Approximation of solutions to irregular equations and attractors of nonlinear dynamical systems in Hilbert spaces

We construct and study a class of methods for approximation of solutions to nonlinear equations in a Hilbert space with an approximately given smooth operator in the absence of regularity of its derivative. The sought-for solution is approximated by the trajectory of the nonlinear dynamical system related to the equation under consideration. The construction of this system is specified by linearization of the original equation with the use of the Gauss–Newton scheme and by different ways of regularization of this scheme. Under several assumptions we establish the existence of a ball which attracts the corresponding domain of the phase space and the existence of a minimal attractor of the system in a small neighborhood of the sought-for solution.