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

Benkhettou, Nadia; Salim, Abdelkrim; Lazreg, Jamal Eddine; Abbas, Saïd; Benchohra, Mouffak

doi:10.2478/awutm-2023-0007

Abstract

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

The result is based on a Dhage fixed point theorem. Further, an example is provided for the justification of our main result.

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

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