Local analyticity in weighted 𝐿¹-spaces and applications to stability problems for Volterra equations

Jordan, G. S.; Staffans, Olof J.; Wheeler, Robert L.

doi:10.1090/S0002-9947-1982-0675078-4

Local analyticity in weighted $L^{1}$-spaces and applications to stability problems for Volterra equations
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by G. S. Jordan, Olof J. Staffans and Robert L. Wheeler PDF

Trans. Amer. Math. Soc. 274 (1982), 749-782 Request permission

Abstract:

We study the qualitative properties of the solutions of linear convolution equations such as $x’ + x ^{\ast } \mu = f, x + a \ast x = f$ and $x \ast \mu = f$. We are especially concerned with finding conditions which ensure that these equations have resolvents which belong to, or are determined up to a term belonging to, certain weighted ${L^1}$-spaces. Our results are obtained as consequences of more general Banach algebra results on functions that are locally analytic with respect to the elements of a weighted ${L^1}$-space. In particular, we derive a proposition of Wiener-Lévy type for weighted ${L^1}$-spaces which underlies all subsequent results. Our method applies equally well to equations more general than those mentioned above. We unify and sharpen the results of several recent papers on the asymptotic behavior of Volterra convolution equations of the types mentioned above, and indicate how many of them can be extended to the Fredholm case. In addition, we give necessary and sufficient conditions on the perturbation term $f$ for the existence of bounded or integrable solutions $x$ in some critical cases when the corresponding limit equations have nontrivial solutions.

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