Existence Uniqueness Theorems for Multi-Term Fractional Delay Differential Equations

Abstract

In this paper we analyze non-linear multi-term fractional delay differential Equation $\begin{array}{l} L\left( D \right)u\left( t \right) = f\left( {t,u\left( t \right),u\left( {t - \tau } \right)} \right),\;t \in \left[ {0,T} \right] > 0, \\ u\left( t \right) = g\left( t \right),\;t \in \left[ { - \tau ,0} \right], \\ \end{array}$ where L(D) = λ_n^cDα_n + λ_n−1^cD^α−1 + ··· + λ₁^cDα₀ + λ₀^cDα₀, λ_i ∈ ∝ (i = 0, 1, ···, n), λ₀, λ_n ≠ 0, 0 ≤ α₀ < α₁ < ··· < λ_n−1 < λ_n < 1, and ^cD^α denotes the Caputo fractional derivative of order a. The Schaefer fixed point theorem and Banach contraction principle are used to investigate the existence and uniqueness of solutions for above equation with periodic/ anti-periodic boundary conditions.