Initial-value / Nonlocal Cauchy Problems for Fractional Differential Equations Involving ψ-Hilfer Multivariable Operators

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 : ℝm → 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 : ℝm → ℝ, m is a positive integer).

Existence and uniqueness of solutions of nonlinear Cauchy‐type problems for first‐order fractional differential equations

“…We define an operator F, called the Nemytskii operator, by The existence of nonnegative continuous solutions for the first-order FDEs (40) subject to another initial condition u(a) = c 0 , where u(a) is required to exist, has been studied, for example, in [15] and [17,18]. When u(a) exists and w 0 = u(a), the existence of nonnegative continuous solutions in C[0, T] of the IVPs for (40) with [a, b] = [0, T] subject to the initial condition u(a) = c 0 was studied in [15], where…”

“…We refer to [15, Definition 2] for the definition of an $$$ {L}^p $$$‐Carathéodory function, where $$$ f $$$ has an $$$ {L}^p $$$ upper bounded function on each bounded subset of $$$ \left[a,b\right]\times J $$$.…”

“…The existence of nonnegative continuous solutions for the first‐order FDEs () subject to another initial condition $$$ u(a)={c}_0 $$$, where $$$ u(a) $$$ is required to exist, has been studied, for example, in [15] and [17, 18]. When $$$ u(a) $$$ exists and $$$ {w}_0=u(a) $$$, the existence of nonnegative continuous solutions in $$$ C\left[0,T\right] $$$ of the IVPs for () with $$$ \left[a,b\right]=\left[0,T\right] $$$ subject to the initial condition $$$ u(a)={c}_0 $$$ was studied in [15], where $$$ f:\left[0,T\right]\times {\mathrm{\mathbb{R}}}_{+}\to {\mathrm{\mathbb{R}}}_{+} $$$ is an $$$ {L}^p $$$‐Carathéodory function for $$$ p\in \left[\frac{1}{1-\alpha },\infty \right) $$$. The existence and uniqueness of local solutions in $$$ C\left[a,a+h\right] $$$ were studi...…”

“…x ∈ [a, b]. (43) A function 𝑓 ∶ [a, b]×J → R is said to be a global L p -Carathéodory function for p ∈ [1, ∞] if 𝑓 satisfies the Carathéodory conditions and the Nemytskii operator F maps L p ([a, b]; J) into L p (a, b).We refer to[15, Definition 2] for the definition of an L p -Carathéodory function, where 𝑓 has an L p upper bounded function on each bounded subset of [a, b] × J.The following result provides necessary and sufficient conditions forF to map L p 1 ((a, b); J) into L p 2 (a, b), where p 1 , p 2 ∈ [1, ∞); see [16, Theorem 2.3]. Let p 1 , p 2 ∈ [1, ∞).…”

Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

“…However, there are few studies on fractional differential equations involving Hilfer fractional derivative with variable coefficient. Due to the variable coefficient function, it is difficult to obtain the analytical solution of such equations ( [12][13][14][15][16][17][18]). The representation of explicit solutions has become a problem, even for the case involving Riemann-Liouville or Caputo derivative, the problem is not completely solved.…”