Generalized penalty method for history-dependent variational–hemivariational inequalities

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Abstract

We consider a history-dependent variational–hemivariational inequality with unilateral constraints in a reflexive Banach space. The unique solvability of the inequality follows from an existence and uniqueness result obtained in Sofonea and Migórski (2016, 2018). In this current paper we introduce and study a generalized penalty method associated to the inequality. To this end we consider a sequence of generalized penalty problems, governed by a parameter λn and an operator Pn. We prove the unique solvability of the penalty problems as well as the convergence of corresponding solutions sequence to the solution of original problem. These results extend the previous results in Sofonea et al. (2018) and Xiao and Sofonea (2019). Finally, we illustrate them in the study of a history-dependent problem with unilateral boundary conditions which describes the quasistatic evolution of a rod–spring system under the action of given applied force.

Introduction

A large number of processes which arise in Mechanics, Physics and Engineering Sciences are described by boundary value problems which, in a weak formulation, lead to mathematical models expressed in terms of variational or hemivariational inequalities. Variational inequalities refer to those inequality problems which have a convex structure. They have been studied extensively for over half a century since 1960s, both theoretically and numerically, by using arguments of convex analysis. Representative references in the field include [1], [2], [3] and, more recently, [4], [5], [6], [7], [8], [9], [10]. The notion of hemivariational inequality was introduced in [11] in the study of engineering problems involving non-smooth, non-monotone and possibly multivalued relations for deformable bodies. Since then, the theory of hemivariational inequalities grew rapidly. It uses as main ingredient the properties of the subdifferential in the sense of Clarke, defined for locally Lipschitz functions which may be nonconvex. Comprehensive references in the area include [12], [13], [14] and, more recently, [15], [16], [17], [18], [19], [20]. Finally, variational–hemivariational inequalities represent a special class of inequalities, driven by both convex and nonconvex functions. They represent a powerful tool in the study of a wide range of nonlinear boundary value problems with or without unilateral constraints, as shown in [21], [22], [23], [24], [25].

History-dependent operators represent a class of nonlinear operators defined on spaces of vector-valued continuous or Lebesgue integrable functions. They arise in Contact Mechanics and describe various memory effects which appear either in the material’s behaviour or in the contact conditions. Variational–hemivariational inequalities involving in their structure a history-dependent operator are called history-dependent variational–hemivariational inequalities. They have been intensively studied in the recent literature. General existence and uniqueness results can be found in [26], [27], together with various applications in Contact Mechanics. A convergence result which shows the continuous dependence of the solution with respect to the data was obtained in [28]. The numerical analysis of history-dependent variational–hemivariational inequalities can be found in [29], [30]. There, numerical schemes have been considered and error estimates have been derived. Additional results on inequality problems with history-dependent operators can be found in [26], [31], [32], [33].

The current paper represents a continuation of [34], [35]. Indeed, the paper [34] was devoted to the study of a penalty method for history-dependent variational–hemivariational inequalities. There, the corresponding unconstrained problems have been constructed with a given penalty operator P and a convergence result was proved. In [35] we studied a generalized penalty method in the study of elliptic variational–hemivariational inequalities, i.e., time-independent variational–hemivariational inequalities. There, in contrast to [34], the unconstrained problems have been constructed with a sequence of penalty operators, denoted {Pn}.

The aim of the current paper is twofold. The first one is to use the generalized penalty method in the study of history-dependent variational–hemivariational inequalities. Thus, our main convergence result, Theorem 16, extends our previous work in [34], [35], since these results can be obtained in the particular cases when Pn=P for each nN and when the time is removed, respectively. Recall also that the proof of Theorem 16 is based on assumptions on the locally Lipschitz function j which are less restrictive than that used in [34], [35]. This ingredient represents one of the traits of novelty of this contribution, as we mention in Remark 20. The second aim of the current paper is to illustrate the use of Theorem 16 in the study of a new mathematical model of contact and to provide the corresponding mechanical interpretations.

The outline of the paper is as follows. Basic notation and preliminary material needed in the rest of the paper are recalled in Section 2. In Section 3 we state the original inequality problem and the penalty problems, together with their unique solvability. Then, in Section 4 we state and prove our main result, Theorem 16, which states that the sequence of solutions of the generalized penalty problems converges to the solution of the original problem. The proof is carried out in several steps, based on arguments of compactness, pseudomonotonicity and the properties of the Clarke subdifferential. Finally, in Section 5 we illustrate the use of this abstract convergence result in the study of a nonlinear boundary value problem which describes the quasistatic evolution of a rod–spring system with unilateral constraints.

Section snippets

Background material

In this section we shortly recall some notation and preliminaries which are needed in the rest of the paper. For more details on the material presented below we refer the reader to [10], [12], [36], [37], [38].

Everywhere in this paper X represents a reflexive Banach space with dual X and , denotes the duality between X and X. We use the notations X and X for the norm on the spaces X and X, respectively, and 0X for the zero element of X. Throughout the paper all the limits, upper

Problem statement and well-posedness results

We now turn to the inequality problem we consider in this paper. The functional framework is the following. Let K be a subset of X and let (Y,Y) be a normed space. Given two operators A:XX and S:C(R+;X)C(R+;Y), a function φ:Y×X×XR, a locally Lipschitz function j:XR and a function f:R+X, we consider the following history-dependent variational–hemivariational inequality.

Problem 11

Find uC(R+;K) such that, for all tR+, Au(t),vu(t)+φ((Su)(t),u(t),v)φ((Su)(t),u(t),u(t))+j0(u(t);vu(t))f(t),vu

A convergence result

In the section we move to the convergence of solution to generalized penalty problem. To this end, besides the assumptions introduced in the previous section, we consider the following assumptions.

(H0)̲ :  λn0 as n.

(H1)̲ : For each vK, there exists a sequence {vn}X such that Pnvn=0X for each nNandvnvinX as n.

(H2)̲ : There exists an operator P:XX such that:

  • (a)

    for any sequence {un} satisfying unu in X and lim supPnun,unu0 we have lim infPnun,unvPu,uv for allvX;

  • (b)

    Pu=0X implies

A spring-rod system with unilateral constraints

The abstract results presented in Sections 3 Problem statement and well-posedness results, 4 A convergence result in this paper are useful in the study of various mathematical models which describe the equilibrium of viscoelastic or viscoplastic bodies in unilateral contact with a foundation. In this section we present a one-dimensional example which illustrates the applicability of these results.

Consider a viscoelastic rod which occupies the interval [0,L] on the Ox axis with L>0. The rod is

Acknowledgements

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement No. 823731 — CONMECH. It is supported by the National Science Centre of Poland under Maestro Project No. UMO-2012/06/A/ST1/00262, National Science Centre of Poland under Preludium Project No. 2017/25/N/ST1/00611, and International Project co-financed by the Ministry of Science and Higher Education of Republic of Poland under Grant No.

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