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An abstract functional differential equation and a related nonlinear Volterra equation

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Abstract

We study the existence, uniqueness, regularity and dependence upon data of solutions of the abstract functional differential equation

$$\frac{{du}}{{dt}} + Au \ni G(u) (0 \leqq t \leqq T), u(0) = x,$$
(1)

, 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

$$u(t) + b * Au(t) \ni F(t) (0 \leqq t \leqq T),$$
(2)

, 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)=∫ 10 (t−s)g(s)ds, andf∈W 1,1(0,T;X).

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Authors and Affiliations

  1. Department of Mathematics and Mathematics Research Center, University of Wisconsin, 53706, Madison, Wisconsin, USA

    M. G. Crandall & J. A. Nohel

Authors
  1. M. G. Crandall
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  2. J. A. Nohel
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Additional information

Sponsored by The United States Army under Contract No. DAAG29-75-C-0024.

Sponsored by The United States Army under Grant No. DAAG29-77-G-0004 and the National Science Foundation Grant No. MCS75-21868.

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Crandall, M.G., Nohel, J.A. An abstract functional differential equation and a related nonlinear Volterra equation. Israel J. Math. 29, 313–328 (1978). https://doi.org/10.1007/BF02761170

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  • DOI: https://doi.org/10.1007/BF02761170

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