Continuous method of second order with constant coefficients for equations of monotone type

Convergence of the second order continuous method with constant coefficients for nonlinear equations is investigated. The cases of a monotone operator equation in Hilbert space and of an accretive operator equation in reflexive Banach space which is strictly convex together with its conjugate, are considered separately. In each case, sufficient conditions for the convergence with respect to the norm of the space specified by the method are obtained. In the accretive case, sufficient conditions for the continuous method convergence include not only the requirements on the operator equation and on the coefficients of the differential equation defining the method, but also on the geometry of space where the equation is solved. Examples of Banach spaces with the desired geometric properties are shown.