Wardowski’s Contraction and Fixed Point Technique for Solving Systems of Functional and Integral Equations

Hammad, Hasanen A.; Bota, Monica-Felicia; Guran, Liliana

doi:https://doi.org/10.1155/2021/7017046

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Abstract Introduction Preliminaries Applications Examples Conclusions Data Availability Additional Points Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Unique and Non-Unique Fixed Points and their Applications

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Research Article | Open Access

Volume 2021 | Article ID 7017046 | https://doi.org/10.1155/2021/7017046

Wardowski’s Contraction and Fixed Point Technique for Solving Systems of Functional and Integral Equations

Hasanen A. Hammad,¹Monica-Felicia Bota,^2,3and Liliana Guran^4,5

Academic Editor: Calogero Vetro

Received28 Jun 2021

Accepted19 Oct 2021

Published16 Nov 2021

Abstract

In this manuscript, some tripled fixed point results are presented in the framework of complete metric spaces. Furthermore, Wardowski’s contraction was mainly applied to discuss some theoretical results with and without a directed graph under suitable assertions. Moreover, some consequences and supportive examples are derived to strengthen the main results. In the last part of the paper, the obtained theoretical results are used to find a unique solution to a system of functional and integral equations.

1. Introduction

Mathematics is one of the most important ways to understand things that happen around us. Mathematics has been divided into branches, and with its help, one can analyze other sciences. Integral and differential equations are very important tools that can be used to build patterns in order to understand the models that happen around us. The fixed point theory also plays a crucial role in integral and differential equations.

A commonly used tool that has a major role in nonlinear analysis is the fixed point technique, which was given by the well-known scientist Banach. The famous “Banach Contraction Principle” [1] can be announced as follows.

Theorem 1. Assume that is a complete metric space (CMS) and is a self-mapping defined on it, such that for all and , the following inequality holds: Then, there exists a unique fixed point (FP) of and the sequence converges to it, for all .

There are more generalizations of the inequality (1) either by replacing the contraction condition or by using more general spaces. For more results, see [2–4].

We construct the present paper as follows: in Section 1, we recall the background of our work; in Section 2, we give essential results, which are useful for understanding the aim of the paper; in Section 3, we discuss the existence of tripled fixed point (TFP) results via -contraction mappings in CMS with and without a directed graph (DG); in Section 4, we prove the existence of a solution of different types of tripled systems of functional integral equations; and in Section 5, illustrative examples are given to support our study.

2. Preliminaries

In 2012, a new type of contraction was given by Wardowski, called Wardowski’s contraction or -contraction (see [5]). He generalized the condition in Banach’s theorem and stated the following definition.

Definition 2 (see [5]). Assume that is a metric space and is a self-mapping defined on it. We say is -contraction, if there is and such that where is the family of all functions such that the conditions below hold:
: for each , if , then ; i.e., is strictly increasing.
: if and only if , where is a sequence of positive numbers.
: for each .
By the inequality (2), the same author introduced some examples of various contractions as follows: for all with and , (i), (ii), (iii), (iv), where all functions .

Remark 3. The inequality (2) implies that is a contractive mapping, that is, for all such that . Hence, every -contraction is continuous.

Remark 4 (see [6]). Let , where and . Then, .

Wardowski states his theorem as follows.

Theorem 5 (see [6]). Assume that the mapping satisfies the contraction condition (2) on a CMS . Then, there is a unique fixed point of and converges to the fixed point for all .

For two mappings, this theorem has been generalized by Isik [7] as follows.

Lemma 6 (see [7]). Suppose that is a CMS and and are self-mappings defined on it. If there is and such that for all and , then there exists a unique common fixed point of and .

A number of papers related to -contraction and related fixed point theorems in the setting of various spaces were published. See, for example, [8–10].

In the paper [11], the concept of the coupled fixed point (CFP) was presented and studied. In partially ordered metric spaces and abstract spaces, some main results in this direction have been considered. See [12, 13].

Definition 7. Assume that and are given mappings; then, the pair is called (i)CFP of if and (ii)a common CFP of and , if and

Using the generalized notion of CFP, Berinde and Borcut [14] defined the notion of a tripled fixed point (TFP) for self-mappings and established some interesting consequences in partially ordered metric spaces. Many other research results were given in this direction, for different spaces and different types of mappings. For additional results, see [4, 15–17].

Definition 8. Assume that and are given mappings; then, the pair is called a TFP of if , , and .
Here, the symbol refers to the set of all TFPs of the mapping , that is,

In [18], Jachymski used the following notations.

Assume that is a MS and is the diagonal of the Cartesian product . Consider a directed graph (DG), where is the set of vertices that coincides with and is the set of edges that contains all loops, i.e., .

The two definitions below were introduced by Chaobankoh and Charoensawa [19].

Definition 9 (see [19]). A mapping is called edge-preserving if implies

Definition 10 (see [19]). A mapping is called -continuous for each and for any sequence of positive integers with as , and Then, for , we have

Definition 11 (see [18]). Let be a CMS and be a directed graph. A triple has the property if for any sequence with and , for , we get .

3. Tripled Fixed Point Technique

Let us start this section by giving the following lemma, which is useful in the proof of the main result.

Lemma 12. Let be a CMS and be a Cartesian product. Define a distance by Then, is also CMS.

Proof. The proof of the lemma is obvious.

Furthermore, let us give the first main theorem of this section.

Theorem 13. Assume that is a CMS and are continuous mappings. If there is and such that implies for each , then and have a unique common TFP.

Proof. Define the mappings by Next, for a CMS (see Lemma 12), we shall show that and justify the inequality (4). For , let Here, if we put then three cases will be discussed for as follows:
: if , then, by relation (12), we obtain : if , then, by (12), we have : if , it follows from (12) that The above cases prove that the condition (4) is fulfilled. Then, and have a unique common FP . This means Hence, Therefore, is a common TFP of and .
The uniqueness follows immediately from the definition of and .

A pivotal result follows below by letting in Theorem 13.

Corollary 14. Assume that is a CMS and is a continuous mapping. If there is and such that implies for all , then has a unique TFP.

Now, we will discuss the existence and uniqueness of a TFP in a CMS with a directed graph.Following the paper [19], we define the set by

Proposition 15. Let be an edge-preserving mapping; then, for all ,
:
:
:

Proof. : consider . Because is a preserving mapping, we get . Using the same property, we can write . It follows that, by induction, . In the same manner, we can prove and .
: assume that By , we get Similarly, one can show that and .: from , we get which is equivalent to .

Definition 16. We say is a -rational contraction mapping if
: Q is edge-preserving.
: there is a positive such that for all , with .

Lemma 17. Assume that is a MS and is a with a DG . Then, for each , we have

Proof. Let . Because is edge-preserving, we have It follows from Proposition 15 that . Because is a , one can obtain Moreover, we have the same result if or . Therefore, the conclusion follows using mathematical induction.

Lemma 18. Let be a on a CMS with a DG . Then, for each , there is such that , , and , as .

Proof. Suppose that ; then, Set , , and in the contractive condition of Lemma 17 and put Then, we have or equivalently, As in (33), we can write Applying condition , we have Using the same steps, we can write From condition to (35), there exists such that For all , the inequality (33) yields Take into account (35) and (37), and taking in (38), we get By (39), there is , such that , for all , or Using (40), for , we get The convergence series leads to . Moreover, we can write This implies that , , and are Cauchy sequences in . The completeness of tells us that there is such that , , and , as . Then, the conclusion follows.

Theorem 19. Assume that is a on a CMS with a DG . Let (a) be -continuous(b)the triple satisfy the property and be continuousThen, if and only if .

Proof. Let ; then, there is so that , , and . So, ; this yields .
Conversely, suppose that ; this means that there is such that Considering a positive integer sequence , by Proposition 15, we obtain Applying Lemma 18 to (44), there are such that (a)Let be -continuous; then, we getFrom triangle inequality, it follows The continuity of and (45) leads to , i.e., . Similarly, one can show that and . Hence, a triple is a TFP of and . (b)If a triple satisfies the property , then we getAgain, by the triangle inequality, we have Using mapping yields As in (50), we obtain that , that is, , i.e., . Similarly, one can prove that and . So .

4. Applications

The fixed point theory is a very important tool in nonlinear analysis, due to its applications in various domains (see [20, 21]).

Before stating the main results of this section, we need the following lemma.

Lemma 20 (see [22]). Assume that is a function defined by for and . Then, (i) is strictly increasing(ii) and is a concave function(iii)for ,

4.1. System of Tripled Functional Equations

The fixed point technique contributes to the study of dynamic programming, which is considered an essential tool in optimization problems such as the study of dynamic economic models. This technique has been studied by many researchers to give a unique solution to a system of functional equations via suitable contraction conditions in various spaces. For more results, we refer to Bhakta and Mitra [23], Liu [24], Pathak et al. [25], Zhang [26], and Bellman and Lee [27].

Consider a system of tripled functional equations below: where and are state and decision spaces, respectively, , , , and .

We denote the set of all bounded real-valued functions on a nonempty set , by . Define for any . Moreover, on , define a distance as follows: for all . Clearly, the pair is a CMS.

Problem (52) will be considered via the two hypotheses below:

: the functions and are bounded.

: for all , , and , for and we have

Theorem 21. Using the hypotheses and on , the problem (52) has a unique bounded common solution.

Proof. On the space , let us define an operator as follows: for each and . The boundedness of the functions and assures that the mapping is well defined.
Suppose that , and take Then, by hypothesis , we have where the nondecreasing character of was used (Lemma 20). Then, Taking on both sides, we have or equivalently, yields and this leads to This confirms that the inequality (21) of Corollary 14 holds with (Remark 4). Then, it follows that the operator has a unique TFP. At the same time, it is a unique bounded solution of the problem (52) on .

4.2. Tripled System of the First Type of Integral Equations

In this subsection, the theoretical results of Corollary 14 will be applied to discuss the existence and uniqueness of a solution of an integral equation tripled system. Let us consider the following system: where is defined for all .

Consider , the set of all real continuous functions defined on , and together with the distance defined above, we can notice that is a CMS.

Now, we discuss the problem (64) according to the assumptions below:

: is a continuous function.

: is a continuous function verifying for each and , and and .

Furthermore, let us present the main theorem of this subsection.

Theorem 22. There is a unique solution of system (64) , as long as the conditions and are satisfied.

Proof. Define a mapping on as follows: for all . In virtue of and , we conclude that for each . Thus, we can write Let ; then, for , we get where the nondecreasing characters of were used (Lemma 20). Thus, By the same approach used at the inequalities (60)–(62), we get Hence, the hypotheses of Corollary 14 are fulfilled on (Remark 4). There is a unique TFP of the mapping . In other words, there is such that

4.3. Tripled System of the Second Type of Integral Equations

Let us consider the following type of system of integral equations: where with .

This subsection is devoted to discussing the influence of the theoretical results of a DG for solving this new type of system of integral equations.

Let endowed with for all . Moreover, define a partial order on a graph as follows, for all and ,

Thus, is a CMS equipped with a directed graph .

Let be a triple with the property and

We can state the main theorem.

Theorem 23. There is at least one solution of the problem (72), if the assumptions below are fulfilled:
: the functions and are continuous such that for all and .
: for all with , , and , we have for all
: there are and so that for any .
: there is such that where

Proof. Let the mapping defined by Next, we show that is -edge-preserving. Let with , , and . Then, we get Using the same steps, we can write and , for all .
Next, from , it follows Ultimately, where the nondecreasing characters of were used (Lemma 20). Thus, Taking on both sides, we get or equivalently, or This leads to Hence, is with (Remark 4). So, it follows from Theorem 19 that the mapping has a TFP, which is a solution of the problem (72).

5. Examples

In this section, some important examples satisfying theoretical consequences are presented, with the role to strengthen our results.

Example 1. Assume that and . Clearly, is a CMS. Define by for all .

Moreover, from the definition, and are continuous. Let be a function defined by for To verify the inequality (12) of Theorem 13, we consider the following cases:

: if and , we can write and ; then,

Taking into account, we can write or which leads to

: if and , we have and ; then, since for all , , one can get and by the same manner of , we get (92).

: if and , then and . Hence, by the same method of , we obtain (92).

: if and , we get and ; it is trivial.

It follows from that the inequality (12) of Theorem 13 with is verified.

Then, is a unique common TFP of and .

Example 2. We consider the following tripled system of functional equations: for .

It is clear that the system (95) is a special form of system (52) with and . The condition of Theorem 21 is clear. For , we can write

where Lemma 20 is used. Hence, is satisfied with and . According to Theorem 21, the system (95) has a unique solution in .

Example 3. Suppose the following tripled system of integral equations: for .

Again, system (97) is a special case of system (64), where .

It is obvious that the condition of Theorem 22 holds. For the condition , we get where Lemma 20 was used. Hence, holds with and . According to Corollary 14, system (97) has a unique solution .

6. Conclusions

The present paper is dedicated to the study of the existence and uniqueness of tripled fixed points in a CMS with and without a directed graph. Common tripled fixed point results are given too. Moreover, some applications of the main results in solving different types of tripled equation systems are presented. Then, using our main results, we study the existence and uniqueness of a solution of some systems of tripled functional and integral equations used in the study of dynamic programming. To sustain our results, the last part of the paper is dedicated to some illustrative examples. Our results come to improve some results from the related literature and give new directions in the study of economic phenomena, using the tripled fixed point technique.

Data Availability

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

Additional Points

Open Questions. (1) A new research direction can be considered the existence of fixed points in the case of multivalued operators. Which conditions can be imposed in order to obtain the uniqueness of the fixed point for the multivalued operators’ case? (2) Moreover, the case of coincidence fixed points and the case of coupled fixed points can be considered for further research proposals.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally in the writing and editing of this article. All authors read and approved the final version of the manuscript.

Acknowledgments

The second author wants to thank the Academy of Romanian Scientists for the support.

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Copyright © 2021 Hasanen A. Hammad et al. 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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