Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients

B. Karpuz, Afyon Kocatepe University, Afyonkarahisar, Turkey

E. J. Qualitative Theory of Diff. Equ., No. 34. (2009), pp. 1-14.

Abstract: In this paper, we present a criterion on the oscillation of unbounded solutions for higher-order dynamic equations of the following form:
\begin{equation}
\big[x(t)+A(t)x(\alpha(t))\big]^{\Delta^{n}}+B(t)F(x(\beta(t)))=\varphi(t)\qquad\text{for}\ t\in[t_{0},\infty)_{\T},\label{asbeq1}\tag{$\star$}
\end{equation}
where $n\in[2,\infty)_{\Z}$, $t_{0}\in\T$, $\sup\{\T\}=\infty$, $A\in\crd([t_{0},\infty)_{\T},\R)$ is allowed to alternate in sign infinitely many times, $B\in\crd([t_{0},\infty)_{\T},\R^{+})$, $F\in\crd(\R,\R)$ is nondecreasing, and $\alpha,\beta\in\crd([t_{0},\infty)_{\T},\T)$ are unbounded increasing functions satisfying $\alpha(t),\beta(t)\leq t$ for all sufficiently large $t$. We give change of order formula for double(iterated) integrals to prove our main result. Some simple examples are given to illustrate the applicability of our results too. In the literature, almost all of the results for \eqref{asbeq1} with $\T=\R$ and $\T=\Z$ hold for bounded solutions. Our results are new and not stated in the literature even for the particular cases $\T=\R$ and/or $\T=\Z$.

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