On totally global solvability of controlled second kind operator equation

We consider the nonlinear evolutionary operator equation of the second kind as follows phi = F [f[u]phi], phi is an element of W [0; T] subset of L-q [0; T]; X), with Volterra type operators F : L-p [0; tau]; Y) -> W [0; T], f[u]: W [0; T] -> L-p [0; T]; Y) of the general form, a control u is an element of D and arbitrary Banach spaces X, Y. For this equation we prove theorems on solution uniqueness and sufficient conditions for totally (with respect to set D) global solvability. Under natural hypotheses associated with pointwise in t is an element of [0; T] estimates the conclusion on univalent totally global solvability is made provided global solvability for a comparison system which is some system of functional integral equations (it could be replaced by a system of equations of analogous type, and in some cases, of ordinary differential equations) with respect to unknown functions [0; T] -> R. As an example we establish sufficient conditions of univalent totally global solvability for a controlled nonlinear nonstationary Navier-Stokes system.