Opuscula Math. 40, no. 2 (2020), 227-239
https://doi.org/10.7494/OpMath.2020.40.2.227

Opuscula Mathematica

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

John R. Graef
Said R. Grace
Ercan Tunç

Abstract. 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.

Keywords: integro-differential equations, fractional differential equations, nonoscillatory solutions, boundedness, Caputo derivative.

Mathematics Subject Classification: 34E10, 34A34.

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• Communicated by Josef Diblík.
• Revised: 2019-10-09.
• Accepted: 2019-11-03.
• Published online: 2020-03-09.