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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An abstract semilinear Volterra integrodifferential equation


Author: G. F. Webb
Journal: Proc. Amer. Math. Soc. 69 (1978), 255-260
MSC: Primary 45K05
DOI: https://doi.org/10.1090/S0002-9939-1978-0467214-4
MathSciNet review: 0467214
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Abstract: The abstract semilinear Volterra integrodifferential equation

$\displaystyle u'(t) = Au(t) + \int_0^t {g(t - s,u(s))ds + f(t),\quad t \geqslant 0,u(0) = x \in X,} $

is investigated, where A is the infinitesimal generator of a semigroup of linear operators in a Banach space X and g is nonlinear and unbounded in its second place. Some results are proved concerning local existence, global existence, continuous dependence upon initial values, and asymptotic stability. The method used treats the equation in the domain of A with the graph norm employing results from linear semigroup theory concerning abstract inhomogeneous linear differential equations.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0467214-4
Keywords: Abstract Volterra integrodifferential equation, semigroup of bounded linear operators, infinitesimal generator, existence, uniqueness, asymptotic behavior
Article copyright: © Copyright 1978 American Mathematical Society