<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper, we study the following fractional differential equation involving the Atangana-Baleanu-Caputo fractional derivative: <disp-formula id="j_awutm-2023-0007_eq_001"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2023-0007_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" id="M1"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mi>B</mi><msub><mi>C</mi><mi>a</mi></msub><msubsup><mi>D</mi><mi>τ</mi><mi>θ</mi></msubsup><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo>,</mo><mi>x</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>G</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo>,</mo><mi>x</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>ϑ</mi><mo>∈</mo><mi>J</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>φ</mi><mi>a</mi></msub><mo>∈</mo><mi>ℝ</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math><tex-math>$$\left\{ {\matrix{   {AB{C_a}D_\tau ^\theta [x(\vartheta ) - F(\vartheta ,x(\vartheta ))] = G(\vartheta ,x(\vartheta )),\;\;\;{\kern 1pt} \vartheta  \in J: = [a,b],} \hfill  \cr    {x(a) = {\varphi _a} \in .} \hfill  \cr  } } \right.$$</tex-math></alternatives></disp-formula></p>
<p>The result is based on a Dhage fixed point theorem. Further, an example is provided for the justification of our main result.</p>
</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. $$\left\{ {\matrix{   {AB{C_a}D_\tau ^\theta [x(\vartheta ) - F(\vartheta ,x(\vartheta ))] = G(\vartheta ,x(\vartheta )),\;\;\;{\kern 1pt} \vartheta  \in J: = [a,b],} \hfill  \cr    {x(a) = {\varphi _a} \in .} \hfill  \cr  } } \right.$$
The result is based on a Dhage fixed point theorem. Further, an example is provided for the justification of our main result.

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

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper, we study the following fractional differential equation involving the Atangana-Baleanu-Caputo fractional derivative: {...

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

Published Online: May 03, 2023

Page range: 79 - 91

DOI: https://doi.org/10.2478/awutm-2023-0007

Keywords
Atangana-Baleanu-Caputo fractional derivative, existence, uniqueness, lower solution, Dhage iteration principle, partially ordered Banach space, nonlinear D-contraction operator

© 2023 Nadia Benkhettou et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: May 03, 2023

Page range: 79 - 91

DOI: https://doi.org/10.2478/awutm-2023-0007

KeywordsAtangana-Baleanu-Caputo fractional derivative, existence, uniqueness, lower solution, Dhage iteration principle, partially ordered Banach space, nonlinear D-contraction operator

© 2023 Nadia Benkhettou et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Atangana-Baleanu-Caputo fractional derivative, existence, uniqueness, lower solution, Dhage iteration principle, partially ordered Banach space, nonlinear D-contraction operator