On a nonlocal implicit problem under Atangana\u2013Baleanu\u2013Caputo fractional derivative

Mohammed S. Abdo,Saeed M. Ali,Abeer S. Alnahdi,S. Saleh,Mdi Begum Jeelani

doi:10.1186/s13661-021-01579-6

Abstract

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.

Highlights

Fractional calculus [1, 2] has persistently magnetized the attention of many researchers in the few past decades
Some interested authors and researchers have realized that innovation for novel fractional derivatives with nonsingular or singular kernels is an urgent necessity to satisfy the need to model more realistic problems in different fields of applied science
(2021) 2021:104 equations (FDEs) and mathematical modeling that incorporate ABC fractional derivatives can be found in the following series of articles [14,15,16,17,18,19,20,21,22,23,24,25,26]

Summary

Introduction

Fractional calculus [1, 2] has persistently magnetized the attention of many researchers in the few past decades. Caputo and Fabrizio in [9] suggested a novel kind of fractional derivatives where the kernel relies on the exponential function. Some properties of this novel operator were studied by Losada and Nieto in [10]. In [11] the authors proposed interesting new fractional operators called Atangana–Baleanu (AB) fractional operators. One of these operators is called Atangana–Baleanu–Caputo (ABC) fractional derivative, and it is basically a generalization of the Caputo operator.

Objectives

Conclusion