Fixed Point Theorems for Generalized Weakly Contractive Condition in Ordered Metric Spaces

Abstract

Fixed point results with the concept of generalized weakly contractive conditions in complete ordered metric spaces are derived. These results generalize the existing fixed point results in the literature.

1. Introduction and Preliminaries

There are a lot of generalizations of the Banach contraction mapping principle in the literature. One of the most interesting of them is the result of Khan et al. [1]. They addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function.

A function is called an altering distance function if is continuous, nondecreasing, and holds.

Khan et al. [1] given the following result.

Theorem 1.1.

Let be a complete metric space, let be an altering distance function, and let be a self-mapping which satisfies the following inequality:

(1.1)

for all and for some . Then has a unique fixed point.

In fact, Khan et al. [1] proved a more general theorem of which the above result is a corollary. Another generalization of the contraction principle was suggested by Alber and Guerre-Delabriere [2] in Hilbert Spaces by introducing the concept of weakly contractive mappings.

A self-mapping on a metric space is called weakly contractive if for each ,

(1.2)

where is positive on and .

Rhoades [3] showed that most results of [2] are still valid for any Banach space. Also, Rhoades [3] proved the following very interesting fixed point theorem which contains contractions as special case .

Theorem 1.2.

Let be a complete metric space. If is a weakly contractive mapping, and in addition, is continuous and nondecreasing function, then has a unique fixed point.

In fact, Alber and Guerre-Delabriere [2] assumed an additional condition on which is . But Rhoades [3] obtained the result noted in Theorem 1.2 without using this particular assumption. Also, the weak contractions are closely related to maps of Boyd and Wong [4] and Reich type [5]. Namely, if is a lower semicontinuous function from the right, then is an upper semicontinuous function from the right, and moreover, (1.2) turns into . Therefore, the weak contraction is of Boyd and Wong type. And if we define for and , then (1.2) is replaced by . Therefore, the weak contraction becomes a Reich-type one.

Recently, the following generalized result was given by Dutta and Choudhury [6] combining Theorem 1.1 and Theorem 1.2.

Theorem 1.3.

Let be a complete metric space, and let be a self-mapping satisfying the inequality

(1.3)

for all , where , are both continuous and nondecreasing functions with if and only if . Then, has a unique fixed point.

Also, Zhang and Song [7] given the following generalized version of Theorem 1.2.

Theorem 1.4.

Let be a complete metric space, and let , be two mappings such that for each ,

(1.4)

where is a lower semicontinuous function with for and ,

(1.5)

Then, there exists a unique point such that .

Very recently, Abbas and Doric [8] and Abbas and Ali Khan [9] have obtained common fixed points of four and two mappings, respectively, which satisfy generalized weak contractive condition.

In recent years, many results appeared related to fixed point theorem in complete metric spaces endowed with a partial ordering in the literature [10–25]. Most of them are a hybrid of two fundamental principle: Banach contraction theorem and the weakly contractive condition. Indeed, they deal with a monotone (either order-preserving or order-reversing) mapping satisfying, with some restriction, a classical contractive condition, and such that for some , either or , where is a self-map on metric space. The first result in this direction was given by Ran and Reurings [22, Theorem 2.1] who presented its applications to matrix equation. Subsequently, Nieto and Rodŕiguez-López [18] extended the result of Ran and Reurings [22] for nondecreasing mappings and applied to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

Further, Harjani and Sadarangani [26] proved the ordered version of Theorem 1.2, Amini-Harandi and Emami [12] proved the ordered version of Rich type fixed point theorem, and Harjani and Sadarangani [27] proved ordered version of Theorem 1.3.

The aim of this paper is to give a generalized ordered version of Theorem 1.4. We will do this using the concept of weakly increasing mapping mentioned by Altun and Simsek [11] (also see [28, 29]).

2. Main Results

We will begin with a single map. The following theorem is a generalized version of Theorems 2.1 and 2.2 of Harjani and Sadarangani [27].

Theorem 2.1.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that

(2.1)

where

(2.2)

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . Also, suppose that there exists with . If

(2.3)

or

(2.4)

holds. Then, has a fixed point.

Proof.

If , then the proof is completed. Suppose that . Now, since , and is nondecreasing, we have

(2.5)

Put , and so . If there exists such that , then it is clear that , and so we are finished. Now, we can suppose that

(2.6)

for all .

First, we will prove that .

From (2.2), we have for

(2.7)

Now, we claim that

(2.8)

for all . Suppose that this is not true; that is, there exists such that . Now, since , we can use the (2.1) for these elements, then we have

(2.9)

This implies , by the property of , we have , which this contradict to (2.6). Therefore, (2.8) is true, and so the sequence is nonincreasing and bounded below. Thus there exists such that . Now suppose that . Therefore from (2.2)

(2.10)

This implies

(2.11)

and so there exist and a subsequence of such that .

By the lower semicontinuity of we have

(2.12)

From (2.1), we have

(2.13)

and taking upper limit as , we have

(2.14)

that is, . Thus, by the property of , we have , which is a contradiction. Therefore, we have .

Next, we show that is Cauchy.

Suppose that this is not true. Then, there is an such that for an integer , there exist integers such that

(2.15)

For every integer , let be the least positive integer exceeding satisfying (2.15) and such that

(2.16)

Now,

(2.17)

Then, by (2.15) and (2.16), it follows that

(2.18)

Also, by the triangle inequality, we have

(2.19)

By using (2.18), we get

(2.20)

Now, by (2.2), we get

(2.21)

and taking upper limit as and using (2.18) and (2.20), we have

(2.22)

This implies that there exist and a subsequence of such that

(2.23)

By the lower semicontinuity of , we have

(2.24)

Now, by (2.1), we get

(2.25)

which is a contradiction. Thus, is a Cauchy sequence. From the completeness of , there exists such that as . If is continuous, then it is clear that . If (2.4) holds, then we have for all . Therefore, for all , we can use (2.1) for and . Since

(2.26)

and so , we have

(2.27)

By the property of , we have . Thus, the proof is complete.

The following corollary is a generalized version of Theorems 1.2 and 1.3 of Harjani and Sadarangani [26].

Corollary 2.2.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that

(2.28)

where

(2.29)

, is a lower semicontinuous functions, and if and only if . Also, suppose that there exists with . If

(2.30)

or

(2.31)

holds. Then, has a fixed point.

Remark 2.3.

In Theorem 1.1 [22], it is proved that if

(2.32)

then for every ,

(2.33)

where is the fixed point of such that

(2.34)

and hence, has a unique fixed point. If condition (2.32) fails, it is possible to find examples of functions with more than one fixed point. There exist some examples to illustrate this fact in [18].

Example 2.4.

Let , and consider a relation on as follows:

(2.35)

It is easy to see that is a partial order on . Let be Euclidean metric on . Now, define a self map of as follows:

(2.36)

Now, we claim that the condition (2.1) of Theorem 2.1 is satisfied with . Indeed, if , then . Therefore, since , then the condition (2.1) is satisfied. Again, if and , then and are not comparative. Now, if , then and

(2.37)

Also, it is easy to see that the other conditions of Theorem 2.1 are satisfied, and so has a fixed point in . Also, note that the weak contractive condition of Theorem 1.3 of this paper and Corollary 2.2 of [7] is not satisfied.

Now, we will give a common fixed point theorem for two maps. For this, we need the following definition, which is given in [28].

Definition 2.5.

Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all .

Note that two weakly increasing mappings need not be nondecreasing. There exist some examples to illustrate this fact in [11].

Theorem 2.6.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , are two weakly increasing mappings such that

(2.38)

for all comparable , where

(2.39)

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . If

(2.40)

or

(2.41)

or

(2.42)

holds. Then, and have a common fixed point.

Remark 2.7.

Note that in this theorem, we remove the condition "there exists an with " of Theorem 2.1. Again, we can consider the result of Remark 2.3 for this theorem.

Proof of Theorem 2.6.

First of all we show that if or has a fixed point, then it is a common fixed point of and . Indeed, let be a fixed point of . Now, assume . If we use (2.38) for , we have

(2.43)

which is a contradiction. Thus, , and so is a common fixed point of and . Similarly, if is a fixed point of , then it is also fixed point of . Now, let be an arbitrary point of . If , the proof is finished, so assume that . We can define a sequence in as follows:

(2.44)

Without lost of generality, we can suppose that the successive term of are different. Otherwise, we are again finished. Note that since and are weakly increasing, we have

(2.45)

and continuing this process, we have

(2.46)

Now, since and are comparable, then we can use (2.38) for these points, then we have

(2.47)

where

(2.48)

Now, if for some , then

(2.49)

and so, from (2.47) we have

(2.50)

which is a contradiction. So, we have for all . Similarly, we have for all . Therefore, we have for all

(2.51)

and so the sequence is nonincreasing and bounded below. Thus, there exists such that . This implies that . Suppose that . Therefore, from (2.39),

(2.52)

This implies , and so there exist and a subsequence of such that .

By the lower semicontinuity of , we have

(2.53)

Now, from (2.38), we have

(2.54)

and taking upper limit as , we have

(2.55)

which is a contradiction. Therefore, we have

(2.56)

Next, we show that is a Cauchy sequence. For this, it is sufficient to show that is a Cauchy sequence. Suppose it is not true. Then, we can find an such that for each even integer , there exist even integers such that

(2.57)

We may also assume that

(2.58)

by choosing to be smallest number exceeding for which (2.57) holds. Now, (2.56), (2.57), and (2.58) imply

(2.59)

and so

(2.60)

Also, by the triangular inequality,

(2.61)

Therefore, we get

(2.62)

On the other hand, since and are comparable, we can use the condition (2.38) for these points. Since

(2.63)

we have

(2.64)

This is a contradiction. Thus, is a Cauchy sequence in , so is a Cauchy sequence. Therefore, there exists a with .

If or is continuous hold, then clearly, . Now, suppose that (2.42) holds and . Since , then from (2.42), for all . Using (2.38), we have

(2.65)

or

(2.66)

so taking upper limit from the last inequality, we have

(2.67)

which is a contradiction. Thus, , and so .

Corollary 2.8.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , be two weakly increasing mappings such that

(2.68)

for all comparable , where is a continuous, nondecreasing, is lower semicontinuous functions, and if and only if . If

(2.69)

or

(2.70)

or

(2.71)

holds. Then, and have a common fixed point.

Corollary 2.9.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , be two weakly increasing mappings such that

(2.72)

for all comparable , where

(2.73)

, is a lower semicontinuous functions, and if and only if . If

(2.74)

or

(2.75)

or

(2.76)

holds. Then, and have a common fixed point.

3. Some Applications

In this section, we present some applications of previous sections, first we obtain some fixed point theorems for single mapping and pair of mappings satisfying a general contractive condition of integral type in complete partially ordered metric spaces. Second, we give an existence theorem for common solution of two integral equations.

Set is a Lebesgue integrable mapping which is summable and nonnegative and satisfies , for each .

Theorem 3.1.

Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that

(3.1)

where

(3.2)

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . Also, suppose that there exists with . If

(3.3)

or

(3.4)

holds. Then, has a fixed point.

Proof.

Define by , then is continuous and nondecreasing with . Thus, (3.1) becomes

(3.5)

which further can be written as

(3.6)

where and . Hence by Theorem 2.1 has unique fixed fixed point.

Theorem 3.2.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , be two weakly increasing mappings such that

(3.7)

for all comparable , where

(3.8)

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . If

(3.9)

or

(3.10)

or

(3.11)

holds. Then, and have a common fixed point.

Proof.

Define by , then is continuous and nondecreasing with . Thus, (3.7) becomes

(3.12)

which further can be written as

(3.13)

where and . Hence, Theorem 2.6 has unique fixed fixed point.

Now, consider the integral equations

(3.14)

Let be a partial order relation on .

Theorem 3.3.

Consider the integral equations (3.14).

(i) and are continuous,

(ii)for each ,

(3.15)

(iii)there exist a continuous function such that

(3.16)

for each and comparable ,

(iv).

Then, the integral equations (3.14) have a unique common solution in .

Proof.

Let with the usual supremum norm; that is, , for . Consider on the partial order defined by

(3.17)

Then, is a partially ordered set. Also, is a complete metric space. Moreover, for any increasing sequence in converging to , we have for any . Also, for every , there exists which is comparable to and [21].

Define , by

(3.18)

Now, from (ii), we have for all ,

(3.19)

Thus, we have and for all . This shows that and are weakly increasing. Also, for each comparable , we have

(3.20)

Hence,

(3.21)

Put . Therefore,

(3.22)

for each comparable . Therefore, all conditions of Corollary 2.8 are satisfied. Thus, the conclusion follows.