Maximal Monotone Research Articles

A class of incrementally scattering-passive nonlinear systems

We investigate a special class of nonlinear infinite dimensional systems. These are obtained by subtracting a nonlinear maximal monotone (possibly multi-valued) operator ℳ from the semigroup generator of a scattering passive linear system. While the linear system may have unbounded linear damping (for instance, boundary damping) which is only densely defined, the nonlinear damping operator ℳ is assumed to be defined on the whole state space. We show that this new class of nonlinear infinite dimensional systems is well-posed and incrementally scattering passive. Our approach uses the theory of maximal monotone operators and the Crandall–Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax–Phillips type nonlinear semigroup that represents the whole system.

Spiking Control Systems

Spikes and rhythms organize control and communication in the animal world, in contrast to the bits and clocks of digital technology. As continuous-time signals that can be counted, spikes have a mixed nature. This article reviews ongoing efforts to develop a control theory of spiking systems. The central thesis is that the mixed nature of spiking results from a mixed feedback principle, and a control theory of mixed feedback can be grounded in the operator theoretic concept of maximal monotonicity. As a nonlinear generalization of passivity, maximal monotonicity acknowledges at once the physics of electrical circuits, the algorithmic tractability of convex optimization, and the feedback control theory of incremental passivity. We discuss the relevance of a theory of spiking control systems in the emerging age of event-based technology.

Renormalized solutions for a p(·)-Laplacian equation with Neumann nonhomogeneous boundary condition involving diffuse measure data and variable exponent

Abstract In this paper we prove the existence of at least one renormalized solution for the p(x)-Laplacian equation associated with a maximal monotone operator and Radon measure data. The functional setting involves Sobolev spaces with variable exponent W 1, p (·)(Ω).

An Inertial Algorithm of Generalized f -Projection for Maximal Monotone Operators and Generalized Mixed Equilibrium Problems in Banach Spaces

In this paper, we study a modified hybrid inertial algorithm of generalized f-projection for approximating maximal monotone operators and solutions of generalized mixed equilibrium problems in Banach spaces. Our results generalize and improve many recent announced results in the literature.

Maximal monotone operators with non-maximal graphical limit

We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator B in terms of existence and maximal monotonicity of the proto-derivative of B.

A study of multivalued variational inequalities via horizon maps and graphical convergence

Multivalued variational inequalities is an active research area that encompasses many important models in mathematics, for example in optimization, fixed point, saddle point and complementary problems. The primary objective of this work is to apply horizon maps and graphical convergence to obtain existence and stability results for these problems. We give an exhaustive list of new results for these problems in terms of horizon maps and graphical convergence. Finally, we apply these results to the pseudomonotone and maximal monotone cases, establishing new formulations of known results in terms of these tools.

Local Continuity of Weak Solutions to the Stefan Problem Involving the Singular $p$-Laplacian

We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partial_t \beta(u)-\Delta_p u\ni 0\quad\text{ for }\tfrac{2N}{N+1}<p<2, \] where $\beta$ is a maximal monotone graph with a jump at zero and $\Delta_p$ is the $p$-Laplacian. Moreover, a logarithmic type modulus of continuity is quantified, which has been conjectured to be optimal.

Existence Results for State-Dependent Maximal Monotone Differential Inclusions: Fixed Point Approach

In this article, we give a new proof of the existence of absolutely continuous solutions for a class of first-order state-dependent maximal monotone differential inclusions. The existence result is obtained by using Schauder’s fixed point theorem. In addition, a stability result is provided. Finally, using a suitable reduction of order technique, we give a new existence result for a general second-order state-dependent maximal monotone differential inclusion.

Forward-reflected-backward splitting method without cocoercivity for the sum of maximal monotone operators in Banach spaces

Let E be a real 2-uniformly convex Banach space with topological dual . We prove weak convergence of the forward-reflected-backward splitting method to a solution of inclusion of sum of two monotone operators. Moreover, we give a variant of the method in which the step size does not depend on the Lipschitz constant of one of the operators. Finally, our results extend and complement several existing results in the literature. We also give some numerical illustrations of the proposed method in comparison with other method in the literature to further demonstrate the applicability and efficiency of our method.

Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory of maximal monotone operators, fixed point arguments and the properties of the subdifferential, both in the sense of Clarke and in the sense of convex analysis. These results lay the background in the study of various classes of inequalities. We use them to prove existence, uniqueness and continuous dependence results for the solution of elliptic and history-dependent variational–hemivariational inequalities. We also present some iterative methods in solving these inequalities, together with various convergence results.

$\Gamma$-Stability of maximal monotone processes

The flow of noncyclic maximal monotone operators is formulated variationally as null-minimization problems, via results of Brezis, Ekeland, Nayroles and Fitzpatrick. By means of De Giorgi’s theory of \Gamma -convergence, the compactness and the stability of these flows are derived under nonparametric perturbations of the operator. These results are here reviewed, and are applied to quasilinear equations arising in electromagnetism.

An optimal transport problem with backward martingale constraints motivated by insider trading

We study a single-period optimal transport problem on R2 with a covariance-type cost function c(x,y)=(x1−y1)(x2−y2) and a backward martingale constraint. We show that a transport plan γ is optimal if and only if there is a maximal monotone set G that supports the x-marginal of γ and such that c(x,y)=minz∈Gc(z,y) for every (x,y)∈suppγ. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading from (Rev. Econ. Stud. 61 (1994) 131–152).

Convergence analysis of the stochastic reflected forward–backward splitting algorithm

We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator. The propose algorithm requires only unbiased estimations of the Lipschitzian operator. We obtain the rate $${\mathcal {O}}(log(n)/n)$$ in expectation for the strongly monotone case, as well as almost sure convergence for the general case. Furthermore, in the context of application to convex–concave saddle point problems, we derive the rate of the primal–dual gap. In particular, we also obtain $${\mathcal {O}}(1/n)$$ rate convergence of the primal–dual gap in the deterministic setting.

Optimal Design Versus Maximal Monge–Kantorovich Metrics

The connection between the minimal elastic compliance problem and Monge transport involving the euclidean metric cost has been evidenced in the year 1997. The aim of this paper is to renew this connection and adapt it to some variants in optimal design, focusing in particular on the optimal pre-stressed membrane problem. We show that the underlying metric cost is associated with an unknown maximal monotone map which maximizes the Monge-Kantorovich distance between two measures. In parallel with the classical duality theory leading to existence and PDE optimality conditions, we present a geometrical approach arising from a two-point scheme in which geodesics with respect to the optimal metric play a central role. It turns out from examples that optimal structures are very often truss-like, i.e. supported by piecewise affine geodesics. In case of a discrete load, we are able to relate the existence of such truss-like solutions to an extension property of maximal monotone maps that is of independent interest.

Nonlinear Nonhomogeneous Obstacle Problems with Multivalued Convection Term

In this paper, a nonlinear elliptic obstacle problem is studied. The nonlinear nonhomogeneous partial differential operator generalizes the notions of p-Laplacian while on the right hand side we have a multivalued convection term (i.e., a multivalued reaction term may depend also on the gradient of the solution). The main result of the paper provides existence of the solutions as well as bondedness and closedness of the set of weak solutions of the problem, under quite general assumptions on the data. The main tool of the paper is the surjectivity theorem for multivalued functions given by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone one.

Analyzing Feedback Interconnections of Maximal Monotone Systems Using Dissipativity Approach

We consider interconnections of two dynamical systems in feedback configuration. The dynamics of the individual systems are modeled by a differential inclusion, and the corresponding set-valued mapping is (anti-) maximal monotone with respect to the state of the system for each fixed value of the external signal that defines the interconnection. We provide conditions on these mappings under which the dynamics of the resulting interconnected system are (anti-) maximal monotone. An interpretation of our main result is provided: firstly, by considering dynamical systems defined by the gradient of a saddle function, and secondly, by considering an interconnection of incrementally passive systems. In the same spirit, when we associate more structure to the individual systems by considering linear complementarity systems, we allow for more flexibility in describing the interconnections and derive more specific sufficient conditions in terms of system matrices that result in the overall system being described by (anti-) maximal monotone operator.

On a second-order functional evolution problem with time and state dependent maximal monotone operators

&lt;p style='text-indent:20px;'&gt;The present paper proposes, in a real separable Hilbert space, to analyze the existence of solutions for a class of perturbed second-order state-dependent maximal monotone operators with a finite delay. The dependence of the operators is -in some sense- absolutely continuous (or bounded continuous) variation in time, and Lipschitz continuous in state. The approach to solve our problem is based on a discretization scheme. The uniqueness result is applied to optimal control.&lt;/p&gt;

A control problem involving time and state-dependent maximal monotone operators via Young measures

The article deals with a minimizing problem related to a differential inclusion governed by time- and state-dependent maximal monotone operators and single-valued perturbations with Young measures. A Bolza-type problem and relaxation are studied.

Finite-time stabilization for bilinear reaction diffusion equation

This paper investigates the finite-time stabilization of the bilinear reaction-diffusion equation. Here the reaction term leads to an unbounded control operator. We will first investigate the well-posedness of the system at hand and then proceed to the question of finite-time extinction with feedback control. Our approach is based on the theory of maximal monotone operators. A numerical example is also presented.

A split common fixed point and null point problem for Lipschitzian $J-$quasi pseudocontractive mappings in Banach spaces

A split common fixed point and null point problem (SCFPNPP) which includes the split common fixed point problem, the split common null point problem and other problems related to the fixed point problem and the null point problem is studied. We introduce a Halpern--Ishikawa type algorithm for studying the split common fixed point and null point problem for Lipschitzian $J-$quasi pseudocontractive operators and maximal monotone operators in real Banach spaces. Moreover, we establish a strong convergence results under some suitable conditions and reduce our main result to the above-mentioned problems. Finally, we applied the study to split feasibility problem (FEP), split equilibrium problem (SEP), split variational inequality problem (SVIP) and split optimization problem (SOP).