Asymptotic and oscillatory behaviour of solutions of a linear differential equation
The linear differential equation \((*)\;y^{(n)}+p(t)y^{(k)}+q(t)y=0\) is considered, where \(n,k\) are integers, \(n>1\), \(1\leq k<n\), \(p(t)\), \(q(t)\) are continuous together with some derivatives. The aim of this paper is to derive the oscillatory properties of a fundamental system of solutions of \((*)\) on the base of the asymptotic behaviour of \((*)\) in the author's previous paper [Arch. Math., Brno 22, 193-202 (1986; Zbl 0607.34051)]. The basic suppositions here are \(q''(t)q(t)^{-1-1/2n}\in L[a,\infty)\), \(p(t)q(t)^{-1+(k+1)/n}\in L[a,b)\) for \(n\) even and similar ones for \(n\) odd. \(L[a,\infty)\) denotes the set of Lebesgue integrable complex-valued functions in \([a,\infty)\).
scientific article; zbMATH DE number 69605
scientific article