Besides existence and uniqueness results, convergence results represent an important topic in Functional Analysis, Numerical Analysis, Differential and Partial Differential Equations Theory. Some elementary examples are the following: the continuous dependence of the solution with respect to the data, the convergence of the solution of a penalty problem to the solution of the original problem as the penalty parameter converges, the convergence of the discrete solution to the solution of the continuous problem as the time step or the discretization parameter converges to zero. Convergence results are important in the study of mathematical models which arise in Mechanics and Engineering Sciences, as well. Thus, the convergence of the solution of a contact problem with a deformable foundation to the solution of a contact problem with a rigid foundation as the stiffness coeficient of the foundation goes to infinity, the convergence of the solution of a viscoelastic problem to the solution of an elastic problem as the viscosity vanishes, the convergence of the solution of a frictional problem to the solution of a frictionless problem as the coefficient of friction converges to zero are simple examples, among others.

For all these reasons, a considerable effort was done to obtain convergence results in the study of various mathematical problems including nonlinear equations, inequality problems, fixed point problems and optimization problems, for instance. Most of the convergence results obtained in the literature provide sufficient conditions which guarantee the convergence of a given sequence $\left\{u_n\right\}$ to the solution of the corresponding problem, denoted in what follows by $\mathcal{P}$. In other words, these results do not describe all the sequences which have this property. Therefore, we naturally arrive to consider the following problem.

Problem ${\mathcal{Q}}_{\mathcal{P}}$.Given a metric space $(X,d)$, a Problem $\mathcal{P}$ which has a unique solution $u\in X$, provide necessary and sufficient conditions which guarantee the convergence of an arbitrary sequence $\left\{u_n\right\}\subset X$ to the solution u.

In other words, Problem ${\mathcal{Q}}_{\mathcal{P}}$ consists to provide a convergence criterion to the solution of Problem $\mathcal{P}$.

Note that the solution of Problem ${\mathcal{Q}}_{\mathcal{P}}$ depends on the structure of the original problem $\mathcal{P}$, cannot be provided in this abstract framework, and requires additional assumptions. Results in solving Problem ${\mathcal{Q}}_{\mathcal{P}}$ have been obtained in [7] in the particular case when $\mathcal{P}$ is a variational inequality, in [16] when $\mathcal{P}$ is a minimization problem as well as in [17], in the case when $\mathcal{P}$ is a fixed point problem and an ordinary differential equation in a Banach or Hilbert space.

In this current paper we continue our research in [7,16,17] with the case when $\mathcal{P}$ is an inclusion problem of the form

Here and below in this paper K is a nonempty subset of a real Hilbert space X, $N_K$ represents the outward normal cone of K, $A:X\to X$ is a nonlinear operator and $f\in X$. Our study is motivated by possible applications in Solid and Contact Mechanics, among others. Indeed, a large number of constitutive laws in nonlinear elasticity and plasticity can be cast in the form (1) and so do a number of mathematical models which describes the contact of a deformable body with a foundation. We shall provide such examples in the last two sections of the current paper. Moreover, we refer the reader to [13] as well as to the recent book [14] where inclusions of the form (1) have been considered, together with various applications in Contact Mechanics.

The rest of the manuscrit is structured as follows. In Section 2 we introduce some preliminary material. Then, in Section 3 we state and prove our main result, Theorem 2. It provides necesssary and sufficient conditions which guarantee the convergence of a sequence $\left\{u_n\right\}$ to the solution u of the inclusion (1). Next, in Section 4 we apply Theorem 2 in order to deduce the continuous dependence of the solution with respect to the data K, A and f, and to obtain a convergence result for a penalty problem, as well. In Section 5 we use these convergence results in the study of a specific inclusion problem which describes the frictional contact of an elastic body with a foundation. Finally, in Section 6 we provide an application of the abstract results in Section 4 in the study of two elastic constitutive laws. We complete the results in Section 5 and Section 6 with various mechanical interpretations.

Most of the preliminary results we present here can be found in many books or surveys. For the convenience of the reader we mention here the books [1,3,9,19,20], for instance. There, details on the framework and notation we use as well as additional results in the field can be found.

Everywhere in this paper, unless it is specified otherwise, we use the functional framework described in Introduction. Therefore, X represents a real Hilbert space endowed with the inner product ${(·,·)}_X$ and its associated norm ${\parallel ·\parallel }_X:=\sqrt{{(·,·)}_X}$. The set of parts of X will be denoted by $2^X$ and notation $0_X$ and $I_X$ will represent the zero element and the identity operator of X, respectively. All the limits below are considered as $n\to \infty$, even if we do not mention it explicitely. The symbols “⇀" and “→" denote the weak and the strong convergence in various spaces which will be specified, except in the case when these convergence take place in $\mathbb{R}$. For a sequence $\left\{{\epsilon }_n\right\}\subset {\mathbb{R}}_{+}$ which converges to zero we use the short hand notation $0\le {\epsilon }_n\to 0$. Finally, we denote by $d(u,K)$ the distance between the element $u\in X$ and the set K, that is

For the convenience of the reader we also recall the following definition.

In the study of Problem (1) we consider the following assumptions on the data.

Recall that in (1) and below $N_K:X\to 2^X$ is the outward normal cone of K in the sense of convex analysis and $P_K:X\to K$ represents the projection operator on K. Then, the following equivalences hold, for all $u,\xi \in X$:

Therefore, using (6) it follows that

This equivalence will be repeatedly used in the rest of the manuscript. Moreover, recall that the projection operator is monotone and nonexpansive, i.e.,

In addition, using assumption (3) we deduce that for each $u\in X$ the following equality holds:

On the other hand, it is well known that conditions (4) implies that the operator is invertible and, moreover, its inverse $A^{-1}:X\to X$ is a strongly monotone Lipschitz continuous operator with constants $\frac{m_A}{L_A^2}$ and $\frac{1}{m_A}$, respectively. A proof of this result can be found in [15, p. 23], for instance. Therefore, under assumption (4) the following inequalities hold:

The unique solvability of the inclusion (1) is provided by the following existence and uniqueness result.

Theorem 1 was proved in [13] by using a fixed point argument. There, various convergence results to the solution of thus inequality have been proved and an example arising in Contact Mechanics has been presended.

We now proceed with the following elementary result which will be used in the next section.

Note that the solution of the inclusion (1) depends on the data A, K and f. For this reasons, sometimes below we shall use the notation $u\left(f\right)$ or $u\left(K\right)$, for instance. This dependence was studied in [14] where the following results have been proved.

Note that Propositions 2 and 3 provide sequences $\left\{u_n\right\}\subset X$ which converge to the solution of the inclusion (1). Nevertheless, these proposition do not describe all the sequences which have this property, as it results from the two elementary examples below.

In this section we state and prove a convergence criterion for the solution of the inclusion (1). To this end, under the assumption of Theorem 1, we denote by u the solution of the inclusion (1). Moreover, given an arbitrary sequence $\left\{u_n\right\}\subset X$, we consider the following statements:

Our main result in this section is the following.

We remark that Theorem 2 provides an answer to Problem ${\mathcal{Q}}_{\mathcal{P}}$ in the particular case when Problem $\mathcal{P}$ is the inclusion problem (1). Indeed, it provides a convergence criterion to the solution of this problem.

Theorem 2 is useful to obtain various convergence results in the study of the inclusion (1). In this section we present two types of such results : results concerning the continuous dependence of the solution with respect to the data and a result concerning the convergence of the solution of a penalty problem.

a) We start with a continuous dependence result of the solution with respect to the data A and f. To this end we consider two sequences $\left\{A_n\right\}$ and $\left\{f_n\right\}$ such that

Then, using Theorem 1 it follows that for each $n\in \mathbb{N}$ the exists a unique solution to the inclusion problem.

Moreover, the solution satisfies

Our first result in this section is the following.

b) We proceed with a result which shows the dependence of the solution with respect to the set of constraints. To this end we consider two sequences of $\left\{a_n\right\}\subset \mathbb{R}$ and $\left\{b_n\right\}\subset X$ such that

Then, we define the set $K_n$ by equality and, using Theorem 1 it follows that for each $n\in \mathbb{N}$ the exists a unique solution $u_n$ to the inclusion problem

Moreover, the solution satisfies

Our second result in this section is the following.

c) We now present a convergence result concening a penalty method. To this end we consider a numerical sequence $\left\{{\lambda }_n\right\}$ such that together with the problem of finding $u_n$ such that

Our third result in this section is the following.

In this section we apply the abstract results in Section 3 and Section 4 in the variational analysis of a mathematical model which describes the bilateral contact between an elastic body and a foundation. The classical formulation of the problem is the following.

Problem $\mathcal{M}$.Find a displacement field $u:\mathrm{\Omega }\to {\mathbb{R}}^d$ and a stress field $\mathbf{\sigma }:\mathrm{\Omega }\to {\mathbb{S}}^d$ such that

Here $\mathrm{\Omega }\subset {\mathbb{R}}^d$ ($d\in \{2,3\}$) is a domain with smooth boundary $\mathrm{\Gamma }$ divided into three measurable disjoint parts ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_3$ such that $meas\left({\mathrm{\Gamma }}_1\right)>0$. It represents the reference configuration of the elastic body. Moreover, $\mathbf{\nu }$ is the unit outward normal to $\mathrm{\Gamma }$, ${\mathbb{S}}^d$ denotes the space of second order symmetric tensors on ${\mathbb{R}}^d$ and, below, we use the notation “$·$”, $\parallel ·\parallel$, 0 for the inner product, the norm and the zero element of the spaces ${\mathbb{R}}^d$ and ${\mathbb{S}}^d$, respectively. A generic point in $\mathrm{\Omega }\cup \mathrm{\Gamma }$ will be denoted by $x=\left(x_i\right)$.

We now provide a short description of the equations and boundary conditions in Problem $\mathcal{M}$ and send the reader to [2,5,6,8,10,12] for more details and comments. First, equation (67) represents the constitutive law of the material in which $\mathcal{A}$ is the elasticity operator and $\mathbf{\epsilon }\left(u\right)$ denotes the linearized strain tensor. Equation (68) is the equilibrium equation in which $f_0$ denotes the density of body forces acting on the body. The boundary condition (69) is the displacement condition which models the fact that the body is held fixed on the part ${\mathrm{\Gamma }}_1$ on its boundary. Condition (70) is the traction boundary condition. It models the fact that a traction of density $f_2$ is acting on the part ${\mathrm{\Gamma }}_2$ of the surface of the body. The boundary conditions (71) and (72) are the interface laws on ${\mathrm{\Gamma }}_3$ where the body is assumed to be in contact with an obstacle, the so-called foundation. Here $u_{\nu }$ and $u_{\tau }$ denote the normal and the tangential displacement, respectively, and ${\mathbf{\sigma }}_{\tau }$ is the tangential part of the stress vector $\mathbf{\sigma }\mathbf{\nu }$. Condition (71) shows that the contact is bilateral, i.e., there is no separation between the body and the foundation. Finally, condition (72) represents the Tresca friction law, in which g denotes the friction bound.

In the analysis of Problem $\mathcal{M}$ we use the standard notation for Sobolev and Lebesgue spaces associated to $\mathrm{\Omega }$ and $\mathrm{\Gamma }$. In particular, we use the spaces $L^2{\left(\mathrm{\Omega }\right)}^d$, $L^2{\left({\mathrm{\Gamma }}_2\right)}^d$ and $H^1{\left(\mathrm{\Omega }\right)}^d$, endowed with their canonical inner products and associated norms. Moreover, for an element $v\in H^1{\left(\mathrm{\Omega }\right)}^d$ we still write v for the trace of v to $\mathrm{\Gamma }$ and we denote by $v_{\nu }$ and $v_{\tau }$ its normal and tangential components on $\mathrm{\Gamma }$ given by $v_{\nu }=v·\mathbf{\nu }$ and $v_{\tau }=v-v_{\nu }\mathbf{\nu }$. In addition, recall that ${\mathbf{\sigma }}_{\tau }=\mathbf{\sigma }\mathbf{\nu }-{\sigma }_{\nu }\mathbf{\nu }$ with ${\sigma }_{\nu }=\mathbf{\sigma }\mathbf{\nu }·\mathbf{\nu }$.

Next, for the displacement field we need the space V and for the stress and strain fields we need the space Q, defined as follows:

The spaces V and Q are real Hilbert spaces endowed with the inner products

Here and below $\mathbf{\epsilon }$ represents the deformation operator, i.e., where an index that follows a comma denotes the partial derivative with respect to the corresponding component of x, e.g., $u_{i,j}=\frac{\partial u_i}{\partial x_j}$. The associated norms on these spaces will be denoted by ${\parallel ·\parallel }_V$ and ${\parallel ·\parallel }_Q$, respectively. Recall that the completeness of the space $(V,\parallel ·{\parallel }_V)$ follows from the assumption $meas\left({\mathrm{\Gamma }}_1\right)>0$, which allows the use of Korn’s inequality. Note also that, by the definition of the inner product in the spaces V and Q, we have

In the study of Problem $\mathcal{M}$ we assume that the operator $\mathcal{A}$ satisfies the following condition.

Moreover, the density of body forces and the friction bound are such that

Assume now that $(u,\mathbf{\sigma })$ represents a couple of regular functions which satisfy (67)–(72). Then, using standard arguments it follows that