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On Volterra equations associated with a linear operator. (English) Zbl 0781.45013

Let \(A\) be a linear operator defined in a Banach space \(X\) with norm \(\|\cdot\|\). The main result:
If \(k\in L^ 1_{\text{loc}}(\mathbb{R}_ +)\) has an absolutely convergent Laplace transform \(\hat k(\lambda):=\int^ \infty_ 0e^{-\lambda t}k(t)dt\), which is nonzero for every \(\text{Re} \lambda>0\), and \((I/\hat k(\lambda)-A)^{-1}\) exists for every \(\lambda>0\), then there exist a linear subspace \(Z_ k\subset X\) and a norm \(|\cdot|_ k>\|\cdot\|\) such that \((Z_ k,|\cdot|_ k)\) is a Banach space, the restriction \(A_ k\) of \(A\) on \(Z_ k\) is a closed linear operator with densely defined domain \(D(A_ k)\) and the Volterra equation of convolution type \[ u(t)=f(t)+\int^ t_ 0k(t-s)A_ ku(s)ds,\;t\in J:=[0,T],\;f\in C(J,X), \] admits a resolvent family of contractions on \(Z_ k\), i.e. a strongly continuous family of bounded linear operators \(\{R(t):t\geq 0\}\) defined in \(Z_ k\), which commutes with \(A_ k\) and satisfies the equation \[ R(t)x=x+\int^ t_ 0k(t- s)A_ kR(s)x ds,\;t\geq 0,\;x\in D(A_ k), \] and inequality \(\| R(t)\|\leq 1\). This Hille-Yosida space \(Z_ k\) is maximal-unique in a certain sense. If \(k\) is, in addition, a positive function, then \(A_ k\) generates a strongly continuous semigroup of contractions on \(Z_ k\).

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
44A10 Laplace transform
47G10 Integral operators
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