Stability Analysis for Differential Equations of the General Conformable Type

Abstract

Fractional calculus is nowadays an efficient tool in modelling many interesting nonlinear phenomena. This study investigates, in a novel way, the Ulam–Hyers (HU) and Ulam–Hyers–Rassias (HUR) stability of differential equations with general conformable derivative (GCD). In our analysis, we employ some version of Banach fixed-point theory (FPT). In this way, we generalize several earlier interesting results. Two examples are given at the end to illustrate our results.

1. Introduction

The stability issue gained a considerable attention in various research fields through applications. There are many kinds of stability, among them is the stability introduced by S. M. Ulam, in his famous talk at a conference held in Wisconsin University in 1940. Since then, it is known as HU stability or simply Ulam stability. Its applications for various types of differential equations have been investigated by many researchers. The readers can see the interesting results in 1–7, for more details. The stability problem of Ulam can be rewritten in the following form.

Consider a group and a metric group . Is it true that given , there exist such that if satisfies

For all , then a homomorphism exists such thatfor every

The problem of Ulam has been extended in many directions for various interesting settings. In particular, Rassias (see [8]) generalized Ulam’s result for Banach spaces.

Initial and boundary value problems with fractional-order derivatives are natural generalization of the classical initial and boundary value problems. It is much more complicated to investigate stability issues of fractional-order problems than their classical analogues; this is because of the singularity and nonlocality in the kernel of fractional differential operators. Fractional derivatives, in general, play negligible roles in a number of fields of science and engineering (see, e.g., [9–13] and the references there in).

In particular, during the last few decades, the area of fractional calculus has been investigated qualitatively by using different tools of functional analysis. These tools include but are not limited to Gronwall Lemma, see, e.g., [14], Pachpatte’s inequality, see, e.g., [15], Schaefer’s FPT, see, e.g., [16], Schauder’s FPT, see, e.g., [17], Banach FPT, see, e.g., [18], and Picard operator, see, e.g., [15]. Various approaches have been used to define fractional derivatives (see, e.g., 19–28, for more details).

It should be remarked that generalized conformable derivative plays an essential role in many applications. For instance, the authors in [29] utilized it to examine some nonlinear evolution equations. Generalized conformable derivative also has been used in [30] to investigate some nonlinear evolution equations. A new generalized version of conformable derivative is given and in [31] with some applications in biological population. In the present study, we generalize several recent interesting works as follows. We use Theorem 2 to generalize the interesting results in [32, 33] by dropping some of the basic assumptions that have been used there. We also use Theorem 3 to generalize the work in [34].

The organization of the study is as follows. In Section 2, we present some preliminaries and some basic definitions. In Section 3, we introduce our stability results in the sense of HU and HUR. In Section 4, two examples are written to show the validity of our results, and in Section 5, we conclude our work.

2. Preliminaries

In this section, some definitions, lemmas, and theorems are given [35–39].

Definition 1. Let us consider a function defined on ; then, the GCD starting from the real of a function is defined byFor all , and is a nonnegative continuous function that satisfiesIf exists, for every ; for some , exists; then, by definition,

Remark 1. To further study the properties of GCD, we suppose that , for all , and is locally integrable.

Definition 2. Let . The conformable fractional integral starting from of a function is defined by

Lemma 1. Suppose that . Thus,

Lemma 2. Suppose that . Thus,

Remark 2. Assume that . If , then and .
The following is the notion of a generalized metric on some set .

Definition 3. (see 40). Consider a mapping . The mapping is called a generalized metric on set iff satisfies: M1 if and only if  M2 , for all  M3 , for all The following theorem (see [40]) represents one of the interesting results of FPT. This theorem plays a fundamental role in our study.

Theorem 1. Suppose that is a metric space that is generalized complete. Let be a strictly contractive operator. If there is an integer with for some , thus(a), where is the unique fixed point of in (b)If , then Define the space as , with ( is some real number).

Lemma 3. Define a metric in such a way thatwhere , , and is positive and continuous. Thus, is a generalized complete metric space.

The goal of this study is to investigate the stability of the following initial value problem:in the sense of HU and HUR. Notice that the solution of the initial value problem (10) is the solution of

3. Ulam–Hyers–Rassias Stability Results

We use this section to present our main results. The theorem below represents the stability of (10) in the sense of HU.

Theorem 2. Suppose is continuous and satisfiesIf an absolutely continuous function satisfiesfor some , therefore, there is a unique solution of (10) withfor every , where is any positive constant and .

Proof. For any , we define the metric in this way:Define the operator such thatSinceso that it is clear that , in addition, we get .
Now, we prove that is strictly contractive:So, it is clear thatwhich implies thatwhich prove that the operator is a strictly contractive one.
We get, from (27),Therefore,Now, according to Theorem 1, there is some solution of (10) satisfyingso thatwhich implies that

Remark 3. It is clear that our findings in the sense of HU are some generalized version of the results obtained in [32, 33] as follows. In our analysis, we do not impose any constrains on unlike equation 5 in Theorem 2 in [32]. In [33], the authors assumed conditions on the function which is not the case in our study.
The following theorem represents the stability of (10) in HUR sense.

Theorem 3. Suppose is continuous and satisfiesIf an absolutely continuous function satisfieswhere is a nondecreasing, continuous function, therefore, there is a unique solution of (10) withfor every , where is any positive constant and .

Proof. For any , we define the metric as follows:Define the operator such thatWe have , for all . In addition, we get .
Now, we prove that is strictly contractive:So,which prove that the operator is a strictly contractive one.
We get, from (13),Hence,Using Theorem 1, there is a solution of (10) withThus,

Remark 4. Notice that the authors in [34] used conformable fractional Laplace transform to study the HUR stability of several kinds of differential equations. They had to assume some specific conditions, see, e.g., condition 12 in Theorem 3.6 is given in [34].

Remark 5. The authors in [41] obtained stability results for differential equations with integer-order derivatives , while in our study it is for GCD. In this sense, we introduce a generalized version of the interesting results [41].

4. Examples

This section uses two examples to show the validity of results.

Example 1. Consider equation (10) for , , , , and .
We haveThen, .
Suppose that satisfiesfor all .
Here, . Using Theorem 2, there is and such that

Example 2. Consider equation (10) for , , , , and .
We haveThen, .
Suppose that satisfiesfor all .
Here, . Using Theorem 3, there is and such that

5. Conclusion

We managed to utilize a version of Banach FPT to present stability results in the sense of HU and HUR for some differential equations involving GCDs. In our analysis, we generalized some interesting results by dropping some of the basic assumptions that have been used in recent known investigations. We used two examples to show the validity of our main findings. We believe that the methodology used in this study can be further applied to many other fractional differential equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was funded by the Deanship of Scientific Research at Jouf University, under Grant no. DSR-2021-03-0324.