Monotone Operators Research Articles

Output finite-time stabilisation of a class of linear and bilinear control systems

Based on the idea of Lyapunov function, we investigate the output finite-time stability of some classes of abstract bilinear systems using feedback control laws. We first design a stabilising feedback control. Then, we study the well-posedness of the system in closed-loop by using the theory of maximal monotone operators. Then we proceed to the question of output finite-time stability in a settling time of the system at hand. The obtained results are also applied to derive output finite time stabilisation results for linear systems. The algorithm of feedback control design is further applied to finite dimension systems and also to PDE of parabolic and hyperbolic types.

Existence of a Generalized Solution for the Fractional Contact Problem

In this paper, we take into consideration the mathematical analysis of time-dependent quasistatic processes involving the contact between a solid body and an extremely rigid structure, referred to as a foundation. It is assumed that the constitutive law is fractional long-memory viscoelastic. The contact is considered to be bilateral and is modeled around Tresca’s law. We establish the existence of the generalized solution’s result. The proof is supported by the surjectivity of the multivalued maximum monotone operator, Rothe’s semidiscretization method, and arguments for evolutionary variational inequality.

Korovkin-Type Theorems for Weakly Nonlinear and Monotone Operators

In this paper we prove analogs of Korovkin’s theorem in the context of weakly nonlinear and monotone operators acting on Banach lattices of functions of several variables. Our results concern the convergence almost everywhere, the convergence in measure and the convergence in \(L^{p}\)-norm. Several results illustrating the theory are also included.

Solving nonlinear implicit variational inclusion problems using S-iteration via relaxed resolvent operator

The goal of the draft is to launch a fresh iteration scheme for nonlinear implicit variational inclusion problems with the help of strongly ()I−-monotone operator. This scheme is known as relaxed S-iteration scheme. Here, we use relaxed resolvent operator into S-iteration and create a new operator and with the help of that operator we will establish the strong convergence result for the nonlinear implicit variational inclusion problems.

Approximating Solutions of Monotone Variational Inclusion, Equilibrium and Fixed Point Problems of Certain Nonlinear Mappings in Banach Spaces

In this paper, motivated by the works of Timnak et al. [Filomat 31(15) (2017), 4673–4693], Ogbuisi and Izuchukwu [Numer. Funct. Anal. 40(13) (2019)] and some other related results in literature, we introduce an iterative algorithm and employ a Bregman distance approach for approximating a zero of the sum of two monotone operators, which is also a common solution of equilibrium problem involving pseudomonotone bifunction and a fixed point problem for an infinite family of Bregman quasi-nonexpansive mappings in the framework of a reflexive Banach space. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution of the aforementioned problems. Furthermore, we give some applications of the consequences of our main result to convex minimization problem and variational inequality problem. Lastly, we display a numerical example to show the applicability of our main result. The result presented in this paper extends and complements many related results in the literature.

Accelerated preconditioning Krasnosel'skiĭ-Mann method for efficiently solving monotone inclusion problems

<abstract><p>In this article, we propose a strongly convergent preconditioning method for finding a zero of the sum of two monotone operators. The proposed method combines a preconditioning approach with the robustness of the Krasnosel'skiĭ-Mann method. We show the strong convergence result of the sequence generated by the proposed method to a solution of the monotone inclusion problem. Finally, we provide numerical experiments on the convex minimization problem.</p></abstract>

Monotonicity and discretization of Hammerstein integrodifference equations

<p style='text-indent:20px;'>The paper provides sufficient conditions for monotonicity, subhomogeneity and concavity of vector-valued Hammerstein integral operators over compact domains, as well as for the persistence of these properties under numerical discretizations of degenerate kernel type. This has immediate consequences on the dynamics of Hammerstein integrodifference equations and allows to deduce a local-global stability principle.</p>

Влияние цифровой революции на развитие искусственного интеллекта и внедрение технологий автоматизации роботизированных процессов в деятельность современной компании

The subject / topic. The state of the information space in the conditions of the digital revolution and the new realities of the digitalization process affecting all spheres of human activity, including the development of artificial intelligence and robotics to displace human labor through special software tools when performing monotonous and routine production operations. The aim of the article. Generalization of the influence of artificial intelligence on the activities of modern companies and their employees. The study substantiates the need to support domestic companies in the field of software development based on the «market in exchange for investment» mechanism. It involves the conclusion of an agreement between the government and software developers with the status of systemically significant organizations to ensure competitive advantages in the digital environment and replace the largest foreign software companies that have left the Russian market after the start of a special military operation in Ukraine. Automation of robotic processes is considered. Conclusions / significance. Conclusions are drawn about the need to develop an effective state policy of Russia on the introduction of artificial intelligence into production using modern logistics chains.

ОЦIНКИ ЕФЕКТИВНОСТI ДЛЯ МЕТОДIВ З ДИВЕРГЕНЦIЄЮ БРЕГМАНА

In this paper, variants of extrapolation from the past algorithm and operator extrapolation algorithm with Bregman divergence for solving variational inequalities with monotone and Lipschitz-continuous operators, which act in a finite-dimensional real linear space, are investigated. Main results: efficiency bounds for the gap function.

Tikhonov regularization of a nonhomogeneous first-order evolution equation

We consider a nonhomogeneous first-order evolution equation governed by a maximal monotone operator $ A $ in a Hilbert space in the presence of a Tikhonov regularization term. We study the existence and strong convergence of weak solutions to such systems. With boundedness conditions on the central path or on the solution trajectory, without assuming the zero set of $ A $ to be nonempty and with suitable assumptions on the Tikhonov regularization coefficient, we prove that the weak solutions to the system converge strongly to the element of least norm in the zero set of $ A $. As a consequence, we provide sufficient conditions where the boundedness of the weak solutions to the nonhomogeneous Tikhonov system is equivalent to the zero set of $ A $ to be nonempty. Our work is motivated by [4,17,22] and extends some results by those authors.

Functional differential inclusions with maximal monotone operators and nonconvex perturbations

In this paper, we study the existence of solutions for functional differential inclusions governed by time-dependent maximal monotone operators with nonconvex perturbations depending on all the variables.

Existence of entropy solutions for some quasilinear anisotropic elliptic unilateral problems with variable exponents

In this paper, we shall be concerned with the study of the following quasilinear anisotropic elliptic Dirichlet problems of the type \begin{document}$\begin{equation} -\mbox{div }\;a(x,u,\nabla u) = f -\mbox{div } F \qquad \mbox{in}\quad \Omega,\quad\quad\quad\quad\quad(1) \end{equation}$ \end{document} where $ f\in L^{1}(\Omega) $ and $ F \in \prod_{i = 1}^{N} L^{p'_{i}(\cdot)}(\Omega), $ and $ a_{i}(x,u,\xi) $ are Carathéodory functions from $ \Omega \times I\!\!R \times I\!\!R^{N} $ into $ I\!\!R $, which satisfy assumptions of growth, coercivity and strict monotonicity. We prove the existence of entropy solutions for the quasilinear elliptic equation associated to the unilateral problem by relying on the penalization method, in the anisotropic variable exponent Sobolev spaces. Our approach is also based on the techniques of monotone operators in Banach spaces, the existence of weak solutions, and some approximations methods. The problems of the type $ (1) $ are very interesting from the purely mathematical point of view. On the other hand, such equations $ (1) $ appear in different contexts, in particular, the mathematical description of motions of the non-newtonien fluids; we quote for instance the electro-rheological fluids; the deformation of membrane constrained by an obstacle, the image processing and other various physical applications.

АЛГОРИТМ ЕКСТРАПОЛЯЦІЇ З МИНУЛОГО ДЛЯ ВАРІАЦІЙНИХ НЕРІВНОСТЕЙ В ГІЛЬБЕРТОВОМУ ПРОСТОРІ

The article considers variational inequalities with operators acting in a Hilbert space. For these problems, variants of the Extrapolation from the Past method have been proposed and studied. A sub-linear efficiency estimate for the gap function is proved. The strong convergence of the Extrapolation from the Past method for variational inequalities with uniformly monotone operators is proved. The linear rate of convergence of the Extrapolation from the Past method for variational inequalities with operators satisfying the generalized strong monotonicity condition is proved. An adaptive version of the algorithm is proposed. Regularized variants of the algorithm are proposed and theorems on their strong convergence are proved.

Pullback attractors and upper semicontinuity for non-autonomous extensible two-beams

In this paper we study the long-time dynamics for non-autonomous extensible non-symmetric two-beams. The governing equations describe the equilibria of an elastically-coupled extensible two-beams system subject to damping terms and evenly compressive axial loads. Under quite general assumptions on nonlinear damping and sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We prove the existence of pullback attractors in the natural space energy. Finally, we prove the regularity of the family of pullback attractors and its upper semicontinuous with respect to the perturbed parameters.

Decay in full von Kármán beam with temperature and microtemperatures effects

In this article we derive the equations that constitute the mathematical model of the full von Kármán beam with temperature and microtemperatures effects. The nonlinear governing equations are derived by using Hamilton principle in the framework of Euler–Bernoulli beam theory. Under quite general assumptions on nonlinear damping function acting on the transversal component and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the derived problem. Then using the multiplier method, we show that solutions decay exponentially. Finally we consider the case of zero thermal conductivity and we show that the dissipation given only by the microtemperatures is strong enough to produce exponential stability.

A coupled problem described by time-dependent subdifferential operator and non-convex perturbed sweeping process

The aim of the present paper is to state the well-posedness of a new dynamical system driven by a differential inclusion involving time-dependent subdifferential operators with integral perturbation and a non-convex perturbed sweeping process. The current study is motivated by recent works on integro-differential sweeping processes and dynamical systems coupled by differential inclusions (involving sweeping processes and maximal monotone operators). In our development, we proceed using a discretization method, combining tools of first-order evolution problems of subdifferential type and those governed by the normal cone of $ r $-prox regular moving sets. We establish a new existence result to the class of dynamical systems under consideration, in the context of infinite dimensional Hilbert setting. This allows us to deal with a Bolza-type problem in optimal control theory.

A modified proximal point algorithm in geodesic metric space

<abstract><p>Proximal point algorithm is one of the most popular technique to find either zero of monotone operator or minimizer of a lower semi-continuous function. In this paper, we propose a new modified proximal point algorithm for solving minimization problems and common fixed point problems in CAT(0) spaces. We prove $ \Delta $ and strong convergence of the proposed algorithm. Our results extend and improve the corresponding recent results in the literature.</p></abstract>

Comparison theorems for evolution inclusions with maximal monotone operators. $L^2$-theory

В сепарабельном гильбертовом пространстве изучается эволюционное включение с зависящим от времени семейством максимально монотонных операторов. Если элементы минимальной нормы семейства максимально монотонных операторов удовлетворяют условиям роста, то области определения семейства максимально монотонных операторов являются замкнутыми выпуклыми множествами. Поэтому будет определен процесс выметания, значениями которого являются нормальные конусы областей определения максимально монотонных операторов. Доказывается, что если процесс выметания при любом однозначном возмущении из пространства интегрируемых функций имеет решение, то этим свойством обладает и эволюционное включение с максимально монотонными операторами и однозначными возмущениями из пространства интегрируемых функций. В терминах свойств семейства максимально монотонных операторов даны самые общие условия, обеспечивающие существование решений процесса выметания. Все полученные результаты, а также предлагаемый подход являются новыми. Они используются для доказательства теоремы существования решений эволюционного включения с многозначным возмущением, значениями которого являются замкнутые невыпуклые множества. Библиография: 19 названий.

Convergence analysis of the generalized Douglas-Rachford splitting method under Hölder subregularity assumptions

<p style='text-indent:20px;'>In solving convex optimization and monotone inclusion problems, operator-splitting methods are often employed to transform optimization and inclusion problems into fixed-point equations as the equations obtained from operator-splitting methods are often easy to be solved by standard techniques. For the inclusion problem involving two maximally monotone operators, under the Hölder metric subregularity of the concerned operator, which is weaker than the strong monotonicity of the operator, we derive relationships between the convergence rate of the generalized Douglas-Rachford splitting algorithm and the order of the Hölder metric subregularity of the concerned operator. Moreover, for general multifunctions in Hilbert spaces, by proximal coderivative, we provided some dual sufficient conditions for Hölder metric subregularity.</p>

Semi-Decentralized Generalized Nash Equilibrium Seeking in Monotone Aggregative Games

We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: (i) a single-layer, fixed-step algorithm with convergence guarantee for general (non cocoercive, non-strictly) monotone aggregative games and (ii) a single-layer proximal-type algorithm for a class of monotone aggregative games with linearly coupled cost functions. We also design novel accelerated variants of the algorithms via (alternating) inertial and over-relaxation steps. Finally, we show via numerical simulations that the proposed algorithms outperform those in the literature in terms of convergence speed.