On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

Opuscula Mathematica

Said R. Grace¹  Ercan Tunç²  John R. Graef³ 

[1] Cairo University, Faculty of Engineering, Department of Engineering Mathematics, Orman, Giza 12221, Egypt;Gaziosmanpasa University, Department of Mathematics, Faculty of Arts and Sciences, 60240, Tokat, Turkey;University of Tennessee at Chattanooga, Department of Mathematics, Chattanooga, TN 37403, USA;

关键词: integro-differential equations;    fractional differential equations;    nonoscillatory solutions;    boundedness;    caputo derivative;   

DOI  :  https://doi.org/10.7494/OpMath.2020.40.2.227

来源: DOAJ

【 摘 要 】

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

【 授权许可】

Unknown