On a nonlocal implicit problem under Atangana–Baleanu–Caputo fractional derivative

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.

On Implicit Atangana–Baleanu–Caputo Fractional Integro-Differential Equations with Delay and Impulses

“…Because of their nonlocal characteristics, AB fractional derivatives are used. Many authors studied the ABC fractional derivative with applications, see for example [1][2][3][4][5][6][7][8][9][10][11]. Te prime reason is the theory of fractional calculus's quick development, which is used extensively in many diferent felds including biology, mathematics, chemistry, physics, mechanics, medicine, environmental science, control theory, image and signal processing, fnance, and others, see reference [12][13][14][15][16][17].…”

Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

“…In view of this, many authors employed ABC derivative to study fractional differential equations and modeling of the infectious diseases, we refer to these works [15][16][17][18][19][20]. Particularly, Alnahdi et al [21], studied the existence, uniqueness and continuous dependence of solutions of the nonlinear implicit fractional differential equation with nonlocal conditions involving the ABC fractional derivative. Furthermore, a lot of excellent materials on the mathematical models with different derivative operators applied to model real-life phenomena such as [22][23][24][25][26][27][28].…”

“…Particularly, Alnahdi et al. [21], studied the existence, uniqueness and continuous dependence of solutions of the nonlinear implicit fractional differential equation with nonlocal conditions involving the ABC fractional derivative. Furthermore, a lot of excellent materials on the mathematical models with different derivative operators applied to model real-life phenomena such as [22–28].…”