\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 158, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/158\hfil Local extrema of positive solutions] {Local extrema of positive solutions of nonlinear functional differential equations} \author[G. E. Chatzarakis, L. Horvat Dmitrovi\'c, M. Pa\v{s}i\'c \hfil EJDE-2018/158\hfilneg] {George E. Chatzarakis, Lana Horvat Dmitrovi\'c, Mervan Pa\v{s}i\'c} \address{George E. Chatzarakis \newline Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), 14121, N. Heraklio, Athens, Greece} \email{geaxatz@otenet.gr, gea.xatz@aspete.gr} \address{Lana Horvat Dmitrovi\'c \newline Department of Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia} \email{Lana.Horvat@fer.hr} \address{Mervan Pa\v{s}i\'c \newline Department of Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000 Zagreb, Croatia} \email{mervan.pasic@fer.hr} \thanks{Submitted May 17, 2018. Published August 31, 2018.} \subjclass[2010]{34A30, 34B30, 34C10, 34C11} \keywords{Functional differential equations; local non-monotonicity; \hfill\break\indent integral criteria; Rayleigh quotient; delay; advance; super-sub linear nonlinearity} \begin{abstract} We study the positive solutions of a general class of second-order functional differential equations, which includes delay, advanced, and delay-advanced equations. We establish integral conditions on the coefficients on a given bounded interval $J$ such that every positive solution has a local maximum in $J$. Then, we use the connection between that integral condition and Rayleigh quotient to get a sufficient condition that is easier to be applied. Several examples are provided to demonstrate the importance of our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} We consider the functional differential equations of the second-order, \begin{equation} \big(r(t)x'(t)\big)'+\sum_{i=1}^n p_i(t)f(x(h_i(t)) +\sum_{j=1}^m q_j(t)|x(\tau_j(t))|^{\alpha_j-1}x(\tau_j(t))=e(t),\label{EqMain} \end{equation} where $r(t)\in C^1(\mathbb{R})$, $n,m\in\mathbb{N}$ and $p_i(t),h_i(t),q_j(t),\tau_j(t),e(t)\in C(\mathbb{R})$. Two auxiliary functions $M_{\rm min}(t)$ and $M_{\rm max}(t)$ are associated to the functional terms $h_i(t)$ and $\tau_j(t)$: \begin{align*} M_{\rm min}(t)=\min\{ h_1(t),\dots ,h_n(t),\tau_1(t),\dots ,\tau_m(t)\},\\ M_{\rm max}(t)=\max\{ h_1(t),\dots ,h_n(t),\tau_1(t),\dots ,\tau_m(t)\}. \end{align*} For $a0,\quad t\in\mathbb{R}\text{ and } e(t)\leq 0, \; t\in J_{a,b},\\ p_i(t)\geq 0 \text{ and } q_j(t)\geq 0, \quad t\in J_{a,b},\; i\in [1,n]_{\mathbb{N}}, \; j\in [1,m]_{\mathbb{N}},\\ \exists i_0,j_0 \text{ such that } p_{i_0}(t)>0 \text{ if } e(t)\equiv 0, \text{ otherwise } q_{j_0}(t)>0. \end{gathered} \end{equation} For each $i\in [1,n]_{\mathbb{N}}$ and $j\in [1,m]_{\mathbb{N}}$ let exist functions $R_{h_i}(t)$ and $R_{\tau_j}(t)$ (depending on $h_i(t)$ and $\tau_j(t)$, respectively) such that for any $x\in C^2(J_{a,b})$ and $x(t)>0$, $t\in J_{a,b}$, we have \begin{equation}\label{Rhit} \text{if } \big(r(t)x'(t)\big)'\leq 0 \text{ in } J_{a,b}, \text{ then } \begin{cases} \frac{x(h_i(t))}{x(t)} \geq R_{h_i}(t)\text{ in } (a,b),\; i\in [1,n]_{\mathbb{N}},\\[3pt] \frac{x(\tau_j(t))}{x(t)} \geq R_{\tau_j}(t)\text{ in } (a,b),\; j\in [1,m]_{\mathbb{N}}. \end{cases} \end{equation} The so-called \textit{generalized concave condition} \eqref{Rhit} is more natural than restrictive, because it is fulfilled in the two most important functional cases, delay and advance: \begin{equation}\label{Rhit-exa} R_{g}(t)= \begin{cases} \frac{g(t)-g(a)}{t-g(a)}, & \text{if $g(t)\leq t$ and $r(t)$ is non-decreasing}, \\[3pt] \frac{g(b)-g(t)}{g(b)-t}, & \text{if $g(t)\geq t$ and $r(t)$ is non-increasing}, \end{cases} \end{equation} where $g(t)$ is an arbitrary continuous functional term (see Proposition \ref{PropAppen} in the appendix). The nonlinear terms in \eqref{EqMain} satisfy \begin{equation}\label{f0} \exists f_0>0,\; f(x)\geq f_0x \quad \text{for all $x\geq 0$,} \end{equation} and \begin{equation} \label{alphai} \begin{gathered} \alpha_j\geq 0,\quad j\in [1,m]_{\mathbb{N}},\\ \text{there exists } (\eta_0,\eta_1,\eta_2,\dots ,\eta_m),\ \eta_j>0,\; j\in [1,m]_{\mathbb{N}} \\ \text{such that } \sum_{j=0}^m\eta_j=1 \text{ and } \sum_{j=1}^m\alpha_j\eta_j=1. \end{gathered} \end{equation} If $q_j(t)\equiv 0$ for all $j\in [1,m]_{\mathbb{N}}$, then the assumption \eqref{alphai} is not required. As to the existence of an $(m+1)$-tuple $(\eta_0,\eta_1,\eta_2,\dots ,\eta_m)$ satisfying \eqref{alphai} with respect to a given $m$-tuple $(\alpha_1,\alpha_2,\dots ,\alpha_m)$ such that $\alpha_1>\dots >\alpha_{j_0}>1>\alpha_{j_0+1}>\dots .> \alpha_m>0$ for some $j_0$, we refer the reader to \cite{SunWong}. Also, if $m=1$ and $\alpha_1>1$, then $\eta_1=1/\alpha_1$ and $\eta_0=1-1/\alpha_1$ satisfy the required conditions in \eqref{alphai}: $\eta_0+\eta_1=1$, $\alpha_1\eta_1=1$ and $\eta_j>0$. Note that \eqref{EqMain} contains several types of nonlinear functional differential equations. Here we consider several special cases. \begin{itemize} \item[(i)] If $q_j(t)\equiv 0$ for all $j\in [1,m]_{\mathbb{N}}$, $f(x)= x$ and $e(t)\equiv 0$, then \eqref{EqMain} is a linear differential equation with several functional arguments. \item[(ii)] If $h_i(t)\leq t$ and $\tau_j(t)\leq t$ (resp. $h_i(t)\geq t$ and $\tau_j(t)\geq t$) for all $i\in [1,n]_{\mathbb{N}}$, $j\in [1,m]_{\mathbb{N}}$, then \eqref{EqMain} is a nonlinear delay (resp. advance) differential equation with several arguments. \item[(iii)] If $h_i(t)\leq t$ and $\tau_j(t)\leq t$ for all $i\in [1,i_0]_{\mathbb{N}}$, $j\in [1,j_0]_{\mathbb{N}}$ as well as $h_i(t)\geq t$ and $\tau_j(t)\geq t$ for all $i\in [i_0+1,n]_{\mathbb{N}}$, $j\in [j_0+1,m]_{\mathbb{N}}$ and $t\in\mathbb{R}$, then \eqref{EqMain} is a nonlinear delayed-advanced differential equation with several arguments. \item[(iv)] If $f(x)\equiv 0$ and $\alpha_1>\dots >\alpha_{j_0}>1>\alpha_{j_0+1} >\dots > \alpha_m>0$ for some $j_0$, then \eqref{EqMain} is a functional differential equation with mixed nonlinearities. \end{itemize} As can be seen from the preceding comments, \eqref{EqMain} includes several types of functional differential equations. \begin{definition}\label{def1.1} \rm A function $x\in C^1(a,b)$ is non-monotonic in $(a,b)$, if there exists $t_*\in (a,b)$ such that $x'(t_*)=0$ and $x'(t)$ changes sign at $t=t_*$. \end{definition} Recently, in \cite{CHP17}, the authors have studied the non-monotonicity of the solutions of the delay differential equation $$ (r(t)x'(t))'+f(x(\tau(t))=0, $$ which is a special case of \eqref{EqMain}. That study used the zero-point analysis of the corresponding dual equation. In the present paper, we follow a different approach. Moreover, we obtain an integral criterion, for the non-monotonicity of solutions, which is a different type of conditions than the point-wise criterion presented in \cite{CHP17}. This aspect will be explained in more detail, in the following sections. For some results on the classic oscillations as a particular case of the non-monotonic behaviour of the functional differential equations, we refer the reader to \cite{BDG15,BCS16,EKZ95,LWCL,Yang,Zafer09}. About the importance of non-monotonic behaviour of some modern mathematical models in applied science, see for instance in \cite{PasicAML15, PasicCNSNS15,Zhang16}. The global non-monotonicity (case where $(a,b)=(a,\infty)$) of the second-order differential equation \[ (r(t)x'(t))'+q(t)f(x(t))=e(t), \] with possible non-homogeneous term and without functional terms, has been recently considered in \cite{PasicRogovch18, PasicTanaka16}. Furthermore, in \cite{CM92}, for a nonlinear functional differential equation \[ (r(t)h(x)x'(t))'+q(t)f(x(g(t))=0, \] the global non-monotonicity of solutions was considered in the form of weakly oscillatory solutions. \section{Main results} \begin{theorem}\label{Theo-main1} Let $a \int_{a'}^{b'} \frac{1}{\Theta(t)}\Big(\frac{d\varphi}{dt}\Big)^2dt, \end{equation} then every positive solution $x(t)$ of \eqref{EqMain} has a local maximum in $(a,b)$ and is non-monotonic in $(a,b)$. \end{theorem} \begin{remark}\rm (i) The restriction from $(a,b)$ to $(a',b')$ in \eqref{AssMain} is necessary, because often $R_{h_i}(a)=0$ or $R_{h_i}(b)=0$ (resp., $R_{\tau_j}(a)=0$ or $R_{\tau_j}(b)=0$), see for instance \eqref{Rhit-exa}. In such a case, $\Theta(a)=0$ or $\Theta(b)=0$; hence, to avoid any singular behaviour in the right integral in \eqref{AssMain}, we use $[a',b']$ for the domain of integration, where $a0$, in order to avoid $\Theta (t)=0$, for some $t\in (a,b)$. \end{remark} In the Sturm-Liouville theory and the variational characterization of lowest eigenvalues, the next {\it Rayleigh quotient} plays a crucial role, \[ \mathcal{R}(\varphi)=\frac{\int_{a'}^{b'} \big(\frac{d\varphi}{dt}\big)^2dt} {\int_{a'}^{b'} \varphi^2(t)dt},\quad \varphi\in C^1(a',b'),\; \varphi\not\equiv 0. \] It is not difficult to see that if $a \mathcal{R}(\varphi), \end{equation} then \eqref{AssMain} holds. Hence, Theorem \ref{Theo-main1} takes the simple form: \begin{theorem}\label{Theo-main} Let $a \Big(\frac{\pi}{b'-a'}\Big)^2 \end{equation} holds, then every positive solution $x(t)$ of \eqref{EqMain} is non-monotonic in $(a,b)$, having a local maximum in $(a,b)$. \end{theorem} Furthermore, it is known that the previous observation can be generalized to the Rayleigh quotient and the corresponding eigenvalue problem, using the weight $\omega(t)=1/r(t)$. In that case, we have \begin{gather*} \mathcal{R}_{1/r}(\varphi)=\frac{\int_{a'}^{b'} \big(\frac{d\varphi}{dt}\big)^2dt}{\int_{a'}^{b'} \frac{\varphi^2(t)}{r(t)}dt},\quad \varphi\in C^1(a',b'),\; \varphi\not\equiv 0,\\ \varphi''+\frac{\lambda}{r(t)}\varphi=0,\quad \varphi(a')=\varphi(b')=0. \end{gather*} Then the set of all eigenvalues $\lambda$ is represented by the sequence $(\lambda_n)_{n\in \mathbb{N}}$ satisfying: $0<\lambda_1 < \lambda_2 \dots < \lambda_n < \dots $, $\lim_{n\to\infty}\lambda_n=\infty$ and \begin{equation}\label{first-eigen-weight} \lambda_1= \min\big\{\mathcal{R}_{1/r}(\varphi):\varphi\in C_0^2(a',b'),\ \varphi\not\equiv 0\big\}=\mathcal{R}_{1/r}(\varphi_1), \end{equation} where $\varphi_1(t)$ is the eigenvector which corresponds to the eigenvalue $\lambda_1$. Hence, each of the following two conditions (C1) and (C2) implies the inequality which constitutes the main assumption \eqref{AssMain}, in Theorem \ref{Theo-main1}. \begin{itemize} \item [(C1)] There exists a test function $\varphi\in C([a',b'])\cap C^1(a',b')$, $\varphi(a')=\varphi(b')=0$ such that $\min_{t\in[a',b']}\Theta(t) > \mathcal{R}_{1/r}(\varphi)$. %\label{AssMainSimpleWeight}. \item [(C2)] It holds that $\min_{t\in[a',b']}\Theta(t) > \lambda_1$, where $\lambda_1$ is given by \eqref{first-eigen-weight}. %\label{AssMainSimpleEigenWeight}$ \end{itemize} Consequently, we have the following theorem. \begin{theorem}\label{Theo-main4} Let $a0$ and $\sum_{j=0}^m\eta_j=1$, then $\sum_{j=0}^m\eta_jA_j\geq \prod_{j=0}^mA_j^{\eta_j}$} \] and use that inequality, taking \[ A_0=\frac{|e(t)|}{\eta_0}\quad \text{and}\quad A_j=\frac{q_j(t)|x(\tau_j(t))|^{\alpha_j}}{\eta_j},\quad j\in [1,m]_{\mathbb{N}}. \] Note that we can use \eqref{Rhit}, because from \eqref{EqMain}, \eqref{rpiqi} and $x(t)\geq 0$, we have $(r(t)x'(t))'\leq 0$ in $J_{a,b}$. Then by means of \eqref{rpiqi}, \eqref{Rhit}, \eqref{f0} and \eqref{alphai}, we have \begin{align*} &\frac{d\omega}{dt}\\ &=\frac{1}{r(t)}-\frac{x(t)}{(r(t)x'(t))^2}(r(t)x'(t))'\\ &=\frac{1}{r(t)}+\frac{x(t)}{(r(t)x'(t))^2} \Big[ \sum_{i=1}^n p_i(t)f(x(h_i(t)) +\sum_{j=1}^m q_j(t)|x(\tau_j(t))|^{\alpha_j-1}x(\tau_j(t))-e(t)\Big]\\ & =\frac{1}{r(t)}+\omega^2(t) \Big\{ \sum_{i=1}^n p_i(t)\frac{f(x(h_i(t))}{x(t)}+\frac{1}{x(t)} \Big[\sum_{j=1}^m q_j(t)|x(\tau_j(t))|^{\alpha_j}+e(t)\Big]\Big\}\\ & \geq\frac{1}{r(t)}+\omega^2(t) \Big\{f_0 \sum_{i=1}^n p_i(t)R_{h_i}(t)+\frac{1}{x(t)} \Big[\sum_{j=1}^m \eta_j\Big(\frac{q_j(t)|x(\tau_j(t))|^{\alpha_j}}{\eta_j}\Big) +\eta_0\Big(\frac{|e(t)|}{\eta_0}\Big)\Big]\Big\}\\ &\geq \frac{1}{r(t)}+\omega^2(t) \Big\{ f_0\sum_{i=1}^n p_i(t)R_{h_i}(t)+\frac{1}{x(t)} \Big(\frac{|e(t)|}{\eta_0} \Big)^{\eta_0}\prod_{j=1}^m\Big(\frac{q_j(t)}{\eta_j} \Big)^{\eta_j}|x(\tau_j(t))|^{\eta_j\alpha_j}\Big\}\\ &= \frac{1}{r(t)}+\omega^2(t) \Big\{ f_0\sum_{i=1}^n p_i(t)R_{h_i}(t)+\Big(\frac{|e(t)|}{\eta_0} \Big)^{\eta_0} \prod_{j=1}^m\Big(\frac{q_j(t)}{\eta_j} \Big)^{\eta_j} \Big(\frac{|x(\tau_j(t))|}{x(t)}\Big)^{\eta_j\alpha_j}\Big\}\\ &\geq \frac{1}{r(t)}+\omega^2(t) \Big\{ f_0\sum_{i=1}^n p_i(t)R_{h_i}(t)+\Big(\frac{|e(t)|}{\eta_0} \Big)^{\eta_0} \prod_{j=1}^m\Big(\frac{q_j(t)}{\eta_j} \Big)^{\eta_j}[R_{\tau_j}(t) ]^{\eta_j\alpha_j}\Big\}\\ &=\frac{1}{r(t)}+\Theta (t)w^2,\ t\in (a,b). \end{align*} Thus, the function $w$ defined in \eqref{wt} satisfies the differential inequality \eqref{Diff-Ineq}, which contradicts the main assumption of this lemma. Therefore, every positive solution $x(t)$ of \eqref{EqMain} has a stationary point in $(a,b)$. \end{proof} \begin{lemma} \label{Lemm-stat-max} Let \eqref{rpiqi} and \eqref{f0} hold and $x(t)$ be a solution of \eqref{EqMain} such that $x(t)>0$ on $[a,b]$. If $t_*\in (a,b)$ is a stationary point of $x(t)$, then $x(t)$ attains a local maximum at $t_*$. \end{lemma} \begin{proof} Let $t_*\in (a,b)$ be a point such that $x'(t_*)=0$. Integrating \eqref{EqMain} over $[t_*,t]$ for all $t\in (a,b)$, we obtain \begin{equation}\label{prec-equa} x'(t)=\frac{-1}{r(t)}\int_{t_*}^{t} \Big[\sum_{i=1}^n p_i(t)f(x(h_i(t))+\sum_{i=j}^m q_j(t) |x(\tau_j(t))|^{\alpha_j-1}x(\tau_j(t))-e(t)\Big]. \end{equation} According to \eqref{rpiqi}, \eqref{f0} and $x(t)>0$, the integral function in \eqref{prec-equa} is positive in $(a,b)$ and hence, the right hand-side in \eqref{prec-equa} is negative for $t>t_*$ and positive for $t0$, $\omega>0$, $\tau\geq 0$ and $e(t)$ is an arbitrary continuous function. The above equation is of the type of \eqref{EqMain} with $r(t)\equiv 1$, $n=1$, $p_1(t)= A\sin(\omega t)$, $f(x)=x$, $h_1(t)=t-\tau$, $m=1$ and $q_1(t)\equiv 0$. Let $(a,b)\subset (\tau,\pi/\omega)$ be an open interval such that \begin{equation}\label{Ass-exa} \tau \frac{9\omega^2\big(\frac{\pi}{6\omega}-a+\tau\big)}{2\big(\frac{\pi}{6\omega} -a\big)},\quad \frac{5\pi}{6\omega}\frac{9\omega^2\big(\frac{\pi}{6\omega} -a+\tau\big)}{2\big(\frac{\pi}{6\omega}-a\big)}. $$ We claim that {\it every positive solution of equation \eqref{LO} has a local maximum in $(a,b)$, provided \eqref{Ass-exa} holds.} Note that this statement cannot easily be derived, even in the homogeneous case ($e(t)\equiv 0$). In that case, from \eqref{LO}, we have $x''=-A\sin(\omega t)x(t-\tau)$. If $x(t)$ is a positive solution, the preceding equality implies that $x''(t)$ is sign-changing, but in general, does not imply that $x'(t)$ is sign-changing. To show the above statement, we use Theorem \ref{Theo-minvalue}. At first, we see that the set $J_{a,b}$ defined in \eqref{Jhab} satisfies $J_{a,b}=(a-\tau,b)\subset (0,\pi/\omega)$, which implies $$ p_1(t)= A\sin(\omega t)>0,\quad t\in J_{a,b}. $$ Hence, \eqref{rpiqi} is satisfied. Since $h_1(t)=t-\tau\leq t$ and $r(t)=1$ is non-decreasing, it follows that \eqref{Rhit} holds because of \eqref{Rhit-exa}, where $$ R_{h_1}(t)=\frac{h_1(t)-h_1(a)}{t-h_1(a)}=\frac{t-a}{t-a+\tau},\quad t>a. $$ From $R_{h_1}(t)$ being an increasing function, we have that in any $[a',b']\subset (0,\infty)$, \begin{equation}\label{min-R} \min_{t\in[a',b']}R_{h_1}(t)=R_{h_1}(a'). \end{equation} Since $f(x)=x$, we have that \eqref{f0} holds with $f_0=1$. Since $q_j(t)\equiv 0$ for all $j\in [1,m]_{\mathbb{N}}$, we do not need assumption \eqref{alphai}. Now, let $[a',b']=[\frac{\pi}{6\omega},\frac{5\pi}{6\omega}]$. Since $r(t)=1$, from \eqref{Theta}, \eqref{Ass-exa} and \eqref{min-R}, we derive that $a\frac{9\omega^2}{4}= \Big(\frac{\pi}{b'-a'}\Big)^2. \] Therefore, \eqref{AssMainSimple2} is satisfied. Consequently, all conditions of Theorem \ref{Theo-minvalue} are fulfilled, thus establishing the main statement of this example. \end{example} \begin{example}\rm Consider the special case of the Duffing equation with time delay feedback, \begin{equation}\label{Duff} x''+\omega_0x+\beta x^3+\lambda\sin(t) x(t-\tau)=-\cos(t/2), \end{equation} where $\omega_0>0$ is natural frequency, $\beta >0$, $\lambda >0$ is the gain parameter and $\tau\geq 0$. Equation \eqref{Duff} is a particular case of the main equation \eqref{EqMain} with \begin{equation}\label{alll} \begin{gathered} r(t)=1,\quad n=2,\quad p_1(t)=\omega_0,\quad h_1(t)=t,\quad p_2(t)=\lambda\sin(t),\quad h_2(t)=t-\tau, \\ f(x)=x, \quad m=1,\quad q_1(t)=\beta,\quad \tau_1(t)=t, \quad \alpha_1=3,\quad e(t)=-\cos(t/2). \end{gathered} \end{equation} Note that the cubic term $\beta x^3$, introducing a strong nonlinearity into the equation, cannot be considered as part of the linear term $\sum_{i=1}^n p_i(t)f(x(h_i(t))$, because $f(x)=\beta x^3$ does not satisfy the required condition \eqref{f0}. Let $(a,b)$ be an open interval such that \[ \tau 0, \] imply $\eta_0=2/3$ and $\eta_1=1/3$. Hence, from \eqref{Rhit-exa}, \eqref{Theta} and \eqref{alll}, we have $R_{h_1}(t)=1$, $R_{\tau_1}(t)=1$, \begin{equation}\label{Theta0} R_{h_2}(t)=\frac{t-a}{t-a+\tau}\quad \Theta (t)=\omega_0+\lambda\sin(t)R_{h_2}(t) +\frac{3}{2^{2/3}}\beta^{1/3}|\cos(t/2)|^{2/3}. \end{equation} Note that $\sin(t)>0$ and $-\cos(t/2)\leq 0$ on $[a-\tau,b]$ as well as $R_{h_2}(a')\leq R_{h_2}(t)$ for all $t\in [a',b']$. Hence from \eqref{Theta0}, we obtain \begin{align*} \frac{\min_{t\in[a',b']}\Theta(t)}{\max_{t\in[a',b']}r(t)} &=\min_{t\in[a',b']}\Theta(t) = \omega_0+\frac{\sqrt{3}\lambda}{2}\frac{\pi-3a}{\pi-3a+3\tau}+ \frac{3}{2^{2/3}}\beta^{1/3}(1/2)^{1/3}\\ &\geq \min\{\omega_0,\lambda, \beta^{1/3}\} \Big(\frac{5}{2}+\frac{\sqrt{3}}{2}\frac{\pi-3a}{\pi-3a+3\tau} \Big)\\ &>9=\Big(\frac{\pi}{b'-a'}\Big)^2, \end{align*} provided \begin{equation}\label{min} \min\{\omega_0,\lambda, \beta^{1/3}\} >\frac{18(\pi-3a+3\tau)}{(5+\sqrt{3})(\pi-3a)+15\tau }. \end{equation} Now, by Theorem \ref{Theo-minvalue}, if \eqref{min} is true, then every positive solution of equation \eqref{Duff} has a local maximum in $(a,b)$. \end{example} \section{Appendix} In this section, we state a proposition that justifies why the generalized condition \eqref{Rhit} is fulfilled both in the delay and the advanced cases, for any functional term $g(t)$, satisfying \eqref{Rhit-exa}. Below, we show this proposition, for the delay case where $R_g(t)$ is defined by the upper branch of \eqref{Rhit-exa}, i.e., \begin{equation}\label{funct-gt} g(a)0$. The proposition can be stated in a corresponding manner, for the advanced case and has a similar proof, for that case. \begin{proposition} \label{PropAppen} Let the functional term $g(t)$ satisfy \eqref{funct-gt}, $J_{a,b}:=(g(a),b)$ and $r(t)$ be a non-decreasing positive function on $J_{a,b}$. If $x\in C^2(J_{a,b})$, $x(s)>0$, $s\in J_{a,b}$ and \begin{equation}\label{leq0} \big(r(s)x'(s)\big)'\leq 0, \ s\in J_{a,b}, \end{equation} then \end{proposition} \begin{equation}\label{**} \frac{x(g(t))}{x(t)} \geq \frac{g(t)-g(a)}{t-g(a)}, \ t\in (a,b). \end{equation} \begin{proof} We will proceed by showing that assumption \eqref{leq0} implies \begin{equation}\label{*pom} \frac{x'(s)}{x(s)} \leq \frac{1}{s-g(a)}, \quad s\in J_{a,b}. \end{equation} Since $(g(t),t)\subseteq J_{a,b}$ for any $t\in (a,b)$, integrating \eqref{*pom} over $[g(t),t]$, we obtain \[ \ln \frac{x(t)}{x(g(t))}\leq \ln \frac{t-g(a)}{g(t)-g(a)}, \] which proves the desired inequality \eqref{**}. Thus, the proof of the proposition reduces to establishing that assumption \eqref{leq0} implies \eqref{*pom}. Since $x(s)>0$ on $J_{a,b}$, let us remark that \eqref{*pom} is trivially satisfied for all $s\in J_{a,b}$ such that $x'(s)\leq 0$. Hence, let $s\in J_{a,b}$ be such that $x'(s)\geq 0$. Now, integrating \eqref{leq0} over $(\sigma ,s)$ for every $\sigma\in J_{a,b}$ such that $\sigma