and Corollary 3.2 also hold if is an
-distance.
An important relaxation of Condition 1 allows the reformulation of Theorem 3.1 and Corollaries 3.2–3.4 except in the result that when
as follows.
Corollary 3.5.
Assume that Condition 1 is reformulated as the -cyclic contractive Condition 2 below.
Condition 2.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ20_HTML.gif)
for a new real sequence under the weaker constraints
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ21_HTML.gif)
and that the finite limit of Corollary 3.2 exists. Then, the following properties hold.
(i)Theorem 3.1 and Corollaries 3.2–3.4 still hold, except that in the case that the distance between adjacent sets is zero (i.e., if all subsets
have a nonempty intersection), the property
is not guaranteed, since the restricted self-maps
can be expansive for some
.
(ii)If then there exists a set
of card
of best proximity points of the self-map
of
such that
, and there are Cauchy sequences
which satisfy
and
as
. The points
are
-close for each
via some existing real constant
in Definition 2.2. Also, the pairs of Cauchy sequences
,
have subsequences
which are
-close via a real constant
in Definition 2.2 for any given
and some integer
.
Proof.
First note that Theorem 3.1(i)–(iii) is independent of the above modification. Note also that now on a subset of
infinite discrete measure so that (3.8)–(3.12) still hold except that
is not guaranteed when
(last part of Theorem 3.1(iv), and Corollary 3.3), since
for
belonging to some proper nonempty subset of
. It still holds that
. Property (i) has been proven. Now, note from Corollary 3.2 that from Theorem 3.1 there is a set
of
points each being a fixed point of the restricted self-map
, for all
under the pairwise constraints
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ22_HTML.gif)
which are necessarily in disjoint adjacent sets since the distances between all the sets are a constant and
. Then the
-distance
of any pair
converges to a constant distance
. Then, there is a convergent sequence
of points in
verifying
as
since
for each
. Those sequences are Cauchy sequences since each convergent sequence in a metric space is a Cauchy sequence. Furthermore,
since
implies that
,
, and
. The remaining parts of Property (ii) concerning closeness according to Definition 2.2 follow the fact that the best proximity points of the self-map
of
are also fixed points of restricted composed maps to which Cauchy sequences of points converge and whose distance is
. Property (ii) has been proven.
Since the validity of Theorem 3.1(iii) is independent of the modification of Condition 1 to the weaker one Condition 2 implying the use of the sequence (see proof of Corollary 3.5), Condition 2 of Corollary 3.5 may be replaced with the following.
Condition 3.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ23_HTML.gif)
The above discussion may be discussed under any of the following replacements of Conditions 1–3.
Condition 4.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ24_HTML.gif)
Condition 5.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ25_HTML.gif)
Condition 6.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ26_HTML.gif)
Condition 7.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F510974/MediaObjects/13663_2010_Article_1295_Equ27_HTML.gif)
where are comparison functions, namely, monotone increasing satisfying
.
Thus, and
as a consequence of their above properties to be comparison functions. In addition,
satisfies the subadditive condition
. As a result of the above properties, note that:
-
(a)
Conditions 4 and 5 imply that
(3.23),
with the equality standing for some
and some
if and only if
, that is, the distance between relevant points in the upper-limits of the integral and between all the adjacent sets are zero.
-
(b)
Conditions 6 and 7 imply that
(3.24)
with the equality standing for some
if and only if
.
The following results follow.
Corollary 3.6.
Theorem 3.1and Corollaries 3.2–3.4 hold "mutatis-mutandis" under any of the -cyclic contractive Conditions 6 and 7.
Corollary 3.7.
Theorem 3.1and Corollaries 3.2–3.4 hold "mutatis-mutandis" under any of the -cyclic contractive Conditions 4 and 5 except that
if the distance between adjacent sets g is zero (i.e., all sets
have a nonempty intersection).
The proofs are direct as that of Theorem 3.1 (see also that of Corollary 3.5) by using the properties (3.24) for that of Corollary 3.6 and (3.23) for that of Corollary 3.7.
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Acknowledgments
The author is grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI2009-07197. He is also grateful to the Basque Government by its support through Grants IT378-10, SAIOTEK S-PE08UN15, and SAIOTEK S-PE09UN12. He thanks the anonymous reviewers by their suggestions to improve the paper.
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De la Sen, M. Fixed Point and Best Proximity Theorems under Two Classes of Integral-Type Contractive Conditions in Uniform Metric Spaces. Fixed Point Theory Appl 2010, 510974 (2010). https://doi.org/10.1155/2010/510974
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DOI: https://doi.org/10.1155/2010/510974