On solvability of a pursuit game with nonlinear dynamics in the Hilbert space

We consider a pursuit differential game in the Hilbert space. The game dynamics is described by two semilinear evolutionary equations with an optionally bounded operator in the Hilbert space; and each of these equations is controlled by its own player. The controls appear linearly in right hand sides of the equations and are restricted by conditions of the norm boundedness with given constants. We establish sufficient conditions for solvability of the determined pursuit game in both linear and nonlinear cases. Here we use the Minty-Browder's theorem and also a chain technology of successive continuation of the solution to a controlled system to intermediate states. As examples of reduction to the abstract operator equation under study we consider Oskolkov's system of equations and a semilinear wave equation.