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Zbl 0603.34064
Ĺ eda, Valter
Nonoscillatory solutions of differential equations with deviating argument.
(English)
[J] Czech. Math. J. 36(111), 93-107 (1986). ISSN 0011-4642; ISSN 1572-9141/e

With the help of the generalized Kiguradze lemmas (the third lemma is taken from {\it U. Elias} [J. Math. Anal. Appl. 97, 277-290 (1983; Zbl 0546.34010)]) the equation (1) $L\sb ny(t)+f[t,y(g(t))]=0$ is investigated, where $L\sb jy(t)$ is the jth quasiderivative of y at the point t, $j=0,1,...,n$, f shows a sign property ($y f(t,y)\ge 0$ or $yf(t,y)\le 0$) and $\lim\sb{t\to \infty}g(t)=\infty.$ Under these conditions for each nonoscillatory solution y(t) of (1) there is an $\ell$, $0\le \ell \le n$, such that $\delta L\sb jy(t)>0$ for $j=0,1,...,\ell -1$, $(-1)\sp{\ell +j}\delta L\sb j\quad y(t)>0,$ for $j=\ell$, $\ell +1,...,n$ for all sufficiently large t, where $\delta =sgn y(t)$ in a neighbourhood of $\infty$. Then y(t) is said to have the property $P\sb{\ell}.$ \par In the paper sufficient conditions are established which guarantee that (i) for the solution y(t) with the property $P\sb{\ell}\lim\sb{t\to \infty}L\sb{\ell}y(t)=0;$ (ii) there is no solution y(t) with the property $P\sb{\ell}$; (iii) the equation (1) has the property A (the property B); (iv) all solutions of (1) are oscillatory.
MSC 2000:
*34K99 Functional-differential equations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34K25 Asymptotic theory of functional-differential equations

Keywords: nonoscillatory solution; Kiguradze lemmas; sign property

Citations: Zbl 0546.34010

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