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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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