The extended fractional Caputo–Fabrizio derivative of order 0 ≤ σ < 1 $0\leq \sigma <1$ on C R [ 0 , 1 ] $C_{\mathbb{R}}[0,1]$ and the existence of solutions for two higher-order series-type differential equations

Abstract

We extend the fractional Caputo–Fabrizio derivative of order $0\leq \sigma <1$ on $C_{\mathbb{R}}[0,1]$ and investigate two higher-order series-type fractional differential equations involving the extended derivation. Also, we provide an example to illustrate one of the main results.