A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

Hammad, Hasanen A.; De la Sen, Manuel

doi:https://doi.org/10.1155/2021/6880478

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6880478 | https://doi.org/10.1155/2021/6880478

A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

Hasanen A. Hammad¹and Manuel De la Sen²

Academic Editor: Maria L. Gandarias

Received12 Apr 2021

Revised16 Jun 2021

Accepted26 Jun 2021

Published12 Jul 2021

Abstract

In this manuscript, exciting fixed point results for a pair of multivalued mappings justifying rational Gupta-Saxena type -contractions in the setting of extended -metric-like spaces are established. The theoretical results have also been strengthened by some nontrivial examples. Finally, the theoretical results are used to study the existence of the solution of Fredholm integral equation which arises from the damped harmonic oscillator, to study initial value problem which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs).

1. Introduction

The importance of fixed-point (FP) theory increased after Banach introduced his principle [1]. It became an essential tool in nonlinear analysis. In particular, there are many important applications in which the FP method is involved such as economics, microbiology, and physical applications. This technique permeates fractional analysis and becomes a basic method in finding solutions to some fractional systems. For more details, we suggest the reader to read the books of Kilbas et al. [2], Samko et al. [3], Wang et al. [4], and Atangana and Baleanu [5]. Furthermore, this methodology is widely used in finding solutions of integral equations, fractional differential equations, and boundary value problems, for example, see [6–11].

Under Hausdorff metric space formalism, Nadler [12] introduced the notion of multivalued mapping as follows: Let be a metric space, represents the set of nonempty closed bounded subsets of , and let be a function given by where .

The function is called also the Hausdorff-Pompeiu metric. Consider then one can deduce that for all , (i)(ii)(iii)(iv)

In 2012, Wardowski [13] generalized the old principle of Banach to a broader kind. He called it -contraction and presented it in the following definition:

Definition 1. A nonlinear mapping described on the metric space is called a -contraction if there exist and so that where is the class of functions so that the assertions below hold:
(₁) for all if then
(₂) for all positive real sequence if and only if as
(₃) there exists so that

Many researchers have turned to generalize ordinary metric spaces to more comprehensive spaces such as -metric, metric-like, and -metric-like spaces. In the other word, Czerwik [14] was able to present -metric spaces, Alghmandi et al. [15] introduced metric-like and -metric-like spaces, and Hammad et al. [16] presented extended -metric like spaces. It should be noted that the definitions of these spaces are mentioned in the next section.

The concept of multivalued mapping and -contraction mapping in abstract spaces are combined, and fruitful FP results in this line are obtained. For instance, see Cosentino and Vetro [17], Sgroi and Vetro [18], and Ali et al. [19].

By the same approach, in this manuscript, some FP results under rational Gupta-Saxena type -contractions (for short, -contractions) are obtained in the setting of extended -metric-like space (for short, -spaces). Also, the results are supported by nontrivial examples. Finally, the theoretical results are applied to study the existence of the solution to the Fredholm integral equation which arises from the damped harmonic oscillator, to study the initial value problem (IVP) which arises from Newton’s law of cooling and to study infinite systems of fractional ODEs.

2. Preliminaries

This section discusses some extended spaces and their topological properties.

Definition 2 [15]. Let be a nonempty set. A function is called a metric-like on , if for all , the conditions below hold: (a)(b)(c)

Here, is named as a metric-like space.

Definition 3 [15]. A -metric-like on a nonempty set is a function so that for all and a constant , the following three conditions hold: (e)(f)(g)

Here, is called a -metric-like space with constant

For more details about the conversion of metric-like and -metric-like spaces, see [20–22].

Very recently, Hammad et al. [16] merged the results of metric-like with -metric-like spaces and formed a -space. Also, topological properties of this space were discussed, and some nice FP results of Wardowski and Suzuki contraction mappings were obtained. This space is introduced as follows:

Definition 4 [16]. Assume that is a nonempty set and is a given function. A function is called a modified -metric-like if for all :
()
()
()

Then, is called a -space.

Note that the class of -spaces is larger than the class of -metric-like spaces by replacing the constant of Definition 3 by a nonconstant function of Definition 4.

For more examples and applications via the mentioned space, see [16].

Definition 5 [16]. Let be a sequence in the -space . (i)If then is convergent to (ii) is called Cauchy if exists and is finite(iii)If for each Cauchy sequence , there is so that ; then, is called complete

Lemma 6 [4]. Suppose that is an extended -metric space and that and , there exists so that

It should be noted that Lemma 6 also satisfied in each of metric spaces [23] and -metric spaces [24]. Hence, it can be easily achieved in -spaces.

3. Main Theorem

We begin this part with the following definition:

Definition 7. Let be a -space with a function . Assume that are two multivalued mappings. We say that the pair is a -contraction if there exist and so that for all then where

Theorem 8. Let be a complete -space and let be a -contraction so that for all Then, and have a unique common FP.

Proof. Let and Assume that and by Lemma 6, there is so that Inductively, we let and by Lemma 6, we select so that Applying (4) and (7), we have If then This is a contradiction according to Therefore, Then, from (8), Similarly, we get Applying (10) and (11), we have Repeating consequently, one can obtain Similarly, By using (13) and (14), we have Taking the limit as in both sides of (15), we get It follows from that By (15), for we get for all Letting in (18) and applying (16) and (17), one can write From (19), there is so that for all It follows that Using (20) for , we have It should be noted that the series converges because which is convergent. Put and Thus, for inequality (20) implies that Passing we conclude that is a Cauchy sequence in but a -space is complete so there exists such that
Now, we want to show that is a common FP of and It is clear that Hence, Since is strictly increasing, therefore By adding to both sides and using (4), we have Since we have Since is strictly increasing, we get Letting in the above inequality, one can write This implies that So, we have Thus, Similarly, one can prove that Therefore, is a common fixed point of and In order to show the uniqueness, let be another common FP of and so that Then, by -contraction (4), one can obtain Based on the properties of and since we conclude that This means i.e., This finishes the proof.☐

Remark 9. The contraction condition of Theorem 8 is reduced to the following: (i)-contraction in -metric-like space, if (ii)-contraction in metric-like space, if (iii)-contraction in -spaces, if (iv)-contraction in -metric-like space, if and (v)-contraction in -metric-like space, if and

Corollary 10. Theorem 8 is still valid if one of the following forms is chosen for in Equation (5): (1)(2)(3)(4)(5)(6)

4. Illustrative Examples

In this section, we give some illustrative examples to support Theorem 8 and its consequences.

Example 1. Let Define by for all ; then, the pair is -space with Define and by and for all respectively. Now, we have Taking log to the base on both sides, we get

This implies that

Hence, the contraction condition (7) is fulfilled with and Thus, all stipulations of Theorem 8 are fulfilled, and is a unique common FP of and

Example 2. Assume that Define a distance function over a set by and by Then, forms a -space. Define multivalued mappings by for all and

Define the function by for all , , and

Now, to fulfill the conditions of Theorem 8, we discuss the following cases:

Case 1. If and consider

Now, consider

Therefore,

Thus, for any value of and Therefore,

Case 2. If and then and
Consider

Now, consider

Therefore,

Thus, for any value of and Therefore,

Thus, all requirements of Theorem 8 are fulfilled, and is a unique common FP of and

Example 3. Assume that Define a distance function over a set by and by Then, forms a -space. Define multivalued mappings by

Define the function by for all , , and

Case 1. If then and Consider

Now, consider

Therefore,

Thus,

Hence, which implies

Case 2. If then and Consider

Now, consider

Therefore,

Thus,

Hence, which implies

Case 3. If and then and Consider

Now, consider

Therefore,

Thus,

Hence, which implies

Case 4. If and then and Consider

Now, consider

Therefore,

Thus,

Hence, which implies

From the four cases, we conclude the contraction condition (iii) of Remark 9 is fulfilled and is a unique FP of

Example 4. Let be a sequence defined by

Suppose that and are functions described by and respectively, for all Then, is space. Define the multivalued mappings and by and for all respectively. Now, we have the following calculations for two cases:

Case 1. For each , and we have

Case 2. For each we get

Multiply on the both cases and taking , we have

Therefore, the condition (iii) of Remark 9 is fulfilled with and . Hence, has a unique FP

5. The Damped Harmonic Oscillator

Assume that we have a body of mass moving back and forth on the -axis around the equilibrium position and has a position at time (see Figure 1). The force subjected to by the spring gives from

In addition to the damping force that impedes the movement of the body, it gives from

From the second law of motion we find that where , and are positive constants. Previously, this force is called the force damping oscillation.

Now suppose that there is an additional time-dependent force that affects the body. Based on Newton’s second law, we can write

By Green’s function, Problem (76) is equivalent to the following Fredholm integral equation: where is a continuous function depends on and the continuous Green’s function for critically damping oscillation is described by and can be found in terms of , and

Suppose that is the set of all bounded real continuous functions defined on , equipped with where Define the function by Then, is a complete -space.

In order to study the existence of a solution to Problem (77), we shall consider the following stipulations: (a)There is a continuous function and for and so that for all where and is a multivalued operator defined on the beginning proof.(b) for some

Now, we present our main result in this part.

Theorem 11. Assume that is a complete -space. Then, Problem (77) has a unique solution provided that the stipulations (a) and (b) are fulfilled.

Proof. Define an operator by It should be noted that the unique FP of the mapping is a solution of the nonlinear Problem (77). Hence, it is a solution to the differential equation of the second degree (76).
For then by the hypotheses (a) and (b), we can get Hence for which yields Taking the for the base to both side, we have Describe the function by in (86), so that and we get where Therefore, all requirements of Theorem 11 under a special case (iii) (-contraction) are fulfilled. Then, there is a unique FP of the mapping which is a unique solution to Problem (77) and then to (76).☐

6. Newton’s Law of Cooling

Newton’s law of cooling is a differential equation that foresees the cooling of somebody that placed in a colder environment; then, the fixed point method can be applied in finding the solution of this differential equation as follows. Consider the IVP in the form of

We say that is a solution of Problem (89) if and only if [25] for all where the functions and are continuous.

Define a distance norm on the set of all real continuous functions defined on , for all where Then, the pair is a complete -space with

Now, we present the main theorem in this section.

Theorem 12. Assume that is a complete -space. Assume also, for all there is a constant so that where . Let so that for all Then, Equation (90) has a unique solution.

Proof. Define an operator by It is clear that, if , then is a solution of the Equation (90) and then (89).
For one can write where Hence, one can write yields, It follows that Consider for and ; then, or Equivalently, Based on Theorem 8 under a special case (iii) of Remark 9 (-contraction), possesses a unique FP, which is a unique solution to Equation (90) and then to (89).
Now, we describe Newton’s law of cooling as follows: Let where and refer to the temperature of the object at the time and the temperature of the environment, respectively, and is proportionality constant. If we put then we obtain the IVP below: Assume that Then, we have According to Theorem 12, the initial value problem (102) has a unique solution. It should be noted that the exact solution of (102) is ☐

7. Infinite Systems of Fractional ODEs

In this part, we study the sufficient condition to find the solution to an infinite nonlinear fractional system from ODEs on the space of all real sequences whose limit is finite which denoted with .

Assume that the system is defined as follows: where is any fixed interval on the real line and is the positive real number.

Suppose that equipped with for all and is a function described by Then, is a complete -space.

System (104) will be studied under the following hypotheses:

(hy₁) and

(hy₂) the function is a continuous for all and

(hy₃) for all we have where , and (hy₄) for all and for

Now, we state and prove our main theorem.

Theorem 13. System (104) has a unique solution under hypotheses (hy₁)-(hy₄).

Proof. Let be mappings described by A sufficient condition for a solution to System (104) to exist is that there is a common FP of the mappings (108). To achieve this, consider Squaring both sides, we get or It follows that after taking the supremum, By taking the of both sides, we find that Letting , one can write where Hence, we conclude by Theorem 12 that System (104) has a unique solution.☐

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no competing interests concerning the publication of this article.

Authors’ Contributions

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

Acknowledgments

This work was supported in part by the Basque Government under Grant IT1207-19. The authors thank the Basque Government for Grant IT1207-19.

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Copyright © 2021 Hasanen A. Hammad and Manuel De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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