Fixed point results for generalized rational Geraghty contractive mappings in extended b-metric spaces

Abstract

In this article, we obtain fixed point results under some rational contractive conditions for mappings in the setup of ordered ρ-metric spaces. Also, three examples and an application to existence of local solutions for first-order periodic problems are provided to illustrate the results presented herein.

Keywords:

• Fixed point
• comparison function
• Geraghty condition
• ordered p-metric space

PUBLIC INTEREST STATEMENT

Metric fixed point theory is a powerful tool for solving several problems in various parts of mathematics and its applications. In this article, we propose a new generalized approach to fixed point problems using so-called p-metric (or extended b-metric) and more general contractive conditions. We show how the obtained results can be applied to prove the existence of solutions of some involved boundary valued problems for differential equations.

1. Introduction

The field of extension of metric space is a growing field these years. The famous extensions of the concept of metric spaces have been done by (Czerwik, 1993) and (Matthews, 1994) where they introduced and studied the concepts of -metric spaces and partial metric spaces, respectively. In this article, we introduce some type of Geraghty contractive mappings and comparison contractive mappings in extended -metric spaces and we establish some fixed point results in complete partially ordered extended -metric spaces. Our results generalize several comparable results in the literature.

2. Preliminaries

Recall (see (Czerwik, 1993)) that a -metric on a set is a generalization of standard metric, where the triangular inequality is replaced by

(2.1)

for some fixed . For more details on fixed point results and their applications in -metric spaces we refer the reader to (Ali, Kamran, & Postolache, 2017; Kamran, Postolache, Ali, & Kiran, 2016; Shatanawi, Pitea, & Lazovic, 2014). Motivated by the concept of -metric spaces, the following further generalization has been recently presented by Parvaneh and Ghoncheh.

Definition 2.1. (Parvaneh & Hosseini Ghoncheh) Let be a (nonempty) set. A function is an extended -metric (-metric, for short) if there exists a strictly increasing continuous function with for all and such that for all , the following conditions hold:

(P₁) iff ,

(P₂) ,

(P₃)

In this case, the pair is called a -metric space, or an extended -metric space.

It should be noted that each -metric is a -metric, with for some , while each metric is a -metric, with . More general examples of -metrics can be constructed using the following easy proposition.

Proposition 2.2. Let be a -metric space with coefficient and let where is a strictly increasing continuous function with for and . Then, is a -metric with .

Taking various functions in the previous proposition, we can obtain a lot of examples of -metrics. We state just a few of them which will be used later in the text.

Example 2.3. (1) If , we get and . Note that .

(2) If , we get and . Note that .

3) If , then and . Note that in this case , for , where is the Lambert -function (see, e.g. (Dence, 2013)).

4) If , then and , for .

Note that such functions and generate -metric spaces which are usually not -metric spaces. For instance, in the case (2) of the previous example (if ), it was shown in (Parvaneh, Dinmohammadi, & Kadelburg) that there is no such that is a -metric with parameter .

Definition 2.4. (Parvaneh & Hosseini Ghoncheh) Let be a -metric space. Then a sequence in is called:

(a) -convergent if there exists such that , as . In this case, we write ;

(b) -Cauchy if as .

(c) The -metric space is -complete if every -Cauchy sequence in -converges.

We will need the following simple lemma about the -convergent sequences.

Lemma 2.5. (Parvaneh & Hosseini Ghoncheh) Let be a -metric space with the function , and suppose that and -converge to, respectively. Then, we have

In particular, if , then . Moreover, for each we have

Some generalizations of Banach Contraction Principle were obtained, starting from (Jaggi, 1977), by using rational contractive conditions in various spaces. On the other hand, Ran and Reurings initiated the studying of fixed point results on partially ordered sets in (Ran & Reurings, 2004), which was widely used in a lot of subsequent papers. Geraghty-type contractive conditions (Geraghty, 1973) were also the matter of investigation of several researchers (see, e.g. (Duki, Kadelburg, & Radenovi, 2011)).

In this article, we introduce the notions of rational -Geraghty contractive mappings (in three versions) and rational - contractive mappings in -metric spaces and we establish some fixed point results in complete partially ordered -metric spaces. Our results generalize several comparable results in the literature. Also, three examples and an application to existence of local solutions for first-order periodic problems are provided to illustrate the results presented herein.

3. Main results

In the rest of the article, will always be a -metric space with function .

3.1. Fixed point results via generalized Geraghty functions

Let denote the class of all functions satisfying the following condition:

Definition 3.1. Let be a partially ordered -metric space. A mapping is called a rational Geraghty contraction of type I if there exists such that

(3.1)

holds for all with , where

A partially ordered -metric space is said to have the sequential limit comparison property (s.l.c. property) if for every non-decreasing sequence in , the convergence of to some yields that for all .

Theorem 3.2. Let be a partially ordered -complete -metric space. Let be a nondecreasing mapping with respect to such that there exists an element with . Suppose that is a rational Geraghty contraction of type . If

(I) is continuous, or

(II) has the s.l.c. property,

then has a fixed point. Moreover, the set of fixed points of is well ordered if and only if has a unique fixed point.

Proof. Let for all . Since and is a nondecreasing mapping, we obtain by induction that

If for some , there is nothing to prove. So, we will assume that for all . We will realize the proof in the following steps.

Step I: We will show that . Since for each , then by (3.1) we have

(3.2)

where

If , then from (3.2) we would have

which is a contradiction. Hence, . So, from (3.2),

(3.3)

Since is a decreasing sequence, there exists such that . We will prove that . Suppose on contrary that . Then, letting in (3.3) we have

which implies that . Now, as we conclude that a contradiction. That is,

(3.4)

Step II: Now, we prove that the sequence is a -Cauchy sequence. Suppose the contrary. Then there exists for which we can find two subsequences and of such that is the smallest index for which

(3.5)

This means that

From (3.4) and using the triangular inequality, we get

By taking the upper limit as , we get

From the definition of and the above limits,

Now, from (3.1) and the above inequalities, we have

which implies that . Now, as we conclude that . Consequently,

a contradiction to (3.5). Therefore, is a -Cauchy sequence. -completeness of yields that -converges to a point .

Step III: is a fixed point of .

First, let be continuous. Then, we have

Now, let (II) hold. Using the assumption on we have . By Lemma 2.5

where

Therefore, from the above relations, we deduce that , so, .

Finally, suppose that the set of fixed point of is well ordered. Assume on contrary, that and are two fixed points of such that . Then, for example , and by (3.1), we have

Since

we get , a contradiction. Hence, , and the fixed point of is unique.

Definition 3.3. Let be a partially ordered -metric space. A mapping is called a rational Geraghty contraction of type II if there exists such that

for all with , where

Theorem 3.4. Let be a partially ordered -complete -metric space. Let be a nondecreasing mapping with respect to such that there exists an element with . Suppose that is a rational Geraghty contractive mapping of type . If

(I) is continuous, or

(II) has the s.l.c. property.

Then has a fixed point. Moreover, the set of fixed points of is well ordered if and only if has a unique fixed point.

Proof. The method of proof is similar to the proof of Theorem 6 of (Zabihi & Razani, 2014) and the previous theorem.

Definition 3.5. Let be a partially ordered -metric space. A mapping is called a rational Geraghty contraction of type III if there exists such that

(3.6)

for all with , where

Theorem 3.6. Let be a partially ordered -complete -metric space. Let be a nondecreasing mapping with respect to such that there exists an element with . Suppose that is a rational Geraghty contractive mapping of type . If

(I) is continuous, or

(II) has the s.l.c. property,

then has a fixed point. Moreover, the set of fixed points of is well ordered if and only if has a unique fixed point.

Proof. Put .

Step I: We will show that . Since for each , then by (3.6) we have

because

Therefore, is decreasing. Similar to what have been done in Theorem 3.2, we have

(3.7)

Step II: Now, we prove that the sequence is a -Cauchy sequence. Suppose the contrary. Then there exists for which we can find two subsequences and of such that is the smallest index for which

(3.8)

This means that

(3.9)

From (3.8) and using the triangular inequality, we get

By taking the upper limit as , we get

Using the triangular inequality, we have

Taking the upper limit as in the above inequality and using (3.7) and (3.9) we get

(3.10)

Again, using the triangular inequality, we have

Taking the upper limit as in the above inequality and using (3.9) we get

From the definition of and the above limits,

Now, from (3.6) and the above inequalities, we have

which implies that . Now, as we conclude that is a p-Cauchy sequence. p-completeness of yields that p-converges to a point .

The rest of the proof is done in a similar manner as in Theorem 3.2.

Choosing different -metrics, as well as functions , we can obtain various corollaries from the above theorems. For example, using the -metric space mentioned in Example 2.3.(3) and an appropriate constant function , we obtain from Theorems 3.2 and 3.4 the following two corollaries.

Corollary 3.7. Let be a partially ordered -complete -metric space with parameter , and let be an increasing mapping with respect to such that there exists an element with . Suppose that

for all with , where , , and

or

If is continuous, or has the s.l.c. property, then has a fixed point.

Corollary 3.8. Let be a partially ordered -complete -metric space with parameter , and let be an increasing mapping with respect to such that there exists an element with . Suppose that

or

for all with , where and , . If is continuous, or has the s.l.c. property, then has a fixed point.

Consider now the -metric defined as in Example 2.3.(1) (for simplicity, take ) and the function given as

where and (recall that , and ). Then we obtain the following corollary of Theorems 3.2 and 3.4.

Corollary 3.9. Let be a partially ordered complete metric space, and let be an increasing mapping with respect to such that there exists an element with . Suppose that

(3.11)

for some and all with , where

(3.12)

or

If is continuous, or has the s.l.c. property, then has a fixed point.

3.2. Fixed point results via comparison functions

Let be the family of all nondecreasing functions such that

for all .

Lemma 3.10. If , then the following are satisfied:

(a) for all;

(b) .

Theorem 3.11. Let be a partially ordered -complete -metric space, and let be an increasing mapping with respect to such that there exists an element with . Suppose that

(3.13)

for some and for all with , where

If is continuous, or has the s.l.c. property, then has a fixed point. Moreover, the set of fixed points of is well ordered if and only if has one and only one fixed point.

Proof. Repeating the beginning of the proof of Theorem 3.2, we obtain the increasing sequence , such that and, this time,

By induction, we get that

As , we conclude that

Now, we prove that the sequence is a -Cauchy sequence. Suppose the contrary, i.e. Then, again as in Theorem 3.2, there exists for which we can find two subsequences and of such that is the smallest index for which

This means that

Obviously,

From the definition of and the above limits we can conclude that

Now, from (3.13) and the above inequalities, we have

which is a contradiction. Consequently, is a -Cauchy sequence.

-completeness of yields that -converges to a point . If is continuous, we have

and is a fixed point of .

Let have the s.l.c. property. Then we have . By (3.13) we have

(3.14)

where

Letting , we get

(3.15)

Again, taking the upper limit as in (3.14) and using Lemma 2.5 and (3.15) we get

So, we get , i.e. .    □

Remark 3.12. In Theorem 3.11, we can replace by the expressions used in Theorems 3.4 or 3.6.

Similarly as for Theorems 3.2, 3.4 and 3.6, we can get several consequences of Theorem 3.11, using various -metrics and/or various functions . We state just the following, based on Example 2.3.(4).

Corollary 3.13. Let be a partially ordered -complete -metric space with parameter , and let be an increasing mapping with respect to such that there exists an element with . Suppose that

where

for some and all with . If is continuous, or has the s.l.c. property, then has a fixed point. Moreover, the set of fixed points of is well ordered if and only if has one and only one fixed point.

Remark 3.14. As any -metric is a -metric with for all , so our results modify the obtained results in (Shahkoohi & Razani, 2014; Zabihi & Razani, 2014) and several other articles.

4. Examples

Example 4.1. Let be equipped with the -metric for all , where , with (Example 2.3.(1)).

Define a relation on by iff , a mapping by

and a function by . For all with , we have:

Therefore,

So, from Theorem 3.2, has a fixed point.

Example 4.2. Let be equipped with the -metric generated from the standard metric as in Corollary 3.9, i.e. take . Define a partial order on by

and consider the mapping given by

Then, is a partially ordered -complete -metric space and is a nondecreasing mapping. Take the function given by

where (i.e. take in Corollary 3.9). We will check the condition (3.11) of this corollary. Considering elements with , the following cases are nontrivial.

(1) , . Then we have , and

(note that is an increasing function for ).

(2) , . Then we have , and

(3) , . Then we have , and

Hence, all the conditions of Corollary 3.9 are fulfilled and the mapping has a (unique) fixed point (which is ).

Note that the same conclusion could not be obtained if the space without partial order were used. Indeed, in this case we should have also the following case to consider.

(4) , . Then and

and it is trivial to check that the condition (3.11) does not hold. In fact, none of the known fixed point results can be used to obtain the conclusion in this case.

Example 4.3. Let be also equipped with the -metric for all , where . Define a relation on by iff , a mapping by

and a function by . It is easy to see that and for all .

For all with , by Mean Value Theorem, we have

So, from Theorem 3.11, has a fixed point.

5. Application to existence of local solutions for first-order periodic problems

In this section, we present an application to existence of a solution for a periodic problem which is a consequence of Theorem 3.11. This kind of application first appeared in (Nieto & Rodríguez-López, 2007).

Let be the set of all real continuous functions on where . Obviously, this space with the -metric given by

for all is a -complete -metric space with . Secondly, can also be equipped with a partial order given by

Moreover, as in (Nieto & Rodríguez-López, 2007), it can be proved that enjoys the s.l.c. property.

Consider the following first-order periodic boundary value problem

(5.1)

where and is a given continuous function. A lower solution for (5.1) is a function such that

where .

Assume that there exists such that, for all and , we have

Problem (5.1) can be rewritten as

where . It is well known that this problem is equivalent to the integral equation

where is the Green’s function given as

Now define an operator by

The mapping is nondecreasing (Harjani & Sadarangani, 2009). Note that if is a fixed point of then is a solution of (5.1).

Let . Then we have

where from , where

Finally, let be a lower solution for (5.1). In (Harjani & Sadarangani, 2009), it was shown that .

Hence, the hypotheses of Theorem 3.11 are satisfied with . Therefore, there exists a fixed point such that . This is then a solution of problem (5.1).

6. Conclusion

Taking , our obtained results coincide with the results in usual -metric spaces and taking , the obtained results coincide with the results in usual metric spaces.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Vahid Parvaneh

Vahid Parvaneh The authors of this article have been working on problems in metric fixed point theory for several years, together or with some other coauthors. Various results were obtained in different setups, generalizing some of the well-known theorems of this theory. In particular, V. Parvaneh, together with Ghoncheh, introduced a new environment, called p-metric spaces, which was further developed to obtain various new fixed point results. It is the area of research in which several new results are expected.