Unique Solvability of the Initial-Value Problem for Fractional Functional Differential Equations—Pantograph-Type Model

Abstract

Contrary to the initial-value problem for ordinary differential equations, where the classical theory of establishing the exact unique solvability conditions exists, the situation with the initial-value problem for linear functional differential equations of the fractional order is usually non-trivial. Here we establish the unique solvability conditions for the initial-value problem for linear functional differential equations of the fractional order. The advantage is the lack of the calculation of fractional derivatives, which is a complicated task. The unique solution is represented by the Neumann series. In addition, as examples, the model with a discrete memory effect and a pantograph-type model from electrodynamics are studied.

Keywords

• fractional order functional differential equations
• unique solvability
• Caputo derivative
• the model with a discrete memory effect
• the pantograph-type model from electrodynamics