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

45N05Abstract integral equations, integral equations in abstract spaces
47D06One-parameter semigroups and linear evolution equations
44A10Laplace transform
47G10Integral operators
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