Novel Stability Results for Caputo Fractional Differential Equations

Modelling some diseases with large mortality rates worldwide, such as COVID-19 and cancer is crucial. Fractional differential equations are being extensively used in such modelling stages. However, exact analytical solutions for the solutions of such kind of equations are not reachable. +erefore, close exact solutions are of interests in many scientific investigations. +e theory of stability in the sense of Ulam and Ulam–Hyers–Rassias provides such close exact solutions. So, this study presents stability results of some Caputo fractional differential equations in the sense of Ulam–Hyers, Ulam–Hyers–Rassias, and generalized Ulam–Hyers–Rassias. Two examples are introduced at the end to show the validity of our results. In this way, we generalize several recent interesting results.


Introduction and Preliminaries
Fractional calculus provides a powerful tool in both theoretical frameworks and practical aspects. In many disciplines, fractional modelling is much more suitable than the classical one. is is because of the nice modelling tools that are available only in fractional calculus (see e.g., [1,2]). In particular, fractional calculus has been used extensively in the modelling stages in the fields of economics, chemistry, aerodynamics, physics, and polymer rheology. It should be remarked also that a certain kind of fractional derivative has been used recently to model Ebola virus (see [3]) and HIV (see [4]). Fractional differential equations with Caputo and Caputo-Fabrizio derivatives are used recently by the authors in [5] for the model of cancer-immune system. e stability issue has gained substantially important attention in several research fields through applications.
ere are many kinds of stability; one of them is the stability introduced by Ulam in 1940. Since then, the problem is known as Ulam-Hyers stability or simply Ulam stability (see e.g., [6], for more details). Its applications for many types of equations have been investigated by many researchers. For more details on this concept, the readers can see the interesting works [7][8][9][10]. e stability problem that is introduced by Ulam can be stated as follows.
Assume that G 1 is a group and (G 2 , χ) a metric group. Given some ε * > 0, does there exist δ * > 0 such that if F: for all x 1 , x 2 ∈ G 1 , then a homomorphism F * : G 1 ⟶ G 2 exists such that for all x 1 ∈ G 1 ? Ulam's problem has been extended in many directions for interesting settings. In particular, Rassias (see [11]) generalized Ulam's result for Banach spaces. e nice result of Rassias reads as follows (see [11]). Theorem 1. Consider Banach spaces B and B * , and suppose a mapping Υ: B ⟶ B * such that the function t↦Υ(tx) from R into B * is continuous for each fixed x ∈ B. Assume that there are some β ≥ 0 and ω ∈ [0, 1), fulfilling Then, a unique solution exists Υ * : B ⟶ B * of the Cauchy equation The theorem of Rassias (see [11]) is nowadays known as the Hyers-Ulam-Rassias stability.
Throughout the study, we denote the set of reals by R, the set of nonzero reals by R * , and the set of complex numbers by C, and we fix an interval I ≔ [], ] + T] for some reals ], T with T > 0. Definition 1. Let λ > 0, χ ∈ C. e Mittag-Leffler function (MLF) (see e.g., [2]) E λ is defined as Similar to the exponential function, the function h(s) � [12][13][14][15] obtained some stability results by using the MLF. Now, we present the notion of generalized metric as follows. Let Z be a nonempty set.

Remark 1. Authors in
e notion of stability in the sense of Ulam-Hyers (UH), Ulam-Hyers-Rassias (UHR), and generalized UHR of fractional differential equations can be introduced as follows (see e.g., [16]). We consider the following fractional differential equation: and the following three inequalities: We define the stability of (6) as follows.
Definition 3. Equation (6) is called stable in the sense of UH if for a given ε > 0 and a function κ which satisfies (7), and there exists a solution κ 0 of (6) such that Definition 4. Equation (6) is called UHR stable if, for some ε > 0 and a function κ satisfying (9), there exists a solution κ 0 of (6) such that where ϱ(t 1 ) is some positive, nondecreasing, and continuous function.
Definition 5. Equation (6) is called generalized UHR stable if, for some ε > 0 and a function κ satisfying (8), there exists a solution κ 0 of (6) such that e theorem below represents a basic well-known fixed point theory (see [17]). is theorem plays a fundamental role in our study.
Define the space X as X ≔ C(I, R).
Remark 2. Note that the authors in [18] proved the existence and uniqueness of global solutions using the norm: is contribution is considered as a generalized version of the interesting results in [19][20][21]. Our contribution is original for many reasons. First, the metric used is a function of the Mittag-Leffler function. Second, the obtained results are in a complete generalized metric space. ird, the tool used is a version of Banach fixed point theory. e main purpose of this study is to study the stability of the following initial value problem: in the sense depicted in Definitions 3-5, where λ ∈ (0, 1), is the Caputo fractional derivative, and G: I × R ⟶ R is a given function. It should be noted that the solution of the initial value problem (16) is given by

Stability Results
In this section, we present our main results. In other words, we prove that, under certain conditions, functions that satisfy (16) approximately (in some sense) are close (in some way) to the solutions of (16). We have done this in both UH sense and also in UHR sense. e following theorem represents the stability of (16) in the sense of UHR.

Theorem 3. Assume G: I × R ⟶ R is continuous and satisfies
for all ω ∈ I, κ i ∈ R, i � 1, 2, and for some L G > 0. If an absolutely continuous function x: I ⟶ R satisfies for all ω ∈ I, where ε > 0 and ϱ(ω) is a positive, nondecreasing, and continuous function, then there is a solution x * of (16) such that where and δ is any positive constant.
Proof. Define the metric d on X in this way: Now, define the operator A: X ⟶ X such that (ω − s) λ−1 G(s, y(s))ds.

Mathematical Problems in Engineering
Since ϱ is nondecreasing, therefore, so that which means that the operator A is a strictly contractive operator. Now, since we have then, which implies that erefore, By employing eorem 2, there is a solution x * of (16) such that so that for all ω ∈ I. e following theorem represents the stability of (16) in the sense of UH. □ Theorem 4. Assume G: I × R ⟶ R is continuous and satisfies If an absolutely continuous function x: I ⟶ R satisfies for all ω ∈ I and some ε > 0, then there is a solution x * of (16) such that where and δ is any positive constant.
Proof. e proof is similar to eorem 3.
□ Remark 3. It should be noted that, in our analysis, we do not assume any condition on the constant L G , unlike the case of eorem 4.1 in [20], where the condition 0 < (L G r λ /(Γ(λ + 1))) < 1 was a basic condition.
Remark 4. Note that our results of the UH stability are some generalizations of the results obtained in [19]. e following theorem represents the stability of (16) in the sense of generalized UHR.

Theorem 5. Assume G: I × R ⟶ R is continuous and satisfies
for all ω ∈ I, κ i ∈ R, i � 1, 2, and for some L G > 0. If an absolutely continuous function x: for all ω ∈ I, where ϱ(ω) is a positive, nondecreasing, and continuous function, then there is a solution x * of (16) such that where and δ is any positive constant.
Proof. e proof is similar to eorem 3.

□
Remark 5. Notice that, in our study of the generalized UHR stability, we do not assume any condition on L G , unlike the case in eorem 3.1 in [20].
Remark 6. Note that, in [19], the authors obtained stability results for differential equations with integer-order derivatives, while in our case, it is for fractional-order derivatives.
In this sense, we generalized the interesting results in [19].

Examples
Two illustrative examples are given to show the validity of results.
Here, ε � 0.01. Using eorem 4, there is a solution x * of the fractional differential equation and M > 0 such that

Conclusion
It is known that, for the majority of fractional differential problems, a widely applicable general approach to determine the analytical solutions is not available. In this paper, we used a version of Banach fixed point theorem to prove that, under certain conditions, functions that satisfy some Caputo fractional differential equations approximately are close in some sense to the exact solutions of such kind of equations. In other words, we presented stability results for some Caputo fractional differential equations in the sense of UH, UHR, and generalized UHR. In our analysis, we used a new metric as a function of the Mittag-Leffler function. We end up with two examples that show the validity of our results.

Data Availability
No data were used to support this study.

Disclosure
An earlier version of this work has been presented as preprint in Authorea.

Conflicts of Interest
e authors declare that they have no conflicts of interest.