Convergence and Parameter Selection for Residual Iterations of Strongly Monotone Nonlinear Operators in Hilbert Spaces

Abstract

Structural assumptions on nonlinear operators are fundamental in determining the existence, uniqueness, stability, and numerical approximation of solutions to nonlinear equations. While monotonicity, coercivity, Lipschitz continuity, compactness, and fixed-point properties are usually studied separately, their direct influence on computational convergence is equally important. In this paper, we develop a structure-guided framework for the approximation of solutions of nonlinear operator equations of the form A(x) = f in real Hilbert spaces. Particular attention is given to strongly monotone and Lipschitz continuous operators. A relaxed residual iteration is analyzed and an explicit contraction factor is derived in terms of the strong monotonicity constant, Lipschitz constant, and relaxation parameter. A two-stage residual correction is subsequently considered, and a convergence estimate based on the product of the individual contraction factors is established. The analysis demonstrates how structural information about an operator can be translated directly into quantitative convergence predictions.

The theoretical results are complemented by numerical experiments for the nonlinear equation x + x3 =

1. The corresponding nonlinear operator is strongly monotone and its local Lipschitz behavior can be

calculated explicitly. Several relaxation parameters are examined, and single-stage, relaxed, and two-stage procedures are compared using approximation errors, residuals, operator evaluations, and convergence ratios. Numerical tables and graphical comparisons show that the choice of relaxation parameter has a substantial influence on computational efficiency and that a two-stage correction can considerably reduce the residual per outer iteration. A second multidimensional example illustrates the extension of the approach to nonlinear operators on Euclidean Hilbert spaces. The results emphasize that structural classification is not only useful for theoretical solvability but can also provide practical information for the design and assessment of numerical algorithms.