On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions

Abstract

Abstract In this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders 0 < ϑ ≤ 1 $0<\vartheta \leq 1$ and 1 < ϑ ≤ 2 $1<\vartheta \leq 2$ . We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.

Keywords

• Implicit fractional differential equations
• Atangana–Baleanu–Caputo fractional derivative
• Integral conditions
• Fixed point theorems