Open Access

Fixed Point and Best Proximity Theorems under Two Classes of Integral-Type Contractive Conditions in Uniform Metric Spaces

Fixed Point Theory and Applications20102010:510974

DOI: 10.1155/2010/510974

Received: 5 July 2010

Accepted: 14 October 2010

Published: 18 October 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., [16]). 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, [79]. 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 [1215] 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
(2.1)

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

(2.2)

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. (1)

    and are symmetrical.

     
  2. (2)

    If is of the form then is symmetrical.

     
  3. (3)

    If is nonempty and with , then is not symmetrical.

     

If, in-addition, then there are no in being -close.

Proof.
  1. (1)

    is symmetrical. is symmetrical. Assertion 2(1) has been proven.

     
  2. (2)

    for some is symmetrical. Assertion 2 (2) is proven.

     
  3. (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
(2.3)

which leads to the contradiction .

(2)If , and since , then
(2.4)

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
(2.5)

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
(3.1)
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
(3.2)

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
(3.3)
such that
(3.4)
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 :
(3.5)
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
(3.6)
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):
(3.7)

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
(3.8)
Note that . Define with , and
(3.9)
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
(3.10)
Then, since the distance between any two adjacent sets is a real constant , one gets the following from (3.8), and (3.10):
(3.11)
where since ,
(3.12)

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
(3.13)
Then, there is a set of card such that
(3.14)

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
(3.15)
for a new real sequence under the weaker constraints
(3.16)

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
(3.17)

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
(3.18)

The above discussion may be discussed under any of the following replacements of Conditions 1–3.

Condition 4.

One has
(3.19)

Condition 5.

One has
(3.20)

Condition 6.

One has
(3.21)

Condition 7.

One has
(3.22)

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:
  1. (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.

     
  2. (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.

Declarations

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.

Authors’ Affiliations

(1)
Institute of Research and Development of Processes, University of the Basque Country

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Copyright

© M. De la Sen. 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.