\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 105, pp. 1--12.\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-2014/105\hfil Oscillatory and asymptotic behavior of solutions] {Oscillatory and asymptotic behavior of solutions for second-order nonlinear integro-dynamic equations on time scales} \author[R. P. Agarwal, S. R. Grace, D. O'Regan, A. Zafer \hfil EJDE-2014/105\hfilneg] {Ravi P. Agarwal, Said R. Grace, Donal O'Regan, A\u{g}acik Zafer} % in alphabetical order \address{Ravi P. Agarwal \newline Department of Mathematics, Texas A\&M University - Kingsville, Kingsville, TX 78363, USA} \email{agarwal@tamuk.edu} \address{Said R. Grace \newline Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt} \email{saidgrace@yahoo.com} \address{Donal O'Regan \newline School of Mathematics, Statistics and Applied mathematics, National University of Ireland, Galway, Ireland} \email{donal.oregan@nuigalway.ie} \address{A\u{g}acik Zafer\newline College of Engineering and Technology, American University of the Middle East, Block 3, Egaila, Kuwait} \email{agacik.zafer@gmail.com} \thanks{Submitted September 11, 2013. Published April 15, 2014.} \subjclass[2000]{34N05, 45D05, 34C10} \keywords{Integro-dynamic equation; oscillation; time scales} \begin{abstract} In this article, we study the asymptotic behavior of non-oscillatory solutions of second-order integro-dynamic equations as well as the oscillatory behavior of forced second order integro-dynamic equations on time scales. The results are new for the continuous and discrete cases. Examples are provided to illustrate the relevance of the results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} \label{s1} We are concerned with the asymptotic behavior of non-oscillatory solutions of the second-order integro-dynamic equation on time scales of the form \begin{equation}\label{11} (r(t)(x^{\Delta } (t))^{\alpha } )^{\Delta } +\int _0^ta(t,s)F(s,x(s))\Delta s=0 \end{equation} and the oscillatory behavior of the second-order forced integro-dynamic equation \begin{equation}\label{12} (r(t)(x^{\Delta } (t)))^{\Delta } +\int _0^ta(t,s)F(s,x(s))\Delta s =e(t). \end{equation} We take $\mathbb{T}\subseteq \mathbb{R_+}=[0,\infty)$ to be an arbitrary time-scale with $0\in \mathbb{T}$ and $\sup \mathbb{T}=$. By $t\ge s$ we mean as usual $t\in [s,\infty )\cap \mathbb{T}$. We shall assume throughout that: \begin{itemize} \item[(i)] $e, r :\mathbb{T}\to \mathbb{R}$ and $a:\mathbb{T}\times \mathbb{T}\to \mathbb{R}$ are rd-continuous and $r(t) >$ 0, and $a(t, s)\ge 0$ for $t>s$, $\alpha$ is the ratio of positive odd integers and \begin{equation} \label{13} \sup_{t\ge t_0 } \int _0^{t_0 }a(t,s)\Delta s:=k<\infty,\quad t_0 \ge 0; \end{equation} \item[(ii)] $F: \mathbb{T}\times \mathbb{R}\to \mathbb{R}$ is continuous and assume that there exist continuous functions $f_1 ,\, f_2 : \mathbb{T}\times \mathbb{R}\to \mathbb{R}$ such that $F(t, x)=f_1 (t, x)-f_2 ( t,x)$ for $t\ge 0$; \item[(iii)] there exist constants $\beta $ and $\gamma $ being the ratios of positive odd integers and functions $p_i \in C_{rd} (\mathbb{T}, (0,\infty ))$, $i = 1,2$, such that \begin{gather*} xf_1 (t, x) \ge p_1 (t) x^{\beta+1 }\quad\text{for $x \neq 0$ and $t \ge 0$},\\ xf_2 (t, x) \le p_2 (t) x^{\gamma+1 }\quad\text{for $x \neq 0$ and $t \ge 0$}. \end{gather*} \end{itemize} We consider only those solutions of equation \eqref{11} (resp, \eqref{12}) which are nontrivial and differentiable for $t\geq 0$. The term solution henceforth applies to such solutions of equation \eqref{11}. A solution $x$ is said to be oscillatory if for every $t_0 > 0$ we have $\inf_{t\ge t_0 } x(t) < 0 <\sup_{t\ge t_0 } x(t)$ and it is said to be non-oscillatory otherwise. Dynamic equations on time-scales is a fairly new topic. For general basic ideas and background, we refer the reader to the seminal book \cite{b1}. Although the oscillation and nonoscillation theory of differential equations and difference equations is well developed, the problem for integro-differential equations of Volterra type was discussed only in a few papers in the literature, see \cite{g1,l1,n1,o1,p1,s1} and their references. We refer the reader to \cite{g2,g3} for some initial papers on the oscillation and nonoscillation of integro-dynamic and integral equations on time scales. To the best of our knowledge, there are no results on the asymptotic behavior of non-oscillatory solutions of \eqref{11} and the oscillatory behavior of \eqref{12}. Therefore, the main goal of this article is to establish some new criteria for the asymptotic behavior of non-oscillatory solutions of equation \eqref{11} and the oscillatory behavior of equation \eqref{12}. \section{Asymptotic behavior of the non-oscillatory solutions of \eqref{11}} \label{s2} In this section we study the asymptotic behavior of all non-oscillatory solutions of equation \eqref{11} with all possible types of nonlinearities. We will employ the following two lemmas, the second of which is actually a consequence of the first. \begin{lemma}[Young inequality \cite{hl}] \label{lem22} Let $X$ and $Y$ be nonnegative real numbers, $n>1$ and $\frac{1}{n}+\frac{1}{m}=1$. Then \begin{equation*} XY\le \frac{1}{n} X^n +\frac{1}{m} Y^m. \end{equation*} Equality holds if and only if $X=Y$. \end{lemma} \begin{lemma}[\cite{a1}] \label{lem21} If $X$ and $Y$ are nonnegative real numbers, then \begin{gather} \label{21} X^\lambda+(\lambda-1)Y^\lambda-\lambda XY^{\lambda-1}\geq 0\quad \text{for }\lambda>1,\\ \label{22} X^\lambda-(1-\lambda)Y^\lambda-\lambda XY^{\lambda-1}\leq 0\quad \text{for }\lambda<1, \end{gather} where the equality holds if and only if $X=Y$. \end{lemma} We define \[ R(t,t_0) =\int _{t_0 }^t\Big(\frac{s}{r(s)} \Big)^{1/\alpha } \Delta s, \quad t> t_0 \ge 0. \] Note that due to monotonicity \begin{equation} \label{nc1} \lim_{t\to\infty}R(t,t_0)\neq 0. \end{equation} Our first result is the following. \begin{theorem} \label{thm21} Let conditions {\rm (i)--(iii)} hold with $\gamma =1$ and $\beta >1$ and suppose \begin{equation} \label{23} \lim_{t\to \infty } \frac{1}{R(t,t_0 )} \int_{t_0 }^t\Big(\frac{1}{r(v)} \int _{t_0 }^v\int _{t_0 }^ua(u,s)p_1^{\frac{1}{1-\beta } } (s)p_2^{\frac{\beta }{\beta -1} } (s) \Delta s\Delta u \Big)^{1/\alpha } \Delta v<\infty \end{equation} for some $t_0 \ge 0$. If $x$ is a non-oscillatory solution of \eqref{11}, then \begin{equation} \label{24} x(t) = O(R(t,t_0 )),\quad \text{as }t\to \infty. \end{equation} \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of equation \eqref{11}. Hence $x$ is either eventually positive or eventually negative. First assume $x$ is eventually positive, say $x (t) >$ 0 for $t\ge t_1$ for some $t_1 \ge t_0$. Using conditions (ii) and (iii) with $\beta >1$ and $ \gamma =1$ in equation \eqref{11}, for $t\ge t_1$, we obtain \begin{equation} \label{25} \big(r(t)(x^{\Delta } (t))^{\alpha }\big)^{\Delta } \le -\int_0^{t_1 } a(t,s)F(s,x(s))\Delta s + \int _{t_1 }^t a(t,s)[p_2 (s)x(s)-p_1 (s)x^{\beta } ]\Delta s. \end{equation} If we apply \eqref{21} with $\lambda =\beta$, $X=p_1^{1/\beta } x$, and $Y=(\frac{1}{\beta } p_2 p_1^{-1/\beta } )^{ \frac{1}{\beta -1}}$ we have \begin{equation} \label{26} p_2 (t)x(t)-p_1 (t)x^{\beta } (t) \le (\beta -1) \beta ^{\frac{\beta }{1-\beta } } p_1^{\frac{1}{1-\beta } } (t)p_2^{\frac{\beta }{\beta -1} } (t),\quad t\ge t_1. \end{equation} Substituting \eqref{26} into \eqref{25} gives \begin{equation}\label{27} \begin{aligned} &\big(r(t)(x^{\Delta } (t))^{\alpha }\big)^{\Delta }\\ &\leq -\int _0^{t_1 }a(t,s)F(s,x(s))\Delta s + (\beta -1)\beta ^{\frac{\beta }{1-\beta } } \int _{t_1 }^ta(t,s)p_1^{\frac{1}{1-\beta } } (s)p_2^{\frac{\beta }{\beta -1} } (s)\Delta s \end{aligned} \end{equation} for all $t\ge t_1 \ge 0$. Let $$ m:=\max \{|F(t,x(t))|:t\in [0,t_1 ]\cap \mathbb{T}\} . $$ By assumption (i), we have \begin{equation} \label{28} \big|-\int _0^{t_1 }a(t,s)F(s,x(s))\Delta s\big| \le \int_0^{t_1 }a(t,s) |F(s,x(s))| \Delta s \le mk:=b. \end{equation} Hence from \eqref{27} and \eqref{28}, we obtain \[ \left(r(t)\left(x^{\Delta } (t)\right)^{\alpha } \right)^{\Delta } \le b+(\beta -1)\beta ^{\frac{\beta }{1-\beta } } \int _{t_1 }^{t}a(t,s)p_1^{\frac{1}{1-\beta } } (s)p_2^{\frac{\beta }{\beta -1} } (s)\Delta s\, . \] Integrating this inequality from ${t}_1$ to $t$ leads to \begin{align*} &\big(x^{\Delta } (t)\big)^{\alpha } \\ &\le \frac{r(t_1 )\left|\left(x^{\Delta } (t_1 )\right)^{\alpha } \right|}{r(t)} +b\frac{t-t_1 }{r(t)} +\frac{(\beta -1)\beta ^{\frac{\beta }{1-\beta } } }{r(t)} \int _{t_1 }^t\int _{t_1 }^ua(u,s)p_1^{\frac{1}{1-\beta } } (s) p_2^{\frac{\beta }{\beta -1} } (s)\Delta s \Delta u \end{align*} or \[ \left(x^{\Delta } (t)\right)^{\alpha } \le \frac{c_0t}{r(t)} +\frac{(\beta -1)\beta ^{\frac{\beta }{1-\beta } } }{r(t)} \int _{t_1 }^{t}\int _{t_1 }^{u}a(t,s)p_1^{\frac{1}{1-\beta } } (s) p_2 ^{\frac{\beta }{\beta -1} } (s)\Delta s\Delta u \] where $$ c_0 =\frac{r(t_1 )|(x^{\Delta } (t_1 ))^{\alpha } |}{t_1 } +b. $$ By employing the well-known inequality \begin{equation} (a_1 +b_1)^{\lambda } \le \sigma _{\lambda } \left(a_1^{\lambda } +b_1^{\lambda } \right) \quad \text{for $ a_1 \ge 0$, $b_1 \ge 0$, and $\lambda >0$,} \end{equation} where $\sigma _{\lambda } =1$ if $\lambda <1$ and $\sigma _{\lambda } =2^{\lambda -1}$ if $\lambda \ge 1$ we see that there exists positive constants $c_1$ and $c_2$ depending on $\alpha$ such that \[ x^{\Delta } (t) \le c_1\big(\frac{t}{r(t)}\big)^{1/\alpha} +c_2\Big(\frac{1 }{r(t)} \int _{t_1 }^{t}\int _{t_1 }^{ u}a(t,s) p_1^{\frac{1}{1-\beta } } (s)p_2 ^{\frac{\beta }{\beta -1} } (s)\Delta s\Delta u\Big)^{1/\alpha}. \] Integrating this inequality from $t_1$ to $ t \ge t_1$, we obtain \begin{equation} \label{210} \begin{aligned} |x(t)| &\le |x(t_1 )| + c_1 R(t,t_1 )\\ &\quad + c_2 \int _{t_1 }^t\Big(\frac{1}{r(v)} \int _{t_1 }^v\int _{t_1 }^ua(u,s)p_1^{\frac{1}{1-\beta } } (s) p_2^{\frac{\beta }{\beta -1} } (s)\Delta s\Delta u \Big)^{1/\alpha } \Delta v \\ &\le |x(t_1 )| + c_1 R(t,t_0 ) \\ &\quad + c_2 \int _{t_0 }^t\Big(\frac{1}{r(v)} \int _{t_0 }^v\int _{t_0 }^ua(u,s)p_1^{\frac{1}{1-\beta } } (s)p_2^{\frac{\beta }{\beta -1} } (s)\Delta s\Delta u \Big)^{1/\alpha } \Delta v. \end{aligned} \end{equation} Dividing both sides of \eqref{210} by $R(t,t_0 )$ and using \eqref{nc1} and \eqref{23}, we see that \eqref{24} holds. The proof is similar if $x$ is eventually negative. \end{proof} Next, we present the following simple result. \begin{theorem} \label{thm22} Let conditions {\rm (i)} and {\rm (ii)} hold with $f_2 $= 0 and $x f_1(t,x) > 0$ for $x\ne 0$ and $t\ge0 $. If $x$ is a non-oscillatory solution of equation \eqref{11}, then \eqref{24} holds. \end{theorem} \begin{proof} Let $x (t)$ be a non-oscillatory solution of equation \eqref{11} with $f_2=0 $. First assume $x$ is eventually positive, say $x (t) >$ 0 for $t\ge t_1$ for some $t_1 \ge t_0$. From \eqref{11} we find that \[ (r(t)(x^{\Delta } (t))^{\alpha } )^{\Delta } =-\int _0^ta(t,s)f_1 (s,x(s))\Delta s \le \int_0^{t_1 } a( t,s) f_ 1 (s,x(s))\Delta s. \] Using \eqref{13} (see \eqref{28}) in the above inequality, we obtain $(r(t)(x^{\Delta } (t))^{\alpha } )^{\Delta } \le b$. The rest of the proof is similar to that of Theorem \ref{thm21} and hence is omitted. \end{proof} \begin{theorem} \label{thm23} Let conditions {\rm (i)--(iii)} hold with $\beta =1$ and $ \gamma <1$ and suppose \begin{equation} \label{211} \lim_{t\to \infty } \frac{1}{R(t,t_0 )} \int_{t_0 }^t \Big(\frac{1}{r(v)} \int _{t_0 }^v\int _{t_0 }^ua(u,s)p_1^{\frac{\gamma}{\gamma-1} } (s) p_2^{\frac{1 }{1-\gamma} } (s) \Delta s\Delta u \Big)^{1/\alpha } \Delta v<\infty \end{equation} for some $t_0 \ge 0$. If $x$ is a non-oscillatory solution of equation \eqref{11}, then \eqref{24} holds. \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of \eqref{11}. First assume $x$ is eventually positive, say $x(t) >$ 0 for $t\ge t_1$ for some $ t_1 \ge t_0$ . From conditions (ii) and (iii) with $\beta =1$ and $ \gamma <1$ in equation \eqref{11} we have \begin{equation} \label{212} \big(r(t)(x^{\Delta } (t))^{\alpha }\big)^{\Delta } \le -\int _{0}^{t_1 }a(t,s)F(s,x(s))\Delta s+ \int _{t_1 }^ta(t,s) [p_2 (s)x^{\gamma } (s)-p_1 (s)x]\Delta s \end{equation} for all t$\ge t_1$. Hence, \[ \big(r(t)(x^{\Delta } (t))^{\alpha }\big)^{\Delta } \le b+\int _{t_1 }^ta(t,s)[p_2 (s)x^{\gamma } (s)-p_1 (s)x]\Delta s, \] where $b$ is as in \eqref{28}. Applying \eqref{22} with $\text{$\lambda =\gamma$, $ X=p_2^{1/ \gamma }x$ and $Y= (\frac{1}{\gamma } p_1 p_2^{\frac{-1}{\gamma } } )^{\frac{1}{\gamma -1} } $}$, we obtain \begin{equation} \label{213} p_2 (t)x^{\gamma } (t)-p_1 (t)x(t) \le (1-\gamma ) \gamma^{\frac{\gamma }{1-\gamma } } p_1^{\frac{\gamma }{\gamma -1} } (t)p_2^{\frac{1}{1-\gamma } } (t),\quad t\ge t_1. \end{equation} Using \eqref{213} in \eqref{212} we have \[ \left(r(t)\left(x^{\Delta } (t)\right)^{\alpha } \right)^{\Delta } \le b+(1-\gamma)\gamma^{\frac{\gamma }{1-\gamma } } \int _{t_1 }^{t}a(t,s)p_1^{\frac{\gamma}{\gamma-1 } } (s)p_2^{\frac{1 }{1-\gamma} } (s)\Delta s \quad t\geq t_1. \] The rest of the proof is similar to that of Theorem \ref{thm21} and hence is omitted. \end{proof} \begin{theorem} \label{thm24} Let conditions {\rm (i)--(iii)} hold with $\beta>1$ and $\gamma <1$ and assume that there exists a positive rd-continuous function $\xi :\mathbb{T}\to \mathbb{T}$ such that \begin{equation} \label{214} \begin{aligned} &\lim_{t\to \infty } \frac{1}{R(t,t_0 )} \int _{t_0 }^t \Big(\frac{1}{r(v)} \int _{t_0 }^v\int _{t_0 }^ua(u,s)\\ &\times \big[c_1 \xi ^{\frac{\beta }{\beta -1} } (s)p_1^{\frac{1}{1-\beta } } (s) +c_2 \xi ^{\frac{\gamma }{\gamma -1} } (s)p_2^{\frac{1}{1-\gamma } } (s)\big] \Delta s\, \Delta u\Big)^{1/\alpha } \Delta v < \infty \end{aligned} \end{equation} for some $t_0 \ge 0,$ where $c_1 =(\beta -1)\beta ^{\frac{\beta }{1-\beta } }$ and $ c_2 =(1-\gamma )\gamma ^{\frac{\gamma }{1-\gamma } }$. If $x$ is a non-oscillatory solution of equation \eqref{11}, then \eqref{24} holds. \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of equation \eqref{11}. First assume $x$ is eventually positive, say $x(t) >$ 0 for $t\ge t_1$ for some $ t_1 \ge t_0$ . Using (ii) and (iii) in equation \eqref{11} we obtain \begin{align*} \big(r(t)(x^{\Delta } (t))^{\alpha }\big)^{\Delta } &\le -\int _0^{t_1 }a(t,s)F(s,x(s))\Delta s +\int_{t_1 }^ta(t,s) [\xi (s)x(s)-p_1 (s)x^{\beta } (s)] \, \Delta s \\ &\quad +\int _{t_1 }^ta(t,s)[p_2 (s)x^{\gamma } (s)-\xi (s)x(s)] \,\Delta s. \end{align*} As in the proof of Theorems \ref{thm21} and \ref{thm23}, one can easily show that \begin{align*} &\big(r(t)(x^{\Delta } (t))^{\alpha }\big)^{\Delta }\\ &\le -\int _0^{t_1 }a(t,s)F(s,x(s))\Delta s \\ &\quad +\int_{t_1 }^ta(t,s) \Big[ (\beta -1)\beta ^{\frac{\beta }{1-\beta } } \xi ^{\frac{\beta }{\beta -1} } (s)p_1^{\frac{1}{1-\beta } } (s) +(1-\gamma )\gamma ^{\frac{\gamma }{1-\gamma } } \xi ^{\frac{\gamma }{1-\gamma } } (s)p_2^{\frac{1}{1-\gamma } } (s) \Big] \, \Delta s. \end{align*} The rest of the proof is similar to that of Theorem \ref{thm21} and hence is omitted. \end{proof} \begin{theorem} \label{thm25} Let conditions {\rm (i)--(iii)} hold with $\beta>1$ and $\gamma<1$ and suppose that there exists a positive rd-continuous function $\xi :\mathbb{T}\to \mathbb{T}$ such that \begin{equation*} \lim_{t\to \infty } \frac{1}{R(t,t_0 )} \int _{t_0 }^t \Big(\frac{1}{r(v)} \int _{t_0 }^v\int _{t_0 }^ua(u,s) \xi^{\frac{\beta}{\beta-1 } } (s) p_1^{\frac{1}{1-\beta} } (s) \, \Delta s\, \Delta u\Big)^{1/\alpha } \Delta v < \infty \end{equation*} and \begin{equation*} \lim_{t\to \infty } \frac{1}{R(t,t_0 )} \int _{t_0 }^t \Big(\frac{1}{r(v)} \int _{t_0 }^v\int _{t_0 }^ua(u,s) \xi^{\frac{\gamma}{\gamma-1 } } (s) p_2^{\frac{1}{1-\gamma } } (s) \, \Delta s\, \Delta u\Big)^{1/\alpha } \Delta v < \infty \end{equation*} for some $t_0 \ge 0$. If $x$ is a non-oscillatory solution of equation \eqref{11}, then \eqref{24} holds. \end{theorem} For the cases when both $f_1$ and $f_2$ are superlinear ($\beta >\gamma >1$) or else sublinear ($1>\beta >\gamma >0$), we have the following result. \begin{theorem} \label{thm26} Let conditions {\rm (i)--(iii)} hold with $\beta>\gamma$ and assume \begin{equation} \label{215} \lim_{t\to \infty } \frac{1}{R(t,t_0 )} \int _{t_0 }^t \Big(\frac{1}{r(v)} \int _{t_0 }^v\int _{t_0 }^ua(u,s) p_1^{\frac{\gamma}{\gamma-\beta } } (s)p_2^{\frac{\beta}{\beta-\gamma } } (s) \, \Delta s\, \Delta u\Big)^{1/\alpha } \Delta v < \infty \end{equation} for some $t_0 \ge 0$. If $x$ is a non-oscillatory solution of equation \eqref{11}, then \eqref{24} holds. \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of \eqref{11}. First assume $x$ is eventually positive, say $x(t) >$ 0 for $t\ge t_1$ for some $ t_1 \ge t_0$. Using conditions (ii) and (iii) in equation \eqref{11} we have \begin{equation} \label{216} \begin{aligned} &\big(r(t)(x^{\Delta } (t))^{\alpha }\big)^{\Delta } \\ &\le -\int _0^{t_1 }a(t,s)F(s,x(s))\Delta s +\int_{t_1 }^ta(t,s) [p_2(s)x^{\gamma}(s)-p_1 (s)x^\beta (s)] \, \Delta s. \end{aligned} \end{equation} By applying Lemma \ref{lem22} with $$ n=\frac{\beta}{\gamma},\quad X=x^\gamma(s),\quad Y=\frac{\gamma p_2(s)}{\beta p_1(s)},\quad m=\frac{m}{\beta-\gamma} $$ we obtain \begin{align*} p_2 (s)x^{\gamma } (s)\,-\,p_1 (s)x^{\beta } (s) &=\frac{\beta}{\gamma } p_1 (s)[x^{\gamma } (s)\frac{\gamma }{\beta } \frac{p_2 (s)}{p_1 (s)} -\frac{\gamma }{\beta } (x^{\gamma } (s))^{\beta /\gamma } ]\\ &= \frac{\beta }{\gamma } p_1 (s)[XY-\frac{1}{n} X^n ] \\ &\le \frac{\beta }{\gamma } p_1 (s)\big(\frac{1}{m} Y^m \big) \\ &=\big(\frac{\beta -\gamma }{\gamma } \big) [\frac{\gamma }{\beta } p_2 (s)]^{\frac{\beta }{\beta -\gamma } } (p_1 (s))^{\frac{\gamma }{\gamma -\beta } }. \end{align*} The rest of the proof is similar to that of Theorem \ref{thm21} and hence is omitted. \end{proof} \begin{remark} \rm If in addition to the hypotheses of Theorems \ref{thm21}--\ref{thm26}, \[ \lim_{t\to \infty } R(t,t_0) <\, \infty, \] then every non-oscillatory solution of \eqref{11} is bounded. \end{remark} \begin{remark} \rm The results given above hold for equations of the form \begin{equation}\label{219} (r(t)(x^{\Delta } (t))^{\alpha } )^{\Delta } +\int _0^ta(t,s)F(s,x(s))\Delta s=e(t) \end{equation} if the additional condition \begin{equation*} \lim_{t\to \infty } \frac{1}{R(t,t_0 )} \int _{t_0 }^t\Big(\frac{1}{r(v)} \int _{t_0 }^v|e(s)|\, \Delta s\Big)^{1/\alpha } \Delta v< \infty \end{equation*} is satisfied. \end{remark} \section{Oscillation results for \eqref{12}} \label{s3} This section we study of the oscillatory properties of \eqref{12}. For this end hypotheses (i) and (ii) are replaced by the assumptions: \begin{itemize} \item[(I)] $e, r: \mathbb{T}\to \mathbb{R}$ and $a: \mathbb{T}\times \mathbb{T}\to \mathbb{R}$ are rd-continuous, $r (t) > 0$ and $ a(t, s)\ge 0$ for $t >s$ and there exist rd-continuous functions $ k,m :\mathbb{T}\to \mathbb{R}^{+} $ such that \begin{eqnarray} \label{31} a(t,s)\le k(t) m(s),\quad t\ge s \end{eqnarray} with \[ k_1:= \sup_{t\ge 0} k(t) <\infty, \quad k_2:= \sup_{t\ge 0} \int_{0}^tm(s)\Delta s <\infty. \] In this case condition \eqref{13} is satisfied with $k = k_1k_2$. \item[(II)] $F: \mathbb{T}\times \mathbb{R}\to \mathbb{R}$ is continuous and assume that there exists rd-continuous function, $q: \mathbb{T}\to (0,\infty )$ and a real number ${\beta }$ with $0 <\beta \le 1$ such that \begin{equation} \label{32} x F(t, x)\le q(t) x^{\beta +1},\quad \text{for $x\ne 0$ and $t\ge 0$}. \end{equation} \end{itemize} In what follows \begin{equation} \label{33} g_{\pm } (t,p) = e(t)\mp k_1 (1- \beta ) \beta^{{\beta /(1-\beta )}} \int_0^t {p}^{{\beta /(\beta -1)}} (s) q(s)^{{1/(1-\beta )}} {m}^{{1/(1-\beta )}}(s)\Delta s, \end{equation} where $0< \beta <1$, $p\in C_{rd} (\mathbb{T}, (0,\infty ))$. We first give sufficient conditions under which non-oscillatory solutions $x$ of equation \eqref{12} satisfy \begin{equation} \label{37} x(t) = O(t),\quad \text{as } t\to \infty. \end{equation} \begin{theorem}\label{thm31} Let $0< \beta < 1$, conditions {\rm (I)} and {\rm (II)} hold, assume the function $t/r(t)$ is bounded, and for some $t_0 \ge 0$, \begin{equation} \label{34} \int _{t_0 }^{\infty }\frac{s}{r(s)} \Delta s<\infty. \end{equation} Let $p\in C_{rd} (\mathbb{T}, (0,\infty )) $ such that \begin{equation} \label{35} \int _{t_0 }^{\infty }sp(s)\, \Delta s <\infty. \end{equation} If \begin{equation} \label{36} \begin{gathered} \limsup_{t\to \infty } \frac{1}{t} \int _{t_0 }^{t}\frac{1}{r(u)} \int_{t_0 }^{u}g_{-} (s,p)\Delta s \Delta u< \infty,\\ \liminf_{t\to \infty } \frac{1}{t} \int _{t_0 }^t\frac{1}{r(u)} \int _{t_0 }^ug_{+} (s,p){\kern 1pt} \Delta s\, \Delta u > - \infty , \end{gathered} \end{equation} then every non-oscillatory solution $x(t)$ of \eqref{12} satisfies $$ \limsup_{t\to \infty } \frac{|x(t)|}{t} < \infty. $$ \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of \eqref{11}. First assume $x$ is eventually positive, say $x(t) >$ 0 for $t\ge t_1 $ for some $t_1 \geq t_0$. Using condition \eqref{32} in \eqref{12} we have \begin{equation} \label{38} \left(r(t)(x^{\Delta } (t))\right)^{\Delta } \le e(t)-\int _0^{t_1 }a(t,s)F(s,x(s))\Delta s + \int _{t_1 }^ta(t,s){q(s) x}^{\beta } (s)\Delta s, \end{equation} for $ t\ge t_1$. Let \[ c:=\mathop{\max }_{0\le t\le t_1 } |F(t,x(t)| < \infty . \] By assumption \eqref{31}, we obtain \[ \big|-\int _0^{t_1 }a(t,s)F(s,x(s))\Delta s \big| \le {c}\int _0^{t_1 }a(t,s)\Delta s\le {ck}_1 k_2 =:b,\quad t\ge t_1 . \] Hence from \eqref{38} we have \begin{equation} \label{39} \begin{aligned} \left(r(t)(x^{\Delta } (t))\right)^{\Delta } &\le e(t)+b+k_1 \int _{t_1 }^t[m(s){q(s) x}^{\beta } (s) -p(s){x(s)}]\Delta s \\ &\quad +k_1 \int _{t_1 }^tp(s){x(s)}\Delta {s}, \quad t \ge {t}_1. \end{aligned} \end{equation} Applying \eqref{22} of Lemma \ref{lem21} with $$ \lambda =\beta, \quad X=(qm)^{1/ \beta } x, \quad Y= \big(\frac{1}{\beta } p(mq)^{-1/\beta} \big)^{\frac{1}{\beta -1}} $$ we have \[ m(s){q(s) x}^{\beta } (s)-p(s){x(s)} \le {(1-}\beta {)}\beta ^{\beta /(1-\beta )} {p}^{\beta /(\beta -1)} (s)m^{1/(1-\beta )} (s)q^{1/(1-\beta )} (s). \] Thus, we obtain \begin{equation} \label{310} \left(r(t)(x^{\Delta } (t))\right)^{\Delta } \le {g}_{+} {(t,p)+b} {+k}_1 \int _{t_1 }^tp(s) {x(s)}\Delta {s} \quad \text{for } t\ge {t}_1 . \end{equation} Integrating \eqref{310} from t${}_1$ to t we have \begin{equation} \label{311} r(t)x^{\Delta } (t)\le r(t_1 )x^{\Delta } (t_1 ) +\int _{t_1 }^t{g}_{+} (s,p) \Delta s + b(t-t_1 ) +k_1 \int _{t_1 }^t\int _{t_1 }^up(s){x(s)} \Delta {s} \, \Delta u, \end{equation} for $t\ge {t}_1$. Employing \cite[Lemma 3]{n1} to interchange the order of integration, we obtain \[ r(t)x^{\Delta } (t)\le {}r(t_1 )x^{\Delta } (t_1 ) +\int _{t_1 }^t{g}_{+} {(s,p)} \Delta s + b(t-t_1 ) {+k}_1 \, t\int _{t_1 }^tp(s){x(s)}\Delta {s},\quad t\ge {t}_1 \] and so, \[ x^{\Delta } (t)\le \frac{r(t_1 )x^{\Delta } (t_1 )}{r(t)} +\frac{1}{r(t)} \int _{t_1 }^t{g}_{+} {(s)} \Delta {s + }\frac{b(t-t_1 )}{r(t)} {+}\frac{{k}_1 \, t}{r(t)} \int _{t_1 }^tp(s){x(s)}\Delta {s}, \quad t\ge {t}_1 . \] Integrating this inequality from $t_{1 }$ to $t$ and using \eqref{34} and the fact that the function $t/r(t)$ is bounded for $t\ge t_1 $, say by $k_{3}$ we see that \begin{align*} x(t) &\le x(t_1 )+r(t_1 )x^{\Delta } (t_1 )\int _{t_1 }^t\frac{1}{r(s)} \Delta s+\int _{t_1 }^t\frac{1}{r(u)} \int_{t_1 }^u{g}_{+} {(s)} \Delta {s}\Delta {u} {} \\ &\quad +b \int _{t_1 }^t\frac{s}{r(s)} \Delta s{+k}_1 k_{3} \int _{t_1 }^t\int _{t_1 }^up(s){x(s)}\Delta {s} \Delta u,\quad t\ge {t}_1. \end{align*} Once again, using \cite[Lemma 3]{n1} we have \begin{equation} \label{312} \begin{aligned} x(t) &\le x(t_1 )+r(t_1 )x^{\Delta } (t_1 )\int _{t_1 }^t\frac{1}{r(s)} \, \Delta s+\int _{t_1 }^t\frac{1}{r(u)} \int _{t_1 }^u{g}_{+} {(s)} \, \Delta {s}\, \Delta {u} \\ &\quad +b \int _{t_1 }^t\frac{s}{r(s)} \Delta s{+k}_1 k_{3} t\, \int _{t_1 }^tp(s){x(s)}\, \Delta {s},\quad t\ge {t}_1 \end{aligned} \end{equation} and so, \begin{equation} \label{313} \frac{x(t)}{t} \le c_1 {+}c_2 \, \int _{t_1 }^tsp(s){ }\Big(\frac{{x(s)}}{s} \Big)\Delta {s},\quad t\ge {t}_1; \end{equation} note \eqref{34} and \eqref{36}, $c_2={k}_1{k}_{3}$ and ${c}_1$ is an upper bound for \[ \frac{1}{t} \Big[{x(t}_1 ) +r(t_1 )x^{\Delta } (t_1 )\int _{t_1 }^t\frac{1}{r(s)} \, \Delta s +\int _{t_1 }^t\frac{1}{r(u)} \, \int _{t_1 }^u{g}_{+} {(s)} \, \Delta {s}\, \Delta {u} {+b }\int _{t_1 }^t\frac{s}{r(s)} \, \Delta s] \] for $t \geq t_1$. Applying Gronwall's inequality \cite[ Corollary 6.7]{b1} to inequality \eqref{313} and then using condition \eqref{35} we have \begin{equation} \label{314} \limsup_{t\to \infty } \frac{x(t)}{t} < \infty . \end{equation} If $x(t)$ is eventually negative, we can set $y = -x$ to see that $y$ satisfies equation \eqref{12} with $e(t)$ replaced by $-e(t)$ and $F(t, x)$ replaced by $-F(t,-y)$. It follows in a similar manner that \begin{equation} \label{315} \limsup_{t\to \infty } \frac{-x(t)}{t} < \infty . \end{equation} The proof is complete. \end{proof} Next, by employing Theorem \ref{thm31} we present the following oscillation result for equation \eqref{12}. \begin{theorem}\label{thm32} Let $0 < \beta < 1$, conditions {\rm (I), (II)}, \eqref{34}, \eqref{35}, and \eqref{36} hold, assume the function $t/r(t)$ is bounded , and there is a function $p\in C_{rd} (\mathbb{T}, (0,\infty )) $ such that \eqref{35} holds. If for every $0$ 0 for $t\ge t_1$ for some $t_1 \ge t_0$. The proof when $x(t)$ is eventually negative is similar. Proceeding as in the proof of Theorem \ref{thm31} we arrive at \eqref{312}. Therefore, \begin{align*} x(t)&\le {x(t}_1 )+r(t_1 )x^{\Delta } (t_1 ) \int _{t_1 }^{\infty}\frac{1}{r(s)} \Delta s +\int _{t_1 }^{t}\frac{1}{r(u)} \int _{t_1 }^{u}{g}_{+} {(s,p)} \Delta {s} \Delta {u} \\ &\quad +b \int _{t_1 }^{\infty}\frac{s}{r(s)} \Delta {\kern 1pt} s{+} {k}_1 k_{3} t \int _{t_1 }^{\infty}s p(s) \big(\frac{{x} {(s)}}{s} \big)\Delta {s} ,\quad t\ge t_1 . \end{align*} Clearly, the conclusion of Theorem \ref{thm31} holds. This together with \eqref{34} imply that \begin{equation} \label{318} x(t)\le M_1+M\,t+ \int _{t_1 }^{t}\frac{1}{r(u)} \int _{t_1 }^{u}{g}_{+} {(s,p)} \Delta {s} \Delta {u}, \end{equation} where $M_1$ and $M$ are positive real numbers. Note that we make $M<1$ possible by increasing the size of $t_1$. Finally, taking liminf in \eqref{318} as t$\to \infty $ and using \eqref{317} result in a contradiction with the fact that $x (t)$ is eventually positive. \end{proof} \begin{corollary} \label{coro31} Let $0 < \beta < 1$ and condition {\rm (I), (II)}, \eqref{34}, and \eqref{35} hold, assume the function $t/r(t)$ is bounded, and for some ${t}_{{0}} \ge 0$ suppose \begin{equation} \label{319} \limsup_{t\to \infty } \frac{1}{t} \int _{t_0 }^t\frac{1}{r(u)} \int _{t_0 }^{u}e(s){\kern 1pt} \Delta s \Delta u <\infty,\quad \liminf _{t\to \infty } \frac{1}{t} \int _{t_0 }^t\frac{1}{r(u)} \int _{t_0 }^ue(s)\Delta s \Delta u > - \infty \end{equation} and \begin{equation} \label{320} \lim_{t\to \infty } \frac{1}{t} \int _{t_0 }^t \frac{1}{r(u)} \int _{t_0}^u p^{\beta /(\beta -1)} (s) q(s)^{1/(1-\beta )} m^{1/(1-\beta )} (s)\Delta s \Delta u < \infty. \end{equation} If for every $00 \end{equation} and \begin{equation} \label{218} \Big(\frac{1}{t^{2} } (x'(t))^{1/3} \Big)' +\int _0^t\frac{t}{t^{2} +s^{2} } [s^{b} {x}^{5/7} (s)- s^{c}x^{3/7} (s)]ds=0,\quad t>0, \end{equation} where $a$, $b$, and $c$ are nonnegative real numbers satisfying $3a< 2$ and $3b-2<5c\leq 3b$. For \eqref{217}, take $\alpha=3$, $r(t)=1/t$, $a(t,s)=t/(t^2+s^2)$, $p_1(t)=t^a$, $p_2(t)=1$, $\beta=5$, $\gamma=3$, $R(t,0)=(3/5) t^{5/3}$. Since \begin{align*} &t^{-5/3} \int _{0 }^t\Big(v \int _{0 }^v\frac{1}{u}\int _{0 }^u \frac{u^2}{u^2+s^2}s^{-3a/2}dsdu\Big)^{1/3 } dv \\ &\leq c_1t^{-5/3} \int _{0 }^t\Big(v \int _{0 }^vu^{-3a/2}du\Big)^{1/3 } dv \\ &= c_2 t^{-a/2}, \end{align*} where $c_1$ and $c_2$ are certain constants, condition \eqref{215} holds. For \eqref{218}, take $\alpha=1/3$, $r(t)=1/t^2$, $a(t,s)=t/(t^2+s^2)$, $p_1(t)=t^b$, $p_2(t)=t^c$, $\beta=5/7$, $\gamma=3/7$, $R(t,0)=(1/10) t^{10}$. Condition \eqref{215} holds, because \begin{align*} & t^{-10} \int _{0 }^t\Big(v^2 \int _{0 }^v\frac{1}{u}\int _{0 }^u \frac{u^2}{u^2+s^2}s^{-3a/2+5c/2}dsdu\Big)^{3 } dv\\ &\leq d_1 t^{-10} \int _{0 }^t\Big(v^2 \int _{0 }^vu^{-3b/2+5c/2}du\Big)^{3 } dv \\ &= d_2 t^{-9b/2+15c/2}, \end{align*} where $d_1$ and $d_2$ are certain constants. As a result, we may conclude from Theorem \ref{thm26} that every non-oscillatory solution of \eqref{217} and of \eqref{218} satisfies $x= O (t^{5/3})$ and $x=O(t^{10})$, respectively, as $t\to \infty$. \end{example} \begin{example} \rm Consider the integro-differential equation \begin{eqnarray} \label{325ab} ((1+t)^{3} x')' +\int _{0}^{t}\frac{x^{\beta } (s)}{(t^{2} +1)(s^{4} +)}ds =t^4\sin t, \end{eqnarray} where $\beta=1/3$ or $\beta=1$. We observe that $r(t)=(1+t)^3$, $k(t)=1/(t^2+1)$, $m(s)=1/(s^4+1)$, $q(t)=1$, $e(t)=t^4\sin t$. Letting $p(t)=m(t)$, we see that the integral appearing in the definition of $g_\pm(t,p)$ given by \eqref{33} becomes bounded. It is then not difficult to show that all conditions of Theorem \ref{thm32} for $\beta=1/3$ are satisfied. On the other hand, all conditions of Theorem \ref{thm34} for $\beta=1$ are also satisfied. Therefore, every solution of equation \eqref{325ab} is oscillatory for $\beta=1/3$ and $\beta=1$. \end{example} \begin{thebibliography}{99} \bibitem{a1} R. P. Agarwal, S. R. Grace, D. O\rq{}Regan; \emph{Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations}, Kluwer, Dordrecht, 2002. MR2091751 (2005i:34001). \bibitem{b1} M. Bohner, A. Peterson; \emph{Dynamic Equations on Time Scales. An Introduction with Applications}, Birkhusser, Boston, 2001. MR1843232 (2002c:34002). \bibitem{g1} K. Gopalsamy; \emph{Stability, instability, oscillation and nonoscillation in scalar integrodifferential systems}, Bull. Austral. Math. Soc. \textbf{28} (1983), 233-246. MR0729010 (85k:45018). \bibitem{g2} S. R. Grace, J. R. Graef, A. Zafer; \emph{Oscillation of integro-dynamic equations on time scales}, Appl. Math. Lett. \textbf{26} (2013), 383--386. \bibitem{g3} S. R. Grace, A. Zafer; \emph{Oscillatory behavior of integro-dynamic and integral equations on time scales} (submitted). \bibitem{hl} G. Hardy, J. E. Littlewood, G.Polya; \emph{Inequalities}, Cambridge University Press,Cabridge,1959. \bibitem{l1} J. J. Levin; \emph{Boundedness and oscillation of some Volterra and delay equations}, J. Differential Equations, \textbf{5} (1969), 369--398. MR0236642 (38 \#4937). \bibitem{o1} H. Onose; \emph{On oscillation of Volterra integral equation and first order functional differential equations}, Hiroshima Math.J. \textbf{20} (1990), 223--229. MR1063361 (91g:45002). \bibitem{p1} N. Parhi, N. Misra; \emph{On oscillatory and nonoscillatory behaviour of solutions of Volterra integral equations}, J. Math. Anal. Appl. \textbf{94} (1983), 137--149. MR0701453 (84g:45003). \bibitem{n1} A. H. Nasr; \emph{Asymptotic behaviour and oscillation of classes of integrodifferential equations}. Proc. Amer. Math. Soc. \textbf{116} (1992), 143--148. MR1094505 (92k:34097). \bibitem{s1} B. Singh; \emph{On the oscillation of a Volterra integral equation}, Czech. Math. J. \textbf{45} (1995), 699--707. MR1354927 (96i:45003). \end{thebibliography} \end{document}