On the initial value problem for fractional Volterra integrodifferential equations with a Caputo–Fabrizio derivative

¹ Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam.
² Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam.
³ School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland.

^* Corresponding author: trivv@tdmu.edu.vn

Received: 30 June 2020
Accepted: 24 January 2021

Abstract

In this paper, a time-fractional integrodifferential equation with the Caputo–Fabrizio type derivative will be considered. The Banach fixed point theorem is the main tool used to extend the results of a recent paper of Tuan and Zhou [J. Comput. Appl. Math. 375 (2020) 112811]. In the case of a globally Lipschitz source terms, thanks to the L^p − L^q estimate method, we establish global in time well-posed results for mild solution. For the case of locally Lipschitz terms, we present existence and uniqueness results. Also, we show that our solution will blow up at a finite time. Finally, we present some numerical examples to illustrate the regularity and continuation of the solution based on the time variable.