Monotone Operators Research Articles

Three-operator reflected forward-backward splitting algorithm with double inertial effects

In this paper, we propose a reflected forward-backward splitting algorithm with two different inertial extrapolations to find a zero of the sum of three monotone operators consisting of the maximal monotone operator, Lipschitz continuous monotone operator, and a cocoercive operator in real Hilbert spaces. One of the interesting features of the proposed algorithm is that both the Lipschitz continuous monotone operator and the cocoercive operator are computed explicitly each with one evaluation per iteration. We then obtain weak and strong convergence results under some mild conditions. We finally give a numerical illustration to show that our proposed algorithm is effective and competitive with other related algorithms in the literature.

Nonemptiness and boundedness of the set of zeros of a monotone operator in Hadamard spaces

Nonemptiness and boundedness of the set of zeros of a monotone operator in Hadamard spaces

A Class of Norm Inequalities for Operator Monotone Functions and Hyponormal Operators

A Class of Norm Inequalities for Operator Monotone Functions and Hyponormal Operators

On a dynamic frictional contact problem with normal damped response and long-term memory

A dynamic frictional contact problem between a viscoelastic body and a foundation is studied. The contact is modeled with normal damped response and a friction law. The constitutive law with long memory is assumed to be nonlinear. The existence result is proved using nonlinear monotone operators, fixed point argument, and extension procedure. Moreover, the exponential stability of the energy solution is established using the multiplier method.

Second-order differential inclusions with two small parameters

Consider in a real Hilbert space H the following problem, denoted (Pɛμ), −ɛu′′(t)+μu′(t)+Au(t)+Bu(t)∋f(t),0<t<T,u(0)=u0,u′(T)=0,where T>0 is a given time instant, ɛ>0,μ≥0 are small parameters, A:D(A)⊂H→H is a maximal monotone operator (possibly multivalued), and B:H→H is a Lipschitz operator (or monotone and Lipschitz on bounded sets). Consider also the following reduced problem, denoted (Pμ), μu′(t)+Au(t)+Bu(t)∋f(t),0<t<T,u(0)=u0,where μ>0, as well as the algebraic equation (inclusion), Au(t)+Bu(t)∋f(t),0≤t≤T.(E00)In this paper we are concerned with the following topics: (a) existence and uniqueness of solutions to the above problems and to equation (E00); (b) continuity of the solution to problem (Pɛμ) with respect to ɛ>0 and μ≥0; (c) convergence of the solution of problem (Pɛμ) to the solution of problem Pμ0, as ɛ→0+ and μ→μ0, where μ0 is a fixed positive number; (d) convergence of the solution of problem (Pɛμ) to the solution of the equation Au+Bu∋f(t) as ɛ→0+ and μ→0+; (e) applications.

Alternated inertial forward-backward-forward splitting algorithm

In this paper, we propose a forward-backward-forward splitting algorithm with alternated inertial step for finding a zero of the sum of two monotone operators in a real Hilbert space. Then, by making use of primal-dual techniques we derive the alternated inertial primal-dual splitting algorithm of forward-backward-forward type for solving structured monotone inclusion problems involving parallel sums and compositions of maximally monotone operators with linear continuous ones and convex minimization problems. Finally, we demonstrate numerically that our proposed algorithm performs better than the existing algorithms.

On the Proximal Point Algorithm of Hybrid-Type in Flat Hadamard Spaces with Applications

In this paper, we introduce a hybrid-type proximal point algorithm for approximating zero of monotone operator in Hadamard-type spaces. We then prove that a sequence generated by the algorithm involving Mann-type iteration converges strongly to a zero of the said operator in the setting of flat Hadamard spaces. To the best of our knowledge, this result presents the first hybrid-type proximal point algorithm in the space. The result is applied to convex minimization and fixed point problems.

On Calculation of the Norm of a Monotone Operator in Ideal Spaces

On Calculation of the Norm of a Monotone Operator in Ideal Spaces

A quasistatic elastic-viscoplastic contact problem with wear and frictionless

We consider here a frictionless contact problem for elastic-viscoplastic materials, in a quasi-static process. The contact with a rigid base is modeled without friction with condition of wear and damage. The damage the elastic deformations of the material is modeled by an internal variable of the body called the damage field. The problem formula is given as a system that includes a variational equation with respect to the displacement field, and a variational inequality of the parabolic type with respect to the damage field. We prove a weak solution existence and uniqueness theorem relating to the problem. The methods utilised are grounded in the concept of monotonic operators, followed by fixed-point arguments.

Modified monotone hybrid inertial algorithm for generalized nonexpansive maps, maximal monotone operators and generalized mixed equilibrium problems

Modified monotone hybrid inertial algorithm for generalized nonexpansive maps, maximal monotone operators and generalized mixed equilibrium problems

Viscosity-type inertial iterative methods for variational inclusion and fixed point problems

&lt;abstract&gt;&lt;p&gt;In this paper, we have introduced some viscosity-type inertial iterative methods for solving fixed point and variational inclusion problems in Hilbert spaces. Our methods calculated the viscosity approximation, fixed point iteration, and inertial extrapolation jointly in the starting of every iteration. Assuming some suitable assumptions, we demonstrated the strong convergence theorems without computing the resolvent of the associated monotone operators. We used some numerical examples to illustrate the efficiency of our iterative approaches and compared them with the related work.&lt;/p&gt;&lt;/abstract&gt;

Local Monotone Operator Learning Using Non-Monotone Operators: MnM-MOL

Local Monotone Operator Learning Using Non-Monotone Operators: MnM-MOL

Approximating zeros of monotone operators in Banach spaces

Approximating zeros of monotone operators in Banach spaces

A study of the multipoint boundary value problem for nonlinear differential equations of arbitrary fractional order based on the fixed point theorem for composite nonlinear operators

Abstract The existence and distinctiveness of solutions, as well as the iterative approximations of the distinctive answer to the initial boundary value issue of nonlinear differential equations, may be solved with the help of the theory of nonlinear operators, which offers a strong theoretical guarantee and a fundamental tool. In this work, we examine the fixed point theory for composites nonlinear operators, which may offer a fresh approach to solving linear differential equations having arbitrary order fractional multipoint boundary value problems. The existing result of the boundary value issue’s solution is confirmed by the fixed point theorem for composites nonlinear operators, and the four-point boundary value problem for fractional-order linear equations of the Riemann-Liouville type is examined. A group of fractional-order nonlinear differential equations with deviating quantities are examined to examine the presence of an unusual positive solution for this boundary value problem. The existence of this unique positive solution is determined by the use of both the mixed monotone operator and the combined nonlinear operator fixed point theorem. Ultimately, the nonlinear Bagley-Torvik equation with a fractional order variable coefficient four-point boundary value problem is converted into a Fredholm-Hammerstein integral the formula of the second type with weakly singular or continuous kernels, and the fixed point principle is used to demonstrate the uniqueness of the solution in the space of continuous functions. Approximate solutions are provided for the second class of nonlinear Fredholm-Hammerstein integral equations with weakly singular kernels. The outcomes demonstrate the applicability and validity of the fixed point theorem to composite nonlinear operators, offering a fresh approach to solving the multipoint boundary value issue for nonlinear differential equations of any fractional order.

Second order dynamical systems governed by maximally monotone operators: Quantitative stability

Second order dynamical systems governed by maximally monotone operators: Quantitative stability

Coupled problems driven by time and state-dependent maximal monotone operators

We are concerned in this paper with a new class of coupled problems by differential inclusions involving time and state-dependent maximal monotone operators with perturbations. The operators vary (in the sense of the pseudo-distance) in an absolutely continuous way in time, and in a Lipschitz-continuous way in state. By a discretization technique, we state the well-posedness (existence and uniqueness of the solution) in a real separable Hilbert space. Applications to optimal control and to fractional theory are provided.

Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Abstract We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$ -spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

A new self-adaptive method for solving resolvent of sum of two monotone operators in Banach spaces

We introduce a Tseng extragradient method for solving monotone inclusion problem in Banach space. A strong convergence result of an Halpern inertial extrapolation method for solving the resolvent of sum of two monotone operators without the knowledge of the Lipschitz constant was established. Lastly, we illustrate some numerical behavior of our iterative scheme to showcase the performance of the proposed method compared to other related results in the literature.

Simulation of distributed network elements in CAS MAXIMA and server online calculations in the educational process

Pressure pipelines are an important part of the water management infrastructure. Water supply networks pump the liquid to the consumer with the help of pumps (or supply it from a water tower) to specified points. The functioning of the entire urban infrastructure depends on the correct calculation and reliable operation of these areas. This paper shows the implementation of a computer calculation of one of the branched network options using numerical methods in the CAS MAXIMA computer algebra system, its graphic subsystem of symbolic mathematics, and the use of the author's online calculation program with a client-server implementation - https://www.k123.org.ua/jgIHjob1.html .&#x0D; The implementation of elements of computer calculations in the educational process allows the generation of an unlimited number of individual variants of initial data. The implementation of computer calculation requires the availability of automatic online generation of answers for each stage of iterative calculations. The task of generating online answers is the basis of the program development.&#x0D; An educational online platform is offered for researching the optimality of numerical, graphic and audio formats for setting the problem. The formats of the web form for entering output data have been studied. Online calculations reduce the time spent on typical monotonous operations. Additional possibilities of instant online calculations and visual graphic visualization of calculation results expand the possibilities of modeling individual elements of a branched network.&#x0D; The online program allows you to use computer calculations, expand the culture of using applied mathematical methods for engineering calculations, introduce algorithms and data structures, and solve the main task of training future engineers - to ensure a modern level of education and readiness to solve new problems.

Two regularized inertial Tseng methods for solving inclusion problems with applications to convex bilevel programming

Combining the inertial technique, regularization, and the Tseng method for approximating a zero of the sum of monotone operators in a real Hilbert space, we propose two regularized inertial Tseng methods for approximating a zero of the sum of two maximal monotone operators in the case where one of them is monotone and Lipschitz continuous. We establish strong convergence of the sequences generated by our algorithms and derive several interesting consequences of our results. We then apply our results to solving convex bilevel programming problems and illustrate the cost efficiency of our methods by numerically comparing them with other methods in the literature.