- Research
- Open access
- Published:
Penalty method for a class of differential nonlinear system arising in contact mechanics
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2022, Article number: 17 (2022)
Abstract
The main goal of this paper is to study a class of differential nonlinear system involving parabolic variational and history-dependent hemivariational inequalities in Banach spaces by using the penalty method. We first construct a penalized problem for such a nonlinear system and then derive the existence and uniqueness of its solution to obtain an approximating sequence for the nonlinear system. Moreover, we prove the strong convergence of the obtained approximating sequence to the solution of the original nonlinear system when the penalty parameter converges to zero. Finally, we apply the obtained convergence result to a long-memory elastic frictional contact problem with wear and damage in mechanics.
-
First part title: Introduction
-
Second part title: Preliminaries
-
Third part title: Convergence result for (1.1)
-
Fourth part title: An application
1 Introduction
Let V, X, Y and W be separable, reflexive Banach spaces, \(V^{*}\) and \(Y^{*}\) be the dual spaces of V and Y, respectively, and \(Y_{1}\) be a separable Hilbert space satisfying \(Y\subset Y_{1}\subset Y^{*}\). Moreover, assume that \(M:V\hookrightarrow X\) is a compact embedding operator, \(K_{V}\) with \(0_{V}\in K_{V}\) and \(K_{Y}\) are nonempty, closed, and convex subsets of V and Y, respectively. Let I be the time interval \([0,T]\) with \(T>0\). Very recently, in order to model an elastic frictional contact problem with long memory, damage, and wear, Chen et al. [4] introduced the following differential nonlinear system driven by a differential equation, a history-dependent hemivariational inequality and a parabolic variational inequality: find \(u:I\rightarrow K_{V}\), \(\zeta :I\rightarrow K_{Y}\) and \(w:I\rightarrow W\) such that, for all \(t\in I=[0,T]\) with \(T>0\),
Moreover, they gave a unique solvability result for (1.1) by using Banach’s fixed-point theorem and applied it to the long-memory elastic frictional contact problem with wear and damage in mechanics.
We would like to mention that (1.1) is an extended model that can be used to describe many real problems such as the long-memory elastic frictional contact problem with wear and damage in mechanics, engineering operation research, network equilibrium problems, and so on [2, 4–8, 23, 30]. Moreover, to choose suitable spaces and maps, many known differential variational inequalities (DVIs) and differential hemivariational inequalities (DHVIs) can be considered as special cases of (1.1) (see, for example, [12–15, 18, 25, 26, 29] and the references therein).
Among the studies on variational inequalities (VIs) and hemivariational inequalities (HVIs), constructing approximating sequences for their solutions and further discussing their convergence analysis are crucially important [10, 11]. It is well known that the penalty method is a kind of efficient approximating method forvarious problems. It is also constantly used for the study of VIs and HVIs (see, for example, [3, 21, 28, 31]). Due to the close relationship with VIs and HVIs, differential variational inequalities (DVIs) and differential hemivariational inequalities (DHVIs) are studied by employing the penalty method, such as Liu and Zeng [16, 17] and Weng et al. [27]. As the generalization of DVIs and DHVIs, the differential variational-hemivariational inequalities (DVHVIs) have drawn the attention of researchers in operations research and contact mechanics. With the penalty method, Tang et al. [25], Liu et al. [16], and Lu et al. [19] recently studied different DVHVIs, obtained their convergence results, and gave the corresponding applications in contact mechanics. However, to the best of our knowledge, there are no results in the literature concerning the penalty method for (1.1). The motivation of the present work is to make an attempt in this direction.
The main goal of this paper is to obtain a convergence result for (1.1) by employing the penalty method. The main contributions of this paper are twofold. First, we construct a penalized problem for (1.1) and show a convergence result, i.e., the solution of (1.1) can be approached as the penalty parameter converges to zero. Secondly, we apply the obtained convergence result to the long-memory elastic frictional contact problem with wear and damage in mechanics.
The rest of the paper is structured as follows. In Sect. 2, we introduce some preliminary materials that will be used in the following sections. In Sect. 3, we construct approximating sequences of solutions to (1.1) by the penalty method and derive its convergence. Finally, in Sect. 4, we apply the obtained convergence result to a long-memory elastic frictional contact problem with wear and damage in mechanics.
2 Preliminaries
Let \((X, \Vert \cdot \Vert _{X})\) be a real Banach space with its dual \(X^{*}\) and \(\langle \cdot ,\cdot \rangle _{X^{*}\times X}\) denote the duality pairing between \(X^{*}\) and X. In this section, we recall some known definitions and lemmas that will be used subsequently (see [20, 22] for more details). Moreover, the symbols “→” and “⇀” represent the strong and weak convergence in various spaces, respectively.
Definition 2.1
A functional \(j:X\rightarrow \mathbb{R}\) is lower semicontinuous if and only if for any convergence sequence \(\{u_{n}\}_{n=1}^{\infty}\subset X\) satisfying \(u_{n}\rightarrow u\in X\), one has \(\liminf_{n\rightarrow \infty} j(u_{n})\geq j(u)\).
Definition 2.2
A functional \(j: X \rightarrow \mathbb{R} \cup \{\infty \}\) is called proper if \(j(v)>-\infty \) for all \(v\in X\) and there exists a point \(u\in X\) such that \(j(u) < +\infty \).
Definition 2.3
Let \(j: X\rightarrow \mathbb{R}\cup \{+\infty \}\) be a proper, convex and lower semicontinuous functional. Define the convex subdifferential of j at u by
Definition 2.4
Let \(j:X\rightarrow \mathbb{R}\) be a locally Lipschitz function. The Clarke directional derivative of j at x in the direction \(v\in X\) is given by
The Clarke subdifferential of j at x is a subset of the dual space \(X^{*}\) defined by
For a set-valued operator \(A:X\rightarrow 2^{X^{*}}\), the graph of A is denoted by \(G(A)\), i.e.,
Definition 2.5
A set-valued operator \(A:X\rightarrow 2^{X^{*}}\) is called monotone if
Moreover, a monotone operator A is called maximal monotone if for any \((u,u^{*}) \in X\times X^{*}\) satisfying
one has \((u,u^{*})\in G(A)\).
For a proper, convex and lower semicontinuous functional \(j: X \rightarrow \mathbb{R} \cup \{\infty \}\), it is well known that \(\partial _{C} j: X\to 2^{X^{*}}\) is maximal monotone.
Definition 2.6
A single-valued operator \(A:X\rightarrow X^{*}\) is said to be
-
(1)
strongly monotone, if there exists \(m_{A}>0\) such that
$$ \langle Au-Av,u-v\rangle _{X^{*}\times X}\geq m_{A} \Vert u-v \Vert _{X}^{2}\quad \text{for all } v\in X ;$$ -
(2)
bounded, if A maps bounded sets of X into bounded sets of \(X^{*}\);
-
(3)
pseudomonotone, if it is bounded and \(u_{n}\rightharpoonup u\) in X with
$$ \limsup_{n\rightarrow \infty}\langle Au_{n},u_{n}-u \rangle _{X^{*} \times X}\leq 0,$$which implies that \(\liminf_{n\rightarrow \infty}\langle Au_{n},u_{n}-v\rangle _{X^{*} \times X}\geq \langle Au,u-v\rangle _{X^{*}\times X}\) for all \(v\in X\);
-
(4)
demicontinuous, if \(u_{n}\rightarrow u\) in X implies that \(Au_{n}\rightharpoonup Au\) in \(X^{*}\);
-
(5)
hemicontinuous at u, if for each \(u,v,w\in X\), \(F(t):=\langle A(u+tv),w\rangle _{X^{*}\times X}\) is continuous on [0,1].
Definition 2.7
An operator \(P:X\rightarrow X^{*}\) is said to be a penalty operator of the set \(K\subset X\) if P is bounded, demicontinuous, monotone, and \(K=\{x\in X\mid Px=0_{X^{*}}\}\), where \(0_{X^{*}}\) represents the zero element of \(X^{*}\).
Lemma 2.1
([20, Proposition 3.23])
If the operator \(A:X\rightarrow X^{*}\) is bounded, demicontinuous, and monotone, then A is pseudomonotone.
Lemma 2.2
([20, Proposition 3.74])
If \(j: X \rightarrow \mathbb{R}\) is a locally Lipschitz function, then for every \(v\in X\), one has
3 Convergence result for (1.1)
In this section, we first use the penalty method to construct a penalized problem of (1.1) and show that the penalized problem has a unique solution by employing Theorem 3.1 of Chen at al. [4]. Then, we show a convergence result that the solution of (1.1) can be approximated by the penalized problem as the penalty parameter converges to 0.
We assume that \((V, H, V^{*} )\) and \((Y, Y_{1}, Y^{*})\) are two Gelfand triplets of Banach spaces that have continuous, compact, and dense embeddings, M is the embedding operator of \(V\hookrightarrow H\), \(M^{*}\) is the adjoint operator of M, and \(\Vert M \Vert \) and \(\Vert M^{*} \Vert \) are the norms of M and \(M^{*}\), respectively. \(K_{V}\) is a convex subset of V. Let \(P:V\rightarrow V^{*}\) be a penalty operator of \(K_{V}\). In order to develop the approximation procedure of (1.1), we need to construct the penalized problem of (1.1). For any given \(\rho >0\), the penalized problem of (1.1) can be constructed as follows.
Problem 3.1
Find \(u_{\rho}:I\rightarrow V\), \(\zeta _{\rho}:I\rightarrow K_{Y}\) and \(w_{\rho}:I\rightarrow W\) such that, for all \(t\in I\),
Remark 3.1
We note that in Problem 3.1, we can consider the penalty operators for both \(K_{V}\) and \(K_{Y}\). Since our main interest is to provide tools in analyzing Problem 3.1, we restrict ourselves to study penalty operators for \(K_{V}\). The case in which \(K_{Y}\) is considered can be solved likewise.
In order to study Problem 3.1, we need the following assumptions on the data.
- H(A)::
-
The operator \(A:I\times V\rightarrow V^{*}\) satisfies
-
(a)
\(A(\cdot , v)\) is continuous on I for any given \(v\in V\);
-
(b)
For any given \(t\in I\), \(A(t,\cdot )\) is hemicontinuous, pseudomonotone, and strongly monotone with \(m_{A}>0\) on V, i.e.,
$$ \bigl\langle A(t, u_{1})-A(t, u_{2}), u_{1}-u_{2} \bigr\rangle _{V^{*} \times V} \geq m_{A} \Vert u_{1}-u_{2} \Vert _{V}^{2},\quad \forall (t,u_{1},u_{2}) \in I\times V\times V;$$ -
(c)
\(A(t,0_{V})=0_{V^{*}}\) for all \(t \in I\).
-
(a)
- H(B)::
-
The operator \(B: I\times V\times Y\rightarrow V^{*}\) satisfies
-
(a)
\(B(\cdot , v,\zeta )\) is continuous on I for any given \(v \in V\) and \(\zeta \in Y\);
-
(b)
\(B(t,\cdot ,\cdot )\) is Lipschitz continuous with \(L_{B}>0\) on \(V\times Y\) for any given \(t\in I\), i.e.,
$$ \begin{aligned} & \bigl\Vert B(t, u_{1},\zeta _{1})-B(t, u_{2},\zeta _{2}) \bigr\Vert _{V^{*}} \leq L_{B} \bigl( \Vert u_{1}-u_{2} \Vert _{V}+ \Vert \zeta _{1}-\zeta _{2} \Vert _{Y} \bigr), \\ &\quad \forall t\in I, \forall u_{1},u_{2} \in V, \forall \zeta _{1},\zeta _{2}\in Y; \end{aligned} $$ -
(c)
There exists \(\varrho \in L^{2}(I;\mathbb{R}^{+})\) such that
$$ \bigl\Vert B(t,u,\zeta ) \bigr\Vert _{V^{*}}\leq \varrho (t) \bigl( \Vert \zeta \Vert _{Y}+ \Vert u \Vert _{V} \bigr), \quad \forall (t,u,\zeta )\in I \times V \times Y.$$
-
(a)
- H(j)::
-
The functional \(j: W\times X\times X\rightarrow \mathbb{R}\) satisfies
-
(a)
\(j(w, u, \cdot )\) is locally Lipschitz on X for any given \((w,u)\in W\times X\);
-
(b)
There exist two constants \(c_{0}, c_{1}>0\) such that
$$ \bigl\Vert \partial j(w,x,y) \bigr\Vert _{X^{*}}\leq c_{1} \bigl(1+ \Vert w \Vert _{W}+ \Vert x \Vert _{X} \bigr)+c_{0} \Vert y \Vert _{X}, \quad \forall (x,y,w)\in X\times X\times W;$$ -
(c)
There exist \(\alpha _{0}>0\) and \(\alpha _{1}>0 \) such that
$$\begin{aligned} & j^{0} (w_{1}, Mu_{1},Mv_{1};Mv_{2}-Mv_{1} )+j^{0} (w_{2}, Mu_{2},Mv_{2};Mv_{1}-Mv_{2} ) \\ &\quad \leq \alpha _{0} \Vert w_{1}-w_{2} \Vert _{W} \Vert v_{1}-v_{2} \Vert _{V}+\alpha _{1} \Vert u_{1}-u_{2} \Vert _{V} \Vert v_{1}-v_{2} \Vert _{V}, \\ & \quad \forall w_{1}, w_{2}\in W, \forall u_{1}, u_{2}, v_{1}, v_{2} \in V. \end{aligned}$$
-
(a)
- H(F)::
-
The operator \(F: I\times W\times V\rightarrow W\) satisfies
-
(a)
\(F(\cdot , w,v)\) is continuous on I for any given \((w,v)\in W\times V\);
-
(b)
\(F(t,\cdot ,\cdot )\) is Lipschitz continuous with \(L_{F} >0\) on \(V\times Y\) for any given \(t\in I\), i.e.,
$$\begin{aligned} & \bigl\Vert F(t,w_{1}, u_{1})-F(t,w_{2}, u_{2}) \bigr\Vert _{W} \leq L_{F} \bigl( \Vert u_{1}-u_{2} \Vert _{V}+ \Vert w_{1}-w_{2} \Vert _{W} \bigr), \\ &\quad \forall t\in I, \forall w_{1},w_{2}\in W, \forall u_{1},u_{2} \in V. \end{aligned}$$
-
(a)
- H(ϕ)::
-
The operator \(\phi : I\times V\times Y\rightarrow Y_{1}\) satisfies
-
(a)
\(\phi (t,\cdot ,\cdot )\) is Lipschitz continuous with \(L_{\phi}>0\) on \(V\times Y\) for any given \(t\in I\), i.e.,
$$ \bigl\Vert \phi (t,u, \zeta )-\phi (t,v, \eta ) \bigr\Vert _{Y_{1}} \leq L_{ \phi} \bigl( \Vert u-v \Vert _{V}+ \Vert \zeta -\eta \Vert _{Y_{1}} \bigr), \quad \forall t\in I, \forall u,v \in V, \forall \zeta ,\eta \in Y;$$ -
(b)
\(\phi (\cdot ,0_{V},0_{Y})\in L^{2}(I; Y_{1})\).
-
(a)
- H(a)::
-
The functional \(a: Y\times Y\rightarrow \mathbb{R}\) is a continuous bilinear symmetric coercive functional and there exist \(a_{1}\in \mathbb{R}\) and \(a_{2}>0\) such that
$$ a(\eta , \eta )+a_{1} \Vert \eta \Vert _{Y_{1}}^{2} \geq a_{2} \Vert \eta \Vert _{Y}^{2}, \quad \forall \eta \in Y.$$
Remark 3.2
By Theorem 3.1 in [4], we know that (1.1) has a unique solution \((\zeta ^{*},u^{*}, w^{*})\in (H^{1}(I;Y_{1})\cap L^{2}(I;Y))\times C(I;K_{Y}) \times C(I;W)\), providing H(A)–H(a) hold with \(m_{A} > \max \{c_{0} \Vert M \Vert _{L(V;X)}^{2},\alpha _{1} \}\).
Remark 3.3
H(j)(c) is equivalent to the following condition
for all \(w_{1},w_{2}\in W\) and all \(u_{1},u_{2},v_{1},v_{2}\in V\) with \(\xi _{i}\in \partial j(w_{i},Mu_{i},Mv_{i})\), \(i=1,2\).
First, to solve the history-dependent hemivariational inequality in Problem 3.1, we consider the following auxiliary problem.
Problem 3.2
For any given \(\rho >0\), \(\zeta \in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\), \(w\in C(I;W)\) and \(f\in C(I;V^{*})\), find \(u_{\rho w\zeta}:I\rightarrow V\) such that, for all \(t\in I\),
Lemma 3.1
Assume that H(A), H(B), and H(j) hold. If \(m_{A} > \max \{c_{0} \Vert M \Vert _{L(V;X)}^{2},\alpha _{1}\}\), then one has the following conclusions:
-
(i)
for any given \(\rho >0\), \(\zeta \in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\), \(w\in C(I;W)\) and \(f\in C(I;V^{*})\), Problem 3.2has a unique solution \(u_{\rho w\zeta}\in C(I;V)\);
-
(ii)
\(u_{\rho w\zeta}\) converges strongly to \(u_{w\zeta}\) as \(\rho \rightarrow 0\), where \(u_{w\zeta}\in C(I;K_{V})\) is the unique solution of the following problem: for any given \(\zeta \in (H^{1}(I;Y_{1})\cap L^{2}(I;Y))\), \(w\in C(I;W)\) and \(f\in C(I;V^{*})\), find \(u_{ w\zeta}:I\rightarrow K_{V}\) such that, for all \(t\in I\),
$$\begin{aligned} & \biggl\langle A \bigl(t,u_{ w\zeta}(t) \bigr) + \int _{0}^{t} B \bigl(t-s,u_{ w\zeta}(s), \zeta (s) \bigr)\,ds,v-u_{ w\zeta}(t) \biggr\rangle _{V^{*}\times V} \\ &\quad\quad {} +j^{0} \bigl(w(t),Mu_{w\zeta}(t),Mu_{w\zeta}(t);Mv(t)-Mu_{w\zeta}(t) \bigr) \\ &\quad \geq \bigl\langle f(t) ,v-u_{ w\zeta}(t) \bigr\rangle _{V^{*} \times V},\quad \forall v\in K_{V}. \end{aligned}$$(3.6)
Proof
(i) Consider a function \(A_{\rho}:I\times V\rightarrow V^{*}\) defined by
Since P is bounded, demicontinuous, monotone, and \(K_{V}=\{u\in V\mid Pu=0_{V^{*}}\}\), it follows from H(A) that \(A_{\rho}(\cdot ,v)\) is continuous for any given \(v\in V\) and \(A_{\rho}(t,\cdot )\) is hemicontinuous, pseudomonotone, and strong monotone with \(A_{\rho}(t,0_{V})=0_{V^{*}}\) for all \(t \in I\). This shows that Problem 3.2 satisfies all the hypotheses of Lemma 3.2 in [4] and so Problem 3.2 has a unique solution \(u_{\rho w\zeta}\in C(I;V)\).
(ii) For fixed \(\eta \in C(I; K_{V} )\), we consider the auxiliary problem for (3.5) as follows: find a map \(u_{\rho w\zeta \eta}:(0, T) \rightarrow V\) such that, for any \(t \in I\) and any \(v \in V\),
where \(f_{\eta}\) is defined by
Let \(u_{0}\in K_{V}\) be fixed. Inserting \(v=u_{0}\) into (3.7), we have
It follows from the strong monotonicity of A that
for all \(t\in [0,T]\). As P is monotone, \(Pv=0\) for all \(v\in K_{V}\) and \(u_{0}\in K_{V}\), one has
for all \(t\in [0,T]\). Thus, Remark 3.3 implies that
From (3.8), (3.9), and H(j)(b), we have
and so
Moreover, A is pseudomonotone, so A is bounded, then there exists a constant N, such that
and H(B)(c) implies that
Combining (3.10), (3.11), and (3.12), one has
which implies that the sequence \(\{u_{\rho w\zeta}(t)\}_{\rho >0}\) is uniformly bounded. Therefore, for any given \(t\in [0,T]\), there exists a subsequence \(\{u_{\rho}(t)\}_{\rho >0}\) such that \(u_{\rho}(t)\rightharpoonup \tilde{u}(t)\) in V as \(\rho \rightarrow 0\) for some \(\tilde{u}(t)\in V\).
Next, we show that \(\tilde{u}\in C(I;K_{V})\). In fact, according to the monotonicity of A, we have
for all \(v\in V\). Taking \(v=\tilde{u}(t)\) into (3.13), one has
Combining H(j), the continuity of \(f_{\eta}\) and the compactness of M, we have
Because of Lemma 2.1, P is pseudomonotone, it follows from (3.13) and (3.14) that
for all \(v\in V\). Since \(v\in V\) is arbitrary, we know that \(P\tilde{u}(t)=0\) and so \(\tilde{u}(t)\in K_{V}\). Moreover, according to (3.7) and \(Pv=0\) for all \(v\in K_{V}\), one has
for all \(v\in K_{V}\). Taking \(v=\tilde{u}(t)\) in (3.15) and passing to the upper limit as \(\rho \rightarrow 0\), we have
Moreover, the pseudomonotonicity of A implies that
Passing to the upper limit as \(\rho \rightarrow 0\) in (3.15), we obtain
Combining (3.16) and (3.17), we have
Since (3.6) has a unique solution, we know that \(\tilde{u}(t)=u_{w\zeta}(t)\) and so \(\tilde{u}\in C(I,K_{V})\).
Finally, we show the strong convergence of \(\{u_{\rho w\zeta}(t)\}\). Indeed, because \(\{u_{\rho w\zeta}(t)\}\) is bounded and for any weakly convergent subsequence of \(\{u_{\rho w\zeta}(t)\}\) converges weakly to the same limit \(u_{w\zeta}(t)\), by Theorem 1.20 in [24], we know that the whole sequence \(\{u_{\rho w\zeta}(t)\}\) converges weakly to \(u_{w\zeta}(t)\) for any \(t\in I\). On the other hand, using the monotonicity of P, one has
Similar to the proof of (3.15), we have
Taking \(v=u_{w\zeta}(t)\) in (3.18) and (3.19), and then passing to the limit as \(\rho \rightarrow 0\), one has
Using \(u_{\rho w\zeta}(t)\rightharpoonup u_{w\zeta}(t)\) in V as \(\rho \rightarrow 0\), it follows from the strong monotonicity of A that
for all \(t\in I\). Consequently, we conclude for each \(t\in I\), \(u_{ \rho w\zeta}(t)\rightarrow u_{w\zeta}(t)\) in V as \(\rho \rightarrow 0\). □
Next, we consider the following auxiliary problem that includes a history-dependent hemivariational inequality and a differential equation in Problem 3.1.
Problem 3.3
For any given \(\rho >0\), \(\zeta \in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\) and \(f\in C(I;V^{*})\), consider the following problem: find \(u_{\rho \zeta}:I\rightarrow V\) and \(w_{\rho \zeta}:I\rightarrow W\) such that, for any \(t\in I\),
Lemma 3.2
Assume that H(A), H(B), H(j), and H(F) hold. If \(m_{A} > \max \{c_{0} \Vert M \Vert _{L(V;X)}^{2},\alpha _{1}\}\), then one has the following conclusions:
-
(i)
for any given \(\rho >0\), \(\zeta \in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\) and \(f\in C(I;V^{*})\), Problem 3.3has a unique solution \((u_{\rho \zeta},w_{\rho \zeta})\in C(I;V)\times C^{1}(I;W)\);
-
(ii)
\((u_{\rho \zeta},w_{\rho \zeta})\) converges strongly to \((u_{\zeta},w_{\zeta})\) as \(\rho \rightarrow 0\), where \((u_{\zeta},w_{\zeta})\) is the unique solution of the following problem: for any given \(\zeta \in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\) and \(f\in C(I;V^{*})\), find \(u_{\zeta}:I\rightarrow K_{V}\) and \(w_{\zeta}:I\rightarrow W\) such that, for any \(t\in I\),
$$ \textstyle\begin{cases} \dot{w}_{\zeta}(t)=F(t,w_{\zeta}(t),u_{\zeta}(t)), \\ \langle A(t,u_{\zeta}(t)) +\int _{0}^{t} B(t-s,u_{\zeta}(s), \zeta (s))\,ds,v-u_{\zeta}(t) \rangle _{V^{*}\times V} \\ \quad {} +j^{0}(w_{\zeta}(t),Mu_{\zeta}(t),Mu_{\zeta}(t);Mv(t)-Mu_{ \zeta}(t))\geq \langle f(t) ,v-u_{\zeta}(t) \rangle _{V^{*}\times V}, \quad \forall v\in K_{V}, \\ w_{\zeta}(0)=w_{0}. \end{cases} $$
Proof
(i) Define an operator \(S:C^{1}(I;W)\rightarrow C(I;V)\) by setting \(S(w_{\rho})(t)=u_{\rho w\zeta}(t)\). From Lemma 3.3 in [4], for each given \(\rho >0\), \(\zeta \in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\),
has a unique solution \(w_{\rho \zeta}\in C^{1}(I;W)\) and \((S(w_{\rho}),w_{\rho \zeta})\in C(I;V)\times C^{1}(I;W)\) is the unique solution of Problem 3.3.
(ii) Consider an operator \(\Lambda : C(I;W)\rightarrow C^{1}(I;K_{V})\) defined as follows:
Then, by the proof of Lemma 3.3 in [4], we know that Λ has a unique fixed point \(w_{\zeta}(t)\). It follows from H(F) that
Now, Gronwall’s inequality yields
Since for each \(s\in I\), \(u_{\rho w\zeta}(s)\rightarrow u_{w\zeta}(s)\) in V as \(\rho \rightarrow 0\) and \(u_{\rho w\zeta},u_{ w\zeta}\in C(I;V)\), one has
Letting \(u_{w\zeta}(t)=u_{\zeta}(t)\) and \(u_{\rho w\zeta}(t)=u_{\rho \zeta}(t)\), we can conclude that
for each \(t\in I\). □
Finally, we only need to solve the following parabolic variational inequality.
Problem 3.4
For any given \(\rho >0\), consider the following problem: find \(\zeta _{\rho}:I\rightarrow K_{Y}\) such that, for all \(t\in I\),
with \(\zeta _{\rho}(0)=\zeta _{0}\).
Lemma 3.3
Suppose that condition H(a) holds. Then, for any given \(\lambda \in L^{2}(I;Y_{1})\), there exists a unique \(\zeta \in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\) such that
with \(\zeta (0)=\zeta _{0}\in K_{Y}\). Moreover, if \(\zeta _{i}\) is the unique solution to problem (3.23) for \(\lambda _{i}\in L^{2}(I;Y_{1})\) with \(i=1,2\), then
with \(d_{1}>0\).
Lemma 3.4
Assume that H(A), H(B), H(j), H(F), H(ϕ), and H(a) hold. If \(m_{A} > \max \{c_{0} \Vert M \Vert _{L(V;X)}^{2},\alpha _{1}\}\), then one has the following conclusions:
-
(i)
for any given \(\rho >0\), Problem 3.4has a unique solution \(\zeta _{\rho}\in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\);
-
(ii)
\(\zeta _{\rho}\) converges strongly to ζ as \(\rho \rightarrow 0\), where ζ is the the unique solution of the following problem: find \(\zeta :I\rightarrow K_{Y}\) such that, for all \(t\in I\),
$$\begin{aligned} & \bigl\langle \dot{\zeta}(t) ,\eta -\zeta (t) \bigr\rangle _{Y^{*}\times Y}+a \bigl( \zeta (t) ,\eta -\zeta (t) \bigr)\geq \bigl\langle \phi \bigl(t,u_{\zeta}(t) ,\zeta (t) \bigr),\eta -\zeta (t) \bigr\rangle _{Y^{*}\times Y}, \\ &\quad \forall \eta \in K_{Y} \end{aligned}$$(3.25)with \(\zeta (0)=\zeta _{0}\).
Proof
(i) Let \(u_{\rho}(t)=u_{\rho \zeta}(t)\) in Lemma 3.2. Then, it follows from Lemma 3.5 in [4] that Problem 3.4 has a unique solution \(\zeta _{\rho}\in H^{1}(I;Y_{1})\cap L^{2}(I;Y)\).
(ii) Let \(\phi _{\zeta}(t):=\phi (t,u_{\zeta}(t),\zeta (t))\). Then, by taking \(\lambda =\phi _{\zeta}\) in Lemma 3.3 and using H(ϕ), we have
Now, Gronwall’s inequality yields
Since for each \(s\in I\), \(u_{\rho \zeta}(s)\rightarrow u_{\zeta}(s)\) in V as \(\rho \rightarrow 0\) and \(u_{\rho \zeta}\in C(I;V)\), \(u_{ w\zeta}\in C(I;K_{V})\), one has
for each \(t\in I\). □
Theorem 3.1
Suppose that the assumptions H(A), H(B), H(j), H(F), H(ϕ), and H(a) hold and \(m_{A} > \max \{c_{0} \Vert M \Vert _{L(V;X)}^{2},\alpha _{1}\}\). Then, one has the following conclusions:
-
(i)
for any given \(\rho >0\), Problem 3.1has a unique solution \((\zeta _{\rho},u_{\rho \zeta},w_{\rho \zeta})\in (H^{1}(I;Y_{1})\cap L^{2}(I;Y)) \times C(I;V)\times C(I;W)\);
-
(ii)
\((\zeta _{\rho},u_{\rho \zeta},w_{\rho \zeta})\) converges strongly to \((\zeta ^{*},u^{*},w^{*})\) as \(\rho \rightarrow 0\), where \((\zeta ^{*},u^{*},w^{*})\) is the unique solution of (1.1).
Proof
Let \((u_{\rho \zeta},w_{\rho \zeta})\) be the same as in Lemma 3.2, \(\zeta _{\rho}\) be the same as in Lemma 3.4, and \((\zeta _{\rho},u_{\rho \zeta},w_{\rho \zeta})=(\zeta ^{*},u^{*},w^{*})\) be the same as in Remark 3.2. Then, it is easy to see that the conclusions (i) and (ii) are true. This finishes the proof. □
4 An application
In this section, we use the abstract results obtained in Sect. 3 to study the long-memory elastic frictional contact problem with wear and damage. To this end, we first recall some notations.
Let \(\mathbb{S}^{d}\) denote the second-order symmetric tensors on \(\mathbb{R}^{d}\). For any given \(\boldsymbol{\sigma}, \boldsymbol{\tau}\in \mathbb{S}^{d}\), define
and \(\Vert \boldsymbol{\tau} \Vert =\sqrt{\boldsymbol{\tau}\cdot \boldsymbol{\tau}}\). We use notations \(\boldsymbol{u}=(u_{i})\), \(\boldsymbol{\sigma}=(\sigma _{ij})\) and \(\boldsymbol{\varepsilon} (\boldsymbol{u})=(\varepsilon _{ij}(\boldsymbol{u}))=(\frac{1}{2}(u_{i,j}+u_{j,i}))\), \(i,j=1,2,\ldots,d\) to denote the displacement vector, the stress tensor and the linearized strain tensor, respectively, where \(u_{i,j}:= \frac{\partial u_{i}}{\partial x_{j}}\). Here and below, the spatial derivative is defined in the sense of distribution. Let Ω be a bounded domain in \(\mathbb{R}^{d}\) (\(d=2,3\)) with Lipschitz continuous boundary \(\Gamma :=\partial \Omega \). Let ν denote the unit outward normal vector defined a.e. on Γ. The normal and tangential components of stress field σ and displacement field u on Γ are denoted by \(\sigma _{\nu}=(\boldsymbol{\sigma \nu})\cdot\boldsymbol{\nu}\), \(u_{\nu}=\boldsymbol{u}\cdot\boldsymbol{\nu}\), \(\boldsymbol{\sigma _{\tau}}=\boldsymbol{\sigma}\boldsymbol{\nu}-\sigma _{\nu}\boldsymbol{\nu}\) and \(\boldsymbol{u_{\tau}}=\boldsymbol{u}-u_{\nu}\boldsymbol{\nu}\), respectively.
Consider a viscoelastic body that occupies Ω. The boundary Γ can be divided into three disjoint measurable parts \(\Gamma _{1}\), \(\Gamma _{2}\), and \(\Gamma _{3}\) with \(\operatorname{meas}(\Gamma _{1})>0\). We are interested in the evolution of the body on the time interval \(I:=[0,T]\) with \(T>0\). We also use the following abbreviations to simplify the notations \(\mathcal{Q}=\Omega \times I\), \(\Sigma =\Gamma \times I\), \(\Sigma _{i}=\Gamma _{i}\times I\), \(i=1,2,3\). The time partial derivative for a function \(f(\boldsymbol{x},t)\) is denoted by \(\dot{f}(\boldsymbol{x},t)\). For the sake of simplicity, we do not mention the dependence of different functions on variable x.
Thus, the long-memory elastic frictional contact problem with wear and damage can be modeled as follows (see [4]).
Problem 4.1
Find a displacement field \(\boldsymbol {u}:\mathcal{Q}\rightarrow \mathbb{R}^{d}\), a stress field \(\boldsymbol{\sigma}:\mathcal{Q}\rightarrow \mathbb{S}^{d}\), a damage field \(\zeta :\mathcal{Q}\rightarrow [0,1]\) and a wear function \(w:\Sigma _{3}\rightarrow \mathbb{R}\) such that
Relations (4.7)–(4.9) show that the body contacts with a rigid foundation covered by a layer of soft material, where \(g>0\) is the thickness of the soft material.
We use the standard Sobolev spaces on Ω and Γ. In particular, let \(H^{1}:=W^{ 1,2}(\Omega ;\mathbb{R}^{d})\) and \(H=L^{2}(\Omega ;\mathbb{R}^{d})\). Let \(V=\{\boldsymbol{v}\in H^{1}\mid \boldsymbol{v}=0\text{ a.e. on }\Gamma _{1}\}\) endowed with the norm
where \(\nabla \boldsymbol{u}=(\frac{\partial u_{i}}{\partial x_{j}})\) for \(i,j=1,\ldots,d\) with \(\boldsymbol{u}\in H^{1}\). Let \(\operatorname{Div}\boldsymbol{\sigma}=(\sigma _{ij,j})=( \frac{\partial \sigma _{ij}}{\partial x_{j}})\) with \(\boldsymbol{\sigma}\in W^{1,2}(\Omega ;\mathbb{S}^{d})\). Then, we have the following Green formula
where
From the assumption of \(\operatorname{meas}(\Gamma _{1}) > 0\), the space V can be endowed with the inner product
which yields the completeness of V and allows us to use Korn’s inequality.
Let \(Y=H^{1}(\Omega ;\mathbb{R})\) and \(Y_{1}=L^{2}(\Omega ;\mathbb{R})\) endowed with the canonical inner products and norms. Denote two convex sets \(K_{V}=\{\boldsymbol{v}\in V\mid v_{\nu}\leq g\text{ a.e. on }\Gamma _{3}\}\) and \(K_{Y}=\{u\in Y\mid 0\leq u\leq 1 \text{ a.e. in }\Omega \}\). We define \(\gamma : V \rightarrow L^{2}(\Gamma _{3};\mathbb{R}^{d})\) as the trace operator and assume that \(j_{\nu}\) and \(j_{\tau}\) admit the regular assumption. Let
Then, the variational formulation of Problem 4.1 can be described as follows (see [4]).
Problem 4.2
Find \(\boldsymbol{u}:I\rightarrow K_{V}\), \(\zeta :I\rightarrow K_{Y}\) and \(w:I\rightarrow L^{2}(\Gamma _{3};\mathbb{R})\) such that, for all \(t \in I\),
Now, we turn to introduce the following penalized problem concerning Problem 4.1.
Problem 4.3
Find a displacement field \(\boldsymbol {u}_{\rho}:\mathcal{Q}\rightarrow \mathbb{R}^{d}\), a stress field \(\boldsymbol{\sigma}_{\rho}:\mathcal{Q}\rightarrow \mathbb{S}^{d}\), a damage field \(\zeta :\mathcal{Q}\rightarrow [0,1]\) and a wear function \(w_{\rho}:\Sigma _{3}\rightarrow \mathbb{R}\) such that
where the operator “+” above a function represents the positive part of it.
It is worth noting that, compared with Problem 4.2, the contact conditions (4.7) and (4.8) are replaced by (4.24) with \(\rho >0\).
Now, we define an operator \(P:V\rightarrow V^{*}\) by
Then, it follows from the arguments in [4] that the variational formulation of Problem 4.3 can be stated as follows.
Problem 4.4
Find \(\boldsymbol{u}_{\rho}:I\rightarrow V\), \(\zeta _{\rho}:I\rightarrow K_{Y}\) and \(w_{\rho}:I\rightarrow L^{2}(\Gamma _{3};\mathbb{R})\) such that, for all \(t \in I\),
In order to solve Problem 4.4, we need the following hypotheses.
\(H(1)\): The elasticity operator \(\mathcal{A}: \Omega \times I \times \mathbb{S}^{d} \rightarrow \mathbb{S}^{d}\) satisfies
\(H(2)\): The relaxation operator \(\mathcal{B}: \Omega \times I \times \mathbb{S}^{d} \times \mathbb{R} \rightarrow \mathbb{S}^{d}\) satisfies
\(H(3)\): The normal compliance function \(p: \Gamma _{3} \times \mathbb{R} \rightarrow \mathbb{R}^{+}\) satisfies
\(H(4)\): The damage source function \(\phi : \Omega \times I\times \mathbb{S}^{d} \times \mathbb{R} \rightarrow \mathbb{R}\) satisfies
\(H(5)\): The normal compliance function \(j_{\nu}: \Gamma _{3} \times \mathbb{R}\times \mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}\) satisfies
\(H(6)\): The normal compliance function \(j_{\tau}: \Gamma _{3} \times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^{d} \rightarrow \mathbb{R}\) satisfies
Moreover, we use the data \(\boldsymbol{f}_{0}\), \(\boldsymbol{f}_{2}\), and α that satisfy the following conditions:
Given Banach spaces \(V=\{\boldsymbol{u}\in H^{1}(\Omega ;\mathbb{R}^{d})\mid \boldsymbol{u}=0 \text{ on }\Gamma _{1}\}\) and \(Y= H^{1}(\Omega ;\mathbb{R})\), and Hilbert spaces \(H=L^{2}(\Omega ;\mathbb{R}^{d})\) and \(Y_{1}=L^{2}(\Omega ;\mathbb{R})\), by the basic embedding theory of Sobolev spaces, we know that \((V,H,V^{*})\) and \((Y,Y_{1},Y^{*})\) form two Gelfand triples. Let \(X=L^{2}(\Gamma _{3};\mathbb{R}^{d}) \) and \(W=L^{2}(\Gamma _{3};\mathbb{R})\).
Now, we give the following theorem.
Theorem 4.1
Suppose that assumptions (4.34)–(4.40) hold and
Then, one has the following conclusions:
-
(i)
Problem 4.2has a unique solution \((\zeta ,\boldsymbol{u}_{\zeta},w_{\zeta})\in (H^{1}(I;Y_{1})\cap L^{2}(I;Y)) \times C(I;K_{V})\times C^{1}(I;W)\); for any given \(\rho >0\), Problem 4.4has a unique solution \((\zeta _{\rho},\boldsymbol {u}_{\rho \zeta},w_{\rho \zeta})\in (H^{1}(I;Y_{1}) \cap L^{2}(I;Y))\times C^{1}(I;K_{V})\times C^{1}(I;W)\);
-
(ii)
\((\zeta _{\rho}\boldsymbol {u}_{\rho \zeta},w_{\rho \zeta})\) converges strongly to \((\zeta ,\boldsymbol{u}_{\zeta},w_{\zeta})\) as \(\rho \rightarrow 0\).
Proof
The unique solvability of Problem 4.2 is the direct conclusion of Theorem 4.1 in [4]. Define operators \(A(t,\cdot ):V\rightarrow V^{*}\), \(B(t,\cdot ,\cdot ):V\times Y\rightarrow V^{*}\), \(P(\cdot ):V\rightarrow V^{*}\), \(F(t,\cdot ,\cdot ):W\times V\rightarrow W\), \(\phi (t,\cdot ,\cdot ):V\times Y\rightarrow Y^{*}\), a functional \(j(\cdot ,\cdot ,\cdot ):W\times X\times X\rightarrow \mathbb{R}\) and a symmetric bilinear form \(a(\cdot ,\cdot ):Y\times Y\rightarrow \mathbb{R}\) by setting
for all \(\boldsymbol{u}, \boldsymbol{v}\in V\), \(w\in W\), \(\zeta , \eta \in Y\) and \(t\in I\). Then, Problem 4.2 can be transformed as follows:
Next, we show that all the conditions of Theorem 3.1 are satisfied. We only need to prove that H(A)(b) is fulfilled and P is a penalty operator because the other assumptions have been testified in [4].
We first show that H(A)(b) is fulfilled. In fact, from (4.34)(c) and Hölder’s inequality, one has
for all \(\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \boldsymbol{v}\in V\) and all \(t\in I\) and so
This shows that \(L_{A}= L_{\mathcal{A}} \) in (4.34)(c), and so condition H(A)(b) holds.
Moreover, we prove that P is a penalty operator of \(K_{V}\). Now, we show that P is monotone. For any given \(r_{1},r_{2},g\in \mathbb{R}\), through simple algebraic calculations we have
and so
This shows that P is monotone. Now, the Sobolev trace theorem states that there exists a positive constant C such that
From Hölder’s inequality, one has
for all \(\boldsymbol{u}_{1}, \boldsymbol{u}_{2}, \boldsymbol{v}\in V\) and so
This shows that P is a bounded and continuous operator. Furthermore, we can see that
and so P is a penalty operator of \(K_{V}\). Thus, we can conclude that Problem 4.4 is equivalent to Problem 3.1 and so Theorem 3.1 ends the proof. □
Availability of data and materials
Not applicable.
References
Barbu, V.: Optimal Control of Variational Inequalities. Pitman, Boston (1984)
Brogliato, B., Tanwani, A.: Dynamical systems coupled with monotone set-valued operators: formalisms applications, well-posedness, and stability. SIAM Rev. 62, 3–129 (2020)
Cai, D.L., Xiao, Y.B.: Convergence results for a class of multivalued variational hemivariational inequality. Commun. Nonlinear Sci. Numer. Simul. 103, 106026 (2021)
Chen, T., Huang, N.J., Li, X.S., Zou, Y.Z.: A new class of differential nonlinear system involving parabolic variational and history-dependent hemi-variational inequalities arising in contact mechanics. Commun. Nonlinear Sci. Numer. Simul. 101, 105886 (2021)
Chen, X.J., Wang, Z.Y.: Differential variational inequality approach to dynamic games with shared constraints. Math. Program. 146(1), 379–408 (2014)
Frémond, M., Nedjar, B.: Damage in concrete: the unilateral phenomenon. Nucl. Eng. Des. 156, 323–335 (1995)
Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual work. Int. J. Solids Struct. 33, 1083–1103 (1996)
Gwinner, J.: On a new class of differential variational inequalities and a stability result. Math. Program. 139(1–2), 205–221 (2013)
Han, J., Migórski, S.: A quasistatic viscoelastic frictional contact problem with multivalued normal compliance, unilateral constraint and material damage. J. Math. Anal. Appl. 443, 57–80 (2016)
Han, W., Migórski, S., Sofonea, M.: A class of variational hemivariational inequalities with applications to frictional contact problems. SIAM J. Math. Anal. 46, 3891–3912 (2014)
Han, W., Zeng, S.: On convergence of numerical methods for variational hemivariational inequalities under minimal solution regularity. Appl. Math. Lett. 93, 105–110 (2019)
Li, W., Wang, X., Huang, N.J.: Differential inverse variational inequalities in finite dimensional spaces. Acta Math. Sci. 35, 407–422 (2015)
Li, X.S., Huang, N.J., O’Regan, D.: Differential mixed variational inequalities in finite dimensional spaces. Nonlinear Anal., Theory Methods Appl. 72, 3875–3886 (2010)
Li, X.W., Liu, Z.H., Sofonea, M.: Unique solvability and exponential stability of differential hemivariational inequalities. Appl. Anal. 99, 2489–2506 (2020)
Liu, Z.H., Motreanu, D., Zeng, S.D.: Nonlinear evolutionary systems driven by mixed variarional inequalities and its applications. Nonlinear Anal., Real World Appl. 42, 409–421 (2018)
Liu, Z.H., Motreanu, D., Zeng, S.D.: Generalized penalty and regularization method for differential variational-hemivariational inequalities. SIAM J. Control Optim. 31, 1158–1183 (2021)
Liu, Z.H., Zeng, S.D.: Penalty method for a class of differential variational inequalities. Appl. Anal. 100, 1574–1589 (2021)
Liu, Z.H., Zeng, S.D., Motreanu, D.: Partial differential hemivariational inequalities. Adv. Nonlinear Anal. 7, 571–586 (2018)
Lu, L., Li, L.J., Sofonea, M.: A generalized penalty method for differential variational-hemivariational inequalities. Acta Math. Sci. 42B, 1–18 (2022)
Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Springer, New York (2013)
Migórski, S., Zeng, S.D.: Penalty and regularization method for variational-hemivariational inequalities with application to frictional contact. Z. Angew. Math. Mech. 98, 1503–1520 (2018)
Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. CRC Press, New York (1994)
Pang, J.S., Stewart, D.E.: Differential variational inequalities. Math. Program. 113, 345–424 (2008)
Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Appications. Monographs and Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton (2018)
Tang, G.J., Cen, J.X., Nguyen, V.T., Zeng, S.D.: Differential variational-hemivariational inequalities: existence, uniqueness, stability, and convergence. J. Fixed Point Theory Appl. 22, 83 (2020)
Wang, X., Qi, Y.W., Tao, C.Q., Xiao, Y.B.: A new class of delay differential variational inequalities. J. Optim. Theory Appl. 172(1), 56–69 (2017)
Weng, Y.H., Chen, T., Huang, N.J.: A new fractional nolinear system driven by a quasi-hemivariational inequality with an application. J. Nonlinear Convex Anal. 22(3), 559–586 (2021)
Xiao, Y.B., Sofonea, M.: Generalized penalty method for elliptic variational-hemivariational inequalities. Appl. Math. Optim. 83, 789–812 (2021)
Zeng, S.D., Liu, Z.H., Migórski, S.: A class of fractional differential hemivariational inequalities with application to contact problem. Z. Angew. Math. Phys. 69, 36 (2018)
Zeng, S.D., Migórski, S., Khan, A.A.: Nonlinear quasi-hemivariational inequalities: existence and optimal control. SIAM J. Control Optim. 59, 1246–1274 (2021)
Zeng, S.D., Migórski, S., Liu, Z.H., Yao, J.C.: Convergence of a generalized penalty method for variational-hemivariational inequalities. Commun. Nonlinear Sci. Numer. Simul. 92, 105476 (2021)
Acknowledgements
The authors are grateful to the editor and the referees for their valuable comments and suggestions.
Funding
This work was supported by the National Natural Science Foundation of China (11671282, 11771067, 12171339, 12171070).
Author information
Authors and Affiliations
Contributions
All authors contributed equally in writing this article. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chu, X., Chen, T., Huang, Nj. et al. Penalty method for a class of differential nonlinear system arising in contact mechanics. Fixed Point Theory Algorithms Sci Eng 2022, 17 (2022). https://doi.org/10.1186/s13663-022-00727-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663-022-00727-6