\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil A classification scheme for positive solutions \hfil EJDE--2000/??} {EJDE--2000/??\hfil Xianling Fan, Wan-Tong Li, \& Chengkui Zhong \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~??, pp.~1--14. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A classification scheme for positive solutions of second order nonlinear iterative differential equations \thanks{{\em Mathematics Subject Classifications:} 34K15. \hfil\break\indent {\em Key words and phrases:} Nonlinear iterative differential equation, oscillation, \hfil\break\indent eventually positive, asymptotic behavior. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil \break\indent Submitted December 11, 1999. Published March 31, 2000. \hfil\break\indent Supported by the NNSF of China and the Foundation for University Key Teacher by \hfil\break\indent Ministry Education.} } \date{} \author{ Xianling Fan, Wan-Tong Li, \& Chengkui Zhong } \maketitle \begin{abstract} This article presents a classification scheme for eventually-positive solutions of second-order nonlinear iterative differential equations, in terms of their asymptotic magnitudes. Necessary and sufficient conditions for the existence of solutions are also provided. \end{abstract} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \section{Introduction} A systematic study of oscillatory properties and asymptotic behavior of solutions of functional differential equations began with the works \cite{ref4, ref11, ref12}. However, a considerable number of papers dealing with these problems are from the last two decades. In 1987, the monograph \cite{ref5} presented a systematic investigation of the oscillatory properties of solutions to ordinary differential equations with deviating arguments. Recently, Bainov, Markova and Simeonov \cite{ref3} studied the equation \begin{equation} \label{1}(r(t)x'(t))'+f(t,x(t),x(\Delta (t,x(t))))=0 \end{equation} with the condition $$ \int_0^\infty \frac{ds}{r(s)}=\infty \,. $$ They provide a classification scheme for non-oscillatory solutions, and provide necessary and sufficient conditions for the existence of solutions. Such schemes are important since further investigations of qualitative behaviors of solutions can then be reduced to only a number of cases. However, a more difficult problem \cite{ref9} is to characterize the case when $$ \int_0^\infty \frac{ds}{r(s)}<\infty \,. $$ This paper concerns with the general class of second order nonlinear differential equations \begin{equation} \label{2}(r(t)(x'(t))^\sigma )'+f(t,x(t),x(\Delta (t,x(t))))=0 \end{equation} with the conditions $\int_0^\infty ds/r(s)^{1/\sigma }=\infty $ and $% \int_0^\infty ds/r(s)^{1/\sigma }<\infty $. We give a classification scheme for eventually-positive solutions of this equation in terms of their asymptotic magnitude, and provide necessary and/or sufficient conditions for the existence of solutions. Our results extend and improve the results in \cite{ref3, ref5}. When $f(t,x(t),x(\Delta (t,x(t))))=f(t,x(t))$, the oscillation and asymptotic behavior of the solutions of (\ref{2}) have been studied by Li \cite{ref6}-\cite{ref10}, Ruan \cite{ref13} and Wong and Agarwal \cite{ref14}. It is known \cite{ref3} that the differential equation of the from (\ref{1}) with delay depending on the unknown function have been investigated only in the papers \cite{ref1}, \cite{ref2}. Let $T\in {\mathbb R}_{+}=[0,\infty )$. Define $T_{-1}=\inf \{\Delta (t,x):t\geq T,x\in R\}$. \paragraph{Definition 1.} The function $x(t)$ is called a solution of the differential equation (\ref {2}) in the interval $[T,+\infty )$, if $x(t)$ is defined for $t\geq T_{-1}$% , it is twice differentiable and satisfies (\ref{2}) for $t\geq T$. \paragraph{Definition 2.} The solution $x(t)$ of (\ref{2}) is called regular, if it is defined on some interval $[T_x,\infty )$ and $\sup\{|x(t)|:t\geq T\}>0$ for $t\geq T_x$. \paragraph{Definition 3.} The solution $x(t)$ of (\ref{2}) is said to be: \noindent(i) eventually positive: if there exists $T\geq 0$ such that $% x(t)>0 $ for all $t\geq T$; \noindent(ii) eventually negative: if there exists $T\geq 0$ such that $% x(t)<0$ for all $t\geq T$; \noindent(iii) non-oscillatory: if it is either eventually positive or eventually negative; \noindent (iv) oscillatory: if it is neither eventually positive nor eventually negative. Throughout this paper, we assume that the following conditions hold: \begin{description} \item {H1)} $r\in C({\mathbb R}_{+},{\mathbb R}_{+})$ and $r(t)>0,t\in {\mathbb R}_{+}$. \item {H2)} $f\in C({\mathbb R}_{+}\times R^2,{\mathbb R})$. \item {H3)} There exists $T\in {\mathbb R}_{+}$ such that $uf(t,u,v)>0$ for $t\geq T$, $uv>0$ and $f(t,u,v)$ is non-decreasing in $u$ and $v$ for each fixed $t\geq T$. \item {H4)} $\Delta \in C({\mathbb R}_{+}\times R,{\mathbb R})$. \item {H5)} There exist a function $\Delta _{*}(t)\in C({\mathbb R}_{+},{% \mathbb R})$ and $T\in {\mathbb R}_{+}$ such that $\lim _{t\to \infty }\Delta _{*}(t)=+\infty $ and $\Delta _{*}(t)\leq \Delta (t,x)$ for $t\geq T$, $x\in {\mathbb R}$. \item {H6)} There exist a function $\Delta ^{*}(t)\in C({\mathbb R}_{+},{% \mathbb R})$ and $T\in {\mathbb R}_{+}$ such that $\Delta ^{*}(t)$ is a nondecreasing function for $t\geq T$ and $\Delta (t,x)\leq \Delta ^{*}(t)\leq t$ for $t\geq T,x\in {\mathbb R}$. \item {H7)} $\sigma $ is a quotient of odd integers. \end{description} \noindent For the sake of convenience, we will employ the following notation $$ R(t)=\int_t^\infty \frac{ds}{r(s)^{1/\sigma }},\quad R(t,T)=\int_T^t\frac{ds% }{r(s)^{1/\sigma }},\quad R_0=\int_0^\infty \frac{ds}{r(s)^{1/\sigma}}\,. $$ In the following section, we give several preparatory lemmas which will be used for later results. In Section 3, we will discuss the case $R_0<\infty $. The case $R_0=\infty $ will be studied in Section 4. \section{Preparatory Lemmas} \begin{lemma} Suppose $x(t)$ is an eventually-positive solution of (\ref {2}). Then $x'(t)$ is of constant sign eventually. \end{lemma} \paragraph{Proof.} Assume that there exists $t_0\geq 0$ such that $x(t)>0$, for $t\geq t_0$. It follows from (H6) that there exists $t_1\geq t_0$ such that $x(\Delta (t,x(t)))>0$ for $t\geq t_1$. From (H4) and (\ref{2}) we conclude that $(r(t)(x'(t))^\sigma )'<0$ for $t\geq t_1$. If $x^{\prime }(t)$ is not eventually positive, then there exists $t_2\geq t_1$ such that $x'(t_2)\leq 0$. Therefore, $r(t_2)(x'(t_2))^\sigma \leq 0$. From (\ref{2}), we have $$ r(t)(x'(t))^\sigma -r(t_2)(x'(t_2))^\sigma +\int_{t_2}^tf(s,x(s),x(\Delta (s,x(s))))ds=0. $$ Thus $$ r(t)(x'(t))^\sigma \leq -\int_{t_2}^tf(s,x(s),x(\Delta (s,x(s))))ds<0, $$ for $t\geq t_2$. This shows that $x'(t)<0$ for $t\geq t_2$. The proof is complete.\hfill$\diamondsuit $\smallskip As a consequence, an eventually positive solution $x(t)$ of (\ref{2}) either satisfies $x(t)>0$ and $x'(t)>0$ for all large $t$, or, $x(t)>0$ and $x'(t)<0$ for all large $t$. \begin{lemma} Suppose that \begin{equation}\label{3} R_0=\int_0^\infty \frac{ds}{r(s)^{1/\sigma }}<\infty \,, \end{equation} and that $x(t)$ is an eventually positive solution of (\ref{2}). Then $\lim _{t\to \infty }x(t)$ exists. \end{lemma} \paragraph{Proof.} If not, then we have $\lim _{t\to \infty }x(t)=\infty $ by Lemma 1. On the other hand, we have noted that $r(t)(x'(t))^\sigma $ is monotone decreasing eventually. Therefore, there exists $t_1\geq 0$ such that $$ r(t)(x'(t))^\sigma \leq r(t_1)(x'(t_1))^\sigma ,\quad \mbox{for }t\geq t_1\,. $$ Then \begin{equation} \label{4}x'(t)\leq (r(t_1))^{1/\sigma }x'(t_1)\frac 1{r(t)^{1/\sigma }}, \end{equation} for $t\geq t_1$, and after integrating, $$ x(t)-x(t_1)\leq (r(t_1))^{1/\sigma }x'(t_1)R(t_1,t), $$ for $t\geq t_1$. But this is contrary to the fact that $\lim _{t\to \infty }x(t)=\infty $ and the assumption that $R_0<\infty $. The proof is complete. \hfill$\diamondsuit$\smallskip \begin{lemma} Suppose that $R_0<\infty $. Let $x(t)$ be an eventually positive solution of (\ref{2}). Then there exist $a_1>0,a_2>0$ and $T\geq 0$ such that $a_1R(t)\leq x(t)\leq a_2$ for $t\geq T$. \end{lemma} \paragraph{Proof.} By Lemma 2, there exists $t_0\geq 0$ such that $x(t)\leq a_2$ for some positive number $a_2$. We know that $x'(t)$ is of constant sign eventually by Lemma 1. If $x'(t)>0$ eventually, then $R(t)\leq x(t)$ eventually because $\lim _{t\to \infty }R(t)=0$. If $x'(t)<0$ eventually, then since $r(t)(x'(t))^\sigma $ is also eventually decreasing, we may assume that $x'(t)<0$ and $r(t)(x'(t))^% \sigma $ is monotone decreasing for $t\geq T$. By (4), we have $$ x(s)-x(t)\leq (r(T))^{1/\sigma }x'(T)R(t,s),\quad s\geq t\geq T. $$ Taking the limit as $s\to \infty $ on both sides of the above inequality, $$ x(t)\geq -(r(T))^{1/\sigma }x'(T)R(t), $$ for $t\geq T$. The proof is complete. \hfill$\diamondsuit$\smallskip Our next result is concerned with necessary conditions for the function $f$ to hold in order that an eventually positive solution of (\ref{2}) exist. \begin{lemma} Suppose that $R_0<\infty $ and $x(t)$ is an eventually positive solution of (\ref{2}). Then $$ \int_0^\infty \frac 1{r(t)^{1/\sigma }}\left( \int_0^tf(s,x(s),x(\Delta (s,x(s))))ds\right) ^{1/\sigma }dt<\infty . $$ \end{lemma} \paragraph{Proof.} In view of Lemma 1, we may assume without loss of generality that $x(t)>0$, and, $x'(t)>0$ or $x'(t)<0$ for $t\geq 0$. From (\ref{2}), we have $$ r(t)(x'(t))^\sigma -r(0)(x'(0))^\sigma +\int_0^tf(s,x(s),x(\Delta (s,x(s))))ds=0\,. $$ Thus, if $x'(t)>0$ for $t\geq 0$, we have $$ \displaylines{ \int_0^u\frac 1{r(t)^{1/\sigma }}\left( \int_0^tf(s,x(s),x(\Delta (s,.x(s))))ds\right) ^{1/\sigma }dt \cr \leq (r(0))^{1/\sigma }x'(0) \int_0^u\frac 1{r(t)^{1/\sigma }}dt\,, }% $$ for $u\geq 0$, and $$ \int_0^u\frac 1{r(t)^{1/\sigma }}\left( \int_0^tf(s,x(s),x(\Delta (s,x(s))))ds\right) ^{1/\sigma }dt\leq (r(0))^{1/\sigma }x'(0)R_0 <\infty \,. $$ If $x'(t)<0$ for $t\geq 0$, we have $$ \int_0^u\frac 1{r(t)^{1/\sigma }}\left( \int_0^tf(s,x(s),x(\Delta (s,x(s))))ds\right) ^{1/\sigma }dt\leq -\int_0^\infty x'(s)ds\leq x(0)<\infty \,. $$ The proof is complete. \hfill$\diamondsuit$\smallskip We now consider the case where $R_0=\infty $. \begin{lemma} Suppose that \begin{equation} \label{5}R_0=\int_0^\infty \frac{ds}{r(s)^{1/\sigma }}=\infty \,. \end{equation} Let $x(t)$ be an eventually positive solution of (\ref{2}). Then $x'(t)$ is eventually positive and there exist $c_1>0$, $c_2>0$ and $T\geq 0$ such that $c_1\leq x(t)\leq c_2R(t,T)$ for $t\geq T$. \end{lemma} \paragraph{Proof.} In view of Lemma 1, $x'(t)$ is of constant sign eventually. If $% x(t)>0$ and $x'(t)<0$ for $t\geq T$, then we have $$ r(t)(x'(t)^\sigma )\leq r(T)(x'(T)^\sigma )<0\,. $$ Thus $$ x'(t)\leq r(T)^{1/\sigma }x'(T)\frac 1{r(t)^{1/\sigma }},\quad t\geq T, $$ which after integrating yields $$ x(t)-x(T)\leq r(T)^{1/\sigma }x'(T)\int_T^t\frac{ds}{r(s)^{1/\sigma }}. $$ The left hand side tends to $-\infty $ in view of (5), which is a contradiction. Thus $x'(t)$ is eventually positive, and thus $% x(t)\geq c_1$ eventually for some positive constant $c_1$. Furthermore, the same reasoning just used also leads to $$ x(t)\leq x(T_0)+r(T_0)^{1/\sigma }x'(T_0)\int_{T_0}^t\frac{ds}{% r(s)^{1/\sigma }}, $$ for $t\geq T_0$, where $T_0$ is a number such that $x(t)>0$ and $x^{\prime }(t)>0$ for $t\geq T_0$. Since $R_0=\infty $, thus there is $c_2>0$ such that $x(t)\leq c_2R(T,t)$ for all large $t$. The proof is complete. \hfill$\diamondsuit$ \section{The case $R_0<\infty $} We have shown in the previous section that when $x(t)$ is an eventually positive solution of (\ref{2}), then $(r(t)(x'(t))^\sigma )'$ is eventually decreasing and $x'(t)$ is eventually of constant sign. We have also shown that under the assumption that $R_0<\infty$, $x(t)$ must converge to some (nonnegative) constant. As a consequence, under the condition $R_0<\infty $, we may now classify an eventually positive solution $x(t)$ of (\ref{2}) according to the limits of the sequences $x(t)$ and $% r(t)(x'(t))^\sigma $. For this purpose, we first denote the set of eventually-positive solutions of (\ref{2}) by $P$. We then single out eventually-positive solutions of (\ref{2}) which converge to zero or to positive constants, and denote the corresponding subsets by $P_0$ and $% P_\alpha $ respectively. But for any $x(t)$ in $P_\alpha $, since $% r(t)(x'(t))^\sigma $ either tends to a finite limit or to $-\infty $% , we can further partition $P_{+}$ into $P_\alpha ^\beta $ and $P_\alpha ^{-\infty }$. \begin{theorem} Suppose $R_0<\infty $. Then any eventually positive solutions of (\ref{2}) must belong to one of the following classes: $$ \displaylines{ P_0=\left\{ x(t)\in P|\lim _{t\to \infty }x(t)=0\right\} , \cr P_\alpha ^\beta =\left\{ x(t)\in P|\lim _{t\to \infty }x(t)=\alpha >0,\quad\lim _{t\to \infty }r(t)(x'(t)^\sigma )=\beta \right\} ,\cr P_\alpha ^{-\infty }=\left\{ x(t)\in P|\lim _{t\to \infty }x(t)=\alpha >0,\quad\lim _{t\to \infty }r(t)(x' (t)^\sigma )=-\infty \right\} . }$$ \end{theorem} To justify the above classification scheme, we will derive several existence theorems. \begin{theorem} Suppose $R_0<\infty $. Then a necessary and sufficient condition for (\ref{2}) to have an eventually positive solution $x(t)$ which belong to $P_\alpha $ is that for some $C>0$, \begin{equation} \label{6}\int_0^\infty \left( \frac 1{r(t)}\int_0^tf(s,C,C)ds\right) ^{1/\sigma }dt<\infty \,. \end{equation} \end{theorem} \paragraph{Proof.} Let $x(t)$ be any eventually positive solution of (\ref{2}) such that \\ $\lim _{t\to \infty }x(t)=c>0$. Thus, in view of (H6), there exist $C_1>0$, $C_2>0$ and $T\geq 0$ such that $C_1\leq x(t)\leq C_2$, $C_1\leq x(\Delta (t,x(t)))\leq C_2$ for $t\geq T$. On the other hand, using Lemma 4 we have $$ \int_T^\infty \left( \frac 1{r(t)}\int_0^tf(s,x(s),x(\Delta (s,x(s))))ds\right) ^{1/\sigma }dt<\infty . $$ Since $f(t,u,v)$ is nondecreasing in $u$ and $v$ for each fixed $t$, thus we have $$ \int_T^\infty \left( \frac 1{r(t)}\int_0^tf(s,C_1,C_1)ds\right) ^{1/\sigma }dt<\infty . $$ Conversely, let $a=C/2$. In view of (\ref{6}), we may choose a $T\geq 0$ so large that \begin{equation} \label{7}\int_T^\infty \left( \frac 1{r(t)}\int_0^tf(s,C,C)ds\right) ^{1/\sigma }dt0$. Choose $M\geq T$ so large that \begin{equation} \label{9}\int_t^\infty \left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds<\frac \epsilon 2\,. \end{equation} Let $\{x^{(n)}\}$ be a sequence in $\Omega $ such that $x^{(n)}\to x $. Since $\Omega $ is closed, $x\in \Omega $. Furthermore, for any $s\geq t\geq M$, \begin{eqnarray*} \lefteqn{ \left| Fx^{(n)}(t)-Fx(t)\right| }\\ &\leq& \int_t^\infty \left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds+\int_t^\infty \left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds \\ &\leq& 2\int_t^\infty \left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds<\epsilon \,. \end{eqnarray*} For $T\leq t\leq s\leq M$, \begin{eqnarray*} \lefteqn{ \left| Fx^{(n)}(t)-Fx(t)\right| }\\ &\leq& \int_M^\infty \left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds+\int_M^\infty \left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds \\ &&+\int_t^M\left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds-\int_s^M\left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds \\ &\leq& \epsilon +\int_t^s\left( \frac 1{r(s)}\int_0^sf(u,C,C)du\right) ^{1/\sigma }ds \\ &\leq& \epsilon +\max _{T\leq u\leq M}\frac 1{r(u)}\int_0^uf(v,C,C)dv\left| s-t\right| \\ &\leq& \epsilon +C_0\left| s-t\right| <2\epsilon ,\quad\mbox{if}\left| s-t\right| <\frac \epsilon {C_0}\,, \end{eqnarray*} where $C_0=\max_{T\leq u\leq M} \int_0^uf(v,C,C)\,dv /r(u)$. And for $% T_{-1}\leq t\leq s0$ and that for some $D>0$, \begin{equation} \label{10}\int_0^\infty f(t,D,D)dt<\infty \,. \end{equation} \end{theorem} \paragraph{Proof.} If $x(t)$ is an eventually-positive solution in $P_\alpha^\beta $, then, in view of Theorem 2, we see that (\ref{6}) holds. Furthermore, as in the proof of Theorem 2, $00 $ and $$ r(t)(u'(t))^\sigma =1+\int_t^\infty f(s,u(s),u(\Delta (s,u(s))))ds, \quad t\geq T\,. $$ Therefore, $\lim _{t\to \infty }r(t)(u'(t))^\sigma =1>0$, and the present proof is complete. \hfill$\diamondsuit$ \smallskip In view of Theorem 3, the following result is obvious. \begin{theorem} \label{thm4} Suppose $R_0<\infty $. A necessary and sufficient condition for (\ref{2}) to have an eventually-positive solution $x(t)$ which belongs to $P_\alpha ^{-\infty }$ is that (6) holds for some $C>0$ and that for any $D>0$, \begin{equation} \label{12} \int_0^\infty f(t,D,D)dt=\infty \end{equation} \end{theorem} Our final result concerns with the existence of eventually-positive solutions in $P_0$ . \begin{theorem} Suppose $R_0<\infty $ and $\sigma =1$. If for some $C>0$, \begin{equation} \label{13} \int_0^\infty f(t,CR(t),CR(\Delta _{*}(t)))dt<\infty , \end{equation} then (\ref{2}) has an eventually-positive solution in $P_0$. Conversely, if (\ref{2}) has an eventually-positive solution $x(t)$ such that $\lim_{t\to \infty }x(t)=0$ and \\ $\lim _{t\to \infty}r(t)(x'(t))^\sigma =d\neq 0$, then for some $C>0$, $$ \int_0^\infty f(t,CR(t),CR(\Delta _{*}(t)))dt<\infty \,. $$ \end{theorem} \paragraph{Proof.} Suppose (\ref{13}) holds. Then there exists a $T\geq 0$ such that $$ \int_t^\infty f(s,CR(s),CR(\Delta _{*}(s)))ds<\frac C2\quad \mbox{for }t\geq T\,. $$ Consider the equation \begin{equation} \label{14}x(t)=\left\{ \begin{array}{ll} R(t)\left( \frac C2+\int_T^tf(s,x(s),x(\Delta (s,x(s))))ds\right) & \\ +\int_t^\infty R(s)f(s,x(s),x(\Delta (s,x(s))))ds & t\geq T, \\ [4pt] Fx(T) & T_{-1}\leq t0$ being similar). Then there exist $C_1>0$, $C_2>0$ and $T\geq 0$ such that $-C_1t\geq T$. Let $s\to \infty $, then $-C_1^{1/\sigma }R(t)<-x(t)<-C_2^{1/\sigma }R(t)$. That is, $C_2^{1/\sigma }R(t)0$ for $t$ larger than or equal to $T$, so that $$ x'(t)\geq d^{1/\sigma }\frac 1{r^{1/\sigma }(t)}, $$ and $$ x(t)\geq x(T)d^{1/\sigma }\int_T^t\frac 1{r^{1/\sigma }(s)}ds\to \infty \,, \mbox{ as } t\to \infty\,, $$ which is a contradiction. \begin{theorem} \label{thm6} Suppose that $R_0=\infty $. Then any eventually-positive solution $x(t)$ of (\ref{2}) must belong to one of the following classes: $$ \displaylines{ P_\alpha ^0=\left\{ x(t)\in P|\lim _{t\to \infty }x(t)\in (0,\infty ),\quad \lim _{t\to \infty }r(t)(x'(t))^\sigma =0\right\}, \cr P_\infty ^0=\left\{ x(t)\in P|\lim _{t\to \infty }x(t)=+\infty , \quad\lim _{t\to \infty }r(t)(x'(t))^\sigma =0\right\} , \cr P_\infty ^\beta =\left\{ x(t)\in P|\lim _{t\to \infty }x(t)=+\infty , \quad\lim _{t\to \infty }r(t)(x'(t))^\sigma =\beta \neq 0\right\} . }$$ \end{theorem} In order to justify our classification scheme, we present the following two results. \begin{theorem} \label{thm7} Suppose that $R_0=\infty $. A necessary and sufficient condition for (\ref{2}) to have an eventually-positive solution $x(t)$ which belongs to $P_\alpha ^0$ is that for some $C>0$, \begin{equation} \label{15} \int_0^\infty \left( \frac 1{r(t)}\int_t^\infty f(s,C,C)ds\right) ^{1/\sigma }dt<\infty \,. \end{equation} \end{theorem} \paragraph{Proof.} Let $x(t)$ be an eventually-positive solution of (\ref{2}) which belong to $% P_\alpha ^0$, i.e., $\lim _{t\to \infty}x(t)=\alpha >0$ and $\lim _{t\to \infty }r(t)(x'(t))^\sigma =0$. Then there exist two positive constants $C_1$, $C_2$ and $T\geq 0$ such that $C_1\leq x(t)\leq C_2$, $% C_1\leq x(\Delta (t,x(t))\leq C_2 $ for $t\geq T$. On the other hand, in view of (\ref{2}) we have $$ r(t)(x'(t))^\sigma =\int_t^\infty f(s,x(s),x(\Delta (s,x(s))))ds\,, $$ for $t\geq T$. After integrating, we see that \begin{eqnarray*} \lefteqn{ \int_T^\infty \left( \frac 1{r(t)}\int_t^\infty f(s,C,C)ds\right) ^{1/\sigma }dt }\\ &\leq& \int_0^\infty \left( \frac 1{r(t)}\int_t^\infty f(s,x(s),x(\Delta (s,x(s))))ds\right) ^{1/\sigma }dt \\ &\leq& \alpha -x(T). \end{eqnarray*} The proof of the converse is similar to that of Theorem 1 and hence is sketched. In view of (\ref{15}), we may choose a $T\geq 0$ so large that \begin{equation} \label{16}\int_T^\infty \left( \frac 1{r(t)}\int_t^\infty f(s,C,C)ds\right) ^{1/\sigma }<\frac C2\,. \end{equation} Define a bounded, convex, and closed subset $\Omega $ of $C([T_{-1},\infty ),% {\mathbb R})$ and an operator $F:\Omega \to \Omega $ as $$ \Omega =\left\{ \begin{array}{l} x\in C([T_{-1},+\infty ), {\mathbb R}) : x(t)=\frac C2 \mbox{ for }T_{-1}\leq t