New concepts and results in stability of fractional differential equations

Abstract

In this paper, some new concepts in stability of fractional differential equations are offered from different perspectives. Hyers–Ulam–Rassias stability as well as Hyers–Ulam stability of a certain fractional differential equation are presented. The techniques rely on a fixed point theorem in a generalized complete metric space. Some applications of our results are also provided.

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 $‖{}^c{D}_{a+}^{\alpha }y(x)-F(x\text{,}y(x))‖⩽\epsilon$ for all x ∈ I and for some ε ⩾ 0, where ${}^c{D}_{a+}^{\alpha }$ 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 equation

$${}^c{D}_{a+}^{\alpha }y(x)=F(x\text{,}y(x))\text{,}$$

such 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.