Ulam-Hyers-Rassias Stability of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

Abstract

Fractional derivatives are used to model the transmission of many real world problems like COVID-19. It is always hard to find analytical solutions for such models. Thus, approximate solutions are of interest in many interesting applications. Stability theory introduces such approximate solutions using some conditions. This article is devoted to the investigation of the stability of nonlinear differential equations with Riemann-Liouville fractional derivative. We employed a version of Banach fixed point theory to study the stability in the sense of Ulam-Hyers-Rassias (UHR). In the end, we provide a couple of examples to illustrate our results. In this way, we extend several earlier outcomes.

1. Introduction

Fractional calculus (FC) has been appearing in a wide range of fields, such as chemistry, economics, polymer rheology, and aerodynamics. This is due to the existence of many nice tools (see e.g., [1, 2]) that are not available in the classical calculus. In particular, FC enables researches to model in an efficient way many complicated real world problems, e.g., COVID-19 (see [3]), Ebola virus (see [4]), and HIV (see [5]). Moreover, it has recent interesting applications in image processing (see [6]) and in diabetes (see [7]).

The stability problem named after Ulam is currently a research trend in many applications (see e.g., [8] for more references and details). It pupped up as a consequence of the famous question asked by Ulam at a conference held in Wisconsin University in the fall term of 1940 (see [9]). The mentioned Ulam’s stability problem can be rewritten as follows:

Let be a metric group and be a group. Is it true that for some , there is a that satisfies if verifies for every ; thus, there exists a homomorphism fulfilling for all .

Answers have been introduced for the question of Ulam by many mathematicians. For instance, in 1941, Hyers gave an exact answer to Ulam’s question. Afterwards, Rassias in 1978 (see [10]) introduced a general form of the result of Hyers. The famous result obtained by Rassias can be rewritten:

Theorem 1. [10]. Assume that are Banach spaces and assume some continuous mapping from into . Suppose that there exists and such that Then, there is a unique solution of the Cauchy equation with

Through the past six decades, the stability subject has been a common issue of investigations in many places (see, e.g., [12, 15, 22, 23, 9, 20, 21, 25, 27, 26, 28]). As a consequence of the interesting results presented in this direction, many articles devoted to this subject have been introduced ([24, 16, 29] and the references therein). In 2010, Jung employed a fixed point technique (FPT) to study the stability of the equation (see [11]). It should be remarked that Jung in [11] generalized the work of Alsina and Ger to the nonlinear case. In 2012, Bojor (see [12]) used different assumptions to study the stability of and improved the result of Jung in [11].

In 2015, Tunç and Biçer in [13] improved the approach of Jung in [11] for the functional differential equation:

In [14], Huang et al. investigated the stability of the following equation:

In [15], Popa and Pugna studied the UH stability (UHS) of Euler’s equation. In [16], Shen introduced Ulam stability for equations on time scales. In [17], the authors employed weakly Picard operator theory to investigate the UHS of some kind of equations in Banach Spaces. Furthermore, they obtained the UHR stability for such kind of equations via Pachpatte’s integral inequalities. FPT has been employed in [18] to study the stability of a nonlinear Volterra integrodifferential equation with delay and in [19] to study the stability of impulsive Volterra integral equation.

The framework of the paper is as follows. In Section 2, we introduce some preliminaries; in Section 3, we present the stability results in UHR sense; in Section 4, we illustrate our results with two examples, and Section 5 is devoted to the conclusion.

2. Preliminaries

From now on, we use to denote real numbers set and to denote the complex numbers set. We define the generalized metric on a nonempty set as follows.

Definition 2. [20]. The mapping is said to be a generalized metric on if and only if fulfills the assertions:
G₁ if and only if ;
G₂ for all ;
G₃ for all .

Now, we present the notion of UHR stability.

Definition 3. The following fractional differential equation is UHR stable if for given and a function which satisfies There is a solution of (8) with , where and are some functions that do not depends on and .

The following theorem represents one of the central results of FPT (see [20]).

Theorem 4. For a generalized complete metric space . Suppose an operator that is strictly contractive with some Lipschitz constant , if there exists an integer that is nonnegative such that for some , then the following are true: (a)The sequence converges to a fixed point of is the unique fixed point of in (c)If , then

The current article is written to study the stability of the following differential equation with right-sided Riemann-Liouville fractional derivative with initial conditions where is some continuous nonlinear function and , where is the well-known greatest integer function.

3. Stability Results

This section is used to present the main findings of this article. In other words, we use it to prove the UHR stability of (10).

Let us first use to denote the space of all continuous functions from the interval into the set of reals . In the next subsections, we investigate the stability of (10) when and when . We start with the case as follows.

3.1. The Case

The following theorem represents the stability of (10) in the sense of UHR.

Theorem 5. Assume that satisfies If a continuous function satisfies , then for all , where is a nonincreasing function. Then, there is a unique function such that for any positive constants .

Proof. We start the proof by defining the metric on in this manner We can prove that the space is a complete generalized metric space (see Lemma in [21]).
Define the operator with Since we have for all and Therefore, . Note also that we have , , and then
In addition, for any we get for all .
Now, using the fact that for all .
Then, which implies that which proves that is a strictly contractive. Following the same way as in the proof of Theorem 7.1 in [22], we get which means that Now, in view of Theorem 4 there is a solution with and then for all . This results prove that in view of Definition 3, (10) is UHR stable.

Remark 6. In the current work, we do not assume any constrains on unlike the case of results in [22] where the assumption is a basic condition.

Now, we investigate the stability of (10) in the case where as follows.

3.2. The Case

Theorem 7. Assume that satisfies If a continuous function satisfies , then for all , where is a nonincreasing function. Then, there is a unique function with where and some positive constant such that .

Proof. We start by defining the metric on by the form and we define the operator such that Since we have for all and so that it is clear that . Note also that we have , , and then
In addition, for any , we get Then, (using ) which proves that the operator is strictly contractive. Following the same way as in the proof of Theorem 5 in [22], we have Then, Now, there is a solution (due to Theorem 4) with and then for all

Remark 8. Notice that in the current work, we do not assume any condition on unlike the case of Theorem 7in [22] where the condition is a basic condition.

4. Examples

The following examples are used to illustrate our findings.

Example 9. Consider equation (10) for , , and .
We have Then, .
Suppose that satisfies and for all .
Here, and . Using Theorem 5, there is a continuous function such that

Example 10. Consider equation (10) for , , and .
We have Then, .
Suppose that satisfies , and for all .
Here, and . Using Theorem 7, there is a continuous function such that

5. Conclusion

A version of Banach’s contraction principle has been successfully utilized in this work to study the UHR stability of nonlinear differential equations with Riemann-Liouville fractional derivatives. In this way, under specific assumptions and conditions, the stability results have been obtained. In our analysis, we get rid of some constrains that have been posed on the lipschitz constants in some interesting recent related works. Two illustrative examples are given at the end to apply our theoretical results and show its validity. Potential future directions of our work can be dedicated to applying our obtained results to some practical applications. Some possible extensions and generalizations of our obtained results can also be our future investigations.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Ethical Approval

The authors declare the ethical standards are taken into consideration.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at Jouf University for funding this work through research grant No. (DSR-2021-03-0210).