Lakshmikantham Monotone Iterative Principle for Hybrid Atangana-Baleanu-Caputo Fractional Differential Equations

Abstract

Abstract In this paper, we study the following fractional differential equation involving the Atangana-Baleanu-Caputo fractional derivative: { A B C a D τ θ [ x ( ϑ ) − F ( ϑ , x ( ϑ ) ) ] = G ( ϑ , x ( ϑ ) ) ,         ϑ ∈ J : = [ a , b ] , x ( a ) = φ a ∈ ℝ . $$\left\{ {\matrix{ {AB{C_a}D_\tau ^\theta [x(\vartheta ) - F(\vartheta ,x(\vartheta ))] = G(\vartheta ,x(\vartheta )),\;\;\;{\kern 1pt} \vartheta \in J: = [a,b],} \hfill \cr {x(a) = {\varphi _a} \in .} \hfill \cr } } \right.$$ The result is based on a Dhage fixed point theorem. Further, an example is provided for the justification of our main result.