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Journals / Results in Nonlinear Analysis / 2022 / Cilt: 5 - Sayı: 3

Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations

Ahmed KAJOUNI
Najat CHEFNAJ
Khalid HİLAL
Pages
360–371
Publication Date
2022-09-30
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

As an application, we have given example to demonstrate the effectiveness of our main result.

📌 3 Key Takeaways

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Objective & Hypothesis:In this work, we prove the existence of a solution for the initial value problem of nonlinear fractional differential equation with quadratic perturbations involving the Caputo fractional derivative ( cDα0+−ρt cDβ0+)(x(t)f(t,x(t)))=g(t,x(t)),t∈J=[0,1],1<α<2,0<β<α( cD0+α−ρt cD0+β)(x(t)f(t,x(t)))=g(t,x(t)),t∈J=[0,1],1<α<2,0<β<α with conditions x0=x(0)f(0,x(0))x0=x(0)f(0,x(0)) and \\x1=x(1)f(1,x(1))x1=x(1)f(1,x(1)).
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Key Findings & Data:Dhage's fixed-point the theorem was used to establish this existence.
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Conclusion & Impact:As an application, we have given example to demonstrate the effectiveness of our main result.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Fractional differential equation · Quadratic perturbations · Dhage fixed point.

Abstract

In this work, we prove the existence of a solution for the initial value problem of nonlinear fractional differential equation with quadratic perturbations involving the Caputo fractional derivative ( cDα0+−ρt cDβ0+)(x(t)f(t,x(t)))=g(t,x(t)),t∈J=[0,1],1<α<2,0<β<α( cD0+α−ρt cD0+β)(x(t)f(t,x(t)))=g(t,x(t)),t∈J=[0,1],1<α<2,0<β<α with conditions x0=x(0)f(0,x(0))x0=x(0)f(0,x(0)) and \\x1=x(1)f(1,x(1))x1=x(1)f(1,x(1)). Dhage's fixed-point the theorem was used to establish this existence. As an application, we have given example to demonstrate the effectiveness of our main result.

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Cite This Article

KAJOUNI, A., CHEFNAJ, N., HİLAL, K. (2022). Existence of Solution for a Nonlinear Fractional Order Differential Equation with a Quadratic Perturbations. Results in Nonlinear Analysis, Cilt: 5 - Sayı: 3, 360–371. https://ajindex.com/articles/870405

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