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On some properties of solutions of the disconjugate equation \(y''=q(t)y\) with an almost periodic coefficient q. (English) Zbl 0644.34039
The disconjugate equation \((q):y''=q(t)y\), where q is an almost periodic function is investigated. Every disconjugate equation (q) is either generally disconjugate (i.e. there exist independent solutions u, v of (q) satisfying \(u(t)v(t)\neq 0\) on R) or specially disconjugate (i.e. there exists a unique (up to the multiplicative constant) solution u of (q) satisfying \(u(t)\neq 0\) on R). It is proved for the specially disconjugate equation that there exists a unique almost periodic solution of the associated Riccati equation \(u'+u\) \(2=q(t)\), and necessary and sufficient conditions are given for (q) to be generally disconjugate or specially disconjugate. These conditions are expressed either through a certain form of a solution of (q) or through a certain form of q or through a certain form of a phase of (q). Next, there are given necessary and sufficient conditions for the existence of an almost periodic solution of the specially disconjugate equation (q). The equation \((\bar q)\) for \(q\in H\{q\}\), where \(H\{q\}\) is the hull generated by q is investigated.
Reviewer: S.Stanek

MSC:
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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