Functional-operator Volterra equations and existence-stability of global solutions of boundary value problems (Q1176886)

The author considers the equation \(z(t)=f(t,(Az)(t))\), \(t\in\Pi\subset\mathbb{R}^ n\), in Lebesgue space of \(m\)-vector functions \(L_{\infty,m}\). Here \(A: L_{\infty,m}\to L_{r,\ell}\) is a linear operator, \(f: \Pi\times\mathbb{R}^ \ell\to\mathbb{R}^ m\). Sufficient conditions for the stability of the existence of global solutions are obtained. These conditions contain some requirements concerning the smoothness of \(f\) in its second argument and the restriction of its growth. Moreover, the operator \(A\) must be contained in the special class of Volterra type operators introduced by the author. As an application a theorem on the stability of the existence of global solutions to control problems is proved.