Hyperbolic integro-differential equations of convolution type

Making use of the theory of Wiener-Hopf operators in the scale of abstract Krein spaces, we prove existence and uniqueness of unbounded solutions for the linear hyperbolic integrodifferential equation (P_o). We extend herewith results obtained in 8] for hyperbolic evolution equations, where the convolution integral was absent. The method utilizes Dunford's functional calculus and permits thus a constructive existence proof for solutions exhibiting an exponential growth rate when time increases. Our approach bases upon the fundamental hypothesis that the spectrum of the time-independent mapping -A shows a parabolic condensation along the negative real axis. This condition completely determines the admissible geometry of the spectral set of the convolution integral operator, and a fortiori the magnitude of the exponential growth rate. The theory works in arbitrary reflexive Banach spaces.