A necessary and sufficient condition for the solutions of a functional differential equation to be oscillatory or tend to zero
It is shown that every solution of the nonhomogeneous functional differential equation \[ {d\over dt} \bigl(x(t) -px(t-\tau) \bigr)+ Q(t)G \bigl(x(t-\sigma) \bigr)=f(t), \] where \(f,Q \in C([T,\infty)\), \((0,\infty))\), \(\sigma,\tau \in(0, \infty)\), \(0\leq p<1\), \(G:\mathbb{R}\to \mathbb{R}\) such that \(xG(x) >0\) for \(x\neq 0\), \(G\) is nondecreasing, Lipschitzian, and satisfies a sublinear condition \[ \int^{\pm k}_0 {dx\over G(x)} <\infty \quad\text{and} \quad\int^\infty f(x)ds <\infty, \] is either oscillatory or tends to zero asymptotically if and only if \[ \int^\infty_T Q(s)ds =\infty. \] Examples and counterexamples are given.
nonhomogeneous functional differential equation
scientific article; zbMATH DE number 1005701