Existence Uniqueness Theorems for Multi-Term Fractional Delay Differential Equations

In this paper we analyze non-linear multi-term fractional delay differential Equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{l} L\left( D \right)u\left( t \right) = f\left( {t,u\left( t \right),u\left( {t - \tau } \right)} \right),\;t \in \left[ {0,T} \right] > 0, \\ u\left( t \right) = g\left( t \right),\;t \in \left[ { - \tau ,0} \right], \\ \end{array}$$\end{document} where L(D) = λncDαn + λn−1cDα−1 + ··· + λ1cDα0 + λ0cDα0, λi ∈ ∝ (i = 0, 1, ···, n), λ0, λn ≠ 0, 0 ≤ α0 < α1 < ··· < λn−1 < λn < 1, and cDα 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.