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.
