On the Asymptotic Behavior of Nonoscillatory Solutions of Certain Fractional Differential Equations

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Abstract

We present the conditions under which every nonoscillator solution x(t) of the forced fractional differential equation
$$\begin{aligned} ^{\mathrm{C}}D_{\mathrm{c}}^{\alpha } y ( t ) = e ( t ) +f ( {t, x ( t )} ), c > 1,\alpha \in ( {0,1} ), \quad \mathrm{{and}} \,\, \delta \ge 1, \end{aligned}$$
where \(y(t)= ( {a(t) ( {{x}'(t)} )^{\delta }})^{\prime },c_0 =\frac{y(c)}{\Gamma (1)}= y(c)\), is a real constant which satisfies
$$\begin{aligned} |x(t)|=O\left( {t^{1/\delta }e^{t}\int _{\mathrm{c}}^{t} {a^{-1/\delta }} (s)\mathrm{d}s} \right) , \quad t \rightarrow \infty \end{aligned}$$
It is shown that the technique can be applied to some related fractional differential equations. Examples are inserted to illustrate the relevance of the obtained results.

Keywords

Integro-differential equation fractional differential equations asymptotic behavior nonoscillatory solutions 

Mathematics Subject Classification

34E10 34A34 

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering Mathematics, Faculty of EngineeringCairo UniversityGizaEgypt

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