Yosida Approximation Research Articles

Cayley inclusion problem with its corresponding generalized resolvent equation problem in uniformly smooth Banach spaces

ABSTRACT A new inclusion problem is introduced using generalized Cayley operator and we call it Cayley inclusion problem. We also study its corresponding resolvent equation problem. By using a generalized resolvent operator and generalized Yosida approximation operator, first we establish a fixed point formulation for Cayley inclusion problem. An algorithm is defined to find the solution of Cayley inclusion problem. An existence and convergence result is proved. Secondly, we have shown the equivalence of Cayley inclusion problem with a resolvent equation. We define an iterative algorithm with some of its equivalent forms for solving resolvent equation problem. A numerical example is constructed and a convergence graph is shown by using MATLAB program.

A dynamic thermo-mechanical actuator system with contact and Barber's heat exchange boundary conditions

AbstractThis work, motivated by the rapid developments in Micro-Electro-Mechanical Systems (MEMS) structures, especially actuators and grippers, analyses the dynamics of a thermo-mechanical system consisting of a horizontal beam joined at one end to a vertical rod. As a result of thermal expansion or vibration of the rod, the other end may come into contact with another part of the MEMS device and that closes an electrical circuit, which is the actuating or switching function of such a beam–rod system. The interaction between the rod's contacting end and the supporting device is described by a normal compliance contact law for the displacements and by an inclusion-type Barber's heat exchange condition for the temperature. The heat-exchange coefficient is a multi-function taking into account the air resistance in the gap when there is no contact and the contact pressure when contact occurs. The model consists of a nonlinear variational inclusion for the temperature coupled with a nonlinear variational equation for the displacements. The existence of a weak solution to the problem is proved by using the Galerkin method, a regularization of Barber's condition with the Yosida approximation of a maximal monotone operator, and a priori estimates.

Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not G\^ateaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

Theoretical and numerical analysis of a class of quasilinear elliptic equations

The purpose of this paper is to give a result of the existence of a non-negative weak solution of a quasilinear elliptic equation in the N-dimensional case, \begin{document}$ N\geq 1 $\end{document} , and to present a novel numerical method to compute it. In this work, we assume that the nonlinearity concerning the derivatives of the solution are sub-quadratics. The numerical algorithm designed to compute an approximation of the non-negative weak solution of the considered equation has coupled the Newton method with domain decomposition and Yosida approximation of the nonlinearity. The domain decomposition is adapted to the nonlinearity at each step of the Newton method. Numerical examples are presented and commented on.

Existence results for a class of semilinear differential variational inequalities with nonlocal boundary conditions

In the paper we study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing evolution equations and generalized variational inequalities. Firstly, we show the properties of the solution set for generalized variational inequalities. Then, the existence results are established and proved mainly by the topological degree theory and the Yosida approximations of the generator of $C_0$-semigroup.

Nonlocal elliptic variational-hemivariational inequalities

We examine the existence of a weak solution to a class of variational-hemivariational inequalities governed by a fractional Laplace operator. Moreover, an existence result is proved for a bilateral obstacle problem for a nonlocal hemivariational inequality. Our approach is based on a surjectivity result for pseudomonotone mappings, properties of Clarke’s subgradient, and the Moreau–Yosida approximation.

Newton-like Inertial Dynamics and Proximal Algorithms Governed by Maximally Monotone Operators

The introduction of the Hessian damping in the continuous version of Nesterov's accelerated gradient method provides, by temporal discretization, fast proximal gradient algorithms where the oscillations are significantly attenuated. We will extend these results to the maximally monotone case. We rely on the technique introduced by Attouch and Peypouquet [Math. Program., 174 (2019), pp. 391--432], where the maximally monotone operator is replaced by its Yosida approximation with an appropriate adjustment of the regularization parameter. In a general Hilbert framework, we obtain the weak convergence of the iterates to equilibria, and the rapid convergence of the discrete velocities to zero. By specializing these algorithms to convex minimization, we obtain the convergence rate $o(1/k^2)$ of the values, and the rapid convergence of the gradients toward zero.

Stability and convergence analysis for a new class of GNYIP involving XOR-operation in ordered positive Hilbert spaces

In the setting of real ordered positive Hilbert spaces, a new class of general nonlinear ordered Yosida inclusion problem involving ? operation has been considered and solved by employing a perturbed two step-iterative algorithm. The stability and convergence analysis of solution of new class of Yosida inclusion problem involving ? operation has been substantiated by applying a new resolvent operator and Yosida approximation operator method with XOR-operation technique. The iterative algorithm and results demonstrated in this article have witnessed, a significant improvement for many previously known results of this domain. Further, we give a numerical example in support of our main result by using MATLprogramming.

Prandtl–Reuss dynamical elasto-perfect plasticity without safe-load conditions

In this article we study the existence theory to the Prandtl–Reuss dynamical model of elasto-perfect plasticity with non-homogeneous mixed boundary conditions. By using only the Yosida approximation of inelastic constitutive multifunction we are able to prove the existence of solutions without assuming any kind of safe-load conditions, which are quite common in the theory of elasto-perfect plasticity.

Solving Yosida inclusion problem in Hadamard manifold

We consider a Yosida inclusion problem in the setting of Hadamard manifolds. We study Korpelevich-type algorithm for computing the approximate solution of Yosida inclusion problem. The resolvent and Yosida approximation operator of a monotone vector field and their properties are used to prove that the sequence generated by the proposed algorithm converges to the solution of Yosida inclusion problem. An application to our problem and algorithm is presented to solve variational inequalities in Hadamard manifolds.

Convergence of a relaxed inertial proximal algorithm for maximally monotone operators

In a Hilbert space $${\mathcal {H}}$$ , given $$A{:}\;{\mathcal {H}}\rightarrow 2^{\mathcal {H}}$$ a maximally monotone operator, we study the convergence properties of a general class of relaxed inertial proximal algorithms. This study aims to extend to the case of the general monotone inclusion $$Ax \ni 0$$ the acceleration techniques initially introduced by Nesterov in the case of convex minimization. The relaxed form of the proximal algorithms plays a central role. It comes naturally with the regularization of the operator A by its Yosida approximation with a variable parameter, a technique recently introduced by Attouch–Peypouquet (Math Program Ser B, 2018. https://doi.org/10.1007/s10107-018-1252-x ) for a particular class of inertial proximal algorithms. Our study provides an algorithmic version of the convergence results obtained by Attouch–Cabot (J Differ Equ 264:7138–7182, 2018) in the case of continuous dynamical systems.

Cayley Inclusion Problem Involving XOR-Operation

In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is shown by using the concept of a resolvent operator as well as the Yosida approximation operator. Finally, an existence and convergence result is proved. An example is constructed for some of the concepts used in this work.

Composite relaxed resolvent operator and Yosida approximation operator for solving a system of Yosida inclusions

Abstract In this paper, we first study a composite relaxed resolvent operator and prove some of its properties. After that, we introduce a Yosida approximation operator based on the composite relaxed resolvent operator and demonstrate some properties of the Yosida approximation operator. Finally, we obtain the solution of a system of Yosida inclusions by applying these concepts. We construct a conjoin example in support of many concepts derived in this paper. Our concepts and results are new in the literature and can be used for further research.

The Yosida approximation iterative technique for split monotone Yosida variational inclusions

In this article, we introduce a new type of split monotone Yosida inclusion problem in the setting of infinite-dimensional Hilbert spaces. To calculate the approximate solutions of split monotone Yosida inclusion problem, first we develop a new iterative algorithm and then study the weak as well strong convergence analysis of iterative sequences generated by the proposed iterative algorithm by using demicontractive property, nonexpansive property, and strongly positive bounded linear property of mappings. A numerical example is formulated to explain our main result through MATLAB programming.

Error estimates for the approximation of multibang control problems

This work is concerned with optimal control problems where the objective functional consists of a tracking-type functional and an additional "multibang" regularization functional that promotes optimal control taking values from a given discrete set pointwise almost everywhere. Under a regularity condition on the set where these discrete values are attained, error estimates for the Moreau--Yosida approximation (which allows its solution by a semismooth Newton method) and the discretization of the problem are derived. Numerical results support the theoretical findings.

Limit Theorems for Stochastic Variational Inequalities with Non-Lipschitz Coefficients

We establish various limit theorems for one-dimensional stochastic variational inequalities with Yamada-Watanabe type conditions on the coefficients, including, the construction of the solution through the Euler scheme, the convergence of the Yosida approximation and stability of the solution. Besides, convergence rates are presented for these two approximations when the coefficients are only Holder continuous.

Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators

We study the behavior of the trajectories of a second-order differential equation with vanishing damping, governed by the Yosida regularization of a maximally monotone operator with time-varying index, along with a new Regularized Inertial Proximal Algorithm obtained by means of a convenient finite-difference discretization. These systems are the counterpart to accelerated forward–backward algorithms in the context of maximally monotone operators. A proper tuning of the parameters allows us to prove the weak convergence of the trajectories to zeroes of the operator. Moreover, it is possible to estimate the rate at which the speed and acceleration vanish. We also study the effect of perturbations or computational errors that leave the convergence properties unchanged. We also analyze a growth condition under which strong convergence can be guaranteed. A simple example shows the criticality of the assumptions on the Yosida approximation parameter, and allows us to illustrate the behavior of these systems compared with some of their close relatives.

Generalized Yosida approximation operator with an application to a system of Yosida inclusions

Generalized Yosida approximation operator with an application to a system of Yosida inclusions

Book Review: Yosida approximations of stochastic differential equations in infinite dimensions and applications

Book Review: Yosida approximations of stochastic differential equations in infinite dimensions and applications

Disjointness preserving [formula omitted]-semigroups and local operators on ordered Banach spaces

We generalize results concerning C0-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving C0-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local C0-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extensions of regular norms.