On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations

Abstract

A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alternative and the Banach contraction principle, the main results are established. In the case of a local Lipschitz right hand side part function, the existence of a bounded mild solution is proved. Some examples illustrating the main results are provided.

Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions

“…We assumed in the paper the initialvalue problem ( 9), (10) has a solution defined for t ≥ t 0 for any initial value x 0 ∈ R n . Some existence results were given in [21,22].…”

Mittag-Leffler-Type Stability of BAM Neural Networks Modeled by the Generalized Proportional Riemann–Liouville Fractional Derivative

“…Similar to classical fractional derivatives, there are two main types of generalized proportional fractional derivatives: Caputo-type and Riemann-Liouville-type. Several results concerning the existence (see, for example, [13,14]), integral presentation of the solutions (see, for example, [15]), stability properties (see, for example, [16,17]) and applications to some models (see, for example, [16]) are considered with the Caputo type of generalized proportional fractional derivatives. Additionally, there are some results concerning the Riemann-Liouville type.…”

A generalized proportional Caputo fractional model of multi-agent linear dynamic systems via impulsive control protocol

“…Motivated by this research, we consider multi-agent linear dynamic systems with the generalized proportional Caputo fractional derivative and an impulsive control protocol. The generalized proportional Caputo fractional derivative was introduced in [34] and subsequently studied in [35][36][37][38] as an undeviating generalization of the existing Caputo fractional derivative. Namely, in this derivative, we have two parameters: α ≥ 0 which is the order of the derivative, and ρ ∈ (0, 1] which could be called the proportionality parameter.…”