Well-posedness for strongly damped abstract Cauchy problems of fractional order

Let X be a complex Banach space and B be a closed linear operator with domain D(B) ⊂ X, a,b,c,d ∈ ℝ and 0<β<α.We prove that the problem (Formula Presented) where gα(t)=tα-1/Γ(α) and hℝ+→ 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 {Rαβ(t)}t≥0 ∈Ⅎ(X).It is shown that such a solution family exists whenever the pair (αβ)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.