A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

On integro‐differential inclusions with operator‐valued kernels

“…[31,32,33]) and the well-posedness and causality was shown under suitable assumptions on the material law M (∂ −1 0 ). As it was already mentioned in [18], the operator M ∂ −1 0 can be a convolution with an operator-valued function and we will point out, which kind of linear material laws yield integro-differential equations. We emphasize that problems of the form (1) and ( 2) are covered by (4), while the converse is false in general, unless additional compatibility constraints on the operators A and M (∂ −1 0 ) are imposed.…”

“…This shows, that for ν > 0 the operator ∂ −1 0,ν is causal, while for ν < 0 we get the anti-causal operator given by ∂ −1 0,ν u = − ∞ • u(s) ds (see[18]). …”

“…Throughout let ν ∈ R. As in [24,22,18] we begin to introduce the exponentially weighted L 2 -space 1 .…”

“…[15,11]). However, following the idea of [18] we can treat this kind of equations as a problem of the form (10) with a modified right-hand side.…”

“…where L : D(L) ⊆ H 0 → H 0 is a selfadjoint strictly positive definite operator and B, C ∈ L 1,µ (R ≥0 ; L(H 0 )). It is remarkable that no further assumptions on the kernel C are imposed, although (19) seems to be of the form (18), where C would have to verify the hypotheses (i) and (iii). The reason for that is that ( 19) can be rewritten as a parabolic system, which is not of the classical form (17).…”