An Existence Study on the Fractional Coupled Nonlinear <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 8.58866 12.3952" width="8.58866pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g></svg>-</span>Difference Systems via Quantum Operators along with Ulam–Hyers and Ulam–Hyers–Rassias Stability

Rezapour, Shahram; Thaiprayoon, Chatthai; Etemad, Sina; Sudsutad, Weerawat; Deressa, Chernet Tuge; Zada, Akbar

doi:https://doi.org/10.1155/2022/4483348

Journal of Function Spaces

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

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Unique and Non-Unique Fixed Points and their Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 4483348 | https://doi.org/10.1155/2022/4483348

An Existence Study on the Fractional Coupled Nonlinear -Difference Systems via Quantum Operators along with Ulam–Hyers and Ulam–Hyers–Rassias Stability

Shahram Rezapour,^1,2,3Chatthai Thaiprayoon,⁴Sina Etemad,⁵Weerawat Sudsutad,⁶Chernet Tuge Deressa,⁷and Akbar Zada⁸

Academic Editor: Anita Tomar

Received01 Jun 2022

Accepted29 Aug 2022

Published24 Sept 2022

Abstract

In this paper, we study the existence of solutions and their uniqueness and different kinds of Ulam–Hyers stability for a new class of nonlinear Caputo quantum boundary value problems. Also, we investigate such properties for the relevant generalized coupled -system involving fractional quantum operators. By using the Banach contraction principle and Leray-Schauder’s fixed–point theorem, we prove the existence and uniqueness of solutions for the suggested fractional quantum problems. The Ulam–Hyers stability of solutions in different forms are studied. Finally, some examples are provided for both -problem and coupled -system to show the validity of the main results.

1. Introduction

Fractional calculus is one of the most important fields in applied mathematics. In recent years, many researchers have studied different branches of this theory and conducted numerous analyses analytically and numerically. Particularly, in recent decades, we can see some papers on the applications of fixed-point theorems to prove the existence of solutions of fractional boundary value problems [1–4]. Because of the quick developments in fractional calculus, many mathematicians discussed on the theory of -calculus that is an equivalent of traditional calculus without defining the concept of limit, and also the parameter refers to quantum. This theory was originally developed by Jackson [5, 6], and it includes many practical aspects in the fields of hypergeometric series, theory of relativity, particle physics, discrete mathematics, quantum mechanics, combinatorics, and complex analysis. For a fundamental introduction of the basic notions of -calculus, one can refer to [7–9]. In the early years, for finding positive solutions of given -difference equations in the nonlinear settings, we lead you to study a work published by both El-Shahed and Al-Askar [10] and also a manuscript by Graef and Kong [11].

So later, various mathematical -difference fractional models of IVPs and BVPs have been presented in which different methods like the lower-upper solutions technique, fixed-point results, and iterative methods have been implemented. For instance, we see -intego-equation on time scales in [12], -delay equations in [13], -integro-equations under the -integral conditions in [14], singular -equations in [15], -sequential symmetric BVPs in [16], -difference equations having -Laplacian in [17], four-point -BVP with different orders in [18], oscillation on -difference inclusions in [19], etc.

Here, we apply similar techniques to discuss the existence property of solutions for given -integro-sum-difference FBVPs depending on the quantum operators. This shows an application of fixed-point theory in relation to -difference theory. This specifies the main contribution of the present reseach.

In 2014, Ahmad et al. [20] studied a -sequential equation in the nonlinear case via four-point -integral conditions given by so that , , , and . As well as, is continuous, and indicates the -RL-integral. These mathematicians extracted different qualitative aspects of solutions for the above -FBVP by means of the classical methods which are available in fixed-point theory.

In 2015, Etemad et al. [21] focused on the new four-point three-term -difference FBVP where , , , , and with .

In 2019, two mathematicians named Ntouyas and Samei [22] devoted their attention to investigate the existence property about the multiterm -integro-difference FBVP where , , , with , , are formulated as for and is continuous with respect to all variables [22].

In 2020, Phuong et al. [23] formulated a novel extended configuration of the Caputo -multi-integro-difference equation with two nonlinearity under -multi-order-integrals conditions where , , , , and .

In this paper, inspired by above -problems, we analyze a structure of the nonlinear Caputo quantum difference fractional boundary problem (or -CFBVP) in the form where , , , , for , and are continuous. As the same way, the operators and denote the -Caputo derivative and the -RL integral, respectively. In the direction of the above problem, we consider a coupled system of nonlinear -CFBVPs with the same -boundary conditions. In other words, the mentioned fractional -system is organized as where , , , , for , and are continuous.

In other words, we extend our -CFBVP to a coupled -difference system and derive the existence and stability results on such a generalized coupled -CFBVP system. In fact, a large number of researchers have devoted their concentration to the discussion on various categories of Ulam-Hyers stabilities for standard systems of FDEs (or refer to [24, 25]), while a few articles can be found in the literature in which the researchers developed the relevant existence and stability theory in relation to nonlinear fractional -difference systems.

The present work is assembled as follows: In Section 2, we state some basic materials required to prove our theoretical results. In both Section 3 and Section 4, several criteria and conditions are presented for the desired uniqueness-existence results, along with different classes of stabilities in relation to the proposed -CFBVPs (6) and (7), respectively, with the help of some known fixed–point theorems. A simulative example, to represent the consistency of our results, is given with each suggested -CFBVP in the relevant section. We give Section 6 to the presentation of the conclusion of this research work.

2. Preliminaries

The basic notions of -calculus are collected in this section by assuming . The -analogue of is given by Rajkovic et al. [26]. Now, if , then

On the other side, by taking , we have [26]. A -number for is defined by

Accordingly, the Gamma function in the quantum settings is defined by and [5, 26].

Definition 1 (see [27]). The -difference-derivative of the given function is defined by where .

Clearly, we have for all and [27].

Definition 2 (see [27]). The -integral of the supposed function is defined as if the series is absolutely convergent.

Similarly, for all and [27].

Definition 3 (see [27]). By letting , the definite -integral of the given function is defined by if the series exists.

By considering as a continuous function at , then [27]. Furthermore, for all .

Definition 4 (see [11, 28]). The -RL--integral of is defined by if integral exists.

One can simply see that the -semi-group property satisfies as for [28]. Also, for , we have

Definition 5 (see [11, 28]). Let , i.e., . The -Caputo -derivative of is defined as if the integral exists.

Note that for , we have

Lemma 6 (see [10]). Let . Then,

By Lemma 6, the general series solution of -difference FDE is given as with and [10]. In this case, we get

3. Analysis of the Cap--Difference FBVP (6)

Let be the space of all real-valued continuous functions on . Clearly, is a Banach space under the norm for all members . In the first step, we provide the following fundamental lemma which presents a characterization of the structure of solutions for the proposed Cap--difference FBVP (6)

Remark 7. For convenience, we consider the following nonzero constants:

Lemma 8. Let , , , , , and for . The solution of the linear Cap--difference FBVP is given by where and are defined in (24).

Proof. Let satisfies the linear Cap--difference FBVP (25). Then . By virtue of and taking -RL--integral, we have where are unknown coefficients that we have to explore them. It is immediately computed that By considering the constants given by (24) and by virtue the given boundary conditions implemented on (29)–(32) and by some straightforward computations, we obtain the following coefficients By inserting (33), (34), and (35) into (28), we derive equation (26) which is the same desired -integral solution of the linear Cap--difference FBVP (25). The proof is completed.

Now, consider the following estimates:

In this paper, for convenience in computation, we set

According to Lemma 8, we define the operator as

Notice that the Cap--difference FBVP (6) has solutions if and only if has fixed points.

To simplify the computations, we set the following notation and the constants

3.1. Uniqueness Result

The uniqueness result for the Cap--difference FBVP (6) is proved by using the Banach’s fixed-point theorem.

Theorem 9. Assume that satisfies the following assumptions.
() There are , such that for every , , , and .
If where is given in (39), and then the Cap--difference FBVP (6) has a unique solution in .

Proof. We convert the Cap--difference FBVP (6) into , where is defined by (38). By the Banach’s contraction principle, we shall guarantee that has exactly one fixed point.
At first, we define a bounded, closed convex subset with where is defined by (39).
Let . The proof will be completed in two steps:
Step 1.
Let and . Estimate By using the property of integral (16), we get From the assumptions and (44), we can estimate From (45) and by the property of integral (16), we obtain Substituting (46)–(51) into (43), we obtain Then, which implies that . Thus, .
Step 2. is a contraction.
Let , . For each , we have By , it follows that Hence, by inserting (55) into (54) and using the property of integral (16), we get which implies that .
In view of (41), , and we conclude that is a contraction. Hence, in accordance with the Banach’s contraction principle, the Cap--difference FBVP (6) has a unique solution .

3.2. Existence Result

The second result is based on the Leray-Schauder’s nonlinear alternative theorem.

Lemma 10 (Leray-Schauder’s nonlinear alternative theorem [29]). Let be a Banach space, be its closed convex subset, and be an open set in such that . Let be a continuous and compact function. Then either (i) there is such that or (ii) there are and such that .

Theorem 11. Let satisfies the following assumptions:
() There is continuous nondecreasing functions : , , such that where , .
() There is such that

Then the Cap--difference FBVP (6) has at least one solution in .

Proof. Consider as (38). In the first step, we will prove that corresponds bounded sets (balls) to bounded ones in . For each positive real constant , is a bounded set (ball) in . Let . We have From () and (44) in Theorem 9, we obtain By the same process in Theorem 9, we can estimate Further, it will be investigated that corresponds bounded sets to equicontinuous sets of
Let , with and , where is a bounded set in . Also, we obtain Obviously, the above inequality goes to zero as , independent of . Hence, by helping the Arzelá-Ascoli theorem, is completely continuous.
Now, we prove that there is an open set such that for and .
Let satisfies for each . So, for , by following the calculations applied in proving the boundedness of , we have It yields that Consequently, we obtain From , there is such that . Let Notice that is completely continuous. For the sake of the choice of , such that for some . Therefore, by Lemma 10, we find out that has the fixed point which implies that the Cap--difference FBVP (6) has at least one solution on .

3.3. On the Stability Property for (6)

Stability analysis is one of the most important parts of each research in the field of existence of solution of fractional boundary value problems. For instances, we can mention to such a stability analysis in some newly published works including [24, 25, 30–32]. In this subsection, we introduce some concepts of stabilities for the Cap--difference FBVP (6). These definitions were extracted from [33].

Let , be continuous and be a nondecreasing mapping. Assume that

Definition 12. The Cap--difference FBVP (6) is called Ulam-Hyers stable if s.t. and every solution of (67), a solution of (6) exists s.t.

Definition 13. The Cap--difference FBVP (6) is called generalized Ulam-Hyers stable if , s.t. fulfilling (67), a solution of (6) exists s.t.

Definition 14. The Cap--difference FBVP (6) is Ulam-Hyers-Rassias stable w.r.t. if s.t. and every solution of (69), a solution of (6) s.t.

Definition 15. The Cap--difference FBVP (6) is termed generalized Ulam-Hyers-Rassias stable w.r.t. if s.t. for every solution of (68), a solution of (6) s.t.

Remark 16. is a solution of (67) if (dependent on ) s.t.

Lemma 17. If satisfies (67), then where is given as in (39) and is introduced in the proof.

Proof. Let satisfie (67). By () of Remark 16, there is (dependent on ) such that Then, the solution of (76) is given as For convenience, consider for the terms that are independent of . That is, Therefore, (77) can be rewritten and by using () of Remark 16, we have This inequality completes the proof.

Theorem 18. Let and to be held. Then, the Cap--difference FBVP (6) is both Ulam-Hyers and generalized Ulam-Hyers stable.

Proof. Let satisfies (67) and fulfills the Cap--difference FBVP (6) given as By the previous lemma, let By using Lemma 17 in (82), we have For , we have After simplification, we get Thus where Thus, the Cap--difference FBVP (6) is Ulam-Hyers stable.
In the sequel, the function implies that the Cap--difference FBVP (6) is generalized Ulam-Hyers stable and .
Now, we add another condition.
() Consider an increasing map . Then, there is such that

Remark 19. Under the hypotheses () and () and (80), the Cap--difference FBVP (6) is the Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stable.

4. Analysis of the Cap--Difference System (7)

Here, we continue to discuss the existence and uniqueness results for the proposed system (7). In view of the assumptions of Section 3 for the Banach space , the norm considered on the product space is which implies that is a Banach space.

Remark 20. For convenience, and based on the given parameters in (7), we have nonzero constants: Keeping in mind Lemma 8, consider the operator as where

Before proceeding, consider the following estimates

To simplify, we also set the following notation and the constants

4.1. Uniqueness Result

In this step, we shall establish the existence of a unique solution to the coupled system of nonlinear -CFBVPs (7), by the Banach’s contraction principle.

Theorem 21. Let , be continuous. Assume that
() There exist positive constants such that for each and , , , , and for Then the coupled system of nonlinear -CFBVPs (7) has a solution on provided that

Proof. We transform the coupled system of nonlinear -CFBVPs (7) into a fixed-point problem , where is an operator as (90).
Let and Next, we set with Note that is a bounded convex closed set in .
Step 1. .
For each and , and by using the condition and (44), we have Then, we get Hence Similarly, we find that Consequently, we have which implies that .
Step 2. We show that is a contraction.
Using condition , for any , and for each , we have and therefore Similarly, we get From (103) and (104), it yields As by (95), the operator is a contraction. The Banach’s contraction principle implies the existence of unique solution for the coupled system of nonlinear -CFBVPs (7) on .

4.2. Existence Result

We get help from Lemma 10 to complete the main result of this subsection.

Theorem 22. Let , be continuous. Assume that
() There exist nonnegative continuous maps , for such that with and .

Then the coupled system of nonlinear -CFBVPs (7) has at least one solution on .

Proof. Here, the process of the proof will be continued during four steps as follows.
Step 1. is continuous.
Let and be two sequences such that and in . Then for each , we get and therefore Similarly, we get From (108) and (109), it yields Since the continuity of and imply that of , , so we have and as ; and is continuous.
Step 2. is uniformly bounded.
We prove that for , there exists such that for every , where we get .
Using the condition and (16), we have Then, we get Hence Similarly, we find that Consequently, we have Then, is uniformly bounded.
Step 3. maps bounded sets into equi-continuous sets of . Let such that and where is defined as in Step 2. Then we have which implies that The right-hand side tends to 0 as , which is independent of . By helping the Arzelá-Ascoli theorem, is completely continuous.
Step 4. The set is bounded.
Let . Then for some . Thus, for any , by using the computations of Step 2, we have This means that is bounded. Consequently, by Lemma 10, has a fixed point and so a solution to the coupled system of nonlinear -CFBVPs (7).

5. Numerical Examples

In this section, we provide some illustrative examples of the exactness and applicability of our main results.

Example 1. Consider the Cap--difference FBVP of the form

Here , , , , , , , , , and . From the given data, we obtain , , and . We consider the functions

For , , and , we can find that

The assumption is satisfied under the values and . Thus,

All assumptions of Theorem 9 are valid. Then the Cap--difference FBVP (120) has a unique solution on . Moreover,

By the conclusions of Theorem 18, the Cap--difference FBVP (120) is both Ulam–Hyers and also generalized Ulam-Hyers stable on . (ii) Set .

By using the property of integral (16) and setting , the implicit solution of the Cap--difference FBVP (120) is given by

Figure 1 displays the solution of the Cap--difference FBVP (120) involving various values of and .

Example 2. Consider the coupled system of nonlinear Cap--difference FBVP under the conditions

Here , , , , , , , , , , , , , , , , and . From all the given data, we obtain , , , , , and . We consider the functions

For , , , , and , we can find that

The assumption is satisfied with , , , and . Hence, and . All assumptions of Theorem 21 are satisfied. Then the coupled system of nonlinear Cap--difference FBVPs (126) has a unique solution on .

6. Conclusion

In this paper, a new category of nonlinear Caputo quantum boundary problems and its relevant generalized coupled -system involving fractional quantum operators was discussed. We presented new -difference equations and system in which we dealt with -integro-sum-difference bundary conditions. Some qualitative aspects of solutions such as the existence, uniqueness, and different classes of stabilities of Ulam-Hyers type were investigated for both given -Cap-difference problems. The results were examined with some examples. As a new idea in the next papers, we aim to extend our method for similar generalized coupled systems under the newly introduced generalized -operators (postquantum operators).

Data Availability

No data were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Acknowledgments

The first and third authors would like to thank Azarbaijan Shahid Madani University.

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Copyright © 2022 Shahram Rezapour 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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