Continuous method of second order with constant coefficients for monotone equations in Hilbert space

Abstract

Convergence of an implicit second-order iterative method with constant coefficients for nonlinear monotone equations in Hilbert space is investigated. For non-negative solutions of a second-order difference numerical inequality, a top-down estimate is established. This estimate is used to prove the convergence of the iterative method under study. The convergence of the iterative method is established under the assumption that the operator of the equation on a Hilbert space is monotone and satisfies the Lipschitz condition. Sufficient conditions for convergence of proposed method also include some relations connecting parameters that determine the specified properties of the operator in the equation to be solved and coefficients of the second-order difference equation that defines the method to be studied. The parametric support of the proposed method is confirmed by an example. The proposed second-order method with constant coefficients has a better upper estimate of the convergence rate compared to the same method with variable coefficients that was studied earlier.