Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0603.34064
Šeda, 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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]