Regularized solutions for abstract Volterra equations. (English) Zbl 0952.45005

Lizama, Carlos

J. Math. Anal. Appl. 243, No. 2, 278-292 (2000).

The author shows the existence, the uniqueness, and some qualitative properties of solutions for the abstract Volterra equation \[ u(t) = f(t) + \int_{0}^{t}a(t-s)Au(s) ds,\quad t\in [0,T] \] on a complex Banach space \(X\) by means of an extended notion of resolvent, where \(A\) is a closed linear unbounded operator with domain \(D(A)\), \(a \not=0\) is a scalar kernel, and \(f \in C([0,T],X)\).

Reviewer: Hu Chuangan (Tianjin)

MSC:

45N05	Abstract integral equations, integral equations in abstract spaces
45D05	Volterra integral equations
47D03	Groups and semigroups of linear operators

Keywords:

abstract linear Volterra equations; convolution; \(k\)-regularized resolvents; semigroups of linear operators; Banach space

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