Solving Quasistatic Contact Problems Using Nonsmooth Optimization Approach

1 Introduction

Currently various engineering applications require scientific analysis of contact phenomena. In order to study such problems a framework in the form of variational and hemivariational inequalities complemented by theory of Finite Element Method emerged. Literature in the field of Contact Mechanics grows rapidly. Main ideas and mathematical tools were introduced in monographs [6, 14, 17, 18, 20]. Other comprehensive studies include [10, 19, 21, 22]. Theory of nonsmooth optimization used in our approach was presented in [1] and applied to a sample static contact problem in [2]. Numerical analysis of contact problems can be found for instance in papers [3, 8, 9, 12, 13, 16] and for a recent study [11].

This paper is a continuation of our previous work [15], where we reformulated time-independent hemivariational inequality as optimization problem and used it to solve a static mechanical contact problem. Here, we consider time-dependent hemivariational inequality, and use similar idea to solve it numerically. In this case we prove existence and uniqueness of a solution using fixed point argument and apply this abstract framework to solve a quasistatic mechanical contact problem. This approach allows us to substitute Coulomb’s law of dry friction by more general, nonmonotone friction law with dependence on tangential component of velocity. Additionally, the friction bound is determined by normal component of displacement, describing penetration of the foundation.

The rest of this paper is organized as follows. In Section 2 we consider an abstract problem in the form of hemivariational inequality and show that under appropriate assumptions it has a unique solution. Section 3 contains a discrete scheme which approximates solution to introduced abstract problem. We also prove the theorem concerning numerical error estimate. An application of presented theory to a mechanical contact model is introduced in Section 4. Finally, in Section 5, we describe computational algorithm used to solve mechanical contact problem. Then we present results of simulations and empirical error estimation for a set of sample data.

2 Hemivariational inequality

We start with basic notation used in this paper. For a normed space $X$ , we denote by $∥ \cdot ∥_{X}$ its norm, by $X^{*}$ its dual space and by $⟨ \cdot, \cdot ⟩_{X^{*} \times X}$ the duality pairing of $X^{*}$ and $X$ . By $c > 0$ we denote a generic constant (value of $c$ may differ in different equations).

Let us now assume that $j : X \to R$ is locally Lipschitz continuous. The generalized directional derivative of $j$ at $x \in X$ in the direction $v \in X$ is defined by

j^{0} (x; v) := limsup y \to x, λ ↘ 0 \frac{j (y + λ v) - j (y)}{λ} .

The generalized subdifferential of $j$ at $x$ is a subset of the dual space $X^{*}$ given by

\partial j (x) := {ξ \in X^{*} | ⟨ ξ, v ⟩_{X^{*} \times X} \leq j^{0} (x; v) for all v \in X} .

If $j : X^{n} \to R$ is a locally Lipschitz function of $n$ variables, then we denote by $\partial_{i} j$ and $j_{i}^{0}$ the Clarke subdifferential and generalized directional derivative with respect to $i$ -th variable of $j$ , respectively.

Let now $V$ be a reflexive Banach space and $X$ be a Banach space. Let $γ \in L (V, X)$ be a linear and continuous operator from $V$ to $X$ , and $c_{γ} := ∥ γ ∥_{L (V, X)}$ . We denote by $γ^{*} : X^{*} \to V^{*}$ the adjoint operator to $γ$ . Let $[0, T]$ be a time interval with $T > 0$ . Let $A, B : V \to V^{*}$ , $J : X \times X \to R$ , $f : [0, T] \to V^{*}$ and let $u_{0} \in V$ . The functional $J$ is assumed to be locally Lipschitz continuous with respect to its second argument. We formulate the considered hemivariational inequality as follows.

Problem $P$ : Find $u : [0, T] \to V$ such that for all $t \in [0, T]$ ,

⟨ A u^{'} (t) + B u (t), w ⟩_{V^{*} \times V} + J_{2}^{0} (γ u (t), γ u^{'} (t); γ w) \geq ⟨ f (t), w ⟩_{V^{*} \times V} \forall w \in V,

and

u (0) = u_{0} .

We introduce the following assumptions.

$H (A) - ---- - :$ Operator $A : V \to V^{*}$ satisfies

$A$ is linear and bounded,
$A$ is symmetric, i.e., $⟨ A u, v ⟩_{V^{*} \times V} = ⟨ A v, u ⟩_{V^{*} \times V}$ for all $u, v \in V$ ,
there exists $m_{A} > 0$ such that $⟨ A u, u ⟩_{V^{*} \times V} \geq m_{A} ∥ u ∥_{V}^{2}$ for all $u \in V$ .

$H (B) - ---- - :$ Operator $B : V \to V^{*}$ is Lipschitz continuous, i.e., there exists $L_{B} > 0$ such that
a $∥ B v - B w ∥_{V^{*}} \leq L_{B} ∥ v - w ∥_{V} for all v, w \in V$ .

$H (J) - ---- - : J : X \times X \to R$ satisfies

$J (v, \cdot)$ is locally Lipschitz continuous for all $v \in V$ ,
there exist $c_{0}, c_{1}, c_{2} \geq 0$ such that $∥ \partial_{2} J (w, v) ∥_{X^{*}} \leq c_{0} + c_{1} ∥ w ∥_{X} + c_{2} ∥ v ∥_{X}$
for all $w, v \in X$ ,
there exist $m_{J 1}, m_{J 2} \geq 0$ such that
$J_{2}^{0} (w_{1}, v_{1}; v_{2} - v_{1}) + J_{2}^{0} (w_{2}, v_{2}; v_{1} - v_{2}) \leq m_{J 1} ∥ v_{1} - v_{2} ∥_{X}^{2} + m_{J 2} ∥ w_{1} - w_{2} ∥_{X} ∥ v_{1} - v_{2} ∥_{X}$
for all $w_{1}, w_{2}, v_{1}, v_{2} \in X$ .

$H (f) - ---- - : f \in C ([0, T]; V^{*})$ .

$H (u_{0}) - ----- - : u_{0} \in V$ .

$(H_{s}) - --- - : m_{A} - 2 m_{J 1} c_{γ}^{2} > 0$ .

We remark that assumption $H (J)$ (c) is equivalent to the following condition

⟨ \partial_{2} J (w_{1}, v_{1}) - \partial_{2} J (w_{2}, v_{2}), v_{1} - v_{2} ⟩_{X^{*} \times X} \geq - m_{J 1} ∥ v_{1} - v_{2} ∥_{X}^{2} - m_{J 2} ∥ w_{1} - w_{2} ∥_{X} ∥ v_{1} - v_{2} ∥_{X}

for all $w_{1}, w_{2}, v_{1}, v_{2} \in X$ . In the special case when $J$ does not depend on its first variable, we obtain a relaxed monotonicity condition, i.e. for all $v_{1}, v_{2} \in X$

⟨ \partial J (v_{1}) - \partial J (v_{2}), v_{1} - v_{2} ⟩_{X^{*} \times X} \geq - m_{J 1} ∥ v_{1} - v_{2} ∥_{X}^{2} .

(2.1)

So, condition $H (J)$ (c) is a generalization of $(???)$ .

Define an operator $K : L^{2} (0, T; V) \to C ([0, T]; V)$ by the formula

K v (t) = \int_{0}^{t} v (s) d s + u_{0} .

We can reformulate Problem $P$ as the following one.

Problem $P_{h v i}$ : Find $v : [0, T] \to V$ such that for all $t \in [0, T]$ ,

⟨ A v (t) + B (K v) (t), w ⟩_{V^{*} \times V} + J_{2}^{0} (γ (K v) (t), γ v (t); γ w) \geq ⟨ f (t), w ⟩_{V^{*} \times V} \forall w \in V .

Let us now present the following theorem.

Theorem 1

Under assumptions $H (A)$ , $H (B)$ , $H (J)$ , $H (f)$ , $H (u_{0})$ and $(H_{s})$ , Problem $P_{h v i}$ has a unique solution $v \in C ([0, T]; V)$ .

Proof. We use a fixed point argument. Given $η \in C ([0, T]; V)$ , define

y_{η} = K η .

Then $y_{η} \in C ([0, T]; V)$ . Consider the auxiliary problem of finding a function $v_{η} : [0, T] \to V$ such that

⟨ A v_{η} (t), w ⟩_{V^{*} \times V} + J_{2}^{0} (γ y_{η} (t), γ v_{η} (t); γ w) \geq ⟨ f (t) - B y_{η} (t), w ⟩_{V^{*} \times V}

(2.2)

for all $w \in V$ . Applying Theorem 4.2 in [11] with $φ \equiv 0$ , we know that there exists a unique element $v_{η} (t) \in V$ which solves this inequality for each $t \in [0, T]$ . Let us show that the function $v_{η} \in C ([0, T]; V)$ . For simplicity, with $t_{1}, t_{2} \in [0, T]$ , denote $v_{η} (t_{i}) = v_{i}$ , $y_{η} (t_{i}) = y_{i}$ , $f (t_{i}) = f_{i}$ for $i = 1, 2$ . We get that

	$⟨ A v_{1}, w ⟩_{V^{*} \times V} + J_{2}^{0} (γ y_{1}, γ v_{1}; γ w)$	$\geq ⟨ f_{1} - B y_{1}, w ⟩_{V^{*} \times V},$		(2.3)
	$⟨ A v_{2}, w ⟩_{V^{*} \times V} + J_{2}^{0} (γ y_{2}, γ v_{2}; γ w)$	$\geq ⟨ f_{2} - B y_{2}, w ⟩_{V^{*} \times V} .$		(2.4)

Taking $w = v_{2} - v_{1}$ in (2.3), $w = v_{1} - v_{2}$ in (2.4) and adding, we get

	$⟨ A v_{1} - A v_{2}, v_{1} - v_{2} ⟩_{V^{*} \times V}$	$\leq J_{2}^{0} (γ y_{1}, γ v_{1}; γ v_{2} - γ v_{1}) + J_{2}^{0} (γ y_{2}, γ v_{2}; γ v_{1} - γ v_{2})$
		$+ ⟨ f_{1} - B y_{1} - f_{2} + B y_{2}, v_{1} - v_{2} ⟩_{V^{*} \times V} .$

Using strong monotonicity of the operator $A$ guaranteed by $H (A)$ (c), assumptions $H (B)$ , $H (J)$ (c) and $H (f)$ we obtain

	$m_{A} ∥ v_{1} - v_{2} ∥_{V}^{2}$	$\leq m_{J 1} ∥ γ v_{1} - γ v_{2} ∥_{X}^{2} + m_{J 2} ∥ γ y_{1} - γ y_{2} ∥_{X} ∥ γ v_{1} - γ v_{2} ∥_{X}$
		$+ (∥ f_{1} - f_{2} ∥_{V^{*}} + L_{B} ∥ y_{1} - y_{2} ∥_{V}) ∥ v_{1} - v_{2} ∥_{V}$
		$\leq m_{J 1} c_{γ}^{2} ∥ v_{1} - v_{2} ∥_{V}^{2} + (∥ f_{1} - f_{2} ∥_{V^{*}} + (m_{J 2} c_{γ}^{2} + L_{B}) ∥ y_{1} - y_{2} ∥_{V}) ∥ v_{1} - v_{2} ∥_{V},$

i.e.,

(m_{A} - m_{J 1} c_{γ}^{2}) ∥ v_{1} - v_{2} ∥_{V} \leq ∥ f_{1} - f_{2} ∥_{V^{*}} + (m_{J 2} c_{γ}^{2} + L_{B}) ∥ y_{1} - y_{2} ∥_{V} .

(2.5)

By the smallness assumption $(H_{s})$ , $m_{A} - m_{J 1} c_{γ}^{2} > 0$ , and from continuity of $y_{η}$ and $f$ , we deduce that $[0, T] ∋ t \mapsto v_{η} (t) \in V$ is a continuous function. This allows us to define an operator $Λ : C ([0, T]; V) \to C ([0, T]; V)$ via the relation

Λη=vηfor\ all\ η∈C([0,T];V).

Let us prove that the operator $Λ$ has a unique fixed point $η^{*} \in C ([0, T]; V)$ . For two arbitrary functions $η_{1}, η_{2} \in C ([0, T]; V)$ , let $v_{i}$ be the solution of (2.2) for $η = η_{i}$ , $i = 1, 2$ . Similarly to (2.5), we have

∥ v_{1} (t) - v_{2} (t) ∥_{V} \leq c ∥ y_{1} (t) - y_{2} (t) ∥_{V} .

Since

y_{1} (t) - y_{2} (t) = \int_{0}^{t} (η_{1} (s) - η_{2} (s)) d s,

we derive from the previous inequality that

∥ Λ η_{1} (t) - Λ η_{2} (t) ∥_{V} \leq c \int_{0}^{t} ∥ η_{1} (s) - η_{2} (s) ∥_{V} d s .

This shows that $Λ$ is a history dependent operator. Applying Theorem 3.20 in [11], we know that the operator $Λ$ has a unique fixed point $η^{*} \in C ([0, T]; V)$ . Moreover, by the definition of $Λ$ , $η^{*}$ is a solution to Problem $P_{h v i}$ . Uniqueness of a solution to Problem $P_{h v i}$ is a consequence of uniqueness of fixed point.

3 Numerical scheme

Let us now fix $h > 0$ and let $V^{h} \subset V$ be a finite dimensional space with a discretization parameter $h > 0$ . For a given $N \in N$ , we introduce the time step $k = T / N$ and the temporal nodes $t_{j} = j k$ , $0 \leq j \leq N$ . Let $u_{0}^{h} \in V^{h}$ be an approximation of element $u_{0}$ in space $V^{h}$ . We write $v^{h k} = {v_{j}^{h k}}_{j = 1}^{N} \subset V^{h}$ and define

(K^{k} v^{h k})_{0} = u_{0}^{h} (K^{k} v^{h k})_{j} = k j \sum i = 1 v_{i}^{h k} + u_{0}^{h}, 1 \leq j \leq N .

We present the following discretized version of Problem $P_{h v i}$ in the form of an operator inclusion problem.

Problem $P_{i n c l}^{h}$ : Find $v^{h k} = {v_{j}^{h k}}_{j = 1}^{N} \subset V^{h}$ such that for all $j \in {1, \dots, N}$

A v_{j}^{h k} + B (K^{k} v^{h k})_{j - 1} + γ^{*} \partial_{2} J (γ (K^{k} v^{h k})_{j - 1}, γ v_{j}^{h k}) ∋ f_{j} .

Now we introduce some preliminary material, namely we recall a special case of the Jensen inequality and the discrete version of Gronwall inequality. On several occasions, we will also apply the elementary inequality

(l \sum i = 1 a_{i})^{2} \leq l l \sum i = 1 a_{i}^{2}, a_{1}, \dots, a_{l} \in R .

(3.1)

Lemma 2

(the Jensen inequality) Let $I \subset R$ be a set of positive measure and let $f : I \to R$ be an integrable function. Then

(\frac{1}{| I |} \int_{I} f (s) d s)^{2} \leq \frac{1}{| I |} \int_{I} (f (s))^{2} d s .

In particular, for $t_{m}, t_{m + 1} \in I$ , $t_{m} < t_{m + 1}$ , $t_{m + 1} - t_{m} = k$ , we have

(\int_{t_{m}}^{t_{m + 1}} f (s) d s)^{2} \leq k \int_{t_{m}}^{t_{m + 1}} (f (s))^{2} d s .

(3.2)

Lemma 3

(the Gronwall inequality, [10, Lemma 7.25]) Let $T$ be given. For $N > 0$ we define $k = T / N$ . Let ${g_{n}}_{n = 1}^{N}$ , ${e_{n}}_{n = 1}^{N}$ be two nonnegative sequences satisfying for $c > 0$ and for all $n \in {1, \dots, N}$

e_{n} \leq c g_{n} + c k n - 1 \sum j = 1 e_{j} .

Then there exists a constant $^c > 0$ such that

max 1 \leq n \leq N e_{n} \leq^c max 1 \leq n \leq N g_{n} .

Let us now prove the following lemma.

Lemma 4

Under assumptions $H (A)$ , $H (B)$ , $H (J)$ , $H (f)$ , $H (u_{0})$ and $(H_{s})$ if Problem $P_{i n c l}^{h}$ has a solution $v^{h k}$ , then it is unique and satisfies

∥ v_{j}^{h k} ∥_{V} \leq c (1 + ∥ f_{j} ∥_{V^{*}})

(3.3)

for all $j \in {1, \dots, N}$ with a positive constant $c$ .

Proof. To simplify the notation in this proof, we write $v$ instead of $v^{h k}$ . Let $v$ be a solution to Problem $P_{i n c l}^{h}$ and let us fix $j \in {1, \dots, N}$ . This means that there exists $ζ_{j} \in \partial_{2} J (γ (K^{k} v)_{j - 1}, γ v_{j})$ such that

A v_{j} + B (K^{k} v)_{j - 1} + γ^{*} ζ_{j} = f_{j} .

From the definition of generalized subdifferential of $J (γ (K^{k} v)_{j - 1}, \cdot)$ we have for all $w^{h} \in V^{h}$ ,

	$⟨ f_{j} - A v_{j} - B (K^{k} v)_{j - 1}, w^{h} ⟩_{V^{*} \times V}$	$= ⟨ γ^{} ζ_{j}, w^{h} ⟩_{V^{} \times V}$
		$= ⟨ ζ_{j}, γ w^{h} ⟩_{X^{*} \times X}$
		$\leq J_{2}^{0} (γ (K^{k} v)_{j - 1}, γ v_{j}; γ w^{h}) .$

After reformulation, we obtain discretized version of hemivariational inequality $P_{h v i}$ ,

⟨ A v_{j} + B (K^{k} v)_{j - 1}, w^{h} ⟩_{V^{*} \times V} + J_{2}^{0} (γ (K^{k} v)_{j - 1}, γ v_{j}; γ w^{h}) \geq ⟨ f_{j}, w^{h} ⟩_{V^{*} \times V} .

(3.4)

Let us now assume that Problem $P_{i n c l}^{h}$ has two solutions $v^{1}$ and $v^{2}$ . We will prove inductively that these solutions are equal. For $j = 1$ we get $(K^{k} v^{1})_{0} = (K^{k} v^{2})_{0} = u_{0}^{h}$ . In inequality (3.4) for a solution $v^{1}$ we set $w^{h} = v_{1}^{2} - v_{1}^{1}$ , and for a solution $v^{2}$ we set $w^{h} = v_{1}^{1} - v_{1}^{2}$ . Then adding these inequalities, we obtain

⟨ A v_{1}^{1} - A v_{1}^{2}, v_{1}^{1} - v_{1}^{2} ⟩_{V^{*} \times V} \leq J_{2}^{0} (γ u_{0}^{h}, γ v_{1}^{1}; γ v_{1}^{2} - γ v_{1}^{1}) + J_{2}^{0} (γ u_{0}^{h}, γ v_{1}^{2}; γ v_{1}^{1} - γ v_{1}^{2}) .

Hence, $H (A)$ (c) and $H (J)$ (c) yield

(m_{A} - m_{J 1} c_{γ}^{2}) ∥ v_{1}^{1} - v_{1}^{2} ∥_{V}^{2} \leq 0,

and consequently from $(H_{s})$ we have $v_{1}^{1} = v_{1}^{2}$ . We now show that if $v_{i}^{1} = v_{i}^{2}$ for $i = 1, \dots, j - 1$ , then $v_{j}^{1} = v_{j}^{2}$ . Similarly, in (3.4) for a solution $v^{1}$ we set $w^{h} = v_{j}^{2} - v_{j}^{1}$ , and for a solution $v^{2}$ we set $w^{h} = v_{j}^{1} - v_{j}^{2}$ . Adding these inequalities we obtain

	$⟨ A v_{j}^{1} - A v_{j}^{2}, v_{j}^{1} - v_{j}^{2} ⟩_{V^{*} \times V}$	$\leq ⟨ B (K^{k} v^{1})_{j - 1} - B (K^{k} v^{2})_{j - 1}, v_{j}^{2} - v_{j}^{1} ⟩_{V^{*} \times V}$
		$+ J_{2}^{0} (γ (K^{k} v^{1})_{j - 1}, γ v_{j}^{1}; γ v_{j}^{2} - γ v_{j}^{1}) + J_{2}^{0} (γ (K^{k} v^{2})_{j - 1}, γ v_{j}^{2}; γ v_{j}^{1} - γ v_{j}^{2}) .$

We observe that $(K^{k} v^{1})_{j - 1} = (K^{k} v^{2})_{j - 1}$ , hence

⟨ A v_{j}^{1} - A v_{j}^{2}, v_{j}^{1} - v_{j}^{2} ⟩_{V^{*} \times V} \leq J_{2}^{0} (γ (K^{k} v^{1})_{j - 1}, γ v_{j}^{1}; γ v_{j}^{2} - γ v_{j}^{1}) + J_{2}^{0} (γ (K^{k} v^{2})_{j - 1}, γ v_{j}^{2}; γ v_{j}^{1} - γ v_{j}^{2}) .

Again $H (A)$ (c) and $H (J)$ (c) yield

(m_{A} - m_{J 1} c_{γ}^{2}) ∥ v_{j}^{1} - v_{j}^{2} ∥_{V}^{2} \leq 0.

Under assumption $(H_{s})$ , we obtain that $v_{j}^{1} = v_{j}^{2}$ . This equality holds for $j = 1, \dots, N$ , hence a solution of $P_{i n c l}^{h}$ is unique.

Now, in order to prove $(???)$ , we set $w^{h} = - v_{j}$ in (3.4) to obtain

⟨ A v_{j} + B (K^{k} v)_{j - 1}, - v_{j} ⟩_{V^{*} \times V} + J_{2}^{0} (γ (K^{k} v)_{j - 1}, γ v_{j}; - γ v_{j} \geq ⟨ f_{j}, - v_{j} ⟩_{V^{*} \times V} .

Hence,

⟨ A v_{j}, v_{j} ⟩_{V^{*} \times V} \leq J_{2}^{0} (γ (K^{k} v)_{j - 1}, γ v_{j}; - γ v_{j}) + ⟨ B (K^{k} v)_{j - 1}, - v_{j} ⟩_{V^{*} \times V} + ⟨ f_{j}, v_{j} ⟩_{V^{*} \times V} .

(3.5)

Using $H (J)$ (c), we get

	$J_{2}^{0} (γ (K^{k} v)_{j - 1}, γ v_{j}; - γ v_{j}) + J_{2}^{0} (0, 0; γ v_{j})$	$\leq m_{J 1} ∥ γ v_{j} ∥_{X}^{2} + m_{J 2} ∥ γ (K^{k} v)_{j - 1} ∥_{X} ∥ γ v_{j} ∥_{X}$
		$\leq m_{J 1} c_{γ}^{2} ∥ v_{j} ∥_{V}^{2} + m_{J 2} c_{γ}^{2} ∥ (K^{k} v)_{j - 1} ∥_{V} ∥ v_{j} ∥_{V} .$

From Proposition 3.23(ii) in [19] and assumption $H (J)$ (b) we have

J_{2}^{0} (0, 0; γ v_{j}) \leq c_{0} ∥ γ v_{j} ∥_{X} \leq c_{0} c_{γ} ∥ v_{j} ∥_{V} .

Hence,

J_{2}^{0} (γ (K^{k} v)_{j - 1}, γ v_{j}; - γ v_{j}) \leq m_{J 1} c_{γ}^{2} ∥ v_{j} ∥_{V}^{2} + m_{J 2} c_{γ}^{2} ∥ (K^{k} v)_{j - 1} ∥_{V} ∥ v_{j} ∥_{V} + c_{0} c_{γ} ∥ v_{j} ∥_{V} .

(3.6)

From $H (B)$ ,

⟨ B (K^{k} v)_{j - 1}, v_{j} ⟩_{V^{*} \times V} \leq ∥ B (K^{k} v)_{j - 1} ∥_{V^{*}} ∥ v_{j} ∥_{V} \leq (L_{B} ∥ (K^{k} v)_{j - 1} ∥_{V} + c) ∥ v_{j} ∥_{V} .

(3.7)

Using $H (A)$ (c), (3.6) and (3.7) in (3.5), we get

m_{A} ∥ v_{j} ∥_{V}^{2} \leq m_{J 1} c_{γ}^{2} ∥ v_{j} ∥_{V}^{2} + (L_{B} + m_{J 2} c_{γ}^{2}) ∥ (K^{k} v)_{j - 1} ∥_{V} ∥ v_{j} ∥_{V} + c ∥ v_{j} ∥_{V} + ∥ f_{j} ∥_{V^{*}} ∥ v_{j} ∥_{V}

Then

(m_{A} - m_{J 1} c_{γ}^{2} - ε) ∥ v_{j} ∥_{V}^{2} \leq c_{ε} ∥ (K^{k} v)_{j - 1} ∥_{V}^{2} + (∥ f_{j} ∥_{V^{*}} + c) ∥ v_{j} ∥_{V} .

Taking sufficiently small $ε > 0$ , using ( $H_{s}$ ) and inequality (3.1), we obtain

∥ v_{j} ∥_{V}^{2} \leq c k^{2} N j - 1 \sum i = 1 ∥ v_{i} ∥_{V}^{2} + c ∥ u_{0} ∥_{V}^{2} + c (∥ f_{j} ∥_{V^{*}} + 1) ∥ v_{j} ∥_{V}

From discrete version of the Gronwall inequality (Lemma 3) with

e_{j} = ∥ v_{j} ∥_{V}^{2}, g_{j} = c ∥ u_{0} ∥_{V}^{2} + c (∥ f_{j} ∥_{V^{*}} + 1) ∥ v_{j} ∥_{V},

we get

∥ v_{j} ∥_{V}^{2} \leq c (∥ f_{j} ∥_{V^{*}} + 1) ∥ v_{j} ∥_{V},

and then

∥ v_{j} ∥_{V} \leq c (1 + ∥ f_{j} ∥_{V^{*}}) .

which concludes the proof of (3.3).

We now consider an optimization problem, which is equivalent to Problem $P_{i n c l}^{h}$ under the stated assumptions. To this end, let us fix $j \in {1, \dots, N}$ and assume $v_{0}, \dots, v_{j - 1}$ to be known, hence also $(K^{k} v)_{j - 1}$ is given. Let a functional $L_{j} : V \to R$ be defined for all $w \in V$ as follows

L_{j} (w) = \frac{1}{2} ⟨ A w, w ⟩_{V^{*} \times V} + ⟨ B (K^{k} v)_{j - 1} - f_{j}, w ⟩_{V^{*} \times V} + J (γ (K^{k} v)_{j - 1}, γ w) .

(3.8)

The next lemma collects some properties of the functional $L_{j}$ .

Lemma 5

Under assumptions $H (A)$ , $H (B)$ , $H (J)$ , $H (f)$ , $H (u_{0})$ and $(H_{s})$ , the functional $L_{j} : V \to R$ defined by (3.8) has the following properties

$L_{j}$ is locally Lipschitz continuous,
$\partial L_{j} (w) \subseteq A w + B (K^{k} v)_{j - 1} - f_{j} + γ^{*} \partial_{2} J (γ (K^{k} v)_{j - 1}, γ w)$ for all $w \in V$ ,
$L_{j}$ is strictly convex,
$L_{j}$ is coercive.

Proof. The proof of (i) is immediate since for $v \in V$ the functional $L_{j}$ is locally Lipschitz continuous as a sum of locally Lipschitz continuous functions with respect to $w$ .

For the proof of (ii), we observe that from $H (A)$ , $H (B)$ and $H (f)$ , the functions

ψ_{1} : V ∋ w \mapsto \frac{1}{2} ⟨ A w, w ⟩_{V^{*} \times V} \in R, ψ_{2} : V ∋ w \mapsto ⟨ B (K^{k} v)_{j - 1} - f_{j}, w ⟩_{V^{*} \times V} \in R

are strictly differentiable with

ψ_{1}^{'} (w) = A w, ψ_{2}^{'} (w) = B (K^{k} v)_{j - 1} - f_{j} .

Now, using the sum and the chain rules for generalized subgradient (cf. Propositions 3.35 and 3.37 in

[19]), we obtain

	$\partial L_{j} (w)$	$= ψ_{1}^{'} (w) + ψ_{2}^{'} (w) + \partial_{2} (J \circ γ) (γ (K^{k} v)_{j - 1}, w)$
		$\subseteq A w + B (K^{k} v)_{j - 1} - f_{j} + γ^{*} \partial_{2} J (γ (K^{k} v)_{j - 1}, γ w),$

which concludes (ii).

In order to prove (iii), let us fix $w^{1}, w^{2} \in V$ . We take $η_{i} \in \partial L_{j} (w^{i})$ for $i = 1, 2$ . From (ii), there exists $ζ_{i} \in \partial_{2} J (γ (K^{k} v)_{j - 1}, γ w^{i})$ such that

η_{i} = A w^{i} + B (K^{k} v)_{j - 1} - f_{j} + γ^{*} ζ_{i} .

From the equivalent condition to $H (J)$ (c), and consequently from (2.1), we have

⟨ \partial_{2} J (γ (K^{k} v)_{j - 1}, γ w^{1}) - \partial_{2} J (γ (K^{k} v)_{j - 1}, γ w^{2}), γ w^{1} - γ w^{2} ⟩_{X^{*} \times X} \geq - m_{J 1} ∥ γ w^{1} - γ w^{2} ∥_{X}^{2} .

Hence and from $H (A)$ (c), we obtain

	$⟨ η_{1} - η_{2}, w^{1} - w^{2} ⟩_{V^{*} \times V}$	$= ⟨ A w^{1} - A w^{2}, w^{1} - w^{2} ⟩_{V^{} \times V} + ⟨ γ^{} ζ_{1} - γ^{} ζ_{2}, w^{1} - w^{2} ⟩_{V^{} \times V}$
		$\geq m_{A} ∥ w^{1} - w^{2} ∥_{V}^{2} + ⟨ ζ_{1} - ζ_{2}, γ w^{1} - γ w^{2} ⟩_{X^{*} \times X}$
		$\geq m_{A} ∥ w^{1} - w$

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