An abstract semilinear Volterra integrodifferential equation

Webb, G. F.

doi:10.1090/S0002-9939-1978-0467214-4

An abstract semilinear Volterra integrodifferential equation
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Proc. Amer. Math. Soc. 69 (1978), 255-260 Request permission

Abstract:

The abstract semilinear Volterra integrodifferential equation \[ u’(t) = Au(t) + \int _0^t {g(t - s,u(s))ds + f(t),\quad t \geqslant 0,u(0) = x \in X,} \] 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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