On admissibility of certain pairs of spaces for abstract integral operators and Volterra equations (Q1311154)

The author studies Volterra equations of the form \[ x(t) = f(t) + \int_{M(t)} Q(t,s) x(s) d \mu_ s, \tag{1} \] where \(f\) and \(Q\) are given continuous functions defined on a nonempty connected locally compact metric space \(\Omega\) and \(\mu\) is a Borel measure on \(\Omega\). The sets \(M(t) \subset \Omega\) satisfy a number of conditions that in some sense make the equation into one of Volterra type. First the author shows that one can generalize to the present setting a result of \textit{V. F. Pulyaev} and \textit{Z. B. Tsalyuk} [Differ. Uravn. 19, No. 4, 684-692 (1983; Zbl 0526.45014)] saying that if equation (1) is stable, then for every subspace \(X \subset BC (\Omega)\) one can find a solution \(x \in X\) when \(f \in X\) if and only if \(\widetilde Q(X) \subset X\) where \((\widetilde Qx) (t) = \int_{M(t)} Q(t,s) x(s)\) \(d \mu_ s\). Then the author studies the problem what can be said about the operator \(\widetilde Q\) and gives for examples necessary and sufficient conditions for \(\widetilde Q(C_ 0 (\Omega)) \subset X\) to be valid. Finally, an existence result, assuming Lipschitz continuity, for a nonlinear version of (1) is given.