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Nonoscillatory solutions of differential equations with deviating argument. (English) Zbl 0603.34064

With the help of the generalized Kiguradze lemmas (the third lemma is taken from U. Elias [J. Math. Anal. Appl. 97, 277-290 (1983; Zbl 0546.34010)]) the equation (1) \(L_ ny(t)+f[t,y(g(t))]=0\) is investigated, where \(L_ jy(t)\) is the jth quasiderivative of y at the point t, \(j=0,1,...,n\), f shows a sign property (\(y f(t,y)\geq 0\) or \(yf(t,y)\leq 0\)) and \(\lim_{t\to \infty}g(t)=\infty.\) Under these conditions for each nonoscillatory solution y(t) of (1) there is an \(\ell\), \(0\leq \ell \leq n\), such that \(\delta L_ jy(t)>0\) for \(j=0,1,...,\ell -1\), \((-1)^{\ell +j}\delta L_ j\quad y(t)>0,\) 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 \(P_{\ell}.\)
In the paper sufficient conditions are established which guarantee that (i) for the solution y(t) with the property \(P_{\ell}\lim_{t\to \infty}L_{\ell}y(t)=0;\) (ii) there is no solution y(t) with the property \(P_{\ell}\); (iii) the equation (1) has the property A (the property B); (iv) all solutions of (1) are oscillatory.

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34K25 Asymptotic theory of functional-differential equations

Citations:

Zbl 0546.34010
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References:

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