Initial-value / Nonlocal Cauchy problems for fractional differential equations involving ψ-Hilfer multivariable operators

Abstract

Abstract In this paper, we investigate two types of problems (the initial-value problem and nonlocal Cauchy problem) for fractional differential equations involving ψ-Hilfer derivative in multivariable case (ψ-m-Hilfer derivative). First we propose and discuss ψ-fractional integral, ψ-fractional derivative and ψ-Hilfer type fractional derivative of a multivariable function f : ℝ m → ℝ (m is a positive integer). Then, using the properties of the ψ-m-Hilfer fractional derivative with m = 1 (the ψ-Hilfer derivative), we derive an equivalent relationship between solutions to the initial-value (Cauchy) problem and solutions to some integral equations, and also present an existence and uniqueness theorem. Based on the equivalency relationship, we establish new and general existence results for the nonlocal Cauchy problem of fractional differential equations involving ψ-Hilfer multivariable operators in the space of weighted continuous functions. Moreover, we obtain a new Gronwall-type inequality with singular kernel, and derive the dependence of the solution on the order and the initial condition for the fractional Cauchy problem with the help of this Gronwall-type inequality. Finally, some examples are given to illustrate our results. Compared with the recent paper [2] and other previous works, the novelties in this paper are in treating the multivariable case of operators (f : ℝ m → ℝ, m is a positive integer).