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

Abstract

In this paper, we study the following fractional differential equation involving the Atangana-Baleanu-Caputo fractional derivative: {ABCaDτθ[x(ϑ)−F(ϑ,x(ϑ))]=G(ϑ,x(ϑ)), ϑ∈J:=[a,b],x(a)=φa∈ℝ.$$\left\{ {\matrix{ {AB{C_a}D_\tau ^\theta [x(\vartheta ) - F(\vartheta ,x(\vartheta ))] = G(\vartheta ,x(\vartheta )),\;\;\;{\kern 1pt} \vartheta \in J: = [a,b],} \hfill \cr {x(a) = {\varphi _a} \in .} \hfill \cr } } \right.$$

Keywords

• atangana-baleanu-caputo fractional derivative
• existence
• uniqueness
• lower solution
• dhage iteration principle
• partially ordered banach space
• nonlinear d-contraction operator
• 26a33
• 34a08
• 34a12