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

MR3052749 Hideaki MATSUNAGA, Satoru MURAKAMI and NGUYEN Van Minh Hideaki MATSUNAGA, Satoru MURAKAMI and NGUYEN Van Minh Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations Funkcialaj Ekvacioj. Serio Internacia 55 2012 479--520 http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-3/55_479.pdf http://www.ams.org/mathscinet-getitem?mr=MR3052749 The present paper is concerned with linear integral equations of the form $x(t)=\int^t_{-\infty}K(t-s)x(s)ds$, 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. Integral equations, Phase space, Solution semigroups, Spectrum, A variation-of-constants formula, Admissibility. Primary: 45D05; Secondary: 45J05, 45M10. 55-479 2012 Decomposition and Variation-of-Constants Formula in the Phase Space for Integral Equations Hideaki MATSUNAGA, Satoru MURAKAMI and NGUYEN Van Minh Hideaki MATSUNAGA, Satoru MURAKAMI and NGUYEN Van Minh

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