Communications in Nonlinear Science and Numerical Simulation
New concepts and results in stability of fractional differential equations☆
Highlights
► New concepts in stability of fractional differential equations are offered. ► A fixed theorem in a generalized complete metric space is used. ► Hyers–Ulam–Rassias stability as well as Hyers–Ulam stability are derived.
Introduction
Let Y be a normed space and I be a given interval. Assume that for a continuously differentiable function f : I → Y satisfying fractional differential inequality for all x ∈ I and for some ε ⩾ 0, where is the Caputo fractional derivative (see Definition 2.3) of order α ∈ (0, 1) with the lower limit a ∈ R, there exists a solution f0 : I → Y of the fractional differential equationsuch that ∥f(x) − f0(x)∥ ⩽ K(ε) for all x ∈ I, where K(ε) is only dependent on ε. Then, we say that the above fractional differential Eq. (1) has the Hyers–Ulam stability.
If the above statement is also true when we replace ε and K(ε) by φ(x) and Φ(x), where φ, Φ : I → [0, ∞) are functions not depending on f and f0 explicitly, then we say that the corresponding fractional differential Eq. (1) has the Hyers–Ulam–Rassias stability.
The first definition of the fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus was mentioned already in 1695 by Leibniz and L’Hospital. Recently, fractional differential equations have been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics and economics. It draws a great application in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic model. Actually, fractional differential equations are considered as an alternative model to integer differential equations. For more details on basic theory of fractional differential equations, one can see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3] and Tarasov [4], and the research papers of Agarwal et al. [5], Ahmad and Nieto [6], Bai [7], Benchohra et al. [8], Chang and Nieto [9], Wang et al. [10], [11], [12], [13], [14], Zhang [15] and Zhou et al. [16], [17]. In these previous works, Cauchy problems, nonlocal problems, impulsive problems and boundary value problems of all kinds of fractional differential equations (inclusions) and optimal control problems are discussed.
In 1940, Ulam proposed the general “Ulam stability problem” (see page 63, [18]): When is it true that by slightly changing the hypothesis of a theorem one can still assert that the thesis of the theorem remains true or approximately true? In 1941, this problem was solved by Hyers [19] in the case of Banach spaces. Thereafter, this type of stability is called the Hyers–Ulam stability. In 1978, Rassias [20] provided a remarkable generalization of the Hyers–Ulam stability of mappings by considering variables. The stability properties of all kinds of equations have attracted the attention of many mathematicians. In particular, the Hyers–Ulam stability and Hyers–Ulam–Rassias stability have been taken up by a number of mathematicians and the study of this area has the grown to be one of the central subjects in the mathematical analysis area. For more details on the recently advanced on the Hyers–Ulam stability and Hyers–Ulam–Rassias stability of differential equations, one can see the monographs of Cădariu [21], Hyers [22] and Jung [23] and the research papers of Jung [24], Miura et al. [25], [26], Rus [27], Obłoza [28], [29] and Takahasi et al. [30].
It is well known that the analysis on stability of fractional differential equations is more complex than that of classical differential equations, since fractional derivatives are nonlocal and have weakly singular kernels. As a result, the development of stability of nonlinear fractional differential equations is a bit slow. Recently, Li and Zhang [31] make a brief overview on the stability results of the fractional differential equations. Particular, Li et al. [32], [33] devote to the Mittag–Leffler stability and the fractional Lyapunov’s second method are first proposed; Deng [34] derive sufficient conditions for the local asymptotical stability of nonlinear fractional differential equations. Although, there are some work on the local stability and Mittag–Leffler stability for fractional differential equations, to the best of my knowledge, there are very rare works on the Ulam stability of fractional differential equations, which maybe provide a new way for the researchers to investigate the stability of fractional differential equations from different perspectives. A pioneering work on the Ulam stability and data dependence for fractional differential equations with Caputo derivative has been reported by Wang et al. [35].
In this paper, we continue to combine these two areas to extend the study to the Hyers–Ulam stability as well as the Hyers–Ulam–Rassias stability of fractional differential equations via a generalized fixed point approach, which provide a new method to study such problems. More precisely, we will adopt some part idea of Wang et al. [35], Caˇdariu and Radu [36] and Jung [37] and prove the Hyers–Ulam–Rassias stability as well as the Hyers–Ulam stability of the fractional differential Eq. (1).
The rest of this paper is organized as follows. In Section 2, some notations and preparation results are given. In Section 3, Hyers–Ulam–Rassias stability results of the fractional differential Eq. (1) are obtained. In Section 4, Hyers–Ulam stability results of the fractional differential Eq. (1) are obtained. At last, three examples are given to demonstrate the applicability of our results.
Section snippets
Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. We need some basic definitions and properties of the fractional calculus theory which are used further in this paper. For more details, see Kilbas et al. [1]. Definition 2.1 Given an interval [a, b] of R. The fractional order integral of a function h ∈ L1([a, b], R) of order α ∈ R+ is defined bywhere Γ(·) is the Gamma function. Definition 2.2 For a function h given on the interval [a, b],
Hyers–Ulam–Rassias stability
In this section, we first investigate the Hyers–Ulam–Rassias stability of the fractional differential Eq. (1) defined on a bounded and closed interval. By using the idea of Caˇdariu and Radu [36] and Jung [37], we will prove the Hyers–Ulam–Rassias stability of the fractional differential Eq. (1) via Theorem 2.6. Theorem 3.1 For given real numbers a and b with a < b, let I = [a, b] be a closed interval and choose a real number c ∈ I. Let K and L be positive constants with 0 < KL < 1. Assume that F : I × R → R is a continuous
Hyers–Ulam stability
In the following theorem, we prove the Hyers–Ulam stability of the fractional differential Eq. (1) defined on a finite and closed interval. Theorem 4.1 Given c ∈ R and r > 0, let I denote a closed ball of radius r and centered at c, that is, I = {x ∈ R∣c − r ⩽ x ⩽ c + r} and let F : I × R → R be a continuous function which satisfies a Lipschitz condition (2) for all x ∈ I and y, z ∈ R, where L is a constant with . If a continuously differentiable function y : I → R satisfies the differential inequality
Examples
In this section we give some examples to illustrate the usefulness of our main results. Example 1 We choose positive constants K and L with KL < 1. For some positive numbers ε, and , let be a closed interval. Given a polynomial p(x), we assume that a continuously differentiable function y : I → R satisfiesfor all x ∈ I. If we set F(x, y) = Ly(x) + p(x) and φ(x) = x−α + ε, then the above inequality has the identical form with (3). Moreover, by
Conclusions
We offer some new concepts in stability of a class of fractional differential equations with Caputo fractional derivative from different perspectives. By applying a fixed point theorem in a generalized complete metric space, we present the Hyers–Ulam–Rassias stability as well as Hyers–Ulam stability of fractional differential equations, which maybe provide a new way for the researchers to discuss such interesting problems in the mathematical analysis area.
The current concepts have applicable
Acknowledgements
The authors thanks the referees for their careful reading of the manuscript and insightful comments, which help to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improve the presentation of the paper.
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The first author acknowledges the support by the Key Projects of Science and Technology Research in the Ministry of Education (211169) and Tianyuan Special Funds of the National Natural Science Foundation of China (11026102); The third author acknowledges the support by National Natural Science Foundation of China (10971173).