On the dominance of roots of characteristic equations for neutral functional differential equations

We present a sufficient condition for a zero of a function that arises typically as the characteristic equation of a linear functional differential equations of neutral type, to be simple and dominant. This knowledge is useful in order to derive the asymptotic behaviour of solutions of such equations. A simple characteristic equation, arisen from the study of delay equations with small delay, is analyzed in greater detail. In the study of the solutions for linear autonomous functional differential equations (FDE), one can derive important information of its asymptotic properties from the spectral properties of the infinitesimal generator of the solution semigroup. Such spectrum is formed by point spectrum only, as the zeros of an entire function, called the characteristic function. Verduyn Lunel [12] showed that the solution of linear autonomous FDE of neutral type can be written down as a series expansion of the solution operator restricted to the generalized eigenspaces, images of the so called spectral projection into such invariant eigenspaces. In Frasson & Verduyn Lunel [5], the large time behaviour of solutions was derived, based on the knowledge of a dominant eigenvalue, that is, an eigenvalue so that its real part is sufficiently larger, uniformly, than the real part of all other eigenvalues. Analogous results can ∗Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo– Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil. Supported by FAPESP 2006/50943-5. E-mail: [email protected].