History-dependent Operators Research Articles

Well-posedness of a class of evolutionary variational–hemivariational inequalities in contact mechanics

A class of evolutionary variational–hemivariational inequalities with a convex constraint is studied in this paper. An inequality in this class involves a first-order derivative and a history-dependent operator. Existence and uniqueness of a solution to the inequality is established by the Rothe method, in which the first-order temporal derivative is approximated by backward Euler’s formula, and the history-dependent operator is approximated by a modified left endpoint rule. The proof of the result relies on basic results in functional analysis only, and it does not require the notion of pseudomonotone operators and abstract surjectivity results for such operators, used in other papers on the Rothe method for other evolutionary variational–hemivariational inequalities. Moreover, a Lipschitz continuous dependence conclusion of the solution on the right-hand side is proved. Finally, a new frictional contact problem for viscoelastic material is discussed, which illustrates an application of the theoretical results.

A class of evolutionary history-dependent variational-hemivariational inequalities with strong and weak solutions

AbstractIn this paper we examine the well-posedness of evolutionary variational-hemivariational inequalities involving a constraint set and history-dependent operators. The strong and weak formulations of such inequalities are studied. First, the existence and uniqueness of solutions to both formulations are proved, and results on the dependence of solution on functional parameters are delivered. Next, the well-posedness is established for a general form of history-dependent variational-hemivariational inequalities with constraints by using a fixed point theorem. Finally, the results are applied to a dynamic frictional contact problem in viscoelasticity in which the contact is described by Signorini-type unilateral boundary condition with a nonmonotone Clarke’s relation.

Second order hemivariational inequality driven by evolution differential inclusion to a dynamic thermoviscoelastic contact problem

In this paper we study an abstract system which consists of a hyperbolic hemivariational inequality coupled with a differential evolution inclusion involving a history-dependent operator in Banach spaces. A hybrid iterative system corresponding to the hemivariational inequality is introduced. Combining the Rothe method, a feedback iterative technique, and a surjectivity result for pseudomonotone operators, we establish existence and a priori estimate for solutions to an approximate problem. Next, through a limiting procedure for solutions of the hybrid iterative system, we show the existence of a mild solution to the original problem. Finally, we apply the main results to a dynamic contact problem in thermoviscoelasticity.

Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems

This paper is devoted to a study of a history-dependent mixed hemivariational-variational inequality arising in contact mechanics. The contact problem concerns the deformation of a viscoelastic body with long memory, subject to a general friction law on one part of the boundary and a frictionless Signorini condition on another part of the boundary. The solution existence and uniqueness of the history-dependent mixed hemivariational-variational inequality based on a Lagrange multiplier approach are proved. Then, a fully discrete scheme is introduced and studied. The trapezoidal rule is used to approximate the integral in the history-dependent operator. For the spatial discretization, the linear finite elements are used to discretize the displacement field, and dual basis functions are used in the approximation of the Lagrange multiplier. Optimal order error estimates are derived for the displacement and the Lagrange multiplier under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence orders.

Convergence Criteria for Fixed Point Problems and Differential Equations

We consider a Cauchy problem for differential equations in a Hilbert space X. The problem is stated in a time interval I, which can be finite or infinite. We use a fixed point argument for history-dependent operators to prove the unique solvability of the problem. Then, we establish convergence criteria for both a general fixed point problem and the corresponding Cauchy problem. These criteria provide the necessary and sufficient conditions on a sequence {un}, which guarantee its convergence to the solution of the corresponding problem, in the space of both continuous and continuously differentiable functions. We then specify our results in the study of a particular differential equation governed by two nonlinear operators. Finally, we provide an application in viscoelasticity and give a mechanical interpretation of the corresponding convergence result.

A class of evolution differential inclusion systems

The main purpose of this paper is to study an abstract system which consists of a non-linear differential inclusion with $C_0$-semigroups and history-dependent operators combined with an evolutionary non-linear inclusion involving pseudomonotone operators, which contains several interesting problems as special cases. We first introduce a hybrid iterative system by using the Rothe method, pseudomonotone operators theory, and a feedback iterative technique. Then, the existence and a priori estimates for solutions to a series of approximating discrete problems are established. Furthermore, through a limiting procedure for solutions of the hybrid iterative system, we show that the existence of solutions to the original problem.

A class of evolution differential inclusion systems

The main purpose of this paper is to study an abstract system which consists of a non-linear differential inclusion with $C_0$-semigroups and history-dependent operators combined with an evolutionary non-linear inclusion involving pseudomonotone operators, which contains several interesting problems as special cases. We first introduce a hybrid iterative system by using the Rothe method, pseudomonotone operators theory, and a feedback iterative technique. Then, the existence and a priori estimates for solutions to a series of approximating discrete problems are established. Furthermore, through a limiting procedure for solutions of the hybrid iterative system, we show that the existence of solutions to the original problem.

A class of Hilfer fractional differential evolution hemivariational inequalities with history-dependent operators

A class of Hilfer fractional differential evolution hemivariational inequalities with history-dependent operators

Error bounds for a class of history-dependent variational inequalities controlled by $\mathcal{D}$-gap~functions

In the present paper, we are concerned with investigating error bounds for history-dependent variational inequalities controlled by the difference gap (for brevity, $\mathcal{D}$-gap) functions. First, we recall a class of elliptic variational inequalities involving the history-dependent operators (for brevity, HDVI). Then, we introduce a new concept of gap functions to the HDVI and propose the regularized gap function for the HDVI via the optimality condition for the concerning minimization problem. Consequently, the $\mathcal{D}$-gap function for the HDVI depends on these regularized gap functions is established. Finally, error bounds for the HDVI controlled by the regularized gap function and the $\mathcal{D}$-gap function are derived under suitable conditions.

Caputo fractional differential variational–hemivariational inequalities involving history-dependent operators: Global error bounds and convergence

In this paper, we introduce and investigate a new differential variational–hemivariational inequality (for brevity, FDVHI) involving history-dependent operators and Caputo fractional order derivative operator. We first propose the new concept of gap functions for the history-dependent variational control system of FDVHI. Furthermore, using the estimate technologies involving Caputo fractional derivatives and history-dependent operators, we establish a RG-function of the Fukushima type (in which RG is a brevity of “regularized gap”) and provide some nice properties of the RG-functions. Then, global error bounds for the FDVHI controlled by the Fukushima RG-function are derived under some suitable assumptions. Finally, by applying penalty methods, we formulate FDVHI to a family penalized problems that are fractional differential variational–hemivariational inequalities without constraints, and establish the convergence result that the solution to the original problem FDVHI can be approached by the solutions of the corresponding penalized problems, as the penalty parameter converges to zero.

Semidiscrete numerical approximation for dynamic hemivariational inequalities with history-dependent operators

In this paper, we are concerned with a class of second-order hemivariational inequalities involving history-dependent operators. For the problem, we first derive a semidiscrete scheme by implicit Euler formula and prove its unique solvability. The existence and uniqueness of a solution to the inequality problem is given by Rothe method. As the core part of the paper, we propose a two-step semidiscrete approximation for the problem, provide its unique solvability and obtain its second-order error estimates. The two-step scheme is more accurate than the standard implicit Euler scheme. Finally, we apply the results to a dynamic frictionless contact problem with long memory.

A general differential quasi variational–hemivariational inequality: Well-posedness and application

In the paper we analyze a general differential quasi variational-hemivariational inequality in Banach spaces which couples the Cauchy problem for an ordinary differential equation with a quasi variational-hemivariational inequality considered with a unilateral constraint and history-dependent operators. The latter appear in all of the data: the governing nonlinear operator, the generalized directional derivative of a locally Lipschitz potential, a convex potential, and the ordinary differential equation. The results on the well-posedness and regularity of solutions are proved by exploiting a fixed point approach. The theory is illustrated by an application to a quasistatic generalized Signorini unilateral frictional contact problem for viscoplastic materials with a nonsmooth multivalued contact condition.

Periodic solutions to history-dependent differential hemivariational inequalities with applications

In this article, we investigate a hybrid model combined by a parabolic differential equation and a parabolic hemivariational inequality (so-called differential hemivariational inequality of parabolic–parabolic type) in general infinite dimensional spaces which includes the history-dependent operator. The solvability of initial value problems as well as the periodic problems of the hemivariational inequality and the differential hemivariational inequality have been proved. In application, we study a contact problem with normal compliance driven by a history-dependent dynamical system.

Second order evolutionary problems driven by mixed quasi-variational–hemivariational inequalities

In this paper we introduce a differential quasi-variational inequality which consists of a second order partial differential equation involving history-dependent operators and a mixed quasi-variational–hemivariational inequality in Banach spaces. At first, by using the KKM theorem and monotonicity arguments, we show that the solution set of the mixed quasi-variational–hemivariational inequality is nonempty, bounded, closed and convex. Then, we establish the measurability and upper semicontinuity of the solution set with respect to the time variable and state variable. Finally, based on the theory of strongly continuous cosine operators and a fixed point theorem for condensing set-valued operators, we build the existence of mild solutions for the differential quasi-variational–hemivariational inequality.

Decay mild solutions of Hilfer fractional differential variational–hemivariational inequalities

The primary objective of this paper is to explore a decay mild solution governed by a class of dynamical systems, called Hilfer fractional differential variational–hemivariational inequality (HFDVHVI, for short), which is composed of a Hilfer fractional evolution differential inclusion and a variational–hemivariational inequality involving two history-dependent operators in the framework of spaces. Our first aim is to investigate the solvability of the mild solutions to (HFDVHVI) by means of fixed point principle. The second step of the paper is to study the existence of decay mild solutions to (HFDVHVI) via giving expression for the Mittag-Leffler function and the Wright function.

Analysis and simulation of an adhesive contact problem governed by fractional differential hemivariational inequalities with history-dependent operator

The main idea of this manuscript is to study an adhesive contact problem with long memory which is governed by a hemivariational inequality and a fractional differential equation. We first prove the existence of a unique solution of the fractional differential hemivariational inequality system. Subsequently, we consider a fully discrete scheme of this system and then focus on deriving error estimates for numerical solutions. To the tail of this manuscript, we present two numerical simulation examples for the adhesive contact problem, which provide numerical evidence to support our theoretical predictions.

Sensitivity analysis of optimal control problems driven by dynamic history-dependent variational-hemivariational inequalities

The aim of this paper is to investigate a nonlinear optimal control problem governed by a complicated dynamic variational-hemivariational inequality (DVHVI, for short) with history-dependent operators in the framework of an evolution triple of spaces. First, we prove a generalized existence and uniqueness theorem to a class of dynamic variational-hemivariational inequalities, which extends the recent results established by Han-Migórski-Sofonea [12, Theorem 9] and Migórski-Bai [23, Theorem 5]. Then, an optimal control problem driven by a (DVHVI) is studied and an existence theorem is delivered. Finally, we explore the sensitivity properties of the optimal control problem under the consideration depending on the initial data ξ∈H and some further parameter λ∈E.

A new class of history-dependent quasi variational–hemivariational inequalities with constraints

In this paper we consider an abstract class of time-dependent quasi variational–hemivariational inequalities which involves history-dependent operators and a set of unilateral constraints. First, we establish the existence and uniqueness of solution by using a recent result for elliptic variational–hemivariational inequalities in reflexive Banach spaces combined with a fixed-point principle for history-dependent operators. Then, we apply the abstract result to show the unique weak solvability to a quasistatic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity and the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip.

Optimal feedback control for a class of fractional evolution equations with history-dependent operators

In this paper, we will study optimal feedback control problems derived by a class of Riemann-Liouville fractional evolution equations with history-dependent operators in separable reflexive Banach spaces. We firstly introduce suitable hypotheses to prove the existence and uniqueness of mild solutions for this kind of Riemann-Liouville fractional evolution equations with history-dependent operators. Then, by introducing a feedback iterative technique and applying Filippov theorem, we show the existence of feasible pairs and optimal control pairs of the optimal feedback control systems with history-dependent operators. Finally, we give some applications to illustrate our main results.

Tykhonov well-posedness of fixed point problems in contact mechanics

We consider a fixed point problem mathcal {S}u=u where {mathcal {S}:C(mathbb{R}_{+};X)to C(mathbb{R}_{+};X)} is an almost history-dependent operator. First, we recall the unique solvability of the problem. Then, we introduce the concept of Tykhonov triple, provide several relevant examples, and prove the corresponding well-posedness results for the considered fixed point problem. This allows us to deduce various consequences which illustrate the stability of the solution with respect to perturbations of the operator mathcal {S}. Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in solid and contact mechanics. To give some examples, we consider two mathematical models which describe the equilibrium of a viscoelastic body in frictionless contact with a foundation. We state the mechanical problems, list the assumptions on the data, and derive their associated fixed point formulation. Then, we illustrate the use of our abstract results in order to deduce the continuous dependence of the solution with respect to the data and parameters. We also provide the corresponding mechanical interpretations.