×

Property (A) of the \(n\)-th order differential equations with deviating argument. (English) Zbl 0830.34057

The author considers the differential equation (1) \(L_n y(t)+ p(t) y(\tau(t))= 0\), where \(n\geq 3\), \(L_n y(t)= r_n(t)^{- 1} [r_{n- 1}(t)^{- 1}\cdots (r_0(t)^{- 1} y(t))']'\), \(r_i(t)\), \(i= 0,1,2,\dots, n\) are positive continuous functions on some ray \([t_0, \infty)\), \(\tau(t)< t\) is an increasing function on \([t_0, \infty)\). A function \(y(t)\) satisfying (2) \(y(t)L_i y(t)> 0\), \(0\leq i\leq 1\), \((- 1)^{i- 1} y(t) L_iy(t)> 0\), \(1\leq i\leq n- 1\), where \(L_i y(t)= r_i(t)^{- 1}(L_{i- 1} y(t))\), \(i= 1,2,\dots, n\), \(L_0 y(t)= r_0(t)^{- 1} y(t)\), is said to be a function of degree 1. If \(\mathcal N\) is the set of all nonoscillatory solutions of (1), one says that (1) enjoys property (A) if \({\mathcal N}= {\mathcal N}_0\) if \(n\) is odd, and \({\mathcal N}= \emptyset\) if \(n\) is even, where \({\mathcal N}_0\) is the set of all nonoscillatory solutions of (1) of degree 0. The aim of this paper is to derive sufficient condition for property (A) of the equation (1).

MSC:

34K10 Boundary value problems for functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: EuDML