Resolvents

Abstract

The concept of the resolvent which is central for the theory of linear Volterra equations is introduced and discussed. It is applied to the inhomogeneous equation to derive various variation of parameters formulas. The main tools for proving existence theorems for the resolvent are described in detail; these methods are the operational calculus in Hilbert spaces, perturbation arguments, and the Laplace-transform method. The generation theorem, the analog of the Hille-Yosida theorem of semigroup theory for Volterra equations, proved in Section 1.5, is of fundamental importance in later chapters. The theory is completed with several counterexamples, and with a discussion of the integral resolvent.