\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 73, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/73\hfil Oscillation Criteria] {Oscillation criteria for forced second-order mixed type quasilinear delay differential equations} \author[S. Murugadass, E. Thandapani, S. Pinelas \hfil EJDE-2010/73\hfilneg] {Sowdaiyan Murugadass, Ethiraju Thandapani, Sandra Pinelas} \address{Sowdaiyan Murugadass \newline Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai - 600005, India} \email{murugadasssm@gmail.com} \address{Ethiraju Thandapani \newline Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai - 600005, India} \email{ethandapani@yahoo.co.in} \address{Sandra Pinelas\newline Departamento de Matem\'atica, Universidade dos A\c cores, Portugal} \email{sandra.pinelas@clix.pt} \thanks{Submitted January 10, 2010. Published May 19, 2010.} \subjclass[2000]{34K11, 34C55} \keywords{Interval oscillation; quasilinear delay differential equation; \hfill\break\indent second order} \begin{abstract} This article presents new oscillation criteria for the second-order delay differential equation \[ (p(t) (x'(t))^{\alpha})' + q(t) x^{\alpha}(t - \tau) + \sum_{i = 1}^{n} q_{i}(t) x^{\alpha_{i}}(t - \tau) = e(t) \] where $\tau \geq 0$, $p(t) \in C^1[0, \infty)$, $q(t),q_{i}(t), e(t) \in C[0, \infty)$, $p(t) > 0$, $\alpha_1 >\dots > \alpha_{m} > \alpha > \alpha_{m+1} > \dots > \alpha_{n} > 0\ (n > m\geq 1)$, $\alpha_1, \dots , \alpha_{n}$ and $\alpha$ are ratio of odd positive integers. Without assuming that $q(t), q_{i}(t)$ and $e(t)$ are nonnegative, the results in \cite{s1,s3} have been extended and a mistake in the proof of the results in \cite{c1} is corrected. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} In this paper, we are concerned with the oscillatory behavior of the quasilinear delay differential equation \begin{equation} \label{e1.1} (p(t) (x'(t))^{\alpha})' + q(t) x^{\alpha}(t - \tau) + \sum_{i = 1}^{n} q_{i}(t) x^{\alpha_{i}}(t - \tau) = e(t) \end{equation} where $\tau \geq 0$, $p(t), q(t), q_{i}(t) \in C[0, \infty)$, $p(t)$ is positive, nondecreasing and differentiable, $\alpha_1, \dots , \alpha_{n}, \alpha$ are ratio of odd positive integers, and $\alpha_1 > \dots > \alpha_{m} > \alpha > \alpha_{m+1} > \dots > \alpha_{n} > 0$. A solution $x(t)$ of \eqref{e1.1} is said to be oscillatory if it is defined on some ray $[T, \infty)$ with $T \geq 0$ and has unbounded set of zeros. Equation \eqref{e1.1} is said to be oscillatory if all solutions extendable throughout $[0, \infty)$ are oscillatory. For $\tau = 0$ and $\alpha = 1$, the oscillatory behavior of \eqref{e1.1} has been studied in Sun and Wong \cite{s3} and Sun and Meng \cite{s1}. When $\alpha = 1$, Chen and Li \cite{c1} extended the results established by Sun and Meng \cite{s1} to \eqref{e1.1}. A close look into the proof of \cite[Theorem 1]{c1} reveals that the authors used $x''(t) \leq 0$ for $t \in [a_1 - \tau, b_1]$ instead of taking $(p(t) x'(t))' \leq 0$ for $t \in [a_1 - \tau, b_1]$. We wish not only to correct the proof of the theorem but also extend the results given in \cite{a1,a2,d1,s3} for ordinary and delay differential equations. In Section 2, we present some new oscillation criteria for the \eqref{e1.1} and in Section 3 we provide some examples to illustrate the results. \section{Oscillation Results} We first present a lemma which is a generalization of Lemma 1 of Sun and Wong \cite{s3}. \begin{lemma} \label{lem2.1} Let $\{\alpha_{i}\}$, $i = 1, 2, \dots, n$ be the $n$-tuple satisfying $\alpha_1 > \dots > \alpha_{m} > \alpha > \alpha_{m+1} > \dots > \alpha_{n} > 0$. Then there is an $n$-tuple $(\eta_1, \eta_2, \dots, \eta_{n})$ satisfying \begin{equation} \label{ea} \sum_{i = 1}^{n} \alpha_{i} \eta_{i} = \alpha \end{equation} which also satisfies \begin{equation} \label{eb} \sum_{i = 1}^{n} \eta_{i} < 1, \ \ 0 < \eta_{i} < 1, \end{equation} or \begin{equation} \label{ec} \sum_{i = 1}^{n} \eta_{i} = 1, \ \ 0 < \eta_{i} < 1. \end{equation} \end{lemma} \begin{lemma} \label{lem2.2} Suppose $X$ and $Y$ are nonnegative, then \[ X^{\gamma} - \gamma \ Y^{\gamma - 1} X + (\gamma - 1) Y^{\gamma} \geq 0, \quad \gamma > 1, \] where the equality holds if and only if $X = Y$. \end{lemma} The proof of the above lemma can be found in \cite{h1}. Following Philos \cite{a1}, we say a continuous function $H(t, s)$ belongs to a function class $D_{a, b}$, denoted by $H \in D_{a, b}$, if $H(b, b) = H(a, a)=0$, $H(b, s) > 0$ and $H(s, a) > 0$ for $b > s > a$, and $H(t, s)$ has continuous partial derivatives with $\frac{\partial H(t, s)}{\partial t}$ and $\frac{\partial H(t, s)}{\partial s}$ in $[a, b] \times [a, b]$. Set \begin{equation} \label{e2.1} \frac{\partial H(t, s)}{\partial t} = (\alpha + 1)h_1(t, s) \sqrt{H(t, s)},\ \frac{\partial H(t, s)}{\partial s} = -(\alpha + 1)h_2(t, s) \sqrt{H(t, s)}. \end{equation} \begin{theorem} \label{thm2.1} If for any $T \geq 0$, there exist $a_1, b_1, c_1, a_2, b_2$ and $c_2$ such that $T \leq a_1 < c_1 < b_1$, $T\leq a_2 < c_2 < b_2$ and \begin{equation} \label{e2.2} \begin{gathered} q_{i}(t) \geq 0, \quad q(t) \geq 0, \quad t \in [a_1 -\tau, b_1] \cup [a_2 - \tau, b_2],\; i= 1, 2, \dots , n , \\ e(t) \leq 0, \quad t \in [a_1 - \tau, b_1], \\ e(t) \geq 0, \quad t \in [a_2 - \tau, b_2], \end{gathered} \end{equation} and there exist $H_{j} \in D_{a_{j},b_{j}}$, $j = 1, 2$, such that \begin{equation} \label{e2.3} \begin{aligned} &\frac{1}{H_{j}(c_{j},a_{j})}\int_{a_{j}}^{c_{j}}H_{j}(s, a_{j}) \Big[Q_{j}(s) - \frac{p(s)}{\alpha^{\alpha}}\Big(\frac{h_{j_1}(s, a_{j})}{\sqrt{H_{j}(s, a_{j})}}\Big)^{\alpha + 1} \Big] ds \\ &+\frac{1}{H_{j}(b_{j},c_{j})}\int_{c_{j}}^{b_{j}}H_{j}(b_{j}, s) \Big[Q_{j}(s) - \frac{p(s)}{\alpha^{\alpha}}\Big(\frac{h_{j_2}(b_{j}, s)}{\sqrt{H_{j}(b_{j}, s)}}\Big)^{\alpha + 1} \Big] ds> 0 \end{aligned} \end{equation} where $h_{j_1}$ and $h_{j_2}$ are defined as in \eqref{e2.1}, \begin{equation} \label{e2.4} Q_{j}(t) = \beta_{j}(t) \Big[ q(t) + k_{0} |e(t)|^{\eta_{0}}\prod_{i = 1}^{n} q_{i}^{\eta_{i}}(t)\Big], \quad k_{0} = \prod_{i = 0}^{n}\eta_{i}^{-\eta_{i}},\quad \eta_{0}=1-\sum_{i=1}^n\eta_i, \end{equation} and $ \eta_1, \eta_2, \dots, \eta_{n} $ are positive constants satisfying (a) and (b) in Lemma 2.1 and $\beta_{j}(t) = \Big(\frac{(t - a_{j})}{(t - a_{j} + \tau)}\Big)^{\alpha}$ then \eqref{e1.1} is oscillatory. \end{theorem} \begin{proof} Suppose that $x(t)$ is a nonoscillatory solution of \eqref{e1.1}. Without loss of generality, we may assume that $x(t) > 0$ for $t \geq t_{0} - 2 \tau > 0$ where $t_{0}$ depends on the solution $x(t)$. When $x(t)$ is eventually negative, the proof follows the same argument by using the interval $[a_2, b_2]$ instead of $[a_1, b_1]$. Choose $a_1, b_1 \geq t_{0}$ such that $q_{i}(t) \geq 0, q(t) \geq 0$ and $e(t) \leq 0$ for $t \in [a_1 - \tau, b_1]$ and $i= 1, 2, \dots , n$. From \eqref{e1.1}, we have $(p(t) (x'(t))^{\alpha})' \leq 0$ for $t \in [a_1 - \tau, b_1]$. Therefore for $a_1 - \tau 0$ and $p(t)$ is nondecreasing, we have \begin{equation} \label{e2.5} \frac{1}{(t - a_1 +\tau)} \geq \frac{x'(t)}{x(t)} , \quad t \in (a_1 - \tau, b_1]. \end{equation} Integrating \eqref{e2.5} from $t - \tau$ to $t > a_1$, we obtain \begin{equation} \label{e2.6} \frac{x(t - \tau)}{x(t)}\geq \frac{t - a_1}{t - a_1 +\tau}, \quad t \in (a_1, b_1]. \end{equation} Define $w(t) = - p(t) \frac{(x'(t))^{\alpha}}{x^{\alpha}(t)}$. From \eqref{e1.1} and \eqref{e2.6} we find that $w(t)$ satisfies the inequality \begin{equation} \label{e2.7} \begin{aligned} w'(t) &\geq q(t) \beta_1(t) + \sum_{i = 1}^{n} q_{i}(t) \beta_1(t) x^{\alpha_{i}- \alpha}(t - \tau)\\ &\quad - e(t)\beta_1(t)x^{-\alpha}(t - \tau) + \alpha\frac{|w(t)|^{1 + \frac{1}{\alpha}}}{p^{1/\alpha}(t)},\quad t \in [a_1, b_1]. \end{aligned} \end{equation} Recall the arithmetic-geometric mean inequality \begin{equation} \label{e2.8} \sum_{i = 0}^{n} \eta_{i} u_{i} \geq \prod_{i = 0}^{n} u_{i}^{\eta_{i}}, \quad u_{i} \geq 0, \end{equation} where $\eta_{0} = 1 - \sum_{i = 1}^{n} \eta_{i}$ and $\eta_{i} > 0, i= 1, 2, \dots , n$, are chosen according to given $\alpha_1, \dots , \alpha_{n}$ as in Lemma 2.1 satisfying (a) and (b). Now return to \eqref{e2.7} and identify $u_{0} = \eta_{0}^{-1} |e(t)|x^{-\alpha}( t - \tau)$ and $u_{i} = \eta_{i}^{-1} q_{i}(t)x^{\alpha_{i} - \alpha}(t - \tau)$ in \eqref{e2.8} to obtain \begin{equation} \label{e2.9} \begin{aligned} w'(t) &\geq \beta_1(t) q(t) +\frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}} {p^{\frac{1}{\alpha}}(t)} + \beta_1(t) \eta_{0}^{-\eta_{0}} |e(t)|^{\eta_{0}} \prod_{i = 1}^{n} \eta_{i}^{-\eta_{i}}q_{i}^{\eta_{i}}(t)\\ & = Q_1(t) + \frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}} {p^{\frac{1}{\alpha}}(t)}, \quad t \in [a_1, b_1], \end{aligned} \end{equation} where $Q_1(t)$ is defined by \eqref{e2.4}. Multiply \eqref{e2.9} by $H_1(b_1, t) \in D_{a_1, b_1}$ and integrating by parts, we find \begin{align*} &-H_1(b_1, c_1) w(c_1)\\ &\geq \int_{c_1}^{b_1} Q_1(s) H_1(b_1, s) ds\\ &\quad+ \int_{c_1}^{b_1} \Big[ -|w(s)|(\alpha+1) h_{12}(b_1, s) \sqrt{H_1(b_1, s)} + \frac{\alpha |w(s)|^{1 + \frac{1}{\alpha}}}{p^{\frac{1}{\alpha}}(s)} H_1(b_1, s) \Big]ds. \end{align*} Using Lemma 2.2 to the right side of the last inequality, we have \[ -H_1(b_1, c_1) w(c_1) \geq \int_{c_1}^{b_1} \Big[Q_1(s) H_1(b_1, s) -\frac{p(s)}{\alpha^{\alpha}} H_1(b_1, s) \Big( \frac{h_{12}(b_1, s)}{\sqrt{H_1(b_1, s)}} \Big)^{\alpha + 1} \Big] ds. \] It follows that \begin{equation} \label{e2.10} - w(c_1) \geq \frac{1}{H_1(b_1, c_1)}\int_{c_1}^{b_1} \Big[Q_1(s) H_1(b_1, s) -\frac{p(s)}{\alpha^{\alpha}} H_1(b_1, s) \Big( \frac{h_{12}(b_1, s)}{\sqrt{H_1(b_1, s)}} \Big)^{\alpha + 1} \Big] ds. \end{equation} On the other hand, multiplying both sides of \eqref{e2.9} by $H_1(t, a_1) \in D_{a_1,b_1}$, integrating by parts, and similar to the above analysis we can easily obtain \begin{equation} \label{e2.11} w(c_1) \geq \frac{1}{H_1(c_1, a_1)}\int_{a_1}^{c_1} \Big[Q_1(s) H_1(s, a_1) -\frac{p(s)}{\alpha^{\alpha}} H_1(s, a_1) \Big( \frac{h_{11}(s, a_1)}{\sqrt{H_1(s, a_1)}} \Big)^{\alpha + 1} \Big] ds. \end{equation} From \eqref{e2.10} and \eqref{e2.11} we have \begin{align*} &\frac{1}{H_1(c_1, a_1)}\int_{a_1}^{c_1} \Big[Q_1(s) H_1(s, a_1) -\frac{p(s)}{\alpha^{\alpha}} H_1(s, a_1) \Big( \frac{h_{11}(s, a_1)}{\sqrt{H_1(s, a_1)}} \Big)^{\alpha + 1} \Big] ds \\ &+\frac{1}{H_1(b_1, c_1)}\int_{c_1}^{b_1} \Big[Q_1(s) H_1(b_1, s) -\frac{p(s)}{\alpha^{\alpha}} H_1(b_1, s) \Big( \frac{h_{12}(b_1, s)}{\sqrt{H_1(b_1, s)}} \Big)^{\alpha + 1} \Big] ds \leq 0 \end{align*} which contradicts \eqref{e2.3} for $j = 1$. The proof is now complete. \end{proof} The following theorem gives an interval oscillation criteria for the unforced \eqref{e1.1} with $e(t) \equiv 0$. \begin{theorem} \label{thm2.2} If for any $T > 0$ there exist $a, b$ and $c$ such that $T \leq a < c < b$ and $q(t) \geq 0, q_{i}(t) \geq 0$ for $t \in [a - \tau, b]$ and $i = 1, 2, \dots, n$, and there exists $H \in D_{a, b}$ such that \begin{align*} &\frac{1}{H(c, a)}\int_{a}^{c} H(s, a) \Big[\overline{Q}(s) -\frac{p(s)}{\alpha^{\alpha}} \Big( \frac{h_1(s, a)}{\sqrt{H(s, a)}} \Big)^{\alpha + 1} \Big]ds \\ &+\frac{1}{H(b, c)}\int_{c}^{b} H(b, s) \Big[\overline{Q}(s) -\frac{p(s)}{\alpha^{\alpha}} \Big( \frac{h_2(b, s)}{\sqrt{H(b, s)}} \Big)^{\alpha + 1} \Big] ds > 0 \end{align*} where $h_1$ and $h_2$ are defined by \eqref{e2.1}, \[ \overline{Q}(t) = \beta(t) \Big[ q(t) + k_1 \prod_{i = 1}^{n} q_{i}^{\eta_{i}}(t)\Big], \quad k_1 = \prod_{i = 1}^{n} \eta_{i}^{-\eta_{i}}, \] and $\eta_1, \eta_2, \dots, \eta_{n}$ are positive constants satisfying (a) and (c) of Lemma 2.1, $\beta(t) = \big(\frac{(t - a)}{(t - a + \tau)}\big)^{\alpha}$, then \eqref{e1.1} with $e(t) \equiv 0$ is oscillatory. \end{theorem} The proof of the above theorem is in fact a particular version of the proof of Theorem \ref{thm2.1}. We need only to note that $e(t) \equiv 0$ and $\eta_{0} = 0$ and apply conditions (a) and (c) of Lemma 2.1. \begin{remark} \label{rmk2.1} \rm When $\tau = 0$, Theorems \ref{thm2.1} and \ref{thm2.2} reduce to the main results in \cite{t1}. Moreover if $\tau = 0$ and $\alpha = 1$, then Theorems \ref{thm2.1} and \ref{thm2.2} reduce to \cite[Theorems 1 and 2]{s1}. \end{remark} Before stating the next result we introduce another function class. Say $u(t) \in E_{a, b}$ if $u \in C^{1}[a, b],\ u^{\alpha + 1}(t) > 0$, and $u(a) = u(b) =0$. \begin{theorem} \label{thm2.3} If for any $T \geq 0$, there exist $a_1, b_1$ and $a_2, b_2$ such that $T \leq a_1 < b_1,\ T \leq a_2 < b_2$ and \eqref{e2.2} holds, and there exists $H_{j} \in E_{a_{j}, b_{j}}$ and a positive nondecreasing function $\phi \in C^{1}([0, \infty), \mathbb{R}) $ such that \begin{equation} \label{e2.12} \int_{a_{j}}^{b_{j}} \phi(t) \Big[ Q_{j}(t) H_{j}^{\alpha + 1}(t) - p(t)\Big( |H_{j}'(t)| + \frac{H_{j}(t) \phi'(t)}{(\alpha + 1) \phi(t)}\Big)^{\alpha + 1} \Big] dt > 0 \end{equation} for $j = 1, 2$, where \begin{equation} \label{e2.13} \begin{gathered} Q_{j}(t) = \beta_{j}(t) \Big[ q(t) + k_{0} |e(t)|^{\eta_{0}} \prod_{i = 1}^{n} q_{i}^{\eta_{i}}(t)\Big], \ k_{0} = \prod_{i = 0}^{n} \eta_{i}^{-\eta_{i}},\\ \beta_{j}(t) = \Big(\frac{(t - a_{j})}{(t - a_{j} + \tau)}\Big)^{\alpha} \end{gathered} \end{equation} then \eqref{e1.1} is oscillatory. \end{theorem} \begin{proof} Suppose that $x(t)$ is a nonoscillatory solution of \eqref{e1.1}. Without loss of generality, we may assume that $x(t) > 0$ for $t \geq t_{0} - 2 \tau > 0$ where $t_{0}$ depends on the solution $x(t)$. When $x(t)$ is eventually negative, the proof follows the same argument by using the interval $[a_2, b_2]$ instead of $[a_1, b_1]$. Choose $q(t) \geq 0, q_{i}(t) \geq 0$ and $e(t) \leq 0$ for $t \in [a_1 - \tau, b_1]$ and $i = 1, 2, \dots, n$. As in the proof of Theorem \ref{thm2.1} \begin{equation} \label{e2.14} \Big( \frac{x(t - \tau)}{x(t)} \Big)^{\alpha} \geq \beta_1(t), \quad t\in (a_1, b_1]. \end{equation} Define $w(t) = -\phi(t) \frac{p(t) (x'(t))^{\alpha}}{x^{\alpha}(t)}$. From \eqref{e1.1} and \eqref{e2.14} we have \begin{align*} w'(t) &\geq \phi(t) q(t) \beta_1(t) + \sum_{i = 1}^{n} \phi(t) q_{i}(t) \beta_1(t) x^{\alpha_{i} - \alpha}(t - \tau) + \frac{w(t) \phi'(t)}{\phi(t)} \\ &\quad - e(t) \beta_1(t) x^{-\alpha}(t - \tau) +\frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}}{(p(t) \phi(t))^{1/\alpha}}. \end{align*} Using Lemma 2.1, we have \begin{equation} \label{e2.15} w'(t) \geq \phi(t) Q_1(t) + \frac{w(t) \phi'(t)}{\phi(t)} +\frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}}{(p(t) \phi(t))^{1/\alpha}}. \end{equation} Multiply \eqref{e2.15} by $H^{\alpha + 1}(t)$ and integrating from $a_1$ to $b_1$ using the fact that $H(a_1) = H(b_1) = 0$, we obtain \begin{equation} \label{e2.16} \begin{aligned} 0 &\geq \int_{a_1}^{b_1} H^{\alpha+1}(t)\phi(t) Q_1(t) dt + \int_{a_1}^{b_1}\Big\{ \frac{\alpha H^{\alpha + 1}(t) |w(t)|^{1 + \frac{1}{\alpha}} }{(p(t) \phi(t))^{1/\alpha}}\\ &\quad -\Big[ (\alpha + 1) H(t)^{\alpha}|H'(t)| + \frac{H^{\alpha+1}(t) \phi'(t) }{ \phi(t)} \Big]|w(t)|\Big\}dt. \end{aligned} \end{equation} Using Lemma 2.2 in \eqref{e2.16}, we have \[ 0 \geq \int_{a_1}^{b_1} \phi(t) \Big[ Q_1(t) H^{\alpha + 1}(t) - p(t)\Big( |H'(t)| + \frac{H(t) \phi'(t)}{(\alpha + 1) \phi(t)}\Big)^{\alpha + 1} \Big] dt \] which contradicts \eqref{e2.12} with $j = 1$. This completes the proof. \end{proof} \begin{corollary} \label{coro2.4} Suppose that $\phi(t) \equiv 1$ in Theorem \ref{thm2.3}, and \eqref{e2.12} is replaced by \[ \int_{a_{j}}^{b_{j}} [ Q_{j}(t) H^{\alpha + 1}(t) - p(t) |H'(t)|^{\alpha + 1} ] dt > 0 \] for $j = 1, 2$. Then \eqref{e1.1} is oscillatory. \end{corollary} \begin{theorem} \label{thm2.5} Assume that for any $T \geq 0$, there exist $a, b$ such that $T \leq a < b$ and $q(t) \geq 0, q_{i}(t) \geq 0$ for $t \in [a, b]$ and $i = 1, 2, \dots, n$. Suppose there exists $H \in E_{a, b}$ and a positive nondecreasing function $\phi \in C'([0, \infty), \mathbb{R})$ such that \[ \int_{a}^{b} \phi(t) \Big[ \overline{Q}(t) H^{\alpha + 1}(t) - p(t)\Big( |H'(t)| + \frac{H(t) \phi'(t)}{(\alpha + 1) \phi(t)}\Big)^{\alpha + 1} \Big] dt > 0 \] where \begin{gather*} \overline{Q}(t) = \beta(t) \Big[ q(t) + k_1 \prod_{i = 1}^{n} q_{i}^{\eta_{i}}(t)\Big], \ k_1 = \prod_{i = 1}^{n} \eta_{i}^{-\eta_{i}}, \\ \beta(t) = \Big(\frac{(t - a)}{(t - a + \tau)}\Big)^{\alpha}. \end{gather*} Then \eqref{e1.1} with $e(t) \equiv 0$ is oscillatory. \end{theorem} The proof of the above theorem is in fact a particular version of the proof of Theorem \ref{thm2.3}. We need only to note that $e(t) \equiv 0$ and $\eta_{0} = 0$ and apply conditions (a) and (c) of Lemma 2.1 \begin{remark} \label{rmk2.2} \rm When $\tau = 0, \alpha = 1, \mbox{and~} \phi(t) \equiv 1$, then Theorem \ref{thm2.3} and \ref{thm2.5} reduced to \cite[Theorems 1 and 2]{s3}. \end{remark} If $n = 1$ and $e(t) \equiv 0$ then we see that Theorems \ref{thm2.1}--\ref{thm2.5} are not valid. Therefore in the following we state and prove some new oscillation criteria for the equation \begin{equation} \label{e2.17} (p(t) (x'(t))^{\alpha})' + q(t) x^{\alpha}(t - \tau) + q_1(t) x^{\alpha_1}(t - \tau) = 0, \quad t \geq 0. \end{equation} \begin{theorem} \label{thm2.6} Assume that for any $T \geq 0$ there exist $a, b$ such that $T \leq a < b$ and $q(t) \geq 0, q_1(t) \geq 0$ for $t \in [a, b]$ . Suppose there exists $H \in E_{a, b}$ and positive nondecreasing function $\phi \in C'([0, \infty), \mathbb{R})$ such that \begin{equation} \label{e2.18} \int_{a}^{b} \phi(t) \Big[ Q_{3}(t) H^{\alpha + 1}(t) - p(t)\Big( |H'(t)| + \frac{H(t) \phi'(t)}{(\alpha + 1) \phi(t)}\Big)^{\alpha + 1} \Big] dt > 0 \end{equation} where \begin{gather*} Q_{3}(t) = \beta(t) [q(t) - M_1 q_1(t)],\quad M_1 = (\alpha_1 -\alpha - 1)\Big( \frac{1}{\alpha_1 - \alpha} \Big)^{\frac{(\alpha_1 - \alpha)}{(\alpha_1 - \alpha - 1)}}, \\ \beta(t) = \Big(\frac{(t - a)}{(t - a + \tau)}\Big)^{\alpha} \end{gather*} and $\alpha_1 > \alpha + 1$, then \eqref{e2.17} is oscillatory. \end{theorem} \begin{proof} Proceeding as in the proof of Theorem \ref{thm2.3}, we obtain \begin{equation} \label{e2.19} \begin{aligned} w'(t) &\geq \frac{w(t) \phi'(t)}{\phi(t)} + \frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}}{(p(t) \phi(t))^{1/\alpha}} + \phi(t) q(t) \beta(t) + \phi(t) q_1(t) \beta(t) x^{\alpha_1 - \alpha}(t - \tau),\\ &\geq \frac{w(t) \phi'(t)}{\phi(t)} +\frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}}{(p(t) \phi(t))^{1/\alpha}} + \phi(t) q(t) \beta(t) \\ &\quad + \phi(t) q_1(t) \beta(t) (x^{\alpha_1 - \alpha}(t - \tau) - x(t - \tau)). \end{aligned} \end{equation} Set $F(x) = x^{\alpha_1 - \alpha} - x$. Using differential calculus, we find that $F(x) \geq - M_1$. From \eqref{e2.19}, we have \[ w'(t) \geq \phi(t) Q_{3}(t) + \frac{\phi'(t)}{\phi(t)} w(t) + \frac{\alpha |w(t)|^{\frac{\alpha + 1}{\alpha}}}{(p(t) \phi(t))^{1/\alpha}}. \] The rest of the proof is similar to that of Theorem \ref{thm2.3}. This completes the proof. \end{proof} \begin{theorem} \label{thm2.7} Assume that for any $T \geq 0$ there exist $a, b$ such that $T \leq a < b$ and $q(t) \geq 0$, $q_1(t) \geq 0$ for $t \in [a, b]$. Suppose there exists $H \in E_{a, b}$ and a positive nondecreasing function $\phi \in C'([0, \infty),\mathbb{R})$ such that \begin{equation} \label{e2.20} \int_{a}^{b} \phi(t) \Big[ Q_{4}(t) H^{\alpha + 1}(t) - p(t)\Big( |H'(t)| + \frac{H(t) \phi'(t)}{(\alpha + 1) \phi(t)}\Big)^{\alpha + 1} \Big] dt > 0 \end{equation} where \[ Q_{4}(t) = \beta(t) [q(t) - M_2 q_1(t)], M_2 = \frac{(\alpha - \alpha_1 - \beta)}{(\alpha - \alpha_1)}\left( \frac{\beta}{\alpha - \alpha_1} \right)^{\frac{\beta}{(\alpha - \alpha_1 - \beta)}}, \] and $\alpha > \alpha_1 + \beta$, then \eqref{e2.17} is oscillatory. \end{theorem} \begin{proof} Proceeding as in the proof of Theorem \ref{thm2.3}, we obtain \begin{equation} \label{e2.21} \begin{aligned} w'(t) &\geq \frac{w(t) \phi'(t)}{\phi(t)} +\frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}}{(p(t) \phi(t))^{1/\alpha}} + \phi(t) q(t) \beta(t)\\ &\quad + \phi(t) q_1(t) \beta(t) [x^{\alpha_1 - \alpha}(t - \tau) - x^{-\beta}(t - \tau)]. \end{aligned} \end{equation} Set $F(x) = x^{\alpha_1 - \alpha} - x^{-\beta}$. Using differential calculus, we find $F(x) \geq -M_2$. From \eqref{e2.21}, we have \[ w'(t) \geq \phi(t) Q_{4}(t) + \frac{w(t) \phi'(t)}{\phi(t)} + \frac{\alpha |w(t)|^{1 + \frac{1}{\alpha}}}{(p(t) \phi(t))^{1/\alpha}}. \] The rest of the proof is similar to that of Theorem \ref{thm2.3}. This completes the proof. \end{proof} \begin{remark} \label{rmk2.3} \rm The results obtained here can also be extended to the following general equation \begin{align*} &(p(t) |(x'(t))|^{\alpha -1 }x'(t))' + q(t) |x(t - \tau_0)|^{\alpha-1}x(t - \tau_0) \\ &+ \sum_{i = 1}^{n} q_{i}(t)|x(t - \tau_i)|^{\alpha_i-1}x(t - \tau_i) = e(t) \end{align*} where $\tau_i \geq 0, ~ i = 0,1,\dots n $ and we left it to interesting readers. \end{remark} \section{Examples} In this section, we present some examples to illustrate the main results. \begin{example} \label{exa3.1} \rm Consider the delay differential equation \begin{equation} \label{e3.1} \begin{aligned} &(t (x'(t))^{3})' + l_1 \cos t (x(t - \pi/8))^{3} \\ &+ l_2 (\sin t)^{20/11} (x(t - \pi/8))^{5} + l_{3} \cos^4 t (x(t - \pi/8)) \\ & = - m \cos^{5} 2t, \end{aligned} \end{equation} where $t \geq 0$, $l_1, l_2, l_{3}, m$ are positive constants. Here $p(t) = t$, $\alpha = 3$, $q(t) = l_1 \cos t$, $q_1(t) = l_2 (\sin t)^{20/11}$, $q_2(t) = l_{3} (\cos t)^{1/4}$, $\alpha_1 = 5$, $\alpha_2 = 1$, $\tau = \frac{\pi}{8}$ and $e(t) = -m \cos 2t$. For any $T \geq 0$, we can choose $a_1 = 2n\pi + \frac{\pi}{8}, b_1 = 2n\pi + \frac{\pi}{4}$, $a_2 = 2n\pi + \frac{3\pi}{8}, b_2 = 2n\pi + \frac{\pi}{2}$ for sufficiently large $n$, where $n$ is a positive integer. It is easy to find that \begin{align*} Q_{j}(t) & = k_{0}\Big[ \frac{(t - a_{j})}{(t - a_{j} + \pi/8)} \Big]^3 (l_1 \cos t + (\cos^{5}2t)^{1/5} (\sin^{20/11}t)^{11/20} (\cos^{4}t)^{1/4})\\ & = k_{0}\Big[ \frac{(t - a_{j})}{(t - a_{j} + \pi/8)} \Big]^3 (l_1 \cos t + (\cos 2t) \sin t \cos t) \end{align*} where $k_{0} = (5m)^{1/5} (\frac{20 l_2}{11})^{11/20} (4l_{3})^{1/4}$. Let $H_1(t) = H_2(t) = \sin 8t$ and $\phi(t) = 1$. Based on Theorem \ref{thm2.3}, we have \eqref{e3.1} is oscillatory if \[ \int_{a_{j}}^{b_{j}} \Big[ k_{0}\Big( \frac{t - a_{j}}{t - a_{j} + \pi/8} \Big)^{3} \Big( l_1 \cos t + \frac{\sin 4t}{4} \Big) \sin ^{4} 8t - 8t \cos^{4} 8t \Big] dt > 0, \ \ j = 1, 2. \] \end{example} \begin{example} \label{exa3.2} \rm Consider the delay differential equation \begin{equation} \label{e3.2} x''(t) + k_1 t^{-\lambda/3} (\sin t ) x(t - \pi/2) + t^{-\delta} x^{3}(t - \pi/2) = 0, \ t \geq 1, \end{equation} where $k_1, \lambda, \delta > 0$ are constants and $\alpha = 1, \alpha_1 = 3, \tau = \frac{\pi}{2}$ in Theorem \ref{thm2.6}. Since $\alpha < \alpha_1$ and $e(t) \equiv 0$, Theorem \ref{thm2.2} and Theorem \ref{thm2.5} are not applicable to this case. However, we can obtain oscillation of \eqref{e3.2} with $H(t) = \sin 2t$ and $\phi(t) = 1. $ For any $t_{0} \geq 1$, we can choose $a = 2k\pi + \pi/2, b = 2k\pi + \pi$ for sufficiently large $k$, where $k$ is a positive integer. It is easy to find that \begin{gather*} Q_{3}(t) = \Big( \frac{t - a}{t - a + \pi/2} \Big) \Big[ k_1 t^{-\lambda/3} \sin t - \frac{t^{-\delta}}{4} \Big], \\ \int_{a}^{b}\Big[ \frac{t - a}{t - a + \pi/2} \Big( k_1 t^{-\lambda/3} \sin t - \frac{t^{-\delta}}{4} \Big) \sin ^{2} 2t -4 \cos^{2} 2t\Big] dt > 0. \end{gather*} So by Theorem \ref{thm2.6}, Equation \eqref{e3.2} is oscillatory if \[ \int_{2k\pi +\pi/2}^{2k\pi +\pi} \Big( \frac{t - a}{t - a + \pi/2} \Big) \Big( k_1 t^{-\lambda/3} \sin t - \frac{t^{-\delta}}{4} \Big) \sin^{2} 2t\, dt > \pi. \] \end{example} \begin{example} \label{exa3.3} \rm Consider the delay differential equation \begin{equation} \label{e3.3} ((x'(t))^{3})' + k_1 t^{-\lambda} (\sin t)\, x^{3}(t - \pi/4) + k_2 t^{-\lambda} x(t - \pi/4) = 0, \end{equation} where $t \geq 1, k_1, k_2$ and $\lambda$ are positive constants and $\alpha = 3, \alpha_1 = 1$ in Theorem \ref{thm2.7}. Since other theorems cannot be applicable to this case but we can obtain oscillation of \eqref{e3.3} with $\beta = 1, H(t) = \sin 4t$ and $\phi(t) = 1$. For any $t_{0} \geq 1$, let $a = 2n\pi + \pi/4, b = 2n\pi + \pi/2$ for $n$ sufficiently large and $n$ is a positive integer. It is easy to see that \begin{align*} &\int_{a}^{b} Q_{4}(t)H^{4}(t) - (H'(t))^{4} \\ & = \int_{a}^{b} \Big[ \Big(\frac{t - a}{t - a + \pi/4}\Big)^{3}\Big(k_1t^{-\lambda}\sin t -\frac{1}{4}k_2t^{-\lambda}\Big)\sin^{4}4t - 256 \cos^{4}4t \Big]dt. \end{align*} So by Theorem \ref{thm2.7}, Equation \eqref{e3.3} is oscillatory if \[ \int_{2n\pi + \pi/4}^{2n\pi + \pi/2} \Big(\frac{t - a}{t - a + \pi/4}\Big)^{3}\Big(k_1t^{-\lambda}\sin t -\frac{1}{4}k_2t^{-\lambda}\Big)\sin^{4}4t\, dt > \frac{3\pi}{32}. \] \end{example} \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal, S. R. Grace and D. O. 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