Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations

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The present paper is concerned with linear integral equations of the form x(t) = ∫t−∞ K(t−s)x(s)ds, where K(s) is a matrix measurable in s, and satisfies some boundedness condition. We propose a dynamical-systems approach to studying the behavior of the equations by treating them as functional equations with infinite delay. As a result we obtain a decomposition of the phase space corresponding to a set of normal eigenvalues of the generator of solution semigroups associated with the equations. Moreover, we establish a representation formula (which is called "a variation-of-constants formula" in the phase space) for solutions of nonhomogeneous integral equations, together with a decomposed formula based on the decomposition of the phase space. Finally, we apply the formula to investigate the admissibility of function spaces with respect to these equations.