Exponential dichotomy and stability of neutral type equations

AbstractA linear functional differential equation of neutral type with unbounded delay Lx(dds)Dx+Bx=0, where D and B are linear bounded retarded operators with exponentially fading memory, is considered. It is shown that if operator L is interpreted as operator from the space C into the special space C−1 of distributions, then its invertibility is equivalent to the presence of exponential dichotomy of the solutions of this equation. As applications, we prove the theorems on stability and instability in the first approximation for neutral functional differential equations of a general form