Existence, Uniqueness, and Convergence of Solutions for General Monotone Inclusions in Hilbert Spaces

الملخص

Monotone inclusions play a central role in applied mathematics, particularly in optimization, variational inequalities, game theory, and physical modeling. This paper generalizes the Newton-like dynamics previously introduced by Abbas et al. for solving inclusions of the form in Hilbert spaces, where  and  are maximal monotone operators, and  is also assumed to be monotone and locally Lipschitz continuous. Unlike earlier works, such as  being a subdifferential or  being a gradient, thus making the method more general. We establish existence and uniqueness of strong solutions using the resolvent operator and demonstrate both weak and strong convergence under appropriate conditions. A Lyapunov function-based analysis is employed to rigorously study the asymptotic behavior. We also explore a discrete version of the proposed dynamics, offering practical algorithms for numerical computation. Applications in mechanical systems, control theory, electrical circuits, and machine learning are discussed, showcasing the method's flexibility and effectiveness. Finally, we highlight the potential for further extensions via adaptive regularization and inertial methods.