Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives

Abstract

The generalized Caputo fractional derivative is a name attributed to the Caputo version of the generalized fractional derivative introduced in Jarad et al. (J. Nonlinear Sci. Appl. 10:2607-2619. Depending on the value of ρ in the limiting case, the generality of the derivative is that it gives birth to two different fractional derivatives. However, the existence and uniqueness of solutions to fractional differential equations with generalized Caputo fractional derivatives have not been proven. In this paper, Cauchy problems for differential equations with the above derivative in the space of continuously differentiable functions are studied. Nonlinear Volterra type integral equations of the second kind corresponding to the Cauchy problem are presented. Using Banach fixed point theorem, the existence and uniqueness of solution to the considered Cauchy problem is proven based on the results obtained.

Global asymptotic stability of the fractional differential equations

“…Many problems related to the differential equations described by a specified derivative can be studied. The first problem consists of proving the existence and the uniqueness of the solution of the fractional differential equations [4]. The second problem is to solve the fractional differential equations.…”

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

“…In this paper, we mainly consider a kind of ψ-Hilfer fractional order differential equation. The addressed equation has time-varying delay terms and non-instantaneous impulsive effects, which are quite different from the related references discussed in the literature [18,19,21,22,[38][39][40][41][42]. The nonlinear fractional order differential system studied in the present paper is more generalized and more practical.…”

“…In [20], Ameen et al studied the Ulam stability and existence theorems for Caputo generalized fractional differential equations where the kernel of the fractional derivative was function dependent so that the result generalized many existing results in history. Further, for more details about some other properties of the solutions, we can see [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations

“…However, little attention has been devoted to FDEs with generalized fractional derivatives. The Hilfer fractional derivative (HFD), a generalization of the RL fractional derivative was first introduced by Hilfer [4,11,15,16]. The existence and uniqueness of general initial and boundary value problems involving HFD were first examined by Fu-rati and Kassim [10] and Wang and Zhang [36], respectively.…”