Accelerated double step-length method for solving monotone nonlinear equations with convex-constraint and application

The hybrid procedure is an efficient technique for improving the global and numerical performance of iterative algorithms for large-scale monotone nonlinear problems. This is achieved by integrating two or more methods into a unified framework. In this study, we present a hybrid of the double step-length method and the Picard-Mann iterative technique for solving monotone nonlinear equations with convex constraints. By combining the Picard-Mann approach with a newly proposed scheme, we obtain an iterative method that reduces computational cost and achieves faster convergence. The acceleration parameter is determined by evaluating the difference between the Broyden update and its approximation using the Frobenius norm. The global convergence of the proposed method is proved, and a Q-linear convergence rate is also established. Numerical experiments demonstrate that the proposed approach is computationally efficient for solving large-scale nonlinear equations compared with existing methods. Finally, the method is applied to signal processing and image restoration problems, highlighting its practical relevance.