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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 ·. The main result:

If kL loc 1 ( + ) has an absolutely convergent Laplace transform k ^(λ):= 0 e -λt k(t)dt, which is nonzero for every Reλ>0, and (I/k ^(λ)-A) -1 exists for every λ>0, then there exist a linear subspace Z k X and a norm |·| k >· such that (Z k ,|·| 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)+ 0 t k(t-s)A k u(s)ds,tJ:=[0,T],fC(J,X),

admits a resolvent family of contractions on Z k , i.e. a strongly continuous family of bounded linear operators {R(t):t0} defined in Z k , which commutes with A k and satisfies the equation

R(t)x=x+ 0 t k(t-s)A k R(s)xds,t0,xD(A k ),

and inequality R(t)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:
45N05Abstract integral equations, integral equations in abstract spaces
47D06One-parameter semigroups and linear evolution equations
44A10Laplace transform
47G10Integral operators