## Asymptotic properties of nonoscillatory solutions to neutral delay differential equations of $$n$$th order.(English)Zbl 0937.34065

The author considers the neutral differential equation $L_n[x(t)+p(t)x(h(t))] + q(t)f(x(g(t))) = b(t), \tag{1}$ with $$L_0z(t) = z(t)$$, $$L_kz(t) = a_k(t)(L_{k-1}z(t))'$$, $$k=1,2,\ldots ,n$$, $$f$$ and $$q$$ are continuous, $$f(u) u >0$$ for $$u\not = 0$$ and $$q$$ can change sign. Under some assumptions imposed on the functions $$p,h,g,b$$ and $$a_k$$ he proves results of the form: every solution (resp.bounded solution) $$x(t)$$ to (1) is either oscillatory or $$\lim_{t\rightarrow \infty} x(t) =0$$ and $$\lim_{t\rightarrow \infty} (x(t)=p(t)x(h(t)))=0$$; every bounded solution $$x(t)$$ to (1) is either oscillatory or $$\lim \inf_{t\rightarrow \infty} |x(t) |= 0$$ and $$\lim_{t\rightarrow \infty}L_k(x(t)+p(t)x(h(t)))=0$$, $$k=1,2,\ldots ,n-1$$. These criteria generalize some criteria given by M. K. Grammatikopoulos and P. Marušiak [Arch. Math., Brno 31, No. 1, 29-36 (1995; Zbl 0832.34066)], T. Kusano and H. Onose [Pacific J. Math. 63, 185-192 (1976; Zbl 0342.34058)] and the author [Asymptotic properties of nonoscillatory solutions to neutral differential equations, Proceedings of the conference on ordinary differential equations, Poprad 1994, 55-61 (1994).

### MSC:

 34K40 Neutral functional-differential equations 34K25 Asymptotic theory of functional-differential equations

### Citations:

Zbl 0832.34066; Zbl 0342.34058
Full Text:

### References:

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