\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 201, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/201\hfil Asymptotic behavior of solutions] {Asymptotic behavior of solutions to second-order differential equations with fractional derivative perturbations} \author[E. Brestovansk\'a, M. Medve\v{d} \hfil EJDE-2014/201\hfilneg] {Eva Brestovansk\'a, Milan Medve\v{d}} % in alphabetical order \address{Eva Brestovansk\'a \newline Department of Economics and Finance, Faculty of Management, Comenius University, Odboj\'arov str., 831 04 Bratislava, Slovakia} \email{Eva.Brestovanska@fm.uniba.sk} \address{Milan Medve\v{d} \newline Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia} \email{Milan.Medved@fmph.uniba.sk} \thanks{Submitted June 16, 2014. Published September 26, 2014.} \subjclass[2000]{34E10, 24A33} \keywords{ Rimann-Liouville derivative; Caputo's derivative; \hfill\break\indent fractional differential equation; asymptotic behavior} \begin{abstract} In this article we study the asymptotic behavior of solutions to nonlinear second-order differential equations having perturbations that involve Caputo's derivatives of several fractional orders. We find sufficient conditions for all solutions to be asymptotic to a straight line. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction}\label{s:1} The aim of this article is to study the asymptotic properties of solutions to scalar second-order ordinary differential equations that are perturbed with a term involving fractional derivatives. In these equations, the fractional derivatives most frequently used are the Riemann-Liouville and the Caputo's fractional derivatives. For basic definitions of fractional calculus and fundamentals of the theory of fractional differential equations, we refer the reader to the monographs \cite{PE,PO}. Fractional derivatives play the role of a damping force in vibrating systems in viscous fluids; which is the case in the well known Bargley-Torvik equation, \begin{equation} u''(t) + A^cD^{3/2}u(t) = au(t) + \phi(t)\,. \end{equation} This equation models the motion of a rigid plate immersing in a viscous liquid with the fractional damping term $A^cD^{3/2}u(t)$ which has Caputo's fractional derivative (see \cite{TOR}). Solutions of the linear fractionally damped oscillator equation with the Caputo's derivative are analyzed in \cite{NA}. Existence results on boundary-value problems for the generalized Bagely-Torvik equation \begin{equation} u''(t) + A\,^cD^\alpha u(t) = f(t, u(t), ^cD^\beta u(t), u'(t)) \end{equation} and for some other fractional differential equations can be found in \cite{ARS1,ARS2,S,OS}. An existence and uniqueness result for the multi-fractional initial-value problem \begin{equation}\label{S} \begin{gathered} Au'' + \sum_{k=1}^NB_k\,^cD^{\alpha_k}u(t) = f(t, u), \\ u(0) = u_0,\quad u'(0) = c_1,\quad 0 < \alpha_k < 2,\; k = 1, 2,\dots, N \end{gathered} \end{equation} can be found in \cite{SE}. Caputo's fractional derivatives in equation \eqref{S} play the role of damping terms. Abstract evolution equations with the Caputo's fractional derivatives in the nonlinearities are studied in \cite{KMT1,KMT2} . Fractionally damped pendulums or oscillators are studied in \cite{NA,SE}. More articles devoted to this type of equations can be found in the list of references. The following equation for a pendulum has the ordinary damping term $\lambda x'(t)$ and the fractional damping terms $\lambda_1\,^cD^{\beta_1}x(t), \dots , \lambda_m\,^cD^{\beta_m}x(t)$: \[ x''(t) + \lambda_1\,^cD^{\beta_1}x(t)+\dots \lambda_m\,^cD^{\beta_m}x(t) + \lambda x'(t) + \omega^2x(t) = g(t, x(t), x'(t)), \] where $t > 0$, $\beta_i \in (0, 1)$, $i =1, 2,\dots, m$. In \cite{NA}, the equation \[ x'' + \lambda_0^cD^\alpha x + \omega^2x = 0,\quad x(0) = x_0, \quad x'(0) = x_1, \quad \lambda > 0. \] is analyzed by using the fractional version of the Laplace transformation. The Laplace image of $x(t)$ is \[ X(s) = \frac{sx_0 + x_1 + \lambda s^{\alpha - 1}x_0}{s^2 + \lambda s^\alpha + \omega^2}, \] and the characteristic equation for the fractional differential equation is \[ s^2 + \lambda s^\alpha + \omega^2 =0. \] When $\alpha = p/q$ this characteristic equation is was analyzed in \cite{PE}. For the linear fractionally damped oscillator with $\alpha = 1/2$ the characteristic equation is \[ s^2 + \lambda s^{1/2} + \omega^2 = 0, \] whose analysis is much more complicated than in the case of the harmonic oscillator with the classical damping term (see \cite{NA}). It is clear that the exact analysis of linear fractional systems is extraordinary difficult. Some analysis and simulations of fractional-order systems can be found in the book \cite{PE}. The form of the equation \eqref{S} enables us to avoid some difficulties in the study of the stability problem by using a desingularization method developed in \cite{MED1,MED2,MED3}. In the asymptotic theory of the $n$-th order nonlinear ordinary differential equations \begin{equation} y^{(n)} = f(t, y, y',\dots,y^{(n-1)}), \end{equation} a classical problem is to establish some conditions for the existence of a solution which approaches a polynomial of degree $1 \leq m \leq n-1$ as $t \to \infty$. The first paper concerning this problem was published by Caligo \cite{CA} in 1941. He proved that if \begin{equation} |A(t)| < \frac{k}{t^{2 +\rho}} \end{equation} for all large $t$, where $k, \rho$ are given, then any solution $y(t)$ of the linear differential equation \begin{equation} y''(t) + A(t)y(t) = 0,\quad t > 0, \end{equation} can be represented asymptotically as $y(t) = c_1t + c_2 + o(1)$ when $t \to +\infty$, with $c_1, c_2 \in \mathbb{R}$ (see \cite{ADMM}). The first article on the nonlinear second-order differential equation \begin{equation}\label{SOD} y''(t) + f(t, y(t)) = 0 \end{equation} was published by Trench \cite{TR} in 1963. Then there are publications by Cohen \cite{COH}, Trench \cite{TR}, Kusano and Trench \cite{KT1} and \cite{KT2}, Dannan \cite{DAN}, Constantin \cite{CON1} and \cite{CON2}, Rogovchenko \cite{R}, Rogovchenko \cite{RR}, Mustafa, Rogovchenko \cite{MR}, Lipovan \cite{LI} and others. In the proofs of their results the key role is played by the Bihari inequality \cite{BI} which is a generalization of the Gronwall inequality. Some results on the existence of solutions of the $n$-th order differential equation approaching a polynomial function of the degree $m$ with $1 \leq m \leq n-1$ are proved by Philos, Purnaras and Tsamatos \cite{PPT}. Their proofs are based on an application of the Schauder Fixed Point Theorem. The paper by Agarwal, Djebali, Moussaoui and Mustafa \cite{ADMM} surveys the literature concerning the topic in asymptotic integration theory of ordinary differential equations. Several conditions under which all solutions of the one-dimensional $p$-Laplacian equation \begin{equation} (|y'|^{p-1}y')' = f(t, y, y'),\quad p > 1 \end{equation} are asymptotic to $a +bt$ as $t \to \infty$ for some real numbers $a, b$ are proved in \cite{MP}. Some sufficient conditions for the existence of such solutions of the equation \begin{equation} (\Phi(y^{(n)})' = f(t, y), \quad n \geq 1, \end{equation} where $\Phi:\mathbb{R} \to \mathbb{R} $ is an increasing homeomorphism with a locally Lipschitz inverse satisfying $\Phi(0) = 0$ are given in the paper \cite{MM}. In the papers \cite{MED3,MED4} the fractional differential equation of the Caputo's type \begin{equation}\label{E1} ^cD^\alpha_ax(t) = f(t, x(t)),\quad a \geq 1,\;\alpha \in (1, 2) \end{equation} is studied. In \cite{MED4} a higher order fractional differential equation is studied. In the both papers sufficient conditions under which all solutions of these equations are asymptotic to $at + b$, is proved. The problem of asymptotic integration of fractional differential equations of the Riemann-Liouville type is studied in \cite{B1,B2}. The obtained results are proved by an application of the fixed point method. The aim of this paper is to give some conditions under which all solutions of a nonlinear second order differential equations perturbed by the Riemann-Liouville integral of a nonlinear function are asymptotic to $at + b$. The proof of this result is based on a desingularization method proposed by the author in the paper \cite{MED1} (see also \cite{MED2}). \section{second-order ODEs perturbed with a fractional derivative}\label{s:2} In this section we study the following fractional initial-value problem \begin{gather}\label{SODE} u''(t) + f(t, u(t), u'(t)) + \sum_{i=1}^mr_i(t)\int_0^t(t-s)^{\alpha_i-1 }h_i(\tau, u(\tau), u'(\tau))d\tau = 0, \\ \label{SODEIV} u(1) = c_1\quad u'(1) = c_2, \end{gather} where $t > 0$ and $0 < \alpha < 1$. \begin{definition}\label{Def1} \rm A function $u:[0, T) \to \mathbb{R}$, $0 < T \leq \infty,$ is called a solution of \eqref{SODE} if $u \in C^2$ on the interval $(0, T)$, $\lim_{\tau \to 0^+}u(t)$ exists and $u(t)$ satisfies \eqref{SODE} on the interval $(0, T)$. This solution is called global if it exists for all $t \in [0, \infty)$. \end{definition} We assume the following hypotheses: \begin{itemize} \item[(H1)] Every solution of the equation \eqref{SODE} is global; \item[(H2)] The functions $f(t, u, v), h_i(t, u, v)$, $i =1, 2,\dots, m$ are continuous on $D = \{(t, u, v): t \in [0, \infty) ,\; u, v \in \mathbb{R}\}$ and the functions $r_i(t)$, $i =1, 2,\dots, m$ are continuous on the interval $[0, \infty)$; \item[(H3)] There exist continuous, nonnegative functions $h_i:[0, \infty) \to \mathbb{R}$, $i =1, 2, 3$ and continuous, positive and nondecreasing functions $g_j:[0, \infty) \to \mathbb{R}$ such that \[ |f(t, u, v)| \leq Se^{-\gamma t}\Big(h_1(t)g_1\big(\frac{|u|}{t}\big) + h_2(t)g_2(|v|) + h_3(t)\Big ),\quad t > 0, \] where $S, \gamma > 0;$ \item[(H4)] There exist continuous, nonnegative functions $h_{ij}:[0, \infty) \to \mathbb{R}$, $i =1, 2,\dots, m$; $j=1, 2, 3$ and continuous positive, nondecreasing functions $G_{ij}:[0, \infty) \to \mathbb{R}$, $i=1, 2,\dots, m$; $j = 1, 2, 3$ such that \[ |f_i(t, u, v)| \leq h_{1i}(t)G_{ij}\big(\frac{|u|}{t}\big) + h_{2i}(t)G_{2i}(|v|) + h_{3i}(t),\quad t > 0; \] for all $(t, u, v) \in D$, $i = 1, 2,\dots, m$; \item[(H5)] $|r_i(t)| \leq S_ie^{-\omega_it}$, $t \geq 0$, where $S_i > 0$, $\omega_i > 1$, $i =1, 2,\dots, m$; \item[(H6)] There exist numbers $p_i > 1$, $i=1, 2,\dots, m$ such that $p_i(\alpha_i -1) + 1 > 0$ with \[ \int_0^\infty h_i(s)^q < \infty, \int_0^\infty h_{ij}(s)^q < \infty,\quad i=1, 2,\dots, m; \; j=1, 2, 3, \] where $q = q_1q_2\dots q_m$, $q_ i = p_i/(p_i - 1)$, $i = 1, 2,\dots, m$; \item[(H7)] \[ \int_0^\infty \frac{\tau^{q-1}d\tau}{\omega(\tau)} = \infty, \] where \[ \omega(w) = g_1(w)^q + g_2(w)^q + \sum_{i=1}^m\sum_{j=1}^2G_{ij}(w)^q. \] \end{itemize} \begin{theorem}\label{thm2.2} If the conditions {\rm (H1)--(H7)} are satisfied then for every global solution $u(t)$ of \eqref{SODE} there exist real numbers $a, b$ such that $u(t) = at + b + o(t)$ as $t \to \infty$. \end{theorem} For the proof of this theorem we use the following lemma, proved in \cite{MED1}. \begin{lemma}\label{lem2.3} Let $p_j, \alpha_j$, $j=1, 2,\dots, m$ satisfy {\rm (H4)}. Then \[ \int_0^t(t-s)^{p_j(\alpha_j - 1)}e^{p_js}ds \leq Q_je^{p_jt},\quad t \geq 0,\,j=1,2,\dots, m, \] where \[ Q_j = \frac{\Gamma(1+p_j(\alpha_j-1))}{p^{1+p_j(\alpha_j-1)}}, \] and \[ \Gamma(x) = \int_0^\infty s^{x-1}e^{-s}ds,\quad x > 0 \] which is the Euler gamma function. \end{lemma} \begin{proof}[Proof of Theorem \ref{thm2.2}] Let $u(t)$ be a solution of \eqref{SODE} corresponding to the initial conditions \eqref{SODEIV}. Then \begin{gather} \begin{aligned} u'(t) &= c_2 - \int_1^tf(s, u(s), u'(s))ds \\ &\quad - \sum_{i=1}^m\int_1^tr_i(s)\int_0^s(s-\tau)^{\alpha_i -1} f_i(\tau, u(\tau), u'(\tau))d\tau ds, \end{aligned} \\ \begin{aligned} u(t) &= c_1 + c_2(t-1) - \int_1^t(t-s)f(s, u(s), u'(s))ds \\ &\quad - \sum_{i=1}^m\int_1^t(t-s)r_i(s) \Big(\int_0^s(s-\tau)^{\alpha_i -1}f_i(\tau, u(\tau), u'(\tau))d\tau\Big )ds. \end{aligned} \end{gather} From conditions (H3)--(H5) it follows that for $t \geq 1$, \begin{align*} |u'(t)| &\leq |c_2| + \int_1^t[h_1(s)g_1\big(\frac{|u(s)|}{s}\big) + h_2(s)g_2(|u'(s)|) + h_3(s)]ds\\ &\quad + \sum_{i=1}^m\int_1^t|r_i(s)|\int_0^s(s-\tau)^{\alpha_i -1} \Big[ h_{1i}(\tau)G_{1i}\big(\frac{|u(\tau)|}{\tau}\big) \\ &\quad+ h_{2i}(\tau)G_{2i}(|u'(\tau)|) + h_{3i}(\tau)\Big]d\tau ds \end{align*} and \begin{align*} \frac{|u(t)|}{t} &\leq C + \int_1^t[h_1(s)g_1\big(\frac{|u(s)|}{s}\big) + h_2(s)g_2(|u'(s)|) + h_3(s)]ds\\ &\quad + \sum_{i=1}^m\int_1^t|r_i(s)| \int_0^s(s-\tau)^{\alpha_i -1}\Big[ h_{1i}(\tau)G_{1i} \big(\frac{|u(\tau)|}{\tau}\big) \\ &\quad + h_{2i}(\tau)G_{2i}(|u'(\tau)|) + h_{3i}(\tau)\Big]d\tau ds, \end{align*} where $C = |c_1| + |c_2|$. If $q_i = p_i/(p_i-1)$ then using Lemma \ref{lem2.3} and the H\"older inequality we estimate \begin{gather*} \begin{aligned} &\int_0^s(s-\tau)^{\alpha_i -1}k_{1i}(\tau)G_{1i} \big(\frac{|u(\tau)|}{\tau}\big)d\tau \\ &\leq \Big(\int_0^s(s-\tau)^{p_i(\alpha_i -1)}e^{p_i\tau}d\tau\Big)^{1/p_i} \Big(\int_0^se^{-q_i\tau}h_{1i}(\tau)^{q_i}G_{1i}\big(\frac{|u(\tau)|}{\tau} \big)^{q_i}d\tau \Big )^{1/q_i} \\ &\leq Q_ie^s\Big(\int_0^se^{-q_i\tau}h_{1i}(\tau)^{q_i}G_{1i} \big(\frac{|u(\tau)|}{\tau}\big)^{q_i}d\tau \Big)^{1/q_i}, \end{aligned} \\ \int_0^s(s-\tau)^{\alpha_i -1}h_{2i}(\tau)G_{2i}(|u'(\tau)|)d\tau \leq Q_ie^s\Big ( \int_0^se^{-q_i\tau}h_{2i}(\tau)^{q_i}G_{2i}(|u'(\tau)|)^{q_i}d\tau \Big)^{1/q_i}, \\ \int_0^s(s-\tau)^{\alpha_i -1}h_{3i}(\tau)d\tau \leq Q_ie^s \Big(\int_0^se^{-q_i\tau} h_{3i}(\tau)^{q_i} d\tau\Big)^{1/q_i}. \end{gather*} These inequalities yield \begin{align*} \frac{|u(t)|}{t} &\leq C + S\int_1^te^{-\gamma s}\Big (h_1(s)g_1 \Big (\frac{|u(s)|}{s}\Big ) + h_2(s)g_2(|u'(s)|) + h_3(s)\Big )ds \\ &\quad + \sum_{i=1}^mS_iQ_i\int_1^te^{-(\omega_i - 1)s} \Big\{\Big( \int_0^se^{-q_i\tau}h_{1i}(\tau)^{q_i}G_{1i} \big(\frac{|u(\tau)|}{\tau}\big)^{q_i}d\tau \Big )^{1/q_i}\\ &\quad + \Big(\int_0^se^{-q_i\tau}h_{2i}(\tau)^{q_i}G_{2i} (|u'(\tau)|)^{q_i}d\tau \Big )^{1/q_i} + \Big(\int_0^se^{-q_i\tau }h_{3i}(\tau)^{q_i}d\tau \Big)^{1/q_i}\Big\}ds \end{align*} Since $\omega_i > 1$ and $\gamma > 0 $, we have the estimate \begin{align*} \frac{|u(t)|}{t} &\leq C + S\int_0^te^{- \gamma s}\Big (h_1(s)g_1\frac{|u(s)|}{s}) + h_2(s)g_2(|u'(s)|) + h_3(s)\Big )ds \\ &\quad + \sum_{i=1}^mS_i\frac{Q_i}{\omega_i -1}\Big\{ \Big(\int_0^te^{-q_i\tau}h_{1i}(\tau)^{q_i}G_{1i} \big(\frac{|u(\tau)|}{\tau}\big)^{q_i}d\tau \Big)^{1/q_i}\\ &\quad + \Big(\int_0^te^{-q_i\tau}h_{2i}(\tau)^{q_i}G_{2i}(|u'(\tau)|)^{q_i} d\tau\Big)^{1/q_i} + \Big(\int_0^te^{-q_i\tau }h_{3i}(\tau)^{q_i}d\tau \Big)^{1/q_i}d\tau \Big\}. \end{align*} Denoting by $z(t)$ the right-hand side of this inequality, we have \[ \frac{|u(t)|}{t} \leq z(t),\quad |u'(t)| \leq z(t),\quad t \geq 0. \] Since $g_1, g_2, G_{1i}, G_{2i}, G_{3i}$ are nondecreasing functions these inequalities yield \begin{align*} z(t) &\leq C + S\int_0^te^{-\gamma s}\Big (h_1(s)g_1(z(s)) + h_2(s)g_2(z(s)) + h_3(s)\Big )ds\\ &\quad + \sum_{i=1}^mS_i\frac{Q_i}{\omega_i -1} \Big\{\Big( \int_0^te^{-q_i\tau}h_{1i}(\tau)^{q_i}G_{1i}(z(\tau))^{q_i} d\tau \Big)^{1/q_i} \\ &\quad + \Big(\int_0^te^{-q_i\tau}h_{2i}(\tau)^{q_i}G_{2i}(z(\tau))^{q_i}d\tau \Big)^{1/q_i} + \Big(\int_0^te^{-q_i\tau }h_{3i}(\tau)^{q_i}d\tau \Big)^{1/q_i}d\tau \Big\}. \end{align*} Let $Q = \max \{\frac{S_iQ_i}{\omega_i-1}, i = 1, 2,\dots, m\}$ and $q = q_1q_2\dots q_m$. Then using the inequality $(\sum_{i=1}^{3m+2}a_i)^q \leq (3m+2)^{q-1}(\sum_{i=1}^{3m+2}a_i^q)$ for any nonnegative numbers $a_i$, $i = 1, 2,\dots, 3m+2$, we obtain the estimate \begin{align*} &z(t)^q\\ &\leq (3m+2)^{q-1}\Big(C^q + S^q\int_1^te^{- \gamma s} \Big(\int_1^t(h_1(s)g_1(z(s)) + h_2(s)g_2(z(s)) + h_3(s) )ds\Big)^q\\ &\quad + Q^q\sum_{i=1}^m\Big\{\Big(\int_0^te^{-q_i\tau}h_{1i}(\tau)^{q_i} G_{1i}(z(\tau))^{q_i} d\tau \Big)^{\hat{q}_i} \\ &\quad + \Big(\int_0^te^{-q_i\tau}h_{2i}(\tau)^{q_i}G_{2i}(z(\tau))^{q_i} d\tau\Big)^{\hat{q}_i} + \Big(\int_0^te^{-q_i\tau }h_{3i}(\tau)^{q_i}d\tau \Big)^{\hat{q}_i}d\tau \Big\}, \end{align*} where $\hat{q}_i = q_1q_2\dots q_{i-1}q_{i+1}\dots q_m$. If $\hat{p}_i = \frac{\hat{q}_i}{\hat{q}_i-1}$ and $p = \frac{q}{q-1}$, then using the H\" older inequality we obtain the following inequalities \begin{gather*} \begin{aligned} &\int_0^te^{-\gamma s}\Big \{\int_1^s\Big (h_1(\tau)g_1(z(\tau)) + h_2(\tau)g_2(z(\tau)) + h_3(\tau) \Big )d\tau\Big \}^q ds\\ & \leq \big(\frac{1}{p\gamma}\big)^{1/p}\int_0^t \Big(h_1(s)g_1(z(s)) + h_2(s)g_2(z(s)) + h_3(s)\Big)^qds \\ &\leq 3^{q-1}\big(\frac{1}{p\gamma}\big)^{1/p} \int_0^t\Big(h_1(s)^qg_1(z(s))^q + h_2(s)^qg_2(z(s))^q + h_3(s)^q\Big )ds, \end{aligned}\\ \begin{aligned} &\Big(\int_0^te^{-q_i\tau}h_{1i}(\tau)^{q_i}G_{1i}(z(\tau))^{q_i} d\tau \Big)^{\hat{q}_i}\\ &\leq\Big(\int_0^te^{-\hat{p}_iq_is}ds\Big)^{\frac{1}{\hat{p}_i}} \Big(\int_0^th_{1i}(s)^qG_{1i}(z(s))^qds\Big) \\ &\leq \frac{1}{(\hat{p}_iq_i -1)^{1/\hat{p}_i}} \int_0^th_{1i}(s)^qG_{1i}(z(s))^qds, \end{aligned} \\ \Big(\int_0^te^{-q_i\tau}h_{2i}(\tau)^{q_i}G_{2i}(z(\tau))^{q_i}d\tau \Big)^{\hat{q}_i} \leq \frac{1}{(\hat{p}_iq_i -1)^{1/ \hat{p}_i}} \int_0^th_{2i}(s)^qG_{2i}(z(s))^qds, \\ \int_0^te^{-q_is}h_{3i}(s)^{q_i}ds \leq \frac{1}{(\hat{p}_iq_i -1)^{1/\hat{p}_i}} \int_0^th_{3i}(s)^qds. \end{gather*} From these inequalities and (H6) it follows that there exist a constant $A > 0$ such that \begin{align*} z(t)^q &\leq A + A \int_0^t [h_1(s)^qg_1(z(s))^q + h_2(s)^qg_2(z(s)) + h_3(s)^q]ds\\ &\quad + A\sum_{i=1}^m\int_0^t h_{1i}(s)^qG_{1i}(z(s))^q ds + A \sum_{i=1}^m\int_0^t h_{2i}(s)^qG_{2i}(z(s))^q ds. \end{align*} This inequality implies that the function $v(t) = z(t)^q$ satisfy the inequality \[ v(t) \leq A + \int_0^tF(s)\omega(v(s)^{\frac{1}{q}})ds, \quad t \geq 0, \] where \begin{gather*} \omega(z) = g_1(z)^q + g_2(z)^q+ \sum_{i=1}^m[G_{1i}(z)^q + G_{2i}(z)^q],\\ F(t) = A\Big(h_1(t)^q + h_2(t)^q + \sum_{i=1}^m[h_{1i}(t)^q + h_{2i}(t)^q]\Big). \end{gather*} From (H6) it follows that $\int_0^\infty F(s)ds < \infty$, and from the Bihari inequality we obtain \[ v(t) \leq K_0 = \Omega^{-1}[\Omega(A) + \int_0^\infty F(s)ds ]< \infty, \quad t \geq 0, \] where \[ \Omega(u) = \int_{v_0}^v\frac{\sigma}{\omega(\sigma)}\,. \] Note that $\Omega(A) + \int_0^\infty F(s)ds$ is always in the range of $\Omega^{-1}$, as $\omega(\infty) = \infty$ by (H7). This implies that there is a constant $K > 0$ such that \[ |u'(t)| \leq z(t) \leq K,\quad \frac{|u(t)|}{t} \leq z(t) \leq K,\quad t \geq 0. \] In conclusion, we obtain the existence of the limit \[ \lim_{t \to \infty}\frac{|u(t)|}{t} = c, \] which completes the proof. \end{proof} \section{Example}\label{s:3} \rm The following example is a fractional modification of the Caligo's example mentioned in the introduction. \begin{equation}\label{EX} \begin{aligned} &u''(t) + Se^{-\gamma t}\Big\{\omega^2\frac{1}{(t+1)^{1+\frac{1}{q}}} \big(\frac{u(t)}{t}\big)+ k_1\frac{1}{(t+1)^{1+\frac{1}{q}}}u'(t) + k_2\frac{1}{t^{1+\frac{1}{q}}}\Big\} \\ &\quad + \sum_{i=1}^m S_ie^{-\omega_it} \int_0^t(t-s)^{\alpha_i-1}\Big\{\frac{\eta_{1i}}{(s+1)^{1+\frac{1}{q_i}}} \ln \big [\big (\frac{u(s)}{s}\big )^{q_i} +2\big ]^{1/q_i} \\ &\quad + \frac{\eta_{2i}}{(s+1)^{1+\frac{1}{q_i}}} \big(\ln\big [u'(s)]^{q_i} + 2\big)^{1/q_i} + \frac{\eta_{3i}}{(s+1)^{1+\frac{1}{q_i}}}\Big\}ds=0, \end{aligned} \end{equation} where $S$, $\gamma$, $\omega$, $k_1$, $k_2$, $\eta_{1i}$, $\eta_{2i}$, $\eta_{3i}$, $i=1,2,\dots, m$ are positive numbers and $\gamma$, $\omega_i$, $q$, $q_i$, $\alpha_i$ satisfy the conditions in Theorem \ref{thm2.2}. Here \[ h_i(t)= \frac{k_i}{(t+1)^{1+\frac{1}{q}}},\quad h_{ji}(t) = \frac{\eta_{ji}}{(t+1)^{1+\frac{1}{q_i}}}, \] $i = 1, 2,\dots, m$, $j=1, 2, 3$, $g_1(u) = g_1(u) = [\ln (u^q + 2)]^{\frac{1}{q}}$, $g_{1i}(u) = g_{2i}(u) = [\ln (u^{q_i} + 2)]^{1/q_i}$. Since \[ \int_0^\infty h_i(s)^qds= \int_0^\infty \frac{1}{(s+1)^{1 + q}}ds = \frac{1}{q} \] and \[ \int_0^\infty\frac{\sigma^{q-1}d\sigma}{g_1(\sigma)^q} = \int_0^\infty\frac{\sigma^{q-1}d\sigma}{[\ln(\sigma^q + 2)]} = \frac{1}{q}\int_0^\infty\frac{d\tau}{\ln (\tau + 2)} = \infty, \] all conditions of Theorem \ref{thm2.2} are satisfied and therefore for any solution of \eqref{EX} there exist constants $a, b \in \mathbb{R}$ such that $u(t) = at + b + o(t)$ as $t \to \infty$. \subsection*{Acknowledgements} This research was supported by the Slovak Grant Agency VEGA-M\v{S}, project No. 1/0071/14. \begin{thebibliography}{00} \bibitem{ADMM} R. P. Agarwal, S. Djebali, T. Moussaoui, O. G. 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