Counterexamples for abstract linear Volterra equations
The main object of this paper is the study of the abstract Volterra equation \(u(t)=\int^ t_ 0a(t-s)Au(s)ds+f(t)\) where \(A\) is an unbounded linear opertor in a Banach space \(X\), \(f\) is an \(X\)-valued function and \(a\) a scalar valued function through the existence and properties of the resolvent operator \(t\mapsto S(t)\in{\mathcal L}(x)\) satisfying \(S(t)Ax=AS(t)x\), \(S(t)x=x+A\int^ t_ 0a(t-s)S(s)xds\). More precisely the authors show that (i) there exists an exponentially bounded \(a\) such that a resolvent operator exists but is not exponentially bounded (ii) there exists a negative self-adjoint unbounded operator \(A\) in a Hilbert space \(X\) and a kernel \(a\) such that the resolvent operator is analytic and \(1/|\hat a|\) grows faster than polynomially \((\hat a\) is the Laplace transform of \(a)\). Finally it is shown that there exists an analytic function \(F\) defined on the right half plane and satisfying \(| F(s)|\leq M/| s|\) on the right half plane, such that \(F\) is not the Laplace transform of an \(L^ \infty\) function on the positive half axis.
scientific article; zbMATH DE number 205193