## 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

### Keywords:

nonoscillatory solution; Kiguradze lemmas; sign property

Zbl 0546.34010
Full Text:

### References:

 [1] Чантурия Т. А.: О монотонных и колеблющихся решениях обыкновенных дифференциальных уравнений высших порядков. Ann. Polon. Math. 37 (1980), 93-111. · Zbl 1170.01312 [2] Elias U.: Generalizations of an Inequality of Kiguradze. J. Math. Anal. Appl. 97 (1983), 277-290. · Zbl 0546.34010 [3] Ohriska J.: Oscillation of Second Order Delay and Ordinary Differential Equation. Czech. Math. J. 34 (109) (1984), 107-112. · Zbl 0543.34054 [4] Philos Ch. G.: Oscillatory and Asymptotic Behavior of the Bounded Solutions of Differential Equations with Deviating Arguments. Hiroshima Math. J. 8 (1978), 31-48. · Zbl 0378.34055 [5] Philos Ch. G.: Oscillatory and asymptotic behaviour of all solutions of differential equations with deviating arguments. Proc. Roy. Soc. Edinburgh 81 A (1978), 195-210. · Zbl 0417.34108 [6] Philos Ch. G., Sficas Y. G., Staikos V. A.: Some Results on the Asymptotic Behavior of Nonoscillatory Solutions of Differential Equations with Deviating Arguments. J. Austral. Math. Soc. (Series A) 32 (1982), 295-317. · Zbl 0499.34053 [7] Švec M.: Behavior of Nonoscillatory Solutions of Some Nonlinear Differential Equations. Acta Math. Univ. Comen. 39 (1980), 115-130. · Zbl 0525.34029
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