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On a nonlinear Volterra integrodifferential equation in Banach spaces. (English) Zbl 1108.45009
The paper deals with the equation: \[ x'(t) = A x(t) + \int_0^t\{ a(t,\, s) f(s,\, x(s)) + g(t,s,x(s))\} ds + f_0(t),\; t\geq 0, \] in a Banach space with graph norm. Here, \(A\) is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators and \(a,\,f,\, g\) and \(f_0\) are continuous functions with some additional properties. Using results from the linear semigroup theory, the authors study here the local existence and uniqueness for the associated initial value problem, the global existence, as well as the asymptotic stability and the continuous dependence of the solution upon initial value. The paper ends with an illustrative example of the main results.

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
45M10 Stability theory for integral equations
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