Abstract

Let X be a complex Banach space and B be a closed linear operator with domain D(B) ? X, a,b,c,d ? ? and 0+? X is given, has a unique solution for any initial condition on D (B) × X as long as the operator B generates an ad-hoc Laplace transformable and strongly continuous solution family {R??(t)}t?0 ??(X).It is shown that such a solution family exists whenever the pair (??)belongs to a subset of the set (1,2] × (0,1] and B is the generator of a cosine family or a C0-semigroup in In any case, it also depends on certain compatibility conditions on the real parameters a,b,c,d that must be satisfied. © The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh.