Maximal Monotone Research Articles

The Proximal Point Method with Remotest Set Control for Maximal Monotone Operators and Quasi-Nonexpansive Mappings

In the present paper, we use the proximal point method with remotest set control for find an approximate common zero of a finite collection of maximal monotone maps in a real Hilbert space under the presence of computational errors. We prove that the inexact proximal point method generates an approximate solution if these errors are summable. Also, we show that if the computational errors are small enough, then the inexact proximal point method generates approximate solutions

Distributed Proximal-Correction Algorithm for the Sum of Maximal Monotone Operators

Distributed Proximal-Correction Algorithm for the Sum of Maximal Monotone Operators

Spectral Analysis of Nonlinear Operators: Theory and Applications to Neural Networks and Optimization

This paper presents a nonlinear spectral framework for analyzing monotone and nonexpansive operators in Banach and Hilbert spaces. We construct a nonlinear spectral resolution for maximal monotone operators using Yosida approximations and Fitzpatrick functions, leading to a family of nonlinear projections and an associated spectral measure. For nonexpansive mappings, we establish an iterative spectral approximation based on Krasnoselskii iterations, with proven convergence and recovery of nonlinear eigenvectors. We further extend this framework to ReLU-based neural networks, analyzing spectral bounds, depth-dependent scaling, and gradient alignment. These results bridge nonlinear operator theory and neural architectures, offering new tools for theoretical analysis and applications in optimization, physics, and machine learning.

A Well-Posed Evolutionary Inclusion in Mechanics

We consider an evolutionary inclusion associated with a time-dependent convex in an abstract Hilbert space. We recall a unique solvability result obtained based on arguments of nonlinear equations with maximal monotone operators combined with a penalty method. Then, we state and prove two well-posedness results. Next, we provide three examples of such inclusions that arise in mechanics. The first one concerns an elastic–perfectly plastic constitutive law, while the last two examples are mathematical models that describe the equilibrium of an elastic body and an elastic–perfectly plastic body, respectively, in frictional contact with an obstacle. The contact is bilateral and the friction is modeled with the Tresca friction law. We use our abstract results in the study of these examples to provide the convergence of the solution with respect to the data.

A Hybrid Inertial S - iteration Algorithm for Quasi -φ- Asymptotically Nonexpansive mappings, Maximal Monotone Operators and Generalized Mixed Equilibrium Problems

In this paper, we propose a hybrid inertial S-iteration algorithm for two quasi -φ- asymptotically nonexpansive mappings, maximal monotone operator and generalized mixed equilibrium problems in a real Banach space. We also, established a strong convergence theorem of the propose iterative scheme. The results present in this paper extend and improve some recent results in the literature.

Stabilization in finite time of a class of evolution equations under multiplicative or additive controls

ABSTRACT Based on the idea of Lyapunov function, we investigate the finite-time stability of certain classes of abstract bounded bilinear systems under feedback control laws. We begin by designing a stabilising feedback control to ensure system stability. Following this, we analyze the well-posedness of the closed-loop system using the theory of maximal monotone operators. We then proceed to investigate finite-time stability, demonstrating that the system reaches equilibrium within a finite settling time. Our approach utilises a space decomposition method in conjunction with Lyapunov functions. These results are subsequently applied to prove the finite-time stabilisation of linear systems. Finally, we present applications to heat and transport equations.

Local Maximal Monotonicity in Variational Analysis and Optimization

The paper is devoted to a systematic study and characterizations of notions of local maximal monotonicity and their strong counterparts for set-valued operators that appear in variational analysis, optimization, and their applications. We obtain novel resolvent characterizations of these notions together with efficient conditions for their preservation under summation in broad infinite-dimensional settings. Further characterizations of these notions are derived by using generalized differentiation of variational analysis in the framework of Hilbert spaces. Funding: This research was supported by the U.S. National Science Foundation [Grants DMS-1808978 and DMS-2204519], the Australian Research Council Discovery Project [Grant DP-190100555], and Project 111 of China [Grant D21024].

Non-potential systems with relativistic operators and maximal monotone boundary conditions

We are concerned with solvability of a non-potential system involving two relativistic operators, subject to boundary conditions expressed in terms of maximal monotone operators. The approach makes use of a fixed point formulation and relies on a priori estimates and convergent to zero matrices.

Zeros of the Sum of a Finite Family of Maximal Monotone Operators

Zeros of the Sum of a Finite Family of Maximal Monotone Operators

Perturbed evolutionary differential hemivariational inequalities involving time-dependent maximal monotone operators

Perturbed evolutionary differential hemivariational inequalities involving time-dependent maximal monotone operators

Diffusive convective elliptic problem in variable exponent space and measure data

Abstract In this article, we study a class of convective diffusive elliptic problem with Dirichlet boundary condition and measure data in variable exponent spaces. We begin by introducing an approximate problem via a truncation approach and Yosida’s regularization. Then, we apply the technique of maximal monotone operators in Banach spaces to obtain a sequence of approximate solutions. Finally, we pass to the limit and prove that this sequence of solutions converges to at least one weak or entropy solution of the original problem. Furthermore, under some additional assumptions on the convective diffusive term, we prove the uniqueness of the entropy solution.

Convection-diffusion problems involving measure data and p(.)-anisotropic operator in variable exponent space

In this paper, we investigate a nonlinear diffusion-convection problem with measure data, involving a general anisotropic operator with variable exponent and a maximal monotone graph. Utilizing Yosida’s regularization, we apply an approximation technique to formulate a regularized problem. We then employ the theory of maximal monotone operators in Banach spaces to demonstrate the existence of at least one solution for the approximate problem. Subsequently, we show that the sequence of solutions to the approximate problem converges, in the limit, to a renormalized and/or entropic solution of the original problem. Finally, we establish the uniqueness of the solution under certain additional assumptions regarding the convection term.

Nonlinear elliptic problem involving natural growth term, L1-data and variable exponent

Using approximation techniques and the theory of maximal monotone operators in Banach spaces, we prove the existence of at least one solution of a wide class of multivalued nonlinear elliptic problems involving natural growth terms and general p(.)-Leray-Lions operator.

Iterative algorithm for zeros of maximal monotone mappings in uniformly smooth and uniformly convex Banach spaces

Let E be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Let $A : E \rightarrow E^*$ be a bounded maximal monotone mappping such that $A^{-1}(0) \ne 0$. Define the algorithm $\{x_n\}$ as follows: for given $x_1 \in E$,\;\; $x_{n+1}= J^{-1}\big(Jx_n-\lambda_nAx_n-\lambda_n\theta_n (Jx_n-Jx_1)\big)$ where $J$ is the normalized duality mapping from $E$ into $E^*$ and $ \{\lambda_n \}$ and $ \theta_n $ are positive real numbers in $(0, 1)$ satisfying suitable conditions. It is proved that $x_n$ converges strongly to some $x^*\in A^{-1}(0)$. The results extend our recent works \cite{mendy-et-al} to larger class of Banach spaces with numerical simulations.

An inertial projective splitting method for the sum of two maximal monotone operators

An inertial projective splitting method for the sum of two maximal monotone operators

Convergence of an inertial reflected–forward–backward splitting algorithm for solving monotone inclusion problems with application to image recovery

Convergence of an inertial reflected–forward–backward splitting algorithm for solving monotone inclusion problems with application to image recovery

Primal-Dual Extrapolation Methods for Monotone Inclusions Under Local Lipschitz Continuity

In this paper, we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone, whereas the other is locally Lipschitz continuous. We propose primal-dual (PD) extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of [Formula: see text] and [Formula: see text], measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an ε-residual solution of strongly and nonstrongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity [Formula: see text]. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an ε-KKT or ε-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under local Lipschitz continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods. Funding: This work was partially supported by the National Science Foundation [Grant IIS-2211491], the Office of Naval Research [Grant N00014-24-1-2702], and the Air Force Office of Scientific Research [Grant FA9550-24-1-0343].

A modified inertial shrinking projection algorithm with adaptive step size for solving split generalized equilibrium, monotone inclusion and fixed point problems

Abstract. In this paper, we study the common solution problem of split generalized equilibrium problem, monotone inclusion problem and common fixed point problem for a countable family of strict pseudo-contractive multivalued mappings. We propose a modified shrinking projection algorithm of inertial form with self-adaptive step sizes for finding a common solution of the aforementioned problem. The self-adaptive step size eliminates the difficulty of computing the operator norm while the inertial term accelerates the rate of convergence of the proposed algorithm. Moreover, unlike several of the existing results in the literature, the monotone inclusion problem considered is a more general problem involving the sum of Lipschitz continuous monotone operators and maximal monotone opera- tors, and knowledge of the Lipschitz constant is not required to implement our algorithm. Under some mild conditions, we establish strong convergence result for the proposed method. Finally, we present some applications and numerical experiments to illustrate the usefulness and applicability of our algorithm as well as comparing it with some related methods. Our results improve and extend corresponding results in the literature. Mathematics Subject Classification (2010): 65K15, 47J25, 65J15. Keywords: Split generalized equilibrium problem, monotone inclusion problem, inertial method, fixed point problem, strict pseudo-contractions, multivalued mappings.

On the local existence and blow-up solutions to a quasi-linear bi-hyperbolic equation with dynamic boundary conditions

On the local existence and blow-up solutions to a quasi-linear bi-hyperbolic equation with dynamic boundary conditions

Outer reflected forward–backward splitting algorithm with inertial extrapolation step

ABSTRACT This paper studies an outer reflected forward–backward splitting algorithm with an inertial step to find a zero of the sum of three monotone operators composing the maximal monotone operator, Lipschitz monotone operator, and a cocoercive operator in real Hilbert spaces. One of the interesting features of the proposed method is that both the Lipschitz monotone operator and the cocoercive operator are computed explicitly each with one evaluation per iteration. We obtain weak and strong convergence results under some easy-to-verify assumptions. We also obtain a non-asymptotic O ( 1 / n ) convergence rate of our proposed algorithm in a non-ergodic sense. We finally give some numerical illustrations arising from compressed sensing and image processing and show that our proposed method is effective and competitive with other related methods in the literature.