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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.

34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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