Abstract

In this paper, we study the existence and uniqueness of common fixed point of generalized α_s-ψ-Geraghty contractive mapping in the framework of b-metric spaces. We give an example to prove the validity of our conclusion.

1. Introduction

The celebrated Banach Contraction Principle ¹ is one of the most important fixed point results in all analysis. In 1993, the concept of b-metric space was firstly introduced by Czerwik in ² and the author got fixed point theorems in this type space. Since then, b-metric space was studied by various researchers in different directions. For instance, Aydi et al. in ³ obtained some common fixed point results for weak φ-contractions on b-metric spaces. Berinde in ⁴ extended a result of usual contractions in quasimetric spaces, to a class of ϕ-contractions. Inspired by ⁴, Pacurar ⁵ proved the existence and uniqueness of fixed point of φ-contractions. In 2018, Zada et al. ⁶ established fixed point results of rational contraction.

In the setting of a complete metric space, Geraghty extended the Banach contraction principle by considering an auxiliary function in ⁷. In 2012, the concept of admissible and contractive mappings were introduced by Samet et al. in ⁸ and the authors presented some fixed point theorems for them. After that, in metric space, Cho et al. ⁹ introduced the concept of Geraghty contraction type mappings and got some fixed point results of these mappings. Recently, Özer et al. ¹⁰ established the existence and uniqueness of the common fixed point theorem for self-maps in -algebra valued metric spaces and they obtained a result on the coupled fixed point theorems in ¹¹. In 2020, Özer et al. ¹² proved a kind of fixed point theorem on the complete -algebra valued -metric spaces. In ¹³, Ullah et al. studied some strong and -convergence results for mapping satisfying condition (E) in the setting of uniformly convex Busemann spaces. Kir et al. ¹⁴ established fixed point theorem for contractive mappings satisfying contraction of Almost Jaggi type.

In metric space, by using altering distance functions, Choudhury et al. ¹⁵ studied a generalization of the weak contraction principle as follows:

Theorem 1.1. ( ¹⁵) Suppose that a mapping , where is a metric space, satisfies the following condition:

for all where is an altering function, that is, is a nondecreasing and continuous function, and if and only if and is a continuous function. Then g has a unique fixed point.

Let be a metric space, and be a lower semicontinuous function. is said to be a generalized weakly contractive mapping if the following condition is satisfied:

where , and

and

Cho ¹⁶ generalized the results of Choudhury et al. ¹⁵ to extend weakly contractive mappings and obtained the following result:

Theorem 1.2. ( ¹⁶) Let be complete. If is a generalized weakly contractive mapping, then there exists a unique such that and .

Considering the contractive conditions which are constructed via auxiliary functions defined with the families , respectively:

and

Guan ¹⁷ proved common fixed point results for a new class of generalized weakly contractive mappings.

Theorem 1.3. ( ¹⁷) Let be a complete metric space with coefficient and be two given self-mappings satisfying that is injective and where is closed. Assume is a lower semicontinuous function. Let be a constant. If there exist functions such that

where

and

then and have a unique coincidence point in . Furthermore, they have a unique common fixed point provided that they are weakly compatible.

Throughout this paper, we aim to obtain common fixed point results for generalized Geraghty contractive mapping in the framework of metric space, which extended the results of Cho. Moreover, we present an example that elaborated the useability of our theorems.

2. Preliminaries

The following definitions and lemmas play the important role in obtaining our results. We state them as follows:

Definition 2.1. ( ²) Let be a nonempty set and be a given constant. A mapping is called a metric if and only if, for , the following conditions are satisfied:

(1) if and only if ;

(2)

(3)

In general, is said to be a metric space with coefficient .

Remark 2.2. Obviously, every metric space should be a metric space with . There are many examples of metric spaces which are not metric spaces. (see ²¹)

Example 2.3. ( ¹⁸) Let be a metric space, and , where is a constan. Then is a metric space with .

Definition 2.4. ( ⁴) Let be a metric space with parameter . Then a sequence in is said to be:

(1) converges to if and only if there exists such that as ;

(2) a Cauchy sequence if and only if when ;

As usual, a metric space is called complete if and only if each Cauchy sequence in this space is convergent.

Definition 2.5. ( ¹⁶) Let be two self-mappings. If , for some , then is called the coincidence point of and and is said to be the point of coincidence of and . Let denote the set of coincidence points of and .

Definition 2.6. ( ¹⁶) Let be two self-mappings. and is called weakly compatible if they commute at every coincidence point.

The following lemma is important for our main results.

Lemma 2.7 ( ¹⁵) Let be a metric space with parameter We assume that and converges to and , respectively. Then we obtain

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

3. Main Results

In this part, we firstly introduce some new definitions and concepts, then we define generalized Geraghty contractions. Moreover, we also provide an example to support our results.

A mapping is called lower semicontinuous if, for and is convergent to , we have

Let denote the class of fuctions and denote the class of the functions satisfying the following conditions:

(1) is non-decreasing,

(2) is continuous,

(3) iff .

Definition 3.1. The self-mappings are said to be orbital admissible and is a constant, if the following condition holds:

Definition 3.2. Let be two self-mappings on . The pair is called triangular orbital admissible and is a constant, if

(i) are orbital admissible;

(ii) and imply .

Lemma 3.3. ²² Let be two self-mappings on such that is triangular orbital admissible. Suppose that there exists such that . Define in by where . The for with , we have .

Definition 3.4. Let be a metric space with coefficient , and let two self-mappings. Assume that and is a lower semicontinuous function and is an arbitrary constant. The mappings is said to be generalized Geraghty contractions, if there exist and satisfying

(1)

for all with and , where

Let be a complete metric space with parameter and be a function. Then

For all , one can get ;

For all , one can get that or .

Theorem 3.5. Let be a complete metric space with coefficient and be generalized Geraghty contractions and or is continuous. If the following conditions are satisfied:

(i) are triangular orbital admissible,

(ii) there is with satisfying ,

(iii) properties and are satisfied.

Then and possess a unique common fixed point. Proof. Let . Define a sequence in by where . Firstly, we show that and have at most one common fixed point. If not, there exist and such that It follows that . According to the property of , we have , applying (1) with and , we obtain

(2)

where

and

It follows from (2) that

which implies that . That is, and . Hence, the pair has at most one common fixed point.

We suppose that for . If not, for some , , by assumption (ii) and Lemma 3.3, we have , and from (1), we obtain

(3)

where

and

By means of definition 3.4, we know that . By virtue of (3) and above inequalities, we have

which implies that

That is, . Thus is a common fixed point of and . If , then the proof is too similar to the case , one can show that is a common fixed point of and .

Now take for each . Letting and in (1), as the same arguments, we obtain

(4)

where

(5)

and

(6)

If for some ,

then it follows from (4), (5) and (6) that

which yields that

That is, , a contradiction. Therefore,

(7)

for all . Using similar arguments, we get

Therefore, is a non-increasing sequence and there exists a such that

If , by virtue of (4), (5), (6) and (7), one can obtain that

(8)

Taking the limit as in (8), we get

which gives a contradiction. It follows that

and which yields that

Next, we shall prove that is a Cauchy sequence in . Obviously, it is sufficient to show that is Cauchy. Assume that is not Cauchy. It follows that there exists for which one can choose and of satisfying is the smallest index for which ,

By the triangle inequality in metric space, we can deduce that

(9)

(10)

(11)

(12)

Letting and in (1), by Lemma 3.3, we know that , so we obtain

(13)

Here,

and

It follows from (9)-(12) that

(14)

and

(15)

By virtue of (13), (14) and (15), we have

which is a contradiction. Hence, is Cauchy. It follows from the completeness of that there exists a in such that

(16)

Considering the definition of , we deduce that

Next we shall prove that if one of the mappings and is continuous, then . Without loss of generality, one can assume that is a continuous mapping. From (16), one can deduce that

That is, is a fixed point of .

Using the property of , we obtian . If , by the contractive conditions (1), we get

(17)

where

and

It follows from (17) that

Hence, , that is, and , a contradiction. This implies that is the unique common fixed point of and . This completes the proof.

Example 3.6. Let and for. Define mappings by

and

Put by

Define mappings and with . Let and .

For such that , we deduce that . So we have imply , that is, are orbital admissible.

According to above inequalities, it suffices to verify that

It is easy to show that all conditions of Theorem 3.5 are satisfied with . Obviously, 0 is the unique common fixed point of and .

If in Theorem 3.5, we obtain immediately Corollary 3.7:

Corollary 3.7. Let be a complete metric space with and be given two self-mappings and one of and is continuous. If the following conditions are fulfilled:

(i) are triangular orbital admissible,

(ii) there is with satisfying ,

(iii) if there are and satisfying

for all with and where

(iv) properties and are satisfied.

Then and possess a unique common fixed point.

If we consider in Theorem 3.5, we get that

Corollary 3.8. Let be a complete metric space with and let be two given self-mappings and one of and is continuous. If the following conditions are satisfied:

(i) are triangular orbital admissible,

(ii) there is with satisfying ,

(iii) if there are and satisfying

for such that and ,

(iv) properties and are satisfied.

Then and possess a unique common fixed point.

Remark 3.9. If and in Corollary 3.8, taking in Theorem 2.1 of ²⁰, Roshan et al. established the existence theorem of common fixed point for mappings satisfying

where is a constant. For , it is easy to calculate that for ,

One can easily to obtain that Theorem 2.1 of ²⁰ can not be applied to get the existence of common fixed points of the mappings and in .

4. Application

In this section, we wish to study the existence of a solution for a pair of boundary value problems. Let denote the space of all continuous function defined on . Consider the following differential equations:

(18)

where are continuous functions. Associated with (18), the Green function is defined by

Define by for It is easy to show that is a complete metric space with coefficient. We define the operators by

and

for all and let be a given function.

Theorem 4.1. Suppose that

(i) are continuous,

(ii) there is with satisfying for all ,

(iii)For all and ,

imply

and

imply

(iv) properties and are satisfied,

(v) For all , ,

Then (18) have a unique solution .

Proof. Define by

It is easy to prove that are triangular orbital admissible. For , by virtue of assumptions (i)-(v), we have

which implies that

Therefore, letting , and , all the conditions of Corollary 3.8 are satisfied. As a result, the mapping and have a unique fixed point , which is a solution of (18).

5. Conclusions

In this manuscript, we introduced a new class of generalized Geraghty contractive mapping and established common fixed point results involving this new class of mappings in the framework of metric spaces. Furthermore, we presented examples that elaborated the useability of our results. Meanwhile, we provided an application to the existence of a solution for a pair of boundary value problems by means of one of our results.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work was supported by the Science and Research Project Foundation of Liaoning Province Education Department (Nos:LQN201902, LJC202003).

References