Maximal Monotone Research Articles

Convergence To Approximate Solutions of Multivalued Operators

The goal of this study is to provide a new explicit iterative process method approach for solving maximal monotone(M.M )operators in Hilbert spaces utilizing a finite family of different types of mappings as( nonexpansive mappings,resolvent mappings and projection mappings. The findings given in this research strengthen and extend key previous findings in the literature. Then, utilizing various structural conditions in Hilbert space and variational inequality problems, we examine the strong convergence to nearest point projection for these explicit iterative process methods Under the presence of two important conditions for convergence, namely closure and convexity. The findings reported in this research strengthen and extend key previous findings from the literature

Existence of Solutions for Inclusion Problems in Musielak-Orlicz-Sobolev Space Setting

In this paper, we mainly prove the existence of (weak) solutions of an inclusion problem with the Dirichlet boundary condition of the following form: L ∈ A x , u , D u + F x , u , D u , in Ω , and u = 0 , on ∂ Ω , in Musielak-Orlicz-Sobolev spaces W 0 1 L Φ Ω by using the surjective theorem, where Ω ⊂ ℝ N is a bounded Lipschitz domain, L belongs to the dual space W 0 1 L Φ Ω ∗ of W 0 1 L Φ Ω , A is a multivalued maximal monotone operator, and F is a multivalued convection term. Some examples for A and F are given in the paper. Then, we give some properties of the solution set of the inclusion problem. We also show the existence of (weak) solutions of the inclusion problem with an obstacle effect.

Bounded perturbation resilience of a regularized forward-reflected-backward splitting method for solving variational inclusion problems with applications

ABSTRACT The forward-reflected-backward splitting method recently introduced for solving variational inclusion problems involves just one forward evaluation and one backward evaluation of the monotone operator and the maximal monotone operator, respectively, per iteration. This structure gives it some advantage over the earlier proposed methods. However, it only provides weak convergence, in general. Our aim in this paper is to improve the forward-reflected-backward splitting method in order to obtain strong convergence. To this end, we first study a regularized variational inclusion problem of finding the zero of the sum of two monotone operators. We then propose a regularized forward-reflected-backward splitting method for approximating a solution to the problem and prove the strong convergence of our iterative scheme under some suitable assumptions on the parameters. Moreover, we show that our algorithm has the bounded perturbation resilience property. Furthermore, we apply our results to convex minimization, split feasibility, split variational inclusion, and image deblurring problems, and illustrate the performance of our algorithm with several numerical examples.

Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators

On the space of continuous functions, we consider a multivalued superposition operator, a Nemytskii operator valued in the space of p-integrable functions with 1≤p<∞. The Nemytskii operator is generated by a multivalued mapping defined on the direct product of a segment of the real line and a separable reflexive Banach space and having closed values in this space. At each point of the phase variable, the mapping either has a closed graph and is convex-valued or it is lower semicontinuous in some neighborhood of this point. In recent years, such mappings are called mappings with mixed semicontinuity properties. We prove that the restriction of the Nemytskii operator onto any compact subset of the space of continuous functions with the sup-norm has a multivalued selector with convex closed values that is upper semicontinuous in the weak topology of the space of p-integrable functions. This result generalizes some well-known results of the author valid in the case when the multivalued mapping generating the Nemytskii operator has compact values. A multivalued selector theorem is used to study an evolution inclusion in a separable Hilbert space with a time-dependent maximal monotone operator and a composite perturbation that is the sum of two multivalued mappings. The values of the first multivalued mapping are closed, bounded, not necessarily convex sets. The second multivalued mapping has closed values and possesses mixed semicontinuity properties. The existence of an absolutely continuous solution is proved. The proof is based on the theorem on an upper semicontinuous convex-valued selector and the author's theorem on selectors continuous in parameter and passing through fixed points of a parameter-dependent multivalued mapping with closed, nonconvex, decomposable values in the space of p-integrable functions as well as on the classical Ky Fan's fixed point theorem. Our approach is fairly simple, concise and clear as compared to other known approaches based on discretization.

On Douglas-Rachford splitting that generally fails to be a proximal mapping: a degenerate proximal point analysis

Based on a degenerate proximal point analysis, we show that the Douglas-Rachford splitting can be reduced to a well-defined resolvent, but generally fails to be a proximal mapping. This extends the recent result of [Bauschke, Schaad and Wang. Math. Program. 2018;168:55–61] to more general setting. The related concepts and consequences are also discussed. In particular, the results regarding the maximal and cyclic monotonicity are instrumental for analysing many operator splitting algorithms.

Double phase implicit obstacle problems with convection term and multivalued operator

This paper is devoted to studying a complicated implicit obstacle problem involving a nonhomogenous differential operator, called double phase operator, a nonlinear convection term (i.e. a reaction term depending on the gradient), and a multivalued term which is described by Clarke’s generalized gradient. We develop a general framework to deliver an existence result for the double phase implicit obstacle problem under consideration. Our proof is based on the Kakutani–Ky Fan fixed point theorem together with the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded pseudomonotone mapping.

General framework to construct local-energy solutions of nonlinear diffusion equations for growing initial data

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations, which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation.

On the Exact Controllability of a Semilinear Evolution Equation with an Unbounded Operator

For the Cauchy problem associated with a controlled semilinear evolution equation with an unbounded maximal monotone operator in a Hilbert space, sufficient conditions are obtained for exact controllability to a given final state. Here a generalization of the Browder–Minty theorem and results on the total global solvability of this equation obtained by the author earlier are used. As an example, a semilinear wave equation is considered.

Strong Convergence of Forward–Reflected–Backward Splitting Methods for Solving Monotone Inclusions with Applications to Image Restoration and Optimal Control

In this paper, we propose and study several strongly convergent versions of the forward–reflected–backward splitting method of Malitsky and Tam for finding a zero of the sum of two monotone operators in a real Hilbert space. Our proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. We also develop inertial versions of our methods and strong convergence results are obtained for these methods when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous and monotone. Finally, we discuss some examples from image restorations and optimal control regarding the implementations of our methods in comparisons with known related methods in the literature.

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 and uniqueness of weak periodic solutions for a coupled parabolic-elliptic system

ased on the maximal monotone mapping theory and applying the Schauder fixed point the- orem, we prove the existence and the uniqueness of weak periodic solution for nonlinear parabolic-elliptic equations in Orlicz-Sobolev spaces, with growth nonlinearity in gradient associated with some appropriate N-functions.

Monotone Inclusions, Acceleration, and Closed-Loop Control

We propose and analyze a new dynamical system with a closed-loop control law in a Hilbert space [Formula: see text], aiming to shed light on the acceleration phenomenon for monotone inclusion problems, which unifies a broad class of optimization, saddle point, and variational inequality (VI) problems under a single framework. Given an operator [Formula: see text] that is maximal monotone, we propose a closed-loop control system that is governed by the operator [Formula: see text], where a feedback law [Formula: see text] is tuned by the resolution of the algebraic equation [Formula: see text] for some [Formula: see text]. Our first contribution is to prove the existence and uniqueness of a global solution via the Cauchy–Lipschitz theorem. We present a simple Lyapunov function for establishing the weak convergence of trajectories via the Opial lemma and strong convergence results under additional conditions. We then prove a global ergodic convergence rate of [Formula: see text] in terms of a gap function and a global pointwise convergence rate of [Formula: see text] in terms of a residue function. Local linear convergence is established in terms of a distance function under an error bound condition. Further, we provide an algorithmic framework based on the implicit discretization of our system in a Euclidean setting, generalizing the large-step hybrid proximal extragradient framework. Even though the discrete-time analysis is a simplification and generalization of existing analyses for a bounded domain, it is largely motivated by the aforementioned continuous-time analysis, illustrating the fundamental role that the closed-loop control plays in acceleration in monotone inclusion. A highlight of our analysis is a new result concerning [Formula: see text]-order tensor algorithms for monotone inclusion problems, complementing the recent analysis for saddle point and VI problems. Funding: This work was supported in part by the Mathematical Data Science Program of the Office of Naval Research [Grant N00014-18-1-2764] and by the Vannevar Bush Faculty Fellowship Program [Grant N00014-21-1-2941].

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.

Revised algorithm for finding a common solution of variational inclusion and fixed point problems

Recent research has uncovered an algorithm for locating the common solution to variational inclusion problems with multivalued maximal monotone mapping and ?-inverse strongly monotone mapping, as well as the points that are invariant under non-expansive mapping. In their algorithm, Zhang et al. [S. Zhang, J. H. W. Lee, C. K. Chan, Algorithms of common solutions to quasi-variational inclusion and fixed point problems, Appl. Math. Mech. 29(5) (2008), 571-581.], ? must satisfy a very strict condition, namely ? ? [0, 2?]; thus, it cannot be used for all Lipschitz continuous mappings, despite the fact that inverse strongly monotone implies Lipschitz continuous. This manuscript aims to define a new algorithm that addresses the flaws of the previously described algorithm. Our algorithm is used to solve minimization problems involving the fixed point set of a non-expansive mapping. In addition, we support all of our claims with numerical examples derived from computer simulation.

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.

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.

Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem

&lt;p style='text-indent:20px;'&gt;In this paper, we first establish a surjectivity result for the sum of a maximal monotone mapping and a generalized pseudomontone mapping by using the generalized Yosida approximation technique. Then, we study a double obstacle quasilinear parabolic variational inequality problem &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ {{\rm{(VIP)}}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; by using the surjectivity result and penalty approximation method. In order to deal with the double obstacle constraints, we construct a quasilinear parabolic partial differential penalty equation, then we obtain the solvability of the quasilinear parabolic partial differential penalty equation. Moreover, we show that the set of the solutions to the penalty equation is bounded and every weak cluster point of this set is a solution of the problem &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ {{\rm{(VIP)}}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. At last, as an application, we obtain numerical solutions of two double obstacle parabolic variational inequality problems by using the power penalty approximation method. Through the figures in the examples, we can intuitively see the different numerical solutions of the problems at different times.&lt;/p&gt;

On fixed point theorems of maximal monotone operators

This paper aims to study the viscosity approximation approach to fixed points and the solution of a variational inequality. Also, we introduce a new iterative technique and establish the convergence theorems under certain conditions in the Hilbert space.

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

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined, whose values are the normal cones of the domains of definition of maximal monotone operators. It is shown that if the sweeping process has a solution for each single-valued perturbation from the space of integrable functions, then the evolution inclusion with the maximal monotone operators and single-valued perturbations from the space of integrable functions is also solvable. Quite general conditions in terms of the properties of the family of maximal monotone operators that ensure the existence of solutions for the sweeping process are presented. All results obtained and the approach presented are new. They are used to prove an existence theorem for evolution inclusions with multivalued perturbations, whose values are closed nonconvex sets. Bibliography: 19 titles.