On a generalization of the Nicoletti problem for functional differential equations with advancing argument (Q1256622)

Let \(B\) be a Banach space and \(\Phi\) the linear space of all continuous functions \(\varphi: [0,+\infty)\to B\). We consider the functional differential equation \[ \varphi'(t)=F(t,\varphi)\quad\text{for}\;t\in [0,+\infty)\tag{1} \] together with a generalized Nicoletti condition (2) \(H(\varphi)=\eta\). The function \(F: [0,+\infty)\times\Phi\to B\), \(H:\Phi \to B\) is a linear operator and \(\eta\in B\). Assume that (1) is an equation with unbounded advancement of argument, i.e. that the value of \(F(t,\varphi)\) may depend on the values of \(\varphi(t)\) for \(t\in[0,+\infty)\). Under suitable assumptions we show that when the solution satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).