Residual spectral algorithm for solving monotone equations on a Hilbert space

A residual algorithm for solving monotone equations on a real Hilbert space is presented. The monotone equations arise in various applications, such as differential equations, as well as economics, engineering, management science, probability theory and other applied sciences. The new scheme uses in a systematic way the residual as search direction combined with a suitable step-length and a nonmonotone line search globalization strategy. A convergence analysis is described. It is also presented the application of a new approach for solving the Lyapunov matricial equation. Numerical experiences are included to highlight the efficacy of the proposed algorithm for the solution of monotone equations.