| Abstract: |
| We present a survey of results that have been obtained over the past years on asymptotic properties of positive solutions of second order nonlinear differential equations
$$y^{\prime\prime} = q(t)\Phi_\gamma(y)\,,$$
difference equations
$$\Delta^2 y(n) = q(n)\Phi_\gamma(y(n+1))\,,\ \ n\in\Bbb N\,,$$
and Caputo fractional differential equations
$${}^C{\cal D}^{\alpha+1} y = q(t)\Phi_\gamma(y)\,,\ \ \alpha\in(0,1)\,,$$
where $\Phi_\gamma(u)=|u|^{\gamma}\,{\rm sgn}\,u$, $\gamma>0$.
Frequently, asymptotic analysis of these equations examines cases in which coefficients and/or solutions are close to power functions, in the sense that $q(t)$ is $= t^{\varrho}$ or $\sim t^{\varrho}$ or $o(t^{\varrho})$ or ${\cal O}(t^{\varrho})$ as $t\to\infty$. An objective is to emphasize how the concept of regular variation can be used to generalize these power behavior settings and, at the same time, provide quite accurate information about the behavior of solutions. Therefore, it will be shown with the application of theory of regular variation, that all positive increasing and all positive decreasing solutions of these equations are regularly varying, providing the coefficient $q$ is regularly varying, as well as that the asymptotic behavior at infinity of these solutions can be determined explicitly. Another objective is to discuss some analogies and discrepancies between the continuous, discrete and fractional case. |
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