A mixed variational formulation of dynamic viscoelastic problems with adhesion and friction

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A mixed variational formulation of dynamic viscoelastic problems with adhesion and friction

Marius Cocou

To cite this version:

Marius Cocou. A mixed variational formulation of dynamic viscoelastic problems with adhesion and friction. Mechanics Research Communications, Elsevier, 2020, pp.103642.

�10.1016/j.mechrescom.2020.103642�. �hal-03085400�

A mixed variational formulation of dynamic viscoelastic problems with adhesion and friction

Marius Cocou

Aix Marseille Univ, CNRS, Centrale Marseille, LMA UMR 7031, Marseille, France

Dedicated to the memory of Professor Nicolaie Dan Cristescu Keywords

Dynamic viscoelastic problems, pointwise contact conditions, adhesion, slip depending friction, mixed variational formulation.

Abstract

The aim of this work is to study a class of dynamic contact prob- lems coupling adhesion and friction between two viscoelastic bodies with nonlinear viscosity operators. The boundary conditions concern relaxed unilateral contact, pointwise friction and adhesion including possible recoverable behavior. A mixed variational formulation of these problems is given as a ve-eld evolution implicit equation coupled with a dierential inclusion describing the evolution of the intensity of adhesion. Based on several estimates and a classical xed point the- orem for multivalued functions, the existence of a strong variational solution is proved.

1 Introduction

This paper deals with the analysis of a class of nonsmooth dynamic contact problems which describe various surface interactions between two viscoelastic bodies with nonlinear viscosity operators. These interactions include some relaxed unilateral contact, friction with slip depending coecient of friction, and complex adhesion conditions.

The quasistatic elastic problems with unilateral contact and local Coulomb friction have been studied in [1, 32, 33] and extensions by taking into account

1

Corresponding author:

Marius Cocou, Laboratoire de Mécanique et d'Acoustique, 4 Impasse Nikola Tesla, CS 40006, 13453 Marseille Cedex 13, France.

Email: cocou@lma.cnrs-mrs.fr

the adhesion, described by the intensity of adhesion introduced in [18, 19], were investigated in [31, 12], see also [34] and references therein.

Dynamic frictional contact problems with normal compliance laws have been studied in [24, 20, 4] and (non)local friction laws were considered in [21, 22, 16, 6, 13], for viscoelastic bodies. Dynamic frictionless problems with adhesion have been studied in [5, 23, 36] and dynamic viscoelastic problems coupling unilateral contact, recoverable adhesion and nonlocal friction have been analyzed in [14, 7].

Using the hemivariational inequalities theory, several nonsmooth qua- sistatic and dynamic contact problems were investigated, see [26, 27, 28, 29]

and references therein. An elastic contact problem with relaxed unilateral conditions and pointwise Coulomb friction in the static case was studied in [30] and the extension to an elastic quasistatic contact problem was investi- gated in [10]. The corresponding viscoelastic dynamic case was analyzed in [7, 8, 9, 11] for dierent contact conditions.

Based on a new mixed variational formulation, given as a ve-eld evo- lution implicit equation coupled with a dierential inclusion that describes the evolution of the intensity of adhesion, this work generalizes and extends the results presented in [7, 8, 9].

The results presented in this paper extend some of those proved in [11]

in the more general case of rate-depending contact interactions, but with a limited adhesion (there is no intensity of adhesion), a linear viscosity operator and under stronger regularity assumptions.

The approach described in this paper enables to consider more general constitutive laws, as, for example, the ones characterizing some elastovis- coplastic materials investigated in [15].

The paper is organized as follows. In Section 2 the classical formulation of the dynamic contact problem is presented. In Section 3 two mixed variational formulations and some auxiliary results are given. Section 4 is devoted to the existence of a strong variational solution which is proved for an equivalent xed point problem for a multi-valued function.

2 Classical formulation

We consider two viscoelastic bodies that occupy the reference domains Ω

of the three-dimensional Euclidean point space E with Lipschitz boundaries denoted by Γ

= ∂Ω

, α = 1, 2 . Let Γ

, Γ

and Γ

denote three open disjoint suciently smooth parts of Γ

such that Γ

= Γ

∪ Γ

∪ Γ

and, to simplify the estimates, meas (Γ

) > 0, α = 1, 2 .

We assume the small deformation hypothesis and we use Cartesian coor-

dinates representations with the summation convention for i, j, k, l = 1, 2, 3 . To simplify the presentation of the functional framework, we shall continue to use the vector and tensor notations for some inclusions.

Let y

(x

, t) denote the position at time t ∈ [0, T ] , where 0 < T < +∞ , of the material point x

represented by the coordinates x

= (x

, x

, x

) in the reference conguration Ω

, and u

(x

, t) = y

(x

, t) − x

denote the displacement vector of x

at time t , with the components u

= (u

, u

, u

) . Let ε

, with the components ε

= (ε

(u

)) , and σ

, with the compo- nents σ

= σ

, be the innitesimal strain tensor and the stress tensor, respectively, corresponding to Ω

, α = 1, 2 .

Denote by ( S , · , k.k

) the space of symmetric second-order tensors with its inner product and the associated norm.

Let A

: Ω

× S → S be the linear elasticity tensor corresponding to Ω

and denote by A

= (A

) its components satisfying the follow- ing classical symmetry and ellipticity conditions: A

= A

= A

∈ L

(Ω

) , ∀ i, j, k, l = 1, 2, 3, ∃ k

> 0 such that A

τ

τ

≥ k

τ

τ

∀ τ = (τ

) ∈ R

verifying τ

= τ

∀ i, j = 1, 2, 3 , α = 1, 2 . Thus, for all τ ∈ S with components τ = (τ

) , A

(x

, τ ) has the components A

(x

)

τ

∀ k, l = 1, 2, 3 , α = 1, 2 , and A

satises the following condi- tions for α = 1, 2 :

(A

(x

, τ )) · τ ≥ k

kτ k

S

a.e. x

∈ Ω

, ∀ τ ∈ S ,

∃ k

> 0 such that

kA

(x

, τ )k

≤ k

kτ k

a.e. x

∈ Ω

, ∀ τ ∈ S .

Let B

: Ω

× S → S denote the nonlinear viscosity tensor corresponding to Ω

, satisfying the following conditions for α = 1, 2 :

∃ k

> 0 such that a.e. x

∈ Ω

, ∀ τ

∈ S ,

(B

(x

, τ

) − B

(x

, τ

)) · (τ

− τ

) ≥ k

kτ

− τ

k

S

, kB

(x

, τ

) − B

(x

, τ

)k

≤ k

kτ

− τ

k

,

B

(·, 0) = 0,

∀ τ ∈ S , B

(·, τ ) is measurable on Ω

.

Assume that the displacements u

= 0 on Γ

× (0, T ), α = 1, 2 , and

that the densities of both bodies are equal to 1 . Let f

= (f

, f

) , f

=

(f

, f

) be the given densities of body forces in Ω

∪ Ω

and of tractions on

Γ

∪ Γ

, respectively. Let u

= (u

, u

) , u

= (u

, u

) denote the initial

displacements and velocities of the bodies, respectively.

Suppose that the solids can be in contact between the potential contact surfaces Γ

and Γ

which are parametrized by two C

functions, ϕ

, ϕ

, dened on an open and bounded subset Ξ of R

, such that ϕ

(ξ) − ϕ

(ξ) ≥ 0 ∀ ξ ∈ Ξ and each Γ

is the graph of ϕ

on Ξ that is Γ

= {(ξ, ϕ

(ξ)) ∈ R

; ξ ∈ Ξ}, α = 1, 2 , see, e.g., [3]. Dene an initial normalized gap between the two contact surfaces by

g

(ξ) = ϕ

(ξ) − ϕ

(ξ)

p 1 + |∇ϕ

(ξ)|

∀ ξ ∈ Ξ.

Let n

denote the unit outward normal vector to Γ

, α = 1, 2 .

We introduce the following notations for the normal and tangential com- ponents of a displacement eld v

, of the relative displacement corresponding to v := (v

, v

) and of the stress vector σ

n

on Γ

, α = 1, 2 , respectively:

v

(ξ, t) := v

(ξ, ϕ

(ξ), t), v

(ξ, t) := v

(ξ, t) · n

(ξ), v

(ξ, t) := v

(ξ, t) + v

(ξ, t),

[v

](ξ, t) := v

(ξ, t) − g

(ξ),

v

(ξ, t) := v

(ξ, t) − v

(ξ, t)n

(ξ), v

(ξ, t) := v

(ξ, t) − v

(ξ, t), σ

(ξ, t) := (σ

(ξ, t)n

(ξ)) · n

(ξ), σ

(ξ, t) = σ

(ξ, t)n

(ξ) − σ

(ξ, t)n

(ξ),

for all ξ ∈ Ξ and for all t ∈ [0, T ] , where we denoted the inner product of two vectors by ” · ” .

In Ξ , we consider an internal state variable β (see [18, 19]) that represents the intensity of adhesion: β = 1 means that the adhesion is total, β = 0 means that there is no adhesion and 0 < β < 1 is the case of partial adhesion.

We assume that the evolution of β is governed, for all t ∈ (0, T ) , by the inclusion β ˙ ∈ ψ([u

], β) in Ξ , where ψ is a given constitutive set-valued mapping. Denote by β

the initial intensity of adhesion.

Let κ, κ : R

→ R be two mappings with κ lower semicontinuous and κ upper semicontinuous, satisfying the following conditions for all s ∈ R

:

κ(s) ≤ κ(s) and 0 ∈ / (κ(s), κ(s)), (2.1)

∃ r

≥ 0 such that max (|κ(s)|, |κ(s)|) ≤ r

, (2.2)

Let µ : Ξ × R

→ R

be the sliding velocity dependent coecient of

friction and assume that µ is a bounded function such that for a.e. ξ ∈ Ξ

µ(ξ, ·) is Lipschitz continuous with the Lipschitz constant independent of ξ ,

and for every v ∈ R

µ(·, v) is measurable. Dene a truncation operator ϑ = ϑ

by ϑ : R → R , ϑ(s) = −l

if s ≤ −l

, ϑ(s) = s if |s| < l

and ϑ(s) = l

if s ≥ l

, where l

> 0 is a given characteristic length, see, e.g., [31, 36].

We choose the following state variables: the innitesimal strain tensor (ε

, ε

) = (ε(u

), ε(u

)) in Ω

∪ Ω

, the normal relative displacement [u

] = u

+ u

− g

, the tangential relative displacement [u

] = u

− u

, and the intensity of adhesion β in Ξ .

Consider the following dynamic viscoelastic contact problem coupling ad- hesion and Coulomb friction.

Problem P

: Find u = (u

, u

) and β such that u(0) = u

, u(0) = ˙ u

, β(0) = β

in Ξ and, for all t ∈ (0, T ) ,

¨

u

− div σ

(u

, u ˙

) = f

in Ω

,

σ

(u

, u ˙

) = A

ε(u

) + B

ε( ˙ u

) in Ω

, u

= 0 on Γ

, σ

n

= f

on Γ

, α = 1, 2, σ

n

+ σ

n

= 0 in Ξ,

κ([u

], β) ≤ σ

≤ κ([u

], β) in Ξ,

|σ

| ≤ µ( ˙ u

) |σ

| in Ξ and

˙

u

6= 0 ⇒ σ

= −µ( ˙ u

)|σ

| u ˙

| u ˙

| , β ∈ [0, 1] and β ˙ ∈ ψ(ϑ([u

]), β) in Ξ, where, for all (x

, t) ∈ Ω

× (0, T ) ,

A

ε(u

)(x

, t) = A

(x

, ε(u

(x

, t))) , B

ε( ˙ u

)(x

, t) = B

(x

, ε( ˙ u

(x

, t))) ,

σ

= σ

(u

, u ˙

) , α = 1, 2 , σ

:= σ

, σ

:= σ

.

The nonlinear constitutive law represents a generalization of the classical Kelvin-Voigt law.

Dierent choices for κ , κ and ψ give various contact and friction condi- tions, including irreversible or recoverable (healing) adhesion, see [31, 12, 14, 9, 7].

We also remark that in this model friction and adhesion are strongly

coupled, as sliding friction can occur even under tensile loads [25], but the

case of the solely compressive loads can be easily considered.

3 Mixed variational formulations and approxi- mation results

We adopt the following notations:

H

(Ω

) := H

(Ω

; R

), α = 1, 2, H

:= H

(Ω

) × H

(Ω

) ∀ s ∈ R ,

V

= {v

∈ H

(Ω

); v

= 0 a.e. on Γ

}, α = 1, 2, V := V

× V

, H := H

= L

(Ω

; R

) × L

(Ω

; R

).

(H, |.|) and (V , k.k) are Hilbert spaces with the associated inner products denoted by (. , .) and by h. , .i , respectively, V ⊂ H ⊂ V

with the inclusion mapping of V into H continuous and densely dened, where (V

, k.k

) is the dual of V and H is identied with its own dual. Let h. , .i

denote the duality pairing between V

and V .

Dene Ξ

= Ξ × (0, T ) , the closed convex cones L

(Ξ) , L

(Ξ

) and the closed convex set L

(Ξ) as follows:

L

(Ξ) := {δ ∈ L

(Ξ); δ ≥ 0 a.e. in Ξ}, L

(Ξ

) := {η ∈ L

(Ξ

); η ≥ 0 a.e. in Ξ

}, L

(Ξ) := {δ ∈ L

(Ξ); δ ∈ [0, 1] a.e. in Ξ}.

Let A : V → V

, B : V → V

be two operators dened by hAv, wi

= X

α=1,2

Z

Ω^α

(A

ε(v

)) · ε(w

) dx,

hBv, wi

= X

α=1,2

Z

Ω^α

(B

ε(v

)) · ε(w

) dx

∀ v = (v

, v

), w = (w

, w

) ∈ V .

Under the above assumptions on A

, B

, α = 1, 2 , it follows that there exist M

, M

, such that, for all v, w ∈ V ,

kAvk

≤ M

kvk

, kBv − Bwk

≤ M

kv − wk

. (3.1) As meas (Γ

) > 0 , by using Korn's inequality it also follows that there exist m

, m

> 0 such that, for all v, w ∈ V ,

hAv, vi

≥ m

kvk

, (3.2)

hBv − Bw, v − wi

≥ m

kv − wk

. (3.3)

Assume u

∈ V , u

∈ H , g

∈ L

(Ξ) , f

∈ L

(0, T ; L

(Ω

; R

)) , f

∈ L

(0, T ; L

(Γ

; R

)) , α = 1, 2 , and dene the mapping f ∈ L

(0, T ; V

) by

hf , vi

= X

α=1,2

Z

Ω^α

f

· v

dx + X

α=1,2

Z

Γ^α_F

f

· v

ds

∀ v = (v

, v

) ∈ V , a.e. t ∈ [0, T ].

Assume also the following initial conditions: β

∈ L

(Ξ) , [u

] ≤ 0 , and κ([u

], β

) = 0 a.e. in Ξ .

For every ζ = (ζ

, ζ

) ∈ L

(0, T ; (L

(Ξ))

) = (L

(Ξ

))

, dene the fol- lowing nonempty, closed, and convex sets:

Λ

(ζ

, ζ

) = {η ∈ L

(Ξ

); κ ◦ (ζ

, ζ

) ≤ η

≤ κ ◦ (ζ

, ζ

) a.e. in Ξ

}, Λ

(ζ

, ζ

) = {η ∈ L

(Ξ

); κ

◦ (ζ

, ζ

) ≤ η

≤ κ

◦ (ζ

, ζ

) a.e. in Ξ

}, Λ

(ζ

, ζ

) = {η ∈ L

(Ξ

); κ

◦ (ζ

, ζ

) ≤ η

≤ κ

◦ (ζ

, ζ

) a.e. in Ξ

},

where, for each r ∈ R, r

:= max (0, r) and r

:= max (0, −r) denote the positive and negative parts, respectively. Also, for every w ∈ W

(0, T ; V ) , υ ∈ L

(Ξ

) , dene the following nonempty and closed sets:

Λ

(w, υ) = {(η, ς) ∈ L

(Ξ

) × (L

(Ξ

))

; η ∈ Λ

([w

], υ), |ς| ≤ µ( ˙ w

) |η|,

ς · w ˙

+ µ( ˙ w

) |η| | w ˙

| = 0 a.e. in Ξ

}, Λ

(w, υ) = {(η, ς) ∈ L

(Ξ

) × (L

(Ξ

))

;

η

∈ Λ

([w

], υ), η

∈ Λ

([w

], υ),

|ς| ≤ µ( ˙ w

) (η

+ η

),

ς · w ˙

+ µ( ˙ w

) (η

+ η

) | w ˙

| = 0 a.e. in Ξ

}, Λ

(w, υ) = {(η

, η

, ς) ∈ (L

(Ξ

))

; η

∈ Λ

([w

], υ),

η

∈ Λ

([w

], υ), |ς| ≤ µ( ˙ w

) (η

+ η

),

ς · w ˙

+ µ( ˙ w

) (η

+ η

) | w ˙

| = 0 a.e. in Ξ

}.

The relations meas (Ξ) < ∞ and (2.2) imply that for all ζ = (ζ

, ζ

) ∈ L

(Ξ

)

the sets Λ

(ζ) , Λ

(ζ) and Λ

(ζ) are bounded in norm in L

(0, T ; L

(Ξ)) = L

(Ξ

)

by R

= r

( meas (Ξ))

T and are bounded in norm in L

(0, T ; L

(Ξ)) by r

.

As the coecient of friction µ is a bounded function, it follows that for all w ∈ W

(0, T ; V ) , υ ∈ L

(Ξ

) the sets Λ

(w, υ) , Λ

(w, υ) , and Λ

(w, υ) are bounded in norm. Thus, there exists R

> 0 such that Λ

(w) ⊂ D

× D

for all w ∈ W

(0, T ; V ) , where D

= {(η

, η

) ∈ (L

(Ξ

))

; kη

k

≤ R

, kη

k

≤ R

} and D

= {ς ∈ (L

(Ξ

))

; kς k

≤ R

} .

A rst variational formulation of the problem P

is the following.

Problem P

: Find u ∈ C

([0, T ]; H) ∩ W

(0, T ; V ) ∩ W

(0, T ; V

) , λ ∈ L

(Ξ

) , γ ∈ (L

(Ξ

))

, β ∈ W

(0, T ; L

(Ξ)) , such that (λ, γ) ∈ Λ

(u, β) , β(t) ∈ L

(Ξ) for all t ∈ (0, T ) , u(0) = u

, u(0) = ˙ u

, β(0) = β

, and for almost all t ∈ (0, T )

h u, ¨ vi

+ hAu, vi

+ hB u, ˙ vi

−(λ, v

)

− (γ, v

)

= hf , vi

∀ v ∈ V , (3.4) β ˙ ∈ ψ(ϑ([u

]), β) a.e. in Ξ

,

where (·, ·)

and (·, ·)

denote the inner products of the correspond- ing spaces.

The formal equivalence between the variational problem P

and the clas- sical problem P

can be proved as usual by Green's formula, where the La- grange multipliers λ , γ satisfy the relations λ = σ

, γ = σ

.

The sets Λ

(ζ

, ζ

) , Λ

(ζ

, ζ

) and Λ

(ζ

, ζ

) have the following useful properties, see [9].

Lemma 3.1. Let (ζ

, ζ

) ∈ (L

(Ξ))

and (η

, η

) ∈ Λ

(ζ

, ζ

) × Λ

(ζ

, ζ

) . Then η

η

= 0 a.e. in Ξ

and there exists η ∈ Λ

(ζ

, ζ

) such that η

= η

, η

= η

a.e. in Ξ

.

Since λ ∈ Λ

([u

], β) if and only if (λ

, λ

) ∈ Λ

([u

], β) × Λ

([u

], β) , the previous lemma enables to consider the following variational problem P

, which has the same solutions u , γ , β as the problem P

and the solutions λ

, λ

satisfy the relation λ = λ

− λ

, where λ is a solution of P

.

Problem P

: Find u ∈ C

([0, T ]; H) ∩ W

(0, T ; V ) ∩ W

(0, T ; V

) ,

(λ

, λ

) ∈ (L

(Ξ

))

, γ ∈ (L

(Ξ

))

, β ∈ W

(0, T ; L

(Ξ)) , such that

(λ

, λ

, γ) ∈ Λ

(u, β) , β(t) ∈ L

(Ξ) for all t ∈ (0, T ) , u(0) = u

,

u(0) = ˙ u

, β(0) = β

, and for almost all t ∈ (0, T ) h u, ¨ vi

+ hAu, vi

+ hB u, ˙ vi

−(λ

− λ

, v

)

− (γ, v

)

(3.5)

= hf , vi

∀ v ∈ V ,

β ˙ ∈ ψ(ϑ([u

]), β) a.e. in Ξ

, (3.6) Assume that the set-valued mapping ψ : (L

(Ξ

))

→ 2

veries the following properties: for each u ∈ W

(0, T ; V ) there exists a unique solution β

∈ W

(0, T ; L

(Ξ)) of the inclusion (3.6), such that β(0) = β

, β(t) ∈ L

(Ξ) for all t ∈ (0, T ) , and if β

, β

are the solutions of (3.6) corresponding to u

, u

∈ W

(0, T ; V ) , respectively, with the same initial condition β

, then the following estimate holds for all t ∈ [0, T ] :

kβ

(t) − β

(t)k

≤ C

Z

0

ku

(s) − u

(s)k

ds, (3.7) where C

is a positive constant independent of u

, u

, β

, β

.

Several choices of ψ , as subdierentials or single-valued mappings satis- fying the previous properties, have been considered, see, e.g., [31, 34, 14, 7].

The existence of strong solutions to problem P

will be established by using some auxiliary results and an equivalent xed point problem.

Similar arguments to those used to prove Theorem 5.2 in [36], now applied to the product space V = V

× V

and to the simple case without adhesion, enable to obtain the following existence and uniqueness result for the strong solution, in the V

sense, of the following intermediate problem.

Lemma 3.2. For each (η

, η

) ∈ (L

(Ξ

))

, ς ∈ (L

(Ξ

))

, there exists a unique function u = u

∈ C

([0, T ]; H)∩W

(0, T ; V )∩W

(0, T ; V

) , such that u(0) = u

, u(0) = ˙ u

, and for almost all t ∈ (0, T )

h¨ u, vi

+ hAu, vi

+ hB u, ˙ vi

−(η

− η

, v

)

− (ς, v

)

(3.8)

= hf , vi

∀ v ∈ V ,

where u ¨ is the second order strong derivative of u considered in V

.

For the involved abstract mathematical result, see, e.g., Theorem 4.10 in

[2].

Lemma 3.3. Let u

, u

be the solutions of variational equation (3.8) corresponding to (η

, η

), (δ

, δ

) ∈ (L

(Ξ

))

, ς

, ς

∈ (L

(Ξ

))

, respectively.

Then there exists a constant C

> 0 , independent of (η

, η

), (δ

, δ

) , and ς

, ς

, such that for all t ∈ [0, T ]

| u ˙

(t) − u ˙

(t)|

+ ku

(t) − u

(t)k

+ Z

0

k u ˙

− u ˙

k

dτ

≤ C

Z

0

{(η

− η

− δ

+ δ

, u ˙

− u ˙

)

+(ς

− ς

, u ˙

− u ˙

)

} dτ. (3.9) Proof. Let (η

, η

), (δ

, δ

) ∈ (L

(Ξ

))

and ς

, ς

∈ (L

(Ξ

))

with u

:=

u

, u

:= u

the corresponding solutions of (3.8) which exist according to Lemma 3.2. Taking in each equation v = ˙ u

− u ˙

, for a.e.

τ ∈ (0, T ) it follows that

h u ¨

− u ¨

, u ˙

− u ˙

i

+ hAu

− Au

, u ˙

− u ˙

i

+hB u ˙

− B u ˙

, u ˙

− u ˙

i

= (η

− η

− δ

+ δ

, u ˙

− u ˙

)

+(ς

− ς

, u ˙

− u ˙

)

.

Since the solutions u

, u

belong to u ∈ C

([0, T ]; H) ∩ W

(0, T ; V ) ∩ W

(0, T ; V

) and verify the same initial conditions, by integrating over (0, t) it follows that for all t ∈ [0, T ]

1

2 | u ˙

(t) − u ˙

(t)|

+ 1

2 hA(u

− u

), u

− u

i

+

Z

0

hB u ˙

− B u ˙

, u ˙

− u ˙

i

dτ

= Z

0

{(η

− η

− δ

+ δ

, u ˙

− u ˙

)

} dτ

+ Z

0

{(ς

− ς

, u ˙

− u ˙

)

} dτ.

By (3.2) and (3.3), the estimate (3.9) follows.

Lemma 3.4. Under the assumptions of Sections 2 and 3, for every (η

, η

) ∈ (L

(Ξ

))

and every ς ∈ (L

(Ξ

))

, let (η

, η

)

be a sequence in (L

(Ξ

))

and (ς

)

be a sequence in (L

(Ξ

))

such that η

* η

, η

* η

in L

(Ξ

) , and ς

* ς in (L

(Ξ

))

. Let u

be the solution of (3.8) corre- sponding to (η

, η

, ς

) according to Lemma 3.2, for every n ∈ N. Then (u

1,ηⁿ₂,ςⁿ)

)

is strongly convergent in C

([0, T ]; H) ∩W

(0, T ; V ) to the so- lution u = u

of (3.8) corresponding to (η

, η

, ς) according to Lemma 3.2.

Proof. We adopt the following notations:

u

:= u

, u

:= u

, so that we have for almost all t ∈ (0, T )

h u ¨

, vi

+ hAu

, vi

+ hB u ˙

, vi

−(η

− η

, v

)

− (ς

, v

)

(3.10)

= hf , vi

∀ v ∈ V .

By Lemma 3.3, for all n ∈ N and t ∈ (0, T ] we have 1

2 | u ˙

(t)|

+ 1

2 ku

(t)k

+ 1 2

Z

0

k u ˙

k

dτ

≤ | u ˙

(t) − u(t)| ˙

+ ku

(t) − u(t)k

+ Z

0

k u ˙

− uk ˙

dτ

+| u(t)| ˙

+ ku(t)k

+ Z

0

k uk ˙

dτ

≤ C

Z

0

{(η

− η

− η

+ η

, u ˙

− u ˙

)

+(ς

− ς, u ˙

− u ˙

)

} dτ + C

, where C

is a positive constant dependent only on u .

Since the sequences (η

, η

)

, (ς

)

are bounded in (L

(Ξ

))

, (L

(Ξ

))

, respectively, by Young's inequality it follows that there exists a positive con- stant C

, depending only on u , C

, C

, the bounds of (η

, η

)

and (ς

)

, such that the following estimates hold for all n ∈ N:

| u ˙

(t)| ≤ C

, ku

(t)k ≤ C

∀t ∈ [0, T ], k u ˙

k

≤ C

. (3.11)

For all v ∈ L

(0, T ; V ) , by (3.10) we have Z

0

h¨ u

, vi

dt + Z

0

hAu

, vi

dt

+ Z

0

hB u ˙

, vi

dt − Z

0

(η

− η

, v

)

dt

− Z

0

(ς

, v

)

dt = Z

0

hf , vi

dt.

This relation and the estimates (3.1), (3.11) imply that there exists a positive constant C

, depending only on C

, M

, and M

, such that

∀ n ∈ N , k u ¨

k

≤ C

. (3.12) From (3.11), (3.12), it follows that there exist a subsequence (u

)

and u ˜ such that

˙

u

*

u ˙˜ in L

(0, T ; H), u

*

u ˜ in L

(0, T ; V ),

˙

u

* u ˙˜ in L

(0, T ; V ), u ¨

* u ¨ ˜ in L

(0, T ; V

).

Since V ⊂ H

⊂ H ⊂ V

with compact embedding from V into H

, according to a classical compactness result, see, e.g., [35], it follows that

˙

u

→ u ˙˜ in L

(0, T ; H

), where 1 > ι > 1

2 , so that, by the trace theorem,

˙

u

→ u ˙˜ in L

(0, T ; (L

(Ξ))

) = (L

(Ξ

))

. (3.13) By Lemma 3.3, for all k ∈ N and t ∈ (0, T ] we have

| u ˙

(t) − u(t)| ˙

+ ku

(t) − u(t)k

+ Z

0

k u ˙

− uk ˙

dτ

≤ C

Z

0

{(η

− η

− η

+ η

, u ˙

− u ˙

)

+(ς

− ς, u ˙

− u ˙

)

} dτ + C

,

Using the weak convergence properties of (η

)

, (η

)

, (ς

)

, and the strong convergence property (3.13), we can pass to limits in the right hand side of the previous estimates for (u

)

and so we obtain that u = ˜ u and

u

→ u in C

([0, T ]; H) ∩ W

(0, T ; V ). (3.14)

As for every subsequence of (u

)

, by using the same arguments as above that enabled to obtain the relation (3.14), one can always nd a (sub)subsequence converging to u in C

([0, T ]; H) ∩ W

(0, T ; V ) , it follows that

u

1,ηⁿ₂,ςⁿ)

→ u

in C

([0, T ]; H) ∩ W

(0, T ; V ). (3.15)

4 An equivalent xed point problem and exis- tence result

Let Φ : (L

(Ξ

))

× (L

(Ξ

))

→ 2

\ {∅} be the set-valued mapping dened by

∀(η

, η

, ς) ∈ (L

(Ξ

))

× (L

(Ξ

))

Φ(η

, η

, ς) = Λ

(u

, β

1,η2,ς)

) (4.1) where u

is the solution of the variational equation (3.8) which corre- sponds to (η

, η

, ς ) according to Lemma 3.2 and β

1,η2,ς)

is the solution of (3.6) corresponding to u = u

.

Since (λ

, λ

, γ) is a xed point of Φ , i.e. (λ

, λ

, γ) ∈ Φ(λ

, λ

, γ) , if and only if (u

, λ

, λ

, γ, β

1,λ2,γ)

) is a solution of the Problem P

, we consider the problem which consists in nding a xed point of the set-valued mapping Φ , called also multivalued function or multifunction.

The existence of a xed point of the multifunction Φ will be proved by using a corollary of the Ky Fan's xed point theorem [17], proved in [30] in the particular case of a reexive Banach space. We recall this result for the reader's convenience.

Denition 4.1. Let Y be a reexive Banach space, D a weakly closed set in Y , and F : D → 2

\ {∅} be a set-valued mapping. F is called sequentially weakly upper semicontinuous if z

* z , y

∈ F (z

) and y

* y imply y ∈ F (z) .

Proposition 4.1. ( [30]) Let Y be a reexive Banach space, D a convex, closed and bounded set in Y , and F : D → 2

\ {∅} a sequentially weakly upper semicontinuous set-valued mapping such that F (z) is convex for every z ∈ D . Then F has a xed point.

Theorem 4.2. Under the assumptions of Sections 2 and 3, there exists

(λ

, λ

, γ) ∈ (L

(Ξ

))

× (L

(Ξ

))

such that (λ

, λ

, γ) ∈ Φ(λ

, λ

, γ) . For

each xed point (λ

, λ

, γ) of the multifunction Φ , (u

, λ, γ, β

1,λ2,γ)

) , where λ = λ

−λ

, is a solution of the Problem P

and (u

, λ

, λ

, γ, β

1,λ2,γ)

) is a solution of the Problem P

.

Proof. We apply Proposition 4.1 to Y = (L

(Ξ

))

, F = Φ and D = [(L

(Ξ

))

∩ D

] × [(L

(Ξ

))

∩ D

] .

The set D ⊂ (L

(Ξ

))

is clearly convex, closed, and bounded.

For all w ∈ W

(0, T ; V ) , υ ∈ L

(Ξ

) , the set Λ

(w, υ) is nonempty, closed, and convex, so that Φ(η

, η

, ς) is a nonempty, closed, and convex subset of D for every (η

, η

, ς) ∈ D .

In order to prove that the multifunction Φ is sequentially weakly upper semicontinuous, let (η

, η

, ς

) ∈ D , (δ

, δ

, $

) ∈ Φ(η

, η

, ς

) ∀ n ∈ N, (η

, η

, ς

) * (η

, η

, ς) , (δ

, δ

, $

) * (δ

, δ

, $) and let us verify that (δ

, δ

, $) ∈ Φ(η

, η

, ς) .

We adopt the following notations:

u

:= u

1,ηⁿ₂,ςⁿ)

, u

:= u

1,η₂ⁿ,ςⁿ)N

, u

:= u

, u

:= u

, β

:= β

1,ηn2,ςn)

, β

:= β

1,η2,ς)

, where β

1,ηn2,ςn)

is the solution of (3.6) corresponding to u

1,η₂ⁿ,ςⁿ)

and β

1,η2,ς)

is the solution of (3.6) corresponding to u

. By Lemma 3.4, we have

u

→ u

in C

([0, T ]; H) ∩ W

(0, T ; V ), (4.2) which implies

u

→ u

, u ˙

→ u ˙

in (L

(Ξ

))

, (4.3) and, by (3.7),

β

(t) → β

(t) in L

(Ξ) for all t ∈ [0, T ]. (4.4) Now, by Lemma 3.1, if (δ

, δ

, $

) ∈ Φ(η

, η

, ς

) = Λ

(u

1,η₂ⁿ,ςⁿ)

, β

1,ηn 2,ςn)

) for all n ∈ N, then

κ([u

], β

) ≤ δ

− δ

≤ κ([u

], β

) a.e. in Ξ

, (4.5)

|$

| ≤ µ( ˙ u

) (δ

+ δ

) a.e. in Ξ

, (4.6)

$

· u ˙

+ µ( ˙ u

) (δ

+ δ

) | u ˙

| = 0 a.e. in Ξ

, (4.7)

for all n ∈ N.

First, the relations (4.5) are equivalent to Z

ω

κ([u

], β

) ≤ Z

ω

(δ

− δ

) ≤ Z

ω

κ([u

], β

), (4.8) for every measurable subset ω ⊂ Ξ

and for all n ∈ N.

By (4.3) and a converse of Lebesgue's dominated convergence theorem, it follows that there exists a subsequence of (u

)

, denoted by (u

)

, such that

[u

] → [u

], u ˙

→ u ˙

a.e. in Ξ

. (4.9) The rst convergence property in (4.9) enables to pass to limits in (4.8) with respect to n

according to Fatou's lemma, by using (4.4), the semi-continuity of κ and κ , the relation (2.2), and the convergence property

Z

ω

(δ

− δ

) → Z

ω

(δ

− δ

).

Thus, we obtain Z

ω

κ([u

], β

) ≤ Z

ω

(δ

− δ

) ≤ Z

ω

κ([u

], β

), for every measurable subset ω ⊂ Ξ

, which implies

κ([u

], β

) ≤ δ

− δ

≤ κ([u

], β

) a.e. in Ξ

. (4.10) Second, the relation (4.6) is equivalent to

Z

ω

|$

| ≤ Z

ω

µ( ˙ u

) (δ

+ δ

),

for every measurable subset ω ⊂ Ξ

and for all n ∈ N. As µ(ξ, ·) is Lipschitz continuous with the Lipschitz constant independent of ξ , by using the second convergence property in (4.9) and passing to limits with respect to n

we obtain

Z

ω

|$| ≤ lim inf Z

ω

|$

| ≤ lim Z

ω

µ( ˙ u

) (δ

+ δ

)

= Z

ω

µ( ˙ u

) (δ

+ δ

).

Thus Z

ω

|$| ≤ Z

ω

µ( ˙ u

) (δ

+ δ

),

for every measurable subset ω ⊂ Ξ

, which implies

|$| ≤ µ( ˙ u

) (δ

+ δ

) a.e. in Ξ

. (4.11) Finally, consider the relation (4.7) which is equivalent to

Z

ω

$

· u ˙

+ Z

ω

µ( ˙ u

) (δ

+ δ

) | u ˙

| = 0, (4.12) for every measurable subset ω ⊂ Ξ

and for all n ∈ N. By the second convergence property in (4.9), we have

µ( ˙ u

) | u ˙

| → µ( ˙ u

) | u ˙

| in L

(Ξ

), (4.13) and, by the relations (2.2), (4.5),

δ

+ δ

*

δ

+ δ

in L

(Ξ

). (4.14) Passing to limits in (4.12) by using (4.13) and (4.14), we obtain

Z

ω

$ · u ˙

+ Z

ω

µ( ˙ u

) (δ

+ δ

) | u ˙

| = 0, for every measurable subset ω ⊂ Ξ

, which implies

$ · u ˙

+ µ( ˙ u

) (δ

+ δ

) | u ˙

| = 0. (4.15) From (4.10), (4.11), (4.15), it follows that (δ

, δ

, $) ∈ Φ(η

, η

, ς ) , so that we may apply the Proposition 4.1 to conclude the proof.

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