An axiomatic theory of Volterra integral equations (Q1907740)

The authors are concerned with an axiomatic theory for Volterra integral equations of the form \[ x(t)= f(t)+ \int^t_0 k(t, s, x(s)) ds, \] with \(x\) taking values in a finite-dimensional vector space. The basic axioms are formulated in accordance with the basic properties of the set of solutions of the integral equation, and they relate to the possibility of extending (continuing) solutions to larger intervals, the uniqueness and the compactness of the set of solutions with graph in a compact set. Two more requirements (axioms) are formulated, one known as the fixed initial value property, and another concerning the possibility of constructing saturated solutions by means of continuable ones. Under adequate properties imposed on the kernel \(k(t, s, x)\), various properties are derived for the set of all solutions of the class of equations of the form indicated above. For instance, one shows that any solution whose graph belongs to a compact set, can be extended to a saturated solution. A Kneser type property is also investigated. That theory is a generalization of Filippov's theory for ordinary differential equations [cf. \textit{V. V. Fedorchuk} and \textit{V. V. Filippov}, ``General topology; basic constructions'' (1988; Zbl 0658.54001)].