Nonoscillatory solutions of linear differential equations with deviating arguments (Q799852)

The author considers the equation (1) \(x^{(n)}(t)+\sigma p(t)x(g(t))=0(t>a)\), where \(\sigma =\pm 1\), \(p(t)>0\) for \(t\geq a\) and g(t)\(\to\infty \) as \(t\to\infty \). A solution of (1) which has arbitrarily large zeros is said to be oscillatory. It is known that a nonoscillatory solution x(t) of (1) satisfies (2) \(x(t)x^{(i)}(t)>0\) (0\(\leq i\leq\ell )\), \((-1)^{i-\ell}x(t)x^{(i)}(t)>0\) (\(\ell\leq i\leq n)\) for some integer \(\ell\), 0\(\leq\ell \leq n\), \((-1)^{n-\ell -1}\sigma =1\). In this paper, for a given \(\ell\) such that \(0<\ell <n\), \((-1)^{n- \ell -1}\sigma =1\), necessary conditions and sufficient conditions are given for (1) to have a solution x(t) which satisfies (2), and a necessary and sufficient condition is established in order that for every \(\lambda >0\) of the equation \(x^{(n)}(t)+\lambda\sigma p(t)x(g(t))=0 (t>a)\) has a solution x(t) which satisfies (2).