An abstract functional differential equation and a related nonlinear Volterra equation |
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Authors: | M G Crandall J A Nohel |
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Institution: | 1. Department of Mathematics and Mathematics Research Center, University of Wisconsin, 53706, Madison, Wisconsin, USA
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Abstract: | We study the existence, uniqueness, regularity and dependence upon data of solutions of the abstract functional differential equation 1 $$\frac{{du}}{{dt}} + Au \ni G(u) (0 \leqq t \leqq T), u(0) = x,$$ , whereT>0 is arbitrary,A is a givenm-accretive operator in a real Banach spaceX, and \(G:C(0,T]; \overline {D(A)} ) \to L^1 (0, T; X)\) is a given mapping. This study provides simple proofs of generalizations of results by several authors concerning the nonlinear Volterra equation 2 $$u(t) + b * Au(t) \ni F(t) (0 \leqq t \leqq T),$$ , for the case in which X is a real Hilbert space. In (2) the kernelb is real, absolutely continuous on 0,T],b*g(t)=∫ 0 1 (t?s)g(s)ds, andf∈W 1,1(0,T;X). |
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