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Properties of solutions of a class of third-order linear differential equations. (English) Zbl 0705.34032
Consider the third-order differential equation \((1)\quad (r(ry')')'+(q_ 1y)'+q_ 2y'=0\) and its adjoint equation \((2)\quad (r(rz')')'+(q_ 2z)'+q_ 1z'=0,\) where r, \(q_ i\) for \(i=1,2\), are real valued continuous functions on \(I=[a,\infty)\) with \(r(t)>0\) on I. Equation (1) is said to be \(C_ 1\) (or \(C_ 2)\) on I, if for each \(c\in (a,\infty)\) and nontrivial solution y(t) of (1) satisfying \(y(c)=y'(c)=0\), we have y(t)\(\neq 0\) for \(t\in [a,c)\) (or \(t\in (c,\infty))\). Obviously, \(C_ 1\) implies nonoscillation. The authors give the following results: If \(\int^{\infty}_{a}(r(s))^{-1}ds=\infty,\) \((q_ 2-q_ 1)'\leq 0\) (or \((q_ 2-q_ 1)'\geq 0)\), and \((q_ 2-q_ 1)'\) does not vanish identically on any interval, then equation (1) is \(C_ 1\) (or \(C_ 2)\). Equation (1) is \(C_ 1\) \((C_ 2)\) if and only if equation (2) is \(C_ 2\) \((C_ 1)\). The authors also give sufficient conditions for all solutions of (1) which have a zero on I to be oscillatory and sufficient conditions for all oscillatory solutions of (1) to be bounded on I.
Reviewer: T.S.Liu

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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