Weakly isotone increasing mappings and endpoints in partially ordered metric spaces

The aim of this work is to extend the notion of weakly isotone increasing mappings to multivalued and present common endpoint theorems for -weakly isotone increasing multivalued mappings satisfying generalized -weak contractive as well as almost contractive inequalities in complete partially ordered metric spaces. Examples are given in support of the new results obtained.

MSC: 47H10, 54H25, 54H10.

The Banach contraction principle [1] is a remarkable result in the metric fixed point theory. Over the years, it has been generalized in different directions and spaces by several mathematicians. In 1997, Alber and Guerre-Delabriere [2] introduced the concept of weak contraction in the following way.

Definition 1.1 Let be a metric space. A mapping is said to be weakly contractive provided that

where and is a continuous nondecreasing function such that if and only if .

Using the concept of weak contractiveness, they succeeded in establishing the existence of fixed points for such mappings in Hilbert spaces. Later on, Rhoades [3] proved that the results in [2] are also valid in complete metric spaces. He also proved the following fixed point theorem which is a generalization of the Banach contraction principle.

Theorem 1.2Letbe a complete metric space, and letbe a weakly contractive mapping. Thenhas a fixed point.

Weak contractive inequalities of the above type have been used to establish fixed point results in a number of subsequent works, some of which are noted in [4,5]. Since then, fixed point theory for single-valued as well as for multivalued weakly contractive type mappings was studied by many authors. Fixed point theorems for multivalued mappings are quite useful in Control theory and have been frequently used in solving problems in Economics and Game theory.

The development of a geometric fixed point theory for multifunctions was initiated by Nadler [6] in 1969. He used the concept of a Hausdorff metric ℋ to establish the multivalued contraction principle containing the Banach contraction principle as a special case as follows.

Theorem 1.3Letbe a complete metric space andbe a mapping fromintosuch that for all,

where. Thenhas a fixed point.

Since then, this discipline has been developed further, and many profound concepts and results have been established with considerable generality; see, for example, the works of Itoh and Takahashi [7], Mizoguchi and Takahashi [8], Hussain and Abbas [9], and references cited therein. Very recently, results on common fixed points for a pair of multivalued operators have been obtained by applying various types of contractive conditions; we refer the reader to [10-14]. In some cases, multivalued mapping defined on a nonempty set assumes a compact value for each x in . There are situations when, for each x in , is assumed to be a closed and bounded subset of . To prove the existence of a fixed point of such mappings, it is essential for mappings to satisfy certain contractive conditions which may involve the Hausdorff metric.

Let be a metric space, and let (resp. ) be the class of all nonempty (resp. nonempty bounded) subsets of . We define functions and as follows:

where denotes the set of all positive real numbers. For and , we write and , respectively. Clearly, . We appeal to the fact that if and only if for and

for . Obviously, for , reduces to the standard notion of the diameter of a set in a metric space :

for any subset .

A point is called a fixed point of a multivalued mapping if . If there exists a point such that , then x is called an endpoint of .

The Fixed Point Theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [[15], Theorem 2.1] who presented its applications to matrix equations. Subsequently, Nieto and Rodríguez-López [16] extended the result of [15] for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed point theorems in ordered metric spaces. In [17], Nashine et al. extended the results in [18] by using -weakly isotone increasing mappings and relaxing other conditions without taking into account any commutativity condition. Beg and Butt [19] studied set-valued mappings and proved common fixed point results for mappings satisfying implicit relation in a partially ordered metric space. Recently, Amini [20] proved endpoint theorems for multivalued mappings in a metric space. More recently, Choudhury and Metiya [21] as well as Nashine and Kadelburg [22] also proved fixed point theorems for multivalued mappings in the framework of a partially ordered metric space.

We will use the following terminology.

Definition 1.4 Let be a nonempty set. Then is called a partially metric space if:

(i) is a metric space,

(ii) is a partially ordered set.

Elements are called comparable if or holds.

Definition 1.5 ([19])

Let A and B be two nonempty subsets of a partially ordered set . The relation ⪯₁ between A and B is defined as follows:

The purpose of this paper is to prove the existence of a common endpoint for a pair of -weakly isotone increasing multivalued mappings under a generalized -weakly contractive condition and under a variant of so-called almost contractive conditions of Berinde [23] without using the continuity of any map and any commutativity condition in a complete ordered metric space. Our results generalize the results of Abbas and Ðorić [24], Choudhury and Metiya [21] and Hussain et al. [10] for more general contractive and weakly contractive conditions for a pair of weakly isotone increasing multivalued mappings. They also extend the results of Babu et al.[25], Berinde [23], Choudhury et al.[26] and Ćirić et al.[27] from single-valued mappings in metric spaces to multivalued mappings in ordered metric spaces. Also, the results on common fixed points of weakly isotone increasing mappings in [22] are modified to the results on common endpoints of -weakly isotone increasing mappings under suitable conditions. Examples are presented to show the usage of the results and, in particular, that the order can be crucial.

In this section, we prove common endpoint theorems for a pair of weakly isotone increasing multivalued mappings under a generalized -weak contractive condition. In order to formulate the results, we extend to multivalued mappings the notion of weakly isotone increasing mappings given by Vetro [[28], Definition 4.2].

Definition 2.1 Let be a partially ordered set and be two maps. The mapping is said to be -weakly isotone increasing if for all , and .

Note that, for single-valued mappings in particular, , is said to be -weakly isotone increasing [[28], Definition 2.2] (see also [29]) if for each we have .

Definition 2.2 ([24])

Two set-valued mappings are said to satisfy the property of generalized -weak contraction if the inequality

holds for all and for given functions , where

The main result of this section is as follows.

Theorem 2.3Letbe a complete partially ordered metric space, and letbe two set-valued mappings that satisfy the property of generalized-weak contraction for all comparable, where

(a) ψis a continuous nondecreasing function withif and only if,

(b) φis a lower semicontinuous function withif and only if.

Also, suppose thatis-weakly isotone increasing and there exists ansuch that. Assume the condition

Then there exists a common endpointofand, i.e. .

Proof First of all, we show that if or has an endpoint, then it is a common endpoint of and . Indeed, let, e.g., z be an endpoint of . If we use the inequality (2.1) for , we have

and we conclude that and . Therefore, z is a common endpoint of and .

We will define a sequence and prove that the limit point of that sequence is a unique common endpoint for and . For a given and a nonnegative integer n, let

and let

If or for some , then the proof is finished. So, assume for all n.

Since , can be chosen so that . Since is -weakly isotone increasing, it is ; in particular, can be chosen so that . Now, (since ); in particular, can be chosen so that .

Continuing this process, we conclude that can be an increasing sequence in :

The sequences and are convergent. Suppose that n is an odd number. Substituting and in (2.1) and using the properties of functions ψ and φ, we obtain

which implies that

Now, from (2.2) and from the triangle inequality for δ, we have

Now, if , then

From (2.5) and (2.6), it follows that

It, furthermore, implies that

a contradiction. So, we have

In a similar way, we can establish the inequality (2.7) when n is an even number. Therefore, the sequence defined in (2.4) is nonincreasing and bounded. Let when . From (2.7), we have

Passing to the (upper) limit as ,

and since φ is lower semicontinuous, we have

a contradiction unless . Hence,

From (2.8) and (2.4), it follows that

Next, we prove that the sequence is a Cauchy sequence. For this, we first prove that for each , there exists such that

We proceed by negation and suppose that the inequality in (2.9) is not true. That is, there exists for which we can find nonnegative integer sequences and such that is the smallest element of the sequence such that for each ,

This means that

From (2.10) and the triangle inequality for δ, we have

Passing to the limit as and using (2.8), we can conclude that

We note that

Using (2.8) and (2.11), we get

and from

using (2.8) and (2.12), we get

Also, from (2.2), (2.8) and (2.12), we have

Putting , in (2.1), we have

Passing to the (upper) limit as and using (2.12), (2.13), we get

a contradiction to . Therefore, the conclusion (2.9) is true. From the construction of the sequence , it follows that the same conclusion holds for . Thus, for each there exists such that

From (2.4) and (2.14), we conclude that is a Cauchy sequence in which is complete. So, there exists such that

To prove that u is an endpoint of , suppose that . From (2.3), we have for all . As the limit point u is independent of the choice of , we also get

From

we have as . Since

passing to the (upper) limit as and using (2.15), we obtain

which implies . Hence, and and this proves that u is an endpoint of and also an endpoint of . The proof is completed. □

If and are two single-valued mappings, then we obtain the following consequence.

Corollary 2.4Letbe a complete partially ordered metric space, and letbe two mappings that satisfy, for all comparable,

whereφ, ψare as in Theorem 2.3 and

Also, suppose thatis-weakly isotone increasing. If the condition (2.3) holds, thenandhave a common fixed point, i.e., .

Putting in Theorem 2.3, we obtain the following

Corollary 2.5Letbe a complete partially ordered metric space, and letbe a set-valued mapping that satisfies

for all comparable, where

and whereφ, ψare as in Theorem 2.3. Also, suppose thatfor alland there issuch that. If the condition (2.3) holds, then there exists an endpointof, i.e., that.

If is a single-valued mapping in Corollary 2.5, then we have the following

Corollary 2.6Letbe a complete partially ordered metric space, and letbe a mapping that satisfies, for all comparable,

whereφ, ψare as in Theorem 2.3 and

Also, suppose thatfor all. If the condition (2.3) holds, thenhas a fixed point, i.e., .

Remark 2.7 In [[15], Corollary 2.5], it was proved that if

then for every ,

where y is a fixed point of such that

and hence has a unique fixed point. If the condition (2.18) fails, it is possible to find examples of mappings with more than one fixed point (cf.[16]).

We illustrate the results of this section with two simple examples. The first one shows how a multivalued variant (Corollary 2.5) can be used. The other shows that (in the single-valued case) the use of order can be crucial.

Example 2.8 Let , where , , . Metric d is defined as so that , and . Order ⪯ is introduced by iff and , so that and , while B and C are incomparable.

Consider the mapping given by

and functions given by , . To prove that the condition (2.16) of Corollary 2.5 holds, it is enough to check that it is satisfied for , and for , (in the case when (2.16) is trivially satisfied).

If , , then and , , so (2.16) holds. If , , then

and

Hence, . Note also that holds for all (only the case is nontrivial, when , , and for , there is such that ). All other conditions of Corollary 2.5 are fulfilled and has an endpoint A.

Example 2.9 Consider the same partially ordered metric space as in the previous example and the mapping defined by

Let again be given by , . It is again easy to show that in the cases , , as well as , , the condition (2.17) of Corollary 2.6 is satisfied, and it follows that has a fixed point A. However, for (incomparable) points , , the condition (2.17) is not satisfied, and so the respective result in the metric space without order cannot be applied to reach the conclusion. Indeed, in this case, , ,

and .

In this section, we prove common endpoint theorems for -weakly isotone increasing multivalued mappings satisfying a variant of an almost contractive condition.

Theorem 3.1Letbe a complete partially ordered metric space. Assume that there is a continuous functionwithfor each, and thatare multivalued mappings such that

for all comparable, where, and

Also, suppose thatis-weakly isotone increasing and there exists ansuch that. If the condition (2.3) holds, thenandhave a common endpoint.

Proof First of all, we show that if or has an endpoint, then it is a common endpoint of and . Indeed, let z be an endpoint of and assume . If we use the inequality (3.1) for and the properties of φ, we have

a contradiction. Thus , and so z is a common endpoint of and .

Let be arbitrary. Define a sequence as follows:

If or for some , then the proof is finished. So, assume for all n.

Since , can be chosen so that . Since is -weakly isotone increasing, it is ; in particular, can be chosen so that . Now, (since ); in particular, can be chosen so that .

Continuing this process, we conclude that can be an increasing sequence in :

If there exists a positive integer N such that , then is a common endpoint of and . Hence, we shall assume that for all .

Now, we claim that for all , we have

From (3.4), we have that for all . Then from (3.1) with , and , , we get

By (3.2), we have

If , by (3.6) and using the fact that for all , we have

a contradiction.

If , we get

On the other hand, by the triangular inequality, we have

Thus, we have

which implies that

If , we get

Thus, in all cases, we have for all , . Similarly, we can prove that for all , . Therefore, we conclude that (3.5) holds.

Now, from (3.5), it follows that the sequence is decreasing. Therefore, there is some such that

We are able to prove that . In fact, by the triangular inequality, we get

By (3.5), we have

From (3.8), taking the upper limit as , we get

If we set

then clearly . As φ is continuous, taking the upper limit on both sides of (3.6), we get

Hence, by (3.7) and (3.9), we deduce

If we suppose that , then we have

a contradiction. Thus , and consequently,

From (3.3) and (3.10), it follows that

Now, we prove that is a Cauchy sequence. To this end, it is sufficient to verify that is a Cauchy sequence. Suppose, on the contrary, that it is not. Then there exists an such that for each even integer 2k there are even integers , with such that

For every even integer 2k, let be the smallest number exceeding satisfying the condition (3.12) for which

From (3.12), (3.13) and the triangular inequality, we have

Hence, by (3.10), it follows that

Now, from the triangular inequality, we have

Passing to the limit as and using (3.10) and (3.14), we get

On the other hand, we have

where

From

taking the upper limit as , using (3.10) and (3.14), we get

On the other hand, we have

and taking the lower limit as , we get

It follows that

and so

Now, using (3.10), (3.14), (3.15), (3.17) and the continuity of φ, we get

Passing to the limit as in (3.16), we obtain

a contradiction. Thus, the assumption (3.12) is wrong. Hence, is a Cauchy sequence. From the completeness of , there exists a such that

As the limit point z is independent of the choice of , we also get

Now, we show that z is a common endpoint of and .

Suppose, to the contrary, that . By the assumption (2.3), for all n. Then using the triangular inequality for δ and taking and in (3.1), we have

Passing to the limit as and using the properties of φ, we have

a contradiction. Hence, , and so . It follows that z is an endpoint of , and also of . This finishes the proof. □

Remark 3.2

(i) The condition

where

implies the condition (3.1).

(ii) The condition (3.19) is equivalent to the condition (3.1) if we suppose that φ is a non-decreasing function.

(iii) From Theorem 3.1 we can derive a corollary involving the condition (3.19).

(iv) Under the hypothesis that φ is a non-decreasing function, we can state many other corollaries using the equivalences established by Jachymski in [30] for single-valued mappings.

Example 3.3 Let be equipped with the standard metric d and order ⪯ given by

Consider the following mappings :

First, we check that is -weakly isotone increasing. Suppose that and . Then implies that and so . This means that for any , we have for all . Similarly, one can prove that for each , we have for all .

Let for and . Now, we check that the condition (3.1) holds for all . Consider the following two possibilities.

1. , i.e., . Denote , . Then

Hence, the condition (3.1) is satisfied.

2. , i.e., and for some . Then

Again, the condition (3.1) is satisfied. Thus, all the conditions of Theorem 3.1 are fulfilled, and and have an endpoint ().

Similar corollaries can be obtained as in the previous section. For example, putting in Theorem 3.1, we obtain immediately the following result.

Corollary 3.4Letbe a complete partially ordered metric space. Assume that there is a continuous functionwithfor each, and thatis a multivalued mapping such that

for all comparable, where, and

Also, suppose thatfor alland that there issuch that. If the condition (2.3) holds, thenhas an endpoint.

To conclude this section, we provide a sufficient condition to ensure the uniqueness of the endpoint in Theorem 3.1,

Theorem 3.5Adding to the hypotheses of Theorem 3.1 the condition

where ∘ denotes the composition of mappings, we obtain the uniqueness of the common endpoint ofand.

Proof Let z and be two common fixed points of and , that is,

It is immediate to show that for all , we have

Then

Hence, and the proof is completed. □

The authors declare they have no competing interests.

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

The first and fourth authors gratefully acknowledge the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University during this research. The third author is thankful to the Ministry of Science and Technological Development of Serbia.

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