A Perron type theorem for functional differential equations (Q819688)

The following nonlinear retarded functional-differential equation \[ x'(t)=Lx_t+f(t,x_t)\,, \tag{1} \] can be considered as a perturbation of the linear retarded functional-differential equation \[ x'(t)=Lx_t\,. \tag{2} \] Here, \(r>0\), \(C=C([-r,0],{\mathbb C}^n)\) is the Banach space of continuous functions from \([-r,0]\) into \({\mathbb C^n}\) and \(f:[\sigma_0,\infty]\times C\rightarrow {\mathbb C}^n\) is a continuous function. As usual \(x_t\in C\) stands for the function \(x_t(\theta)=x(t+\theta)\), \(-r\leq\theta\leq 0\). The following theorem is the main result of the paper. It generalizes a Perron-type theorem from ordinary differential equations to retarded functional-differential equations: Let \(x\) be a solution of (1), such that \(| f(t,x_t)| \leq\gamma(t)\,| x_t| \), \(t\geq\sigma_0\), holds with a continuous function \(\gamma:[\sigma_0,\infty]\to[0,\infty)\) satisfying \(\int_t^{t+1} \gamma(s)\,ds\rightarrow\infty\) as \(t\to\infty\). Then, either (i) the limit \(\mu=\mu(x)=\lim_{t\rightarrow\infty} (\ln | x_t| /t)\) exists and is equal to the real part of one of the eigenvalues of (2), or (ii) for each \(b\in \mathbb R\), we have \(\lim_{t\rightarrow\infty} e^{bt} x(t)=0\).