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On Kneser-type solutions of sublinear ordinary differential equations. (English) Zbl 0696.34034

Summary: A nontrivial solution u: [a,\(+\infty [\to {\mathbb{R}}\) of an ordinary differential equation of n-th order is called a Kneser-type solution (KS) if \((-1)^ iu^{(i)}(t)\geq 0\) for \(t\geq a\) \((i=0,...,n-1)\). A KS is called degenerate (singular) if it is constant (zero) in some neighbourhood of \(+\infty\), and nondegenerate otherwise. In the paper a class of equations admitting sufficiently many singular KSs is introduced and studied. For the equations from this class a sufficient condition for the existence of a nondegenerate KS with a prescribed limit at \(+\infty\) is established. Two-sided a priori asymptotic estimates of such solutions are obtained.

MSC:

34C99 Qualitative theory for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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