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  • Research Article
  • Open Access

Some Fixed-Point Theorems for Multivalued Monotone Mappings in Ordered Uniform Space

Fixed Point Theory and Applications20112011:186237

https://doi.org/10.1155/2011/186237

  • Received: 22 September 2010
  • Accepted: 8 March 2011
  • Published:

Abstract

We use the order relation on uniform spaces defined by Altun and Imdad (2009) to prove some new fixed-point and coupled fixed-point theorems for multivalued monotone mappings in ordered uniform spaces.

Keywords

  • Distance Function
  • Order Relation
  • Lower Semicontinuous
  • Multivalued Mapping
  • Uniform Space

1. Introduction

There exists considerable literature of fixed-point theory dealing with results on fixed or common fixed-points in uniform space (e.g., between [114]). But the majority of these results are proved for contractive or contractive type mapping (notice from the cited references). Also some fixed-point and coupled fixed-point theorems in partially ordered metric spaces are given in [1520]. Recently, Aamri and El Moutawakil [2] have introduced the concept of -distance function on uniform spaces and utilize it to improve some well-known results of the existing literature involving both -contractive or -expansive mappings. Lately, Altun and Imdad [21] have introduced a partial ordering on uniform spaces utilizing -distance function and have used the same to prove a fixed-point theorem for single-valued nondecreasing mappings on ordered uniform spaces. In this paper, we use the partial ordering on uniform spaces which is defined by [21], so we prove some fixed-point theorems of multivalued monotone mappings and some coupled fixed-point theorems of multivalued mappings which are given for ordered metric spaces in [22] on ordered uniform spaces.

Now, we recall some relevant definitions and properties from the foundation of uniform spaces. We call a pair to be a uniform space which consists of a nonempty set together with an uniformity wherein the latter begins with a special kind of filter on whose all elements contain the diagonal . If and , then and are said to be -close. Also a sequence in , is said to be a Cauchy sequence with regard to uniformity if for any , there exists such that and are -close for . An uniformity defines a unique topology on for which the neighborhoods of are the sets when runs over .

A uniform space is said to be Hausdorff if and only if the intersection of all the reduces to diagonal of , that is, for implies . Notice that Hausdorffness of the topology induced by the uniformity guarantees the uniqueness of limit of a sequence in uniform spaces. An element of uniformity is said to be symmetrical if . Since each contains a symmetrical and if then and are both and -close and then one may assume that each is symmetrical. When topological concepts are mentioned in the context of a uniform space , they are naturally interpreted with respect to the topological space .

2. Preliminaries

We will require the following definitions and lemmas in the sequel.

Definition 2.1 (see [2]).

Let be a uniform space. A function is said to be an -distance if

for any , there exists , such that and for some imply ,

, for all .

The following lemma embodies some useful properties of -distance.

Lemma 2.2 (see [1, 2]).

Let be a Hausdorff uniform space and be an -distance on . Let and be arbitrary sequences in and , be sequences in converging to 0. Then, for , the following holds:

(a)if and for all , then . In particular, if and , then ,

(b)if and for all , then converges to ,

(c)if for all , then is a Cauchy sequence in .

Let be a uniform space equipped with -distance . A sequence in is -Cauchy if it satisfies the usual metric condition. There are several concepts of completeness in this setting.

Definition 2.3 (see [1, 2]).

Let be a uniform space and be an -distance on . Then

(i) said to be -complete if for every -Cauchy sequence there exists with ,

(ii) is said to be -Cauchy complete if for every -Cauchy sequence there exists with with respect to ,

(iii) is -continuous if implies
(2.1)

(iv) is -continuous if with respect to implies with respect to .

Remark 2.4 (see [2]).

Let be a Hausdorff uniform space and let be a -Cauchy sequence. Suppose that is -complete, then there exists such that . Then Lemma 2.2(b) gives that with respect to the topology which shows that -completeness implies -Cauchy completeness.

Lemma 2.5 (see [15]).

Let be a Hausdorff uniform space, be -distance on and . Define the relation " " on as follows:
(2.2)

Then " " is a (partial) order on induced by .

3. The Fixed-Point Theorems of Multivalued Mappings

Theorem 3.1.

Let a Hausdorff uniform space and is an -distance on , be a function which is bounded below and " " the order introduced by . Let be also a -Cauchy complete space, be a multivalued mapping, and . Suppose that:

(i) is upper semicontinuous, that is, and with and , implies ,

(ii) ,

(iii)for each , .

Then has a fixed-point and there exists a sequence with
(3.1)

such that . Moreover if is lower semicontinuous, then for all .

Proof.

By the condition (ii), take . From (iii), there exist and . Again from (iii), there exist . Thus .

Continuing this procedure we get a sequence satisfying
(3.2)
So by the definition of " ", we have , that is, the sequence is a nonincreasing sequence in . Since is bounded from below, is convergent and hence it is Cauchy, that is, for all , there exists such that for all we have . Since , we have or . Therefore,
(3.3)

which shows that (in view of Lemma 2.2(c)) that is -Cauchy sequence. By the -Cauchy completeness of , converges to . Since is upper semicontinuous, .

Moreover, when is lower semicontinuous, for each
(3.4)

So , for all .

Similarly, we can prove the following.

Theorem 3.2.

Let a Hausdorff uniform space and an -distance on , be a function which is bounded above and " " the order introduced by . Let be also a -Cauchy complete space, be a multivalued mapping, and . Suppose that

(i) is upper semicontinuous, that is, and with and , implies ,

(ii) ,

(iii)for each , .

Then has a fixed-point and there exists a sequence with
(3.5)

such that . Moreover, if is upper semicontinuous, then for all .

Corollary 3.3.

Let a Hausdorff uniform space and is an -distance on , be a function which is bounded below and " " the order introduced by . Let be also a -Cauchy complete space, be a multivalued mapping and . Suppose that:

(i) is upper semicontinuous, that is, and with and , implies ,

(ii) satisfies the monotonic condition: for any , with and any , there exists such that ,

(iii)there exists an such that .

Then has a fixed-point and there exists a sequence with
(3.6)

such that . Moreover if is lower semicontinuous, then for all .

Proof.

By (iii), . For , take and . By the monotonicity of , there exists such that . So , and . The conclusion follows from Theorem 3.1.

Corollary 3.4.

Let a Hausdorff uniform space and is an -distance on , be a function which is bounded above and " " the order introduced by . Let be also a -Cauchy complete space, be a multivalued mapping and . Suppose that:

(i) is upper semicontinuous,

(ii) satisfies the monotonic condition; for any with and any , there exists such that ,

(iii)there exists an such that .

Then has a fixed-point and there exists a sequence with
(3.7)

such that . Moreover if is upper semicontinuous, then for all .

Corollary 3.5.

Let a Hausdorff uniform space and is an -distance on , be a function which is bounded below and " " the order introduced by . Let be also a -Cauchy complete space, be a map and . Suppose that:

(i) is -continuous,

(ii) ,

(iii)for each , .

Then has a fixed-point and the sequence
(3.8)

converges to . Moreover if is lower semicontinuous, then for all .

Corollary 3.6.

Let be a Hausdorff uniform space, is an -distance on , be a function which is bounded above, and " " the order introduced by . Let be also a -Cauchy complete space, be a map and . Suppose that:

(i) is -continuous,

(ii) ,

(iii)for each , .

Then has a fixed-point . And the sequence
(3.9)

converges to . Moreover, if is upper semicontinuous, then for all .

Corollary 3.7.

Let be a Hausdorff uniform space, is an -distance on , be a function which is bounded below, and " " the order introduced by . Let be also a -Cauchy complete space, be a map and . Suppose that:

(i) is -continuous,

(ii) is monotone increasing, that is, for we have ,

(iii)there exists an , with .

Then has a fixed-point and the sequence
(3.10)

converges to . Moreover if is lower semicontinuous, then for all .

Example 3.8.

Let and . Define as for all , , ve . Since definition of , and this show that the uniform space is a Hausdorff uniform space. On the other hand, , and for and thus is an -distance as it is a metric on . Next define , , . Since , therefore . But as therefore and . Again similarly and which show that this ordering is partial and hence is a partially ordered uniform space. Define as , and , then by a routine calculation one can verify that all the conditions of Corollary 3.7 are satisfied and has a fixed-point. Notice that which shows that is neither -contractive nor expansive, therefore the results of [2] are not applicable in the context of this example. Thus, this example demonstrates the utility of our result.

Corollary 3.9.

Let be a Hausdorff uniform space, is an -distance on , be a function which is bounded above and " " the order introduced by . Let be also a -Cauchy complete space and be a map. Suppose that

(i) is -continuous,

(ii) is monotone increasing, that is, for we have ,

(iii)there exists an with .

Then has a fixed-point . And the sequence
(3.11)

converges to . Moreover if is upper semicontinuous, then for all .

Theorem 3.10.

Let be a Hausdorff uniform space, is an -distance on , be a continuous function bounded below and " " the order introduced by . Let be also a -Cauchy complete space, be a multivalued mapping and . Suppose that

(i) satisfies the monotonic condition: for each and each there exists such that ,

(ii) is compact for each ,

(iii) .

Then has a fixed-point .

Proof.

We will prove that has a maximum element. Let be a totally ordered subset in , where is a directed set. For and , one has , which implies that for . Since is bounded below, is a convergence net in . From , we get that is a -cauchy net in . By the -Cauchy completeness of , let converge to in .

For given . So for all .

For , by the condition (i), for each , there exists a such that . By the compactness of , there exists a convergence subnet of . Suppose that converges to . Take such that implies .

We have
(3.12)
So for all and
(3.13)

So and this gives that . Hence we have proven that has an upper bound in .

By Zorn's Lemma, there exists a maximum element in . By the definition of , there exists a such that . By the condition (i), there exists a such that . Hence . Since is the maximum element in , it follows that and . So is a fixed-point of .

Theorem 3.11.

Let be a Hausdorff uniform space, is an -distance on , be a continuous function bounded above and " " the order introduced by . Let be also a -Cauchy complete space, be a multivalued mapping and . Suppose that

(i) satisfies the following condition; for each and , there exists such that ,

(ii) is compact for each ,

(iii) .

Then has a fixed-point.

Corollary 3.12.

Let be a Hausdorff uniform space, is an -distance on , be a continuous function bounded below and " " the order introduced by . Let be also a -Cauchy complete space and be a map. Suppose that;

(i) is monotone increasing, that is for , ,

(ii)there is an such that .

Then has a fixed-point.

Corollary 3.13.

Let be a Hausdorff uniform space, is an -distance on , be a continuous function bounded above and " " the order introduced by . Let be also a -Cauchy complete space and be a map. Suppose that;

(i) is monotone increasing, that is, for , ;

(ii)there is an such that .

Then has a fixed-point.

4. The Coupled Fixed-Point Theorems of Multivalued Mappings

Definition 4.1.

An element is called a coupled fixed-point of the multivalued mapping if , .

Theorem 4.2.

Let be a Hausdorff uniform space, is an -distance on , be a function bounded below and " " be the order in introduced by . Let be also a -Cauchy complete space, be a multivalued mapping, , , and , and . Suppose that:

(i) is upper semicontinuous, that is, , and , with , and implies ,

(ii) ,

(iii)for each , there is such that and .

Then has a coupled fixed-point , that is, and . And there exist two sequences and with
(4.1)

such that and .

Proof.

By the condition (ii), take . From (iii), there exist such that , and , . Again from (iii), there exist such that , and , .

Continuing this procedure we get two sequences and satisfying and
(4.2)
So
(4.3)
Hence,
(4.4)

From this we get that and are convergent sequences. By the definition of " " as in the proof of Theorem 3.1, it is easy to prove that and are -Cauchy sequences. Since is -Cauchy complete, let converge to and converge to . Since is upper semicontinuous, and . Hence is a coupled fixed-point of .

Corollary 4.3.

Let be a Hausdorff uniform space, is an -distance on , be a function bounded below, and " " be the order in introduced by . Let be also a -Cauchy complete space, be a mapping and and and . Suppose that;

(i) is -continuous,

(ii) ,

(iii)for each , and .

Then has a coupled fixed-point , that is, and . And there exist two sequences and with , , such that and .

Corollary 4.4.

Let be a Hausdorff uniform space, is an -distance on , be a function bounded below, and " " be the order in introduced by . Let be also a -Cauchy complete space, be a mapping and and and . Suppose that;

(i) is -continuous,

(ii) ,

(iii) is mixed monotone, that is for each and , .

Then has a coupled fixed-point . And there exist two sequences and with , , such that and .

Theorem 4.5.

Let be a Hausdorff uniform space, is an -distance on , be a continuous function, and " " be the order in introduced by . Let be also a -Cauchy complete space, be a multivalued mapping, , , and , and . Suppose that;

(i) is mixed monotone, that is, for , and , , there exist , such that , ,

(ii) ,

(iii) is compact for each .

Then has a coupled fixed-point.

Proof.

By (ii), there exists with , and . Let , , and . Then . Define the order relation " " in by
(4.5)

It is easy to prove that becomes an ordered space.

We will prove that has a maximum element. Let be a totally ordered subset in , where is a directed set. For and , one has . So and , which implies that
(4.6)

for .

Since and are convergence nets in . From
(4.7)
we get that and are -Cauchy nets in . By the -Cauchy completeness of , let convergence to and convergence to in . For given ,
(4.8)

So and for all .

For , by the condition (i), for each with and with , there exist and such that and . By the compactness of and , there exist convergence subnets of and of . Suppose that converges to and converges to ). Take , such that implies . We have
(4.9)
So and for all . And
(4.10)

So and , this gives that . Hence we have proven that has an upper bound in .

By Zorn's lemma, there exists a maximum element in . By the definition of , there exist , , such that , and , . By the condition (i) there exist , such that and . Hence and . Since is maximum element in , it follows that , and it follows that and . So is a coupled fixed-point of .

Corollary 4.6.

Let be a Hausdorff uniform space, is an -distance on , be a continuous function, and " " be the order in introduced by . Let be also a -Cauchy complete space and be a mapping. Suppose that;

(i) is mixed monotone, that is for , and ,

(ii)there exist such that and .

Then has a coupled fixed-point.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, University of Gazi, Teknikokullar, 06500 Ankara, Turkey

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