Exact Solvability Conditions for the Non-Local Initial Value Problem for Systems of Linear Fractional Functional Differential Equations

Abstract

The exact conditions sufficient for the unique solvability of the initial value problem for a system of linear fractional functional differential equations determined by isotone operators are established. In a sense, the conditions obtained are optimal. The method of the test elements intended for the estimation of the spectral radius of a linear operator is used. The unique solution is presented by the Neumann’s series. All theoretical investigations are shown in the examples. A pantograph-type model from electrodynamics is studied.

General exact solvability conditions for the initial value problems for linear fractional functional differential equations

“…and thus coincides with the sequence studied, e.g., in [4]. Formula (3.6) defines the standard iteration sequence used in studies of the uniqueness of the trivial solution…”

“…, m the inequality (4.10) 2) appearing in the Theorem 4.1 presented are unimprovable in the sense that, generally speaking, that condition can not be assumed with ρ = 1. To check this, one can use, e.g., example 1 from [4].…”

“…By the σ-positivity of the operator (5.2) (see condition (5.3) and Lemma 9 from [4]), inequality (5.4) and properties (3.4), (3.5), (4.1) for increasing functions y 0 , y 1 we get that continuous function ω(t) from (5.6) implies the condition (4.2) from Theorem 4.1. The application of that theorem to the initial-value problem (5.1) and corresponding homogeneous problem implies the assertions required.…”

“…The fractional differential equations (FDEs) get a significant interest in modern literature on differential equations and are represented by numerous papers. Here referred to a few of them only [1,2,3,4,5,6,7,8,9].…”

“…The main goal of our investigation is the exact conditions lookup of the unique solvability of the boundary value problem for the fractional functional differential equations (FFDEs). Some recent results [3,4,5,6,8] motivated us to continue in this direction.…”