# zbMATH — the first resource for mathematics

Comparison theorems for deviated differential equations with property $$A$$. (English) Zbl 0935.34060
The equation $u^{(n)}(t)+p(t)u(\tau(t))=0\tag{1}$ with $$p\in L_{\text loc}({\mathbb{R}}_+,{\mathbb{R}}_+)$$ and $$\tau\in C({\mathbb{R}}_+,{\mathbb{R}}_+)$$ is said to be having property A if any of its proper solutions is oscillatory when $$n$$ is even, and either is oscillatory or satisfies $$|u^{(i)}|(t)\downarrow 0$$ as $$t\uparrow+\infty$$, $$i=1,\ldots,n$$, when $$n$$ is odd. Four comparison theorems for equation (1) and the equation $u^{(n)}(t)+q(t)u(\sigma(t))=0\tag{2}$ with $$q\in L_{\text{loc}}({\mathbb{R}}_+,{\mathbb{R}}_+)$$ and $$\sigma\in C({\mathbb{R}}_+,{\mathbb{R}}_+)$$ are formulated. For example, it is stated that if $$\sigma(t)\geq\tau(t)\geq t$$, for sufficiently large $$t_0$$ $\int_t^{+\infty}\tau^{n-2}(s)p(s) ds\geq \int_t^{+\infty}\sigma^{n-2}(s)q(s) ds,\quad t\geq t_0,$ and (2) has property A, then (1) has property A. Moreover, six theorems on effective criteria for property A are formulated.

##### MSC:
 34K12 Growth, boundedness, comparison of solutions to functional-differential equations
Full Text: