Preservation of the global solvability of a first-kind operator equation with controlled additional nonlinearity

For the Cauchy problem associated with a first-kind evolutionary operator equation in a Banach space supplemented by a controlled term that depends nonlinearly on the phase variable, we obtain conditions for the preservation of unique global solvability under small variations of control (in other words, conditions for the stability of the existence of global solutions) and also a uniform estimate of the increment of solutions with respect to the norm of the space. As an example, we consider the initial-boundary-value problem for the Oskolkov system.