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

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.

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 [1–14]). 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 [15–20]. 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.