A two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations in Hilbert Space

Abstract

The current research work extends spectral gradient projection methods for solving systems of nonlinear monotone equations in Hilbert spaces. Opial lemma is utilized in extending the current space utilized (Euclidean space) to a more general infinite dimensional space (Hilbert space). first-order optimization methods have long been established as effective methods for solving large-scale unconstrained optimization problems. Over time, these methods have been extended to address the task of finding zeros of single-valued monotone maps within the framework of finite-dimensional Hilbert spaces. Notably, Awwal a.m. et al., 2020 [2] recently introduced a two-step spectral gradient projection method to find the zeros of a single-valued monotone map. Motivated by their work, the current research extend the work of Awwal A. M. et al., 2020 [2] from the setting of the finite dimensional Hilbert spaces  to a more general setting of infinite dimensional Hilbert spaces. We also establish weak convergence of the generated sequence iterates. This area has garnered significant attention due to its application in various fields, including image processing and control theory. It also contribute to the development of efficient algorithms for solving complex optimization problems.