Vol. 20, No. 1, 1967

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On integro-differential equations in Banach spaces

Jürg Thomas Marti

Vol. 20 (1967), No. 1, 99–108

The abstract integro-differential equation

du(t)∕dt = Au (t) + 0 B (t− s)u(s)ds+ f(t)

is studied, where u(t) and f(t) are functions of [0,) to a Banach space #,A and B(t) are linear operators on χ to itself, A is closed with domain 𝒟(A) and B(t) and f(t) are strongly continuous on [0,). Let A be the infinitesimal generator of a semi-group of linear operators of class (C0) and let u(t) ∈𝒟(A) on [0,), where u(O) is a prescribed initial value. It is then shown that there exists a unique strongly continuously differentiable solution of both the homogeneous and inhomogeneous problem. By the method of successive approximations, absolutely convergent series expansions of the solutions are obtained. Further it is proved that the solution operator of the -adjoint homogeneous problem equals the -adjoint of the solution operator of the homogeneous equation.

Mathematical Subject Classification
Primary: 45.40
Received: 2 August 1965
Published: 1 January 1967
Jürg Thomas Marti