A history-dependent contact problem with unilateral constraint

Flavius Patrulescu
(original), Numerical Analysis, paper

Abstract

We consider a mathematical model which describes the quasistatic contact between a viscoplastic body and a foundation. The contact is frictionless and is modelled with a new and nonstandard condition which involves both normal compliance, unilateral constraint and memory effects. We derive a variational formulation of the problem then we prove its unique weak solvability. The proof is based on arguments on history-dependent variational inequalities

Authors

Anca Farcaş
(Babeş-Bolyai University Faculty of Mathematics and Computer Sciences)

Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Mircea Sofonea
(Laboratoire de Mathématiques et Physique, Université de Perpignan)

Keywords

viscoplastic material; frictionless contact; unilateral constraint; history-depdendent variational inequality; weak solution

Cite this paper as

A. Farcaş, F. Pătrulescu, M. Sofonea, A history-dependent contact problem with unilateral constraint, Ann. Acad. Rom. Sci. Ser. Math. Appl., vol 4, no. 1 (2012), pp. 90-96

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Annals of the Academy of Romanian Scientists. Mathematics and its Applications

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Publishing House of Romanian Scientists (Editura Academiei Oamenilor de Ştiinţă din România), Bucharest

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2066-5997

Online ISSN

2066-6594

MR

2959899

ZBL

1284.74088

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https://scholar.google.ro/scholar?oi=bibs&hl=ro&cites=9695790534973251397

[1] Jarusek, M. Sofonea, On the solvability of  dynamic elastic-visco-plastic contact problems, Zeitschrift fur Angewandte Matematik und Mechanik  (ZAMM), 88 (2008), 3-22.
[2] Shillor, M. Sofonea,  J.J. Telega, Models and Analysis of Quasistatic Contact. Lect. Notes Phys., Springer, Berlin Heidelberg (2004).
[3] Sofonea, A. Matei. History-dependent quasivariational inequalities arising in Contact Mechanics, Eur. J. Appl. Math., 22 (2011), 471-491.

 

Paper (preprint) in HTML form

2012_Patrulescu_AABSMA_History_dependent_contact_pb

A HISTORY-DEPENDENT CONTACT PROBLEM WITH UNILATERAL CONSTRAINT

ANCA FARCAS* ∗ ∗ ^(**){ }^{*}∗ FLAVIUS PATRULESCU † † ^(†){ }^{\dagger}†MIRCEA SOFONEA ‡ ‡ ^(‡){ }^{\ddagger}‡

Abstract

We consider a mathematical model which describes the quasistatic contact between a viscoplastic body and a foundation. The contact is frictionless and is modelled with a new and nonstandard condition which involves both normal compliance, unilateral constraint and memory effects. We derive a variational formulation of the problem then we prove its unique weak solvability. The proof is based on arguments on history-dependent variational inequalities.

Keywords: viscoplastic material, frictionless contact, unilateral constraint, history-depdendent variational inequality, weak solution.

1 The model

We consider a viscoplastic body which occupies the domain Ω ⊂ R d Ω ⊂ R d Omega subR^(d)\Omega \subset \mathbb{R}^{d}Ω⊂Rd ( d = 1 , 2 , 3 ) d = 1 , 2 , 3 ) d=1,2,3)d= 1,2,3)d=1,2,3) with a Lipschitz continuous boundary Γ Γ Gamma\GammaΓ, divided into three measurable parts Γ 1 , Γ 2 Γ 1 , Γ 2 Gamma_(1),Gamma_(2)\Gamma_{1}, \Gamma_{2}Γ1,Γ2 and Γ 3 Γ 3 Gamma_(3)\Gamma_{3}Γ3, such that meas ( Γ 1 ) > 0 Γ 1 > 0 (Gamma_(1)) > 0\left(\Gamma_{1}\right)>0(Γ1)>0. We use the notation x = ( x i ) x = x i x=(x_(i))\boldsymbol{x}=\left(x_{i}\right)x=(xi) for a typical point in Ω ∪ Γ Ω ∪ Γ Omega uu Gamma\Omega \cup \GammaΩ∪Γ and we denote by ν = ( ν i ) ν = ν i nu=(nu_(i))\boldsymbol{\nu}=\left(\nu_{i}\right)ν=(νi) the outward unit normal at Γ Γ Gamma\GammaΓ. Here and below the indices i , j , k , l i , j , k , l i,j,k,li, j, k, li,j,k,l run between 1
and d d ddd and an index that follows a comma represents the partial derivative with respect to the corresponding component of the spatial variable, e.g. v i , j = ∂ v i / ∂ x j v i , j = ∂ v i / ∂ x j v_(i,j)=delv_(i)//delx_(j)v_{i, j}=\partial v_{i} / \partial x_{j}vi,j=∂vi/∂xj. The body is subject to the action of body forces of density f 0 f 0 f_(0)\boldsymbol{f}_{0}f0, is fixed on Γ 1 Γ 1 Gamma_(1)\Gamma_{1}Γ1, and surface tractions of density f 2 f 2 f_(2)\boldsymbol{f}_{2}f2 act on Γ 2 Γ 2 Gamma_(2)\Gamma_{2}Γ2. On Γ 3 Γ 3 Gamma_(3)\Gamma_{3}Γ3, the body is in frictionless contact with a deformable obstacle, the so-called foundation. We assume that the problem is quasistatic and the time interval of interest is R + = [ 0 , ∞ ) R + = [ 0 , ∞ ) R_(+)=[0,oo)\mathbb{R}_{+}=[0, \infty)R+=[0,∞). Everywhere in this paper the dot above a variable represents derivative with respect to the time variable, S d S d S^(d)\mathbb{S}^{d}Sd denotes the space of second order symmetric tensors on R d R d R^(d)\mathbb{R}^{d}Rd and r + r + r^(+)r^{+}r+is the positive part of r r rrr, i.e. r + = max { 0 , r } r + = max { 0 , r } r^(+)=max{0,r}r^{+}=\max \{0, r\}r+=max{0,r}. The classical formulation of the problem is the following.
Problem P P P\mathcal{P}P. Find a displacement field u : Ω × R + → R d u : Ω × R + → R d u:Omega xxR_(+)rarrR^(d)\boldsymbol{u}: \Omega \times \mathbb{R}_{+} \rightarrow \mathbb{R}^{d}u:Ω×R+→Rd and a stress field σ : Ω × R + → S d σ : Ω × R + → S d sigma:Omega xxR_(+)rarrS^(d)\sigma: \Omega \times \mathbb{R}_{+} \rightarrow \mathbb{S}^{d}σ:Ω×R+→Sd such that, for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+,
(4) σ ˙ ( t ) = E ε ( u ˙ ( t ) ) + G ( σ ( t ) , ε ( u ( t ) ) ) in Ω , Div σ ( t ) + f 0 ( t ) = 0 in Ω , u ( t ) = 0 on Γ 1 , σ ( t ) ν = f 2 ( t ) on Γ 2 , (4) σ ˙ ( t ) = E ε ( u ˙ ( t ) ) + G ( σ ( t ) , ε ( u ( t ) ) )  in  Ω , Div σ ( t ) + f 0 ( t ) = 0  in  Ω , u ( t ) = 0  on  Γ 1 , σ ( t ) ν = f 2 ( t )  on  Γ 2 , {:(4){:[sigma^(˙)(t)=Eepsi(u^(˙)(t))+G(sigma(t)","epsi(u(t)))," in ",Omega","],[Div sigma(t)+f_(0)(t)=0," in ",Omega","],[u(t)=0," on ",Gamma_(1)","],[sigma(t)nu=f_(2)(t)," on ",Gamma_(2)","]:}:}\begin{array}{rll} \dot{\boldsymbol{\sigma}}(t)=\mathcal{E} \varepsilon(\dot{\boldsymbol{u}}(t))+\mathcal{G}(\boldsymbol{\sigma}(t), \varepsilon(\boldsymbol{u}(t))) & \text { in } & \Omega, \\ \operatorname{Div} \boldsymbol{\sigma}(t)+\boldsymbol{f}_{0}(t)=\mathbf{0} & \text { in } & \Omega, \\ \boldsymbol{u}(t)=\mathbf{0} & \text { on } & \Gamma_{1}, \\ \boldsymbol{\sigma}(t) \boldsymbol{\nu}=\boldsymbol{f}_{2}(t) & \text { on } & \Gamma_{2}, \tag{4} \end{array}(4)σ˙(t)=Eε(u˙(t))+G(σ(t),ε(u(t))) in Ω,Divσ(t)+f0(t)=0 in Ω,u(t)=0 on Γ1,σ(t)ν=f2(t) on Γ2,
u ν ( t ) ≤ g , ( u ν ( t ) − g ) ( σ ν ( t ) + p ( u ν ( t ) ) + ξ ( t ) ≤ 0 , 0 ≤ ξ ( t ) ≤ ∫ 0 t b ( t − s ) u ν + ( s ) d s , ξ ( t ) = 0 if u ν ( t ) < 0 , ξ ( t ) = ∫ 0 t b ( t − s ) u ν + ( s ) d s if u ν ( t ) > 0 } on Γ 3 , u ν ( t ) ≤ g , u ν ( t ) − g σ ν ( t ) + p u ν ( t ) + ξ ( t ) ≤ 0 , 0 ≤ ξ ( t ) ≤ ∫ 0 t   b ( t − s ) u ν + ( s ) d s , ξ ( t ) = 0  if  u ν ( t ) < 0 , ξ ( t ) = ∫ 0 t   b ( t − s ) u ν + ( s ) d s  if  u ν ( t ) > 0  on  Γ 3 , {:[{:[u_(nu)(t) <= g","],[(u_(nu)(t)-g)(sigma_(nu)(t)+p(u_(nu)(t))+xi(t) <= 0,:}],[0 <= xi(t) <= int_(0)^(t)b(t-s)u_(nu)^(+)(s)ds","],[xi(t)=0" if "u_(nu)(t) < 0","]:},,],[xi(t)=int_(0)^(t)b(t-s)u_(nu)^(+)(s)ds" if ",u_(nu)(t) > 0]}quad" on "quadGamma_(3),\left.\begin{array}{rll} \begin{array}{l} u_{\nu}(t) \leq g, \\ \left(u_{\nu}(t)-g\right)\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t) \leq 0,\right. \\ 0 \leq \xi(t) \leq \int_{0}^{t} b(t-s) u_{\nu}^{+}(s) d s, \\ \xi(t)=0 \text { if } u_{\nu}(t)<0, \end{array} & & \\ \xi(t)=\int_{0}^{t} b(t-s) u_{\nu}^{+}(s) d s \text { if } & u_{\nu}(t)>0 \end{array}\right\} \quad \text { on } \quad \Gamma_{3},uν(t)≤g,(uν(t)−g)(σν(t)+p(uν(t))+ξ(t)≤0,0≤ξ(t)≤∫0tb(t−s)uν+(s)ds,ξ(t)=0 if uν(t)<0,ξ(t)=∫0tb(t−s)uν+(s)ds if uν(t)>0} on Γ3,
Equation (1) represents the viscoplastic constitutive law of the material in which ε ( u ) ε ( u ) epsi(u)\boldsymbol{\varepsilon}(\boldsymbol{u})ε(u) denotes the linearized stress tensor, E E E\mathcal{E}E is the elasticity tensor and G G G\mathcal{G}G is a given constitutive function. Equation (2) is the equilibrium equation in which Div denotes the divergence operator for tensor valued functions. Conditions (3) and (4) are the displacement and traction boundary conditions, respectively, and condition (5) represents the contact condition with normal compliance, unilateral constraint and memory term, in which σ ν σ ν sigma_(nu)\sigma_{\nu}σν denotes the normal stress, u ν u ν u_(nu)u_{\nu}uν is the normal displacement, g ≥ 0 g ≥ 0 g >= 0g \geq 0g≥0 and p , b p , b p,bp, bp,b
are given functions. In the case when b b bbb vanishes, this condition was used in [1, 3, for instance. Condition (6) shows that the tangential stress on the contact surface, denoted σ τ σ τ sigma_(tau)\boldsymbol{\sigma}_{\tau}στ, vanishes. We use it here since we assume that the contact process is frictionless. Finally, (7) represents the initial conditions in which u 0 u 0 u_(0)\boldsymbol{u}_{0}u0 and σ 0 σ 0 sigma_(0)\boldsymbol{\sigma}_{0}σ0 denote the initial displacement and the initial stress field, respectively.
Quasistatic frictionless and frictional contact problems for viscoplastic materials with a constitutive law of the form (1) have been studied in various papers, see 22 for a survey. There, various models of contact were stated and their variational analysis, including existence and uniqueness results, was provided. The novelty of the current paper arises on the contact condition (5); it describes a deformable foundation which becomes rigid when the penetration reaches the critical bound g g ggg and which develops memory effects. Considering such condition leads to a new and nonstandard mathematical model which, in a variational formulation, is governed by a history-dependent variational inequality for the displacement field.
The rest of the paper is structured as follows. In Section 2 we list the assumptions on the data and introduce the variational formulation of the problem. Then, in Section 3 we state our main result, Theorem (1) and provide a sketch of the proof.

2 Variational formulation

In the study of problem P P P\mathcal{P}P we use the standard notation for Sobolev and Lebesgue spaces associated to Ω Ω Omega\OmegaΩ and Γ Γ Gamma\GammaΓ. Also, we denote by "." and ‖ ⋅ ‖ ‖ ⋅ ‖ ||*||\|\cdot\|‖⋅‖ the inner product and norm on R d R d R^(d)\mathbb{R}^{d}Rd and S d S d S^(d)\mathbb{S}^{d}Sd, respectively. For each Banach space X X XXX we use the notation C ( R + ; X ) C R + ; X C(R_(+);X)C\left(\mathbb{R}_{+} ; X\right)C(R+;X) for the space of continuously functions defined on R + R + R_(+)\mathbb{R}_{+}R+with values on X X XXX and, for a subset K ⊂ X K ⊂ X K sub XK \subset XK⊂X, we still use the symbol C ( R + ; K ) C R + ; K C(R_(+);K)C\left(\mathbb{R}_{+} ; K\right)C(R+;K) for the set of continuous functions defined on R + R + R_(+)\mathbb{R}_{+}R+with values on K K KKK. We also consider the spaces
V = { v ∈ H 1 ( Ω ) d : v = 0 on Γ 1 } , Q = { τ = ( τ i j ) ∈ L 2 ( Ω ) d : τ i j = τ j i } . V = v ∈ H 1 ( Ω ) d : v = 0  on  Γ 1 , Q = τ = τ i j ∈ L 2 ( Ω ) d : τ i j = τ j i . V={v inH^(1)(Omega)^(d):v=0" on "Gamma_(1)},Q={tau=(tau_(ij))inL^(2)(Omega)^(d):tau_(ij)=tau_(ji)}.V=\left\{\boldsymbol{v} \in H^{1}(\Omega)^{d}: \boldsymbol{v}=\mathbf{0} \text { on } \Gamma_{1}\right\}, Q=\left\{\boldsymbol{\tau}=\left(\tau_{i j}\right) \in L^{2}(\Omega)^{d}: \tau_{i j}=\tau_{j i}\right\} .V={v∈H1(Ω)d:v=0 on Γ1},Q={τ=(τij)∈L2(Ω)d:τij=τji}.
These are Hilbert spaces together with the inner products ( ⋅ , ⋅ ) V , ( ⋅ , ⋅ ) Q ( ⋅ , ⋅ ) V , ( ⋅ , ⋅ ) Q (*,*)_(V),(*,*)_(Q)(\cdot, \cdot)_{V},(\cdot, \cdot)_{Q}(⋅,⋅)V,(⋅,⋅)Q,
( u , v ) V = ∫ Ω ε ( u ) ⋅ ε ( v ) d x , ( σ , τ ) Q = ∫ Ω σ ⋅ τ d x ( u , v ) V = ∫ Ω   ε ( u ) ⋅ ε ( v ) d x , ( σ , τ ) Q = ∫ Ω   σ ⋅ τ d x (u,v)_(V)=int_(Omega)epsi(u)*epsi(v)dx,quad(sigma,tau)_(Q)=int_(Omega)sigma*tau dx(\boldsymbol{u}, \boldsymbol{v})_{V}=\int_{\Omega} \boldsymbol{\varepsilon}(\boldsymbol{u}) \cdot \boldsymbol{\varepsilon}(\boldsymbol{v}) d x, \quad(\boldsymbol{\sigma}, \boldsymbol{\tau})_{Q}=\int_{\Omega} \boldsymbol{\sigma} \cdot \boldsymbol{\tau} d x(u,v)V=∫Ωε(u)⋅ε(v)dx,(σ,τ)Q=∫Ωσ⋅τdx
and the associated norms ‖ ⋅ ‖ V , ‖ ⋅ ‖ Q ‖ ⋅ ‖ V , ‖ ⋅ ‖ Q ||*||_(V),||*||_(Q)\|\cdot\|_{V},\|\cdot\|_{Q}‖⋅‖V,‖⋅‖Q, respectively. For an element v ∈ V v ∈ V v in V\boldsymbol{v} \in Vv∈V we still write v v v\boldsymbol{v}v for the trace of V V VVV and we denote by v ν v ν v_(nu)v_{\nu}vν the normal component of v v v\boldsymbol{v}v on Γ Γ Gamma\GammaΓ given by v ν = v ⋅ ν v ν = v ⋅ ν v_(nu)=v*nuv_{\nu}=\boldsymbol{v} \cdot \boldsymbol{\nu}vν=v⋅ν.
We assume that the elasticity tensor E E E\mathcal{E}E, the nonlinear constitutive function G G G\mathcal{G}G and the normal compliance function p p ppp satisfy the following conditions.
(8) { (a) E = ( E i j k l ) : Ω × S d → S d . (b) E i j k l = E k l i j = E j i k l ∈ L ∞ ( Ω ) , 1 ≤ i , j , k , l ≤ d . (c) There exists m E > 0 such that E τ ⋅ τ ≥ m E ‖ τ ‖ 2 ∀ τ ∈ S d , a.e. in Ω . (9) { (a) G : Ω × S d × S d → S d . (b) There exists L G > 0 such that ‖ G ( x , σ 1 , ε 1 ) − G ( x , σ 2 , ε 2 ) ‖ ≤ L G ( ‖ σ 1 − σ 2 ‖ + ‖ ε 1 − ε 2 ‖ ) ∀ σ 1 , σ 2 , ε 1 , ε 2 ∈ S d , a.e. x ∈ Ω . (c) The mapping x ↦ G ( x , σ , ε ) is measurable on Ω , for any σ , ε ∈ S d . (d) The mapping x ↦ G ( x , 0 , 0 ) belongs to Q . { (a) p : Γ 3 × R → R + . (b) There exists L p > 0 such that | p ( x , r 1 ) − p ( x , r 2 ) | ≤ L p | r 1 − r 2 | ∀ r 1 , r 2 ∈ R , a.e. x ∈ Γ 3 . (c) p ( x , r 1 ) − p ( x , r 2 ) ) ( r 1 − r 2 ) ≥ 0 ∀ r 1 , r 2 ∈ R , a.e. x ∈ Γ 3 . (d) The mapping x ↦ p ( x , r ) is measurable on Γ 3 , for any r ∈ R . (e) p ( x , r ) = 0 for all r ≤ 0 , a.e. x ∈ Γ 3 . (8)  (a)  E = E i j k l : Ω × S d → S d .  (b)  E i j k l = E k l i j = E j i k l ∈ L ∞ ( Ω ) , 1 ≤ i , j , k , l ≤ d .  (c) There exists  m E > 0  such that  E τ ⋅ τ ≥ m E ‖ τ ‖ 2 ∀ τ ∈ S d ,  a.e. in  Ω . (9)  (a)  G : Ω × S d × S d → S d .  (b) There exists  L G > 0  such that  G x , σ 1 , ε 1 − G x , σ 2 , ε 2 ≤ L G σ 1 − σ 2 + ε 1 − ε 2 ∀ σ 1 , σ 2 , ε 1 , ε 2 ∈ S d ,  a.e.  x ∈ Ω .  (c) The mapping  x ↦ G ( x , σ , ε )  is measurable on  Ω ,  for any  σ , ε ∈ S d .  (d) The mapping  x ↦ G ( x , 0 , 0 )  belongs to  Q .  (a)  p : Γ 3 × R → R + . (b) There exists  L p > 0  such that  p x , r 1 − p x , r 2 ≤ L p r 1 − r 2 ∀ r 1 , r 2 ∈ R ,  a.e.  x ∈ Γ 3 .  (c)  p x , r 1 − p x , r 2 r 1 − r 2 ≥ 0 ∀ r 1 , r 2 ∈ R ,  a.e.  x ∈ Γ 3 .  (d) The mapping  x ↦ p ( x , r )  is measurable on  Γ 3 ,  for any  r ∈ R .  (e)  p ( x , r ) = 0  for all  r ≤ 0 ,  a.e.  x ∈ Γ 3 . {:[(8){[" (a) "E=(E_(ijkl)):Omega xxS^(d)rarrS^(d).],[" (b) "E_(ijkl)=E_(klij)=E_(jikl)inL^(oo)(Omega)","1 <= i","j","k","l <= d.],[" (c) There exists "m_(E) > 0" such that "],[Etau*tau >= m_(E)||tau||^(2)AA tau inS^(d)","" a.e. in "Omega.]:}],[(9){[" (a) "G:Omega xxS^(d)xxS^(d)rarrS^(d).],[" (b) There exists "L_(G) > 0" such that "],[||G(x,sigma_(1),epsi_(1))-G(x,sigma_(2),epsi_(2))|| <= L_(G)(||sigma_(1)-sigma_(2)||+||epsi_(1)-epsi_(2)||)],[AAsigma_(1)","sigma_(2)","epsi_(1)","epsi_(2)inS^(d)","" a.e. "x in Omega.],[" (c) The mapping "x|->G(x","sigma","epsi)" is measurable on "Omega","],[" for any "sigma","epsi inS^(d).],[" (d) The mapping "x|->G(x","0","0)" belongs to "Q.]:}],[{[" (a) "p:Gamma_(3)xxRrarrR_(+).],[(b) There exists L_(p) > 0" such that "],[|p(x,r_(1))-p(x,r_(2))| <= L_(p)|r_(1)-r_(2)|AAr_(1)","r_(2)inR","" a.e. "x inGamma_(3).],[" (c) "{:p(x,r_(1))-p(x,r_(2)))(r_(1)-r_(2)) >= 0AAr_(1)","r_(2)inR","" a.e. "x inGamma_(3).],[" (d) The mapping "x|->p(x","r)" is measurable on "Gamma_(3)","],[" for any "r inR.],[" (e) "p(x","r)=0" for all "r <= 0","" a.e. "x inGamma_(3).]:}]:}\begin{align*} & \left\{\begin{array}{l} \text { (a) } \mathcal{E}=\left(\mathcal{E}_{i j k l}\right): \Omega \times \mathbb{S}^{d} \rightarrow \mathbb{S}^{d} . \\ \text { (b) } \mathcal{E}_{i j k l}=\mathcal{E}_{k l i j}=\mathcal{E}_{j i k l} \in L^{\infty}(\Omega), 1 \leq i, j, k, l \leq d . \\ \text { (c) There exists } m_{\mathcal{E}}>0 \text { such that } \\ \mathcal{E} \boldsymbol{\tau} \cdot \boldsymbol{\tau} \geq m_{\mathcal{E}}\|\boldsymbol{\tau}\|^{2} \forall \boldsymbol{\tau} \in \mathbb{S}^{d}, \text { a.e. in } \Omega . \end{array}\right. \tag{8}\\ & \left\{\begin{array}{l} \text { (a) } \mathcal{G}: \Omega \times \mathbb{S}^{d} \times \mathbb{S}^{d} \rightarrow \mathbb{S}^{d} . \\ \text { (b) There exists } L_{\mathcal{G}}>0 \text { such that } \\ \left\|\mathcal{G}\left(\boldsymbol{x}, \boldsymbol{\sigma}_{1}, \boldsymbol{\varepsilon}_{1}\right)-\mathcal{G}\left(\boldsymbol{x}, \boldsymbol{\sigma}_{2}, \boldsymbol{\varepsilon}_{2}\right)\right\| \leq L_{\mathcal{G}}\left(\left\|\boldsymbol{\sigma}_{1}-\boldsymbol{\sigma}_{2}\right\|+\left\|\boldsymbol{\varepsilon}_{1}-\boldsymbol{\varepsilon}_{2}\right\|\right) \\ \forall \boldsymbol{\sigma}_{1}, \boldsymbol{\sigma}_{2}, \boldsymbol{\varepsilon}_{1}, \boldsymbol{\varepsilon}_{2} \in \mathbb{S}^{d}, \text { a.e. } \boldsymbol{x} \in \Omega . \\ \text { (c) The mapping } \boldsymbol{x} \mapsto \mathcal{G}(\boldsymbol{x}, \boldsymbol{\sigma}, \boldsymbol{\varepsilon}) \text { is measurable on } \Omega, \\ \text { for any } \boldsymbol{\sigma}, \boldsymbol{\varepsilon} \in \mathbb{S}^{d} . \\ \text { (d) The mapping } \boldsymbol{x} \mapsto \mathcal{G}(\boldsymbol{x}, \mathbf{0}, \mathbf{0}) \text { belongs to } Q . \end{array}\right. \tag{9}\\ & \left\{\begin{array}{l} \text { (a) } p: \Gamma_{3} \times \mathbb{R} \rightarrow \mathbb{R}_{+} . \\ \text {(b) There exists } L_{p}>0 \text { such that } \\ \left|p\left(\boldsymbol{x}, r_{1}\right)-p\left(\boldsymbol{x}, r_{2}\right)\right| \leq L_{p}\left|r_{1}-r_{2}\right| \forall r_{1}, r_{2} \in \mathbb{R}, \text { a.e. } \boldsymbol{x} \in \Gamma_{3} . \\ \text { (c) } \left.p\left(\boldsymbol{x}, r_{1}\right)-p\left(\boldsymbol{x}, r_{2}\right)\right)\left(r_{1}-r_{2}\right) \geq 0 \forall r_{1}, r_{2} \in \mathbb{R}, \text { a.e. } \boldsymbol{x} \in \Gamma_{3} . \\ \text { (d) The mapping } \boldsymbol{x} \mapsto p(\boldsymbol{x}, r) \text { is measurable on } \Gamma_{3}, \\ \text { for any } r \in \mathbb{R} . \\ \text { (e) } p(\boldsymbol{x}, r)=0 \text { for all } r \leq 0, \text { a.e. } \boldsymbol{x} \in \Gamma_{3} . \end{array}\right. \end{align*}(8){ (a) E=(Eijkl):Ω×Sd→Sd. (b) Eijkl=Eklij=Ejikl∈L∞(Ω),1≤i,j,k,l≤d. (c) There exists mE>0 such that Eτ⋅τ≥mE‖τ‖2∀τ∈Sd, a.e. in Ω.(9){ (a) G:Ω×Sd×Sd→Sd. (b) There exists LG>0 such that ‖G(x,σ1,ε1)−G(x,σ2,ε2)‖≤LG(‖σ1−σ2‖+‖ε1−ε2‖)∀σ1,σ2,ε1,ε2∈Sd, a.e. x∈Ω. (c) The mapping x↦G(x,σ,ε) is measurable on Ω, for any σ,ε∈Sd. (d) The mapping x↦G(x,0,0) belongs to Q.{ (a) p:Γ3×R→R+.(b) There exists Lp>0 such that |p(x,r1)−p(x,r2)|≤Lp|r1−r2|∀r1,r2∈R, a.e. x∈Γ3. (c) p(x,r1)−p(x,r2))(r1−r2)≥0∀r1,r2∈R, a.e. x∈Γ3. (d) The mapping x↦p(x,r) is measurable on Γ3, for any r∈R. (e) p(x,r)=0 for all r≤0, a.e. x∈Γ3.
Moreover, the densities of body forces and surface tractions, the memory function and the initial data are such that
(11) f 0 ∈ C ( R + ; L 2 ( Ω ) d ) , f 2 ∈ C ( R + ; L 2 ( Γ 2 ) d ) . (12) b ∈ C ( R + ; L ∞ ( Γ 3 ) ) , b ( t , x ) ≥ 0 a.e. x ∈ Γ 3 , (13) u 0 ∈ V , σ 0 ∈ Q . (11) f 0 ∈ C R + ; L 2 ( Ω ) d , f 2 ∈ C R + ; L 2 Γ 2 d . (12) b ∈ C R + ; L ∞ Γ 3 , b ( t , x ) ≥ 0  a.e.  x ∈ Γ 3 , (13) u 0 ∈ V , σ 0 ∈ Q . {:[(11)f_(0)in C(R_(+);L^(2)(Omega)^(d))","quadf_(2)in C(R_(+);L^(2)(Gamma_(2))^(d)).],[(12)b in C(R_(+);L^(oo)(Gamma_(3)))","quad b(t","x) >= 0quad" a.e. "x inGamma_(3)","],[(13)u_(0)in V","quadsigma_(0)in Q.]:}\begin{align*} & \boldsymbol{f}_{0} \in C\left(\mathbb{R}_{+} ; L^{2}(\Omega)^{d}\right), \quad \boldsymbol{f}_{2} \in C\left(\mathbb{R}_{+} ; L^{2}\left(\Gamma_{2}\right)^{d}\right) . \tag{11}\\ & b \in C\left(\mathbb{R}_{+} ; L^{\infty}\left(\Gamma_{3}\right)\right), \quad b(t, \boldsymbol{x}) \geq 0 \quad \text { a.e. } \boldsymbol{x} \in \Gamma_{3}, \tag{12}\\ & \boldsymbol{u}_{0} \in V, \quad \boldsymbol{\sigma}_{0} \in Q . \tag{13} \end{align*}(11)f0∈C(R+;L2(Ω)d),f2∈C(R+;L2(Γ2)d).(12)b∈C(R+;L∞(Γ3)),b(t,x)≥0 a.e. x∈Γ3,(13)u0∈V,σ0∈Q.
Consider now the subset U ⊂ V U ⊂ V U sub VU \subset VU⊂V, the operators P : V → V , B P : V → V , B P:V rarr V,BP: V \rightarrow V, \mathcal{B}P:V→V,B : C ( R + ; V ) → C ( R + ; L 2 ( Γ 3 ) ) C R + ; V → C R + ; L 2 Γ 3 C(R_(+);V)rarr C(R_(+);L^(2)(Gamma_(3)))C\left(\mathbb{R}_{+} ; V\right) \rightarrow C\left(\mathbb{R}_{+} ; L^{2}\left(\Gamma_{3}\right)\right)C(R+;V)→C(R+;L2(Γ3)) and the function f : R + → V f : R + → V f:R_(+)rarr V\boldsymbol{f}: \mathbb{R}_{+} \rightarrow Vf:R+→V defined by
(14) U = { v ∈ V : v ν ≤ g on Γ 3 } (15) ( P u , v ) V = ∫ Γ 3 p ( u ν ) v ν d a ∀ u , v ∈ V (16) ( B u ( t ) , ξ ) L 2 ( Γ 3 ) = ( ∫ 0 t b ( t − s ) u ν + ( s ) d s , ξ ) L 2 ( Γ 3 ) ∀ u ∈ C ( R + ; V ) , ξ ∈ L 2 ( Γ 3 ) , t ∈ R + (17) ( f ( t ) , v ) V = ∫ Ω f 0 ( t ) ⋅ v d x + ∫ Γ 2 f 2 ( t ) ⋅ v d a ∀ v ∈ V , t ∈ R + (14) U = v ∈ V : v ν ≤ g  on  Γ 3 (15) ( P u , v ) V = ∫ Γ 3   p u ν v ν d a ∀ u , v ∈ V (16) ( B u ( t ) , ξ ) L 2 Γ 3 = ∫ 0 t   b ( t − s ) u ν + ( s ) d s , ξ L 2 Γ 3 ∀ u ∈ C R + ; V , ξ ∈ L 2 Γ 3 , t ∈ R + (17) ( f ( t ) , v ) V = ∫ Ω   f 0 ( t ) ⋅ v d x + ∫ Γ 2   f 2 ( t ) ⋅ v d a ∀ v ∈ V , t ∈ R + {:[(14)U={v in V:v_(nu) <= g" on "Gamma_(3)}],[(15)(Pu","v)_(V)=int_(Gamma_(3))p(u_(nu))v_(nu)da quad AA u","v in V],[(16)(Bu(t)","xi)_(L^(2)(Gamma_(3)))=(int_(0)^(t)b(t-s)u_(nu)^(+)(s)ds,xi)_(L^(2)(Gamma_(3)))],[quad AA u in C(R_(+);V)","xi inL^(2)(Gamma_(3))","t inR_(+)],[(17)(f(t)","v)_(V)=int_(Omega)f_(0)(t)*vdx+int_(Gamma_(2))f_(2)(t)*vda quad AA v in V","t inR_(+)]:}\begin{align*} & U=\left\{\boldsymbol{v} \in V: v_{\nu} \leq g \text { on } \Gamma_{3}\right\} \tag{14}\\ & (P \boldsymbol{u}, \boldsymbol{v})_{V}=\int_{\Gamma_{3}} p\left(u_{\nu}\right) v_{\nu} d a \quad \forall \boldsymbol{u}, \boldsymbol{v} \in V \tag{15}\\ & (\mathcal{B} \boldsymbol{u}(t), \xi)_{L^{2}\left(\Gamma_{3}\right)}=\left(\int_{0}^{t} b(t-s) u_{\nu}^{+}(s) d s, \xi\right)_{L^{2}\left(\Gamma_{3}\right)} \tag{16}\\ & \quad \forall \boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right), \xi \in L^{2}\left(\Gamma_{3}\right), t \in \mathbb{R}_{+} \\ & (\boldsymbol{f}(t), \boldsymbol{v})_{V}=\int_{\Omega} \boldsymbol{f}_{0}(t) \cdot \boldsymbol{v} d x+\int_{\Gamma_{2}} \boldsymbol{f}_{2}(t) \cdot \boldsymbol{v} d a \quad \forall \boldsymbol{v} \in V, t \in \mathbb{R}_{+} \tag{17} \end{align*}(14)U={v∈V:vν≤g on Γ3}(15)(Pu,v)V=∫Γ3p(uν)vνda∀u,v∈V(16)(Bu(t),ξ)L2(Γ3)=(∫0tb(t−s)uν+(s)ds,ξ)L2(Γ3)∀u∈C(R+;V),ξ∈L2(Γ3),t∈R+(17)(f(t),v)V=∫Ωf0(t)⋅vdx+∫Γ2f2(t)⋅vda∀v∈V,t∈R+
Then, the variational formulation of Problem P P P\mathcal{P}P is the following.
Problem P V P V P_(V)\mathcal{P}_{V}PV. Find a displacement field u : R + → U u : R + → U u:R_(+)rarr U\boldsymbol{u}: \mathbb{R}_{+} \rightarrow Uu:R+→U and a stress field σ : R + → Q σ : R + → Q sigma:R_(+)rarr Q\boldsymbol{\sigma}: \mathbb{R}_{+} \rightarrow Qσ:R+→Q such that, for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+,
(18) σ ( t ) = E ε ( u ( t ) ) + ∫ 0 t G ( σ ( s ) , ε ( u ( s ) ) ) d s + σ 0 − E ε ( u 0 ) (19) ( σ ( t ) , ε ( v ) − ε ( u ( t ) ) ) Q + ( P u ( t ) , v − u ( t ) ) V + ( B u ( t ) , v ν + − u ν + ( t ) ) L 2 ( Γ 3 ) ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ U (18) σ ( t ) = E ε ( u ( t ) ) + ∫ 0 t   G ( σ ( s ) , ε ( u ( s ) ) ) d s + σ 0 − E ε u 0 (19) ( σ ( t ) , ε ( v ) − ε ( u ( t ) ) ) Q + ( P u ( t ) , v − u ( t ) ) V + B u ( t ) , v ν + − u ν + ( t ) L 2 Γ 3 ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ U {:[(18)sigma(t)=Eepsi(u(t))+int_(0)^(t)G(sigma(s)","epsi(u(s)))ds+sigma_(0)-Eepsi(u_(0))],[(19)(sigma(t)","epsi(v)-epsi(u(t)))_(Q)+(Pu(t)","v-u(t))_(V)],[quad+(Bu(t),v_(nu)^(+)-u_(nu)^(+)(t))_(L^(2)(Gamma_(3))) >= (f(t)","v-u(t))_(V)quad AA v in U]:}\begin{align*} & \boldsymbol{\sigma}(t)=\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(t))+\int_{0}^{t} \mathcal{G}(\boldsymbol{\sigma}(s), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))) d s+\boldsymbol{\sigma}_{0}-\mathcal{E} \boldsymbol{\varepsilon}\left(\boldsymbol{u}_{0}\right) \tag{18}\\ & (\boldsymbol{\sigma}(t), \boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}+(P \boldsymbol{u}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V} \tag{19}\\ & \quad+\left(\mathcal{B} \boldsymbol{u}(t), v_{\nu}^{+}-u_{\nu}^{+}(t)\right)_{L^{2}\left(\Gamma_{3}\right)} \geq(\boldsymbol{f}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V} \quad \forall \boldsymbol{v} \in U \end{align*}(18)σ(t)=Eε(u(t))+∫0tG(σ(s),ε(u(s)))ds+σ0−Eε(u0)(19)(σ(t),ε(v)−ε(u(t)))Q+(Pu(t),v−u(t))V+(Bu(t),vν+−uν+(t))L2(Γ3)≥(f(t),v−u(t))V∀v∈U
Note that (18) is a consequence of (11) and (17), while (19) can be easily obtained by using integrations by parts, (20)-(6) and notation (14)-(17).

3 Existence and uniqueness

The unique solvability of Problem P V P V P_(V)\mathcal{P}_{V}PV is given by the following result.
Theorem 1 Assume that (8)-(13) hold. Then Problem P V P V P_(V)\mathcal{P}_{V}PV has a unique solution, which satisfies u ∈ C ( R + ; U ) u ∈ C R + ; U u in C(R_(+);U)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; U\right)u∈C(R+;U) and σ ∈ C ( R + ; Q ) σ ∈ C R + ; Q sigma in C(R_(+);Q)\boldsymbol{\sigma} \in C\left(\mathbb{R}_{+} ; Q\right)σ∈C(R+;Q).
Proof. The proof is carried out in several steps which we describe below.
(i) We use the Banach fixed point argument to prove that for each function u ∈ C ( R + ; V ) u ∈ C R + ; V u in C(R_(+);V)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right)u∈C(R+;V) there exists a unique function S u ∈ C ( R + ; Q ) S u ∈ C R + ; Q Su in C(R_(+);Q)\mathcal{S} \boldsymbol{u} \in C\left(\mathbb{R}_{+} ; Q\right)Su∈C(R+;Q) such that
S u ( t ) = ∫ 0 t G ( S u ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε ( u 0 ) ∀ t ∈ R + S u ( t ) = ∫ 0 t   G ( S u ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε u 0 ∀ t ∈ R + Su(t)=int_(0)^(t)G(Su(s)+Eepsi(u(s)),epsi(u(s)))ds+sigma_(0)-Eepsi(u_(0))quad AA t inR_(+)\mathcal{S} \boldsymbol{u}(t)=\int_{0}^{t} \mathcal{G}(\mathcal{S} \boldsymbol{u}(s)+\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(s)), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))) d s+\boldsymbol{\sigma}_{0}-\mathcal{E} \boldsymbol{\varepsilon}\left(\boldsymbol{u}_{0}\right) \quad \forall t \in \mathbb{R}_{+}Su(t)=∫0tG(Su(s)+Eε(u(s)),ε(u(s)))ds+σ0−Eε(u0)∀t∈R+
(ii) Next, we note that ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) is a solution of Problem P V P V P_(V)\mathcal{P}_{V}PV iff
(20) σ ( t ) = E ε ( u ( t ) ) + S u ( t ) ∀ t ∈ R + (21) ( E ε ( u ( t ) ) , ε ( v ) − ε ( u ( t ) ) ) Q + ( S u ( t ) , ε ( v ) − ε ( u ( t ) ) ) Q + ( B u ( t ) , v ν + − u ν + ( t ) ) L 2 ( Γ 3 ) + ( P u ( t ) , v − u ( t ) ) V ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ U , ∀ t ∈ R + (20) σ ( t ) = E ε ( u ( t ) ) + S u ( t ) ∀ t ∈ R + (21) ( E ε ( u ( t ) ) , ε ( v ) − ε ( u ( t ) ) ) Q + ( S u ( t ) , ε ( v ) − ε ( u ( t ) ) ) Q + B u ( t ) , v ν + − u ν + ( t ) L 2 Γ 3 + ( P u ( t ) , v − u ( t ) ) V ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ U , ∀ t ∈ R + {:[(20)sigma(t)=Eepsi(u(t))+Su(t)quad AA t inR_(+)],[(21)(Eepsi(u(t))","epsi(v)-epsi(u(t)))_(Q)+(Su(t)","epsi(v)-epsi(u(t)))_(Q)],[quad+(Bu(t),v_(nu)^(+)-u_(nu)^(+)(t))_(L^(2)(Gamma_(3)))+(Pu(t)","v-u(t))_(V)],[quad >= (f(t)","v-u(t))_(V)quad AA v in U","AA t inR_(+)]:}\begin{align*} & \boldsymbol{\sigma}(t)=\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(t))+\mathcal{S} \boldsymbol{u}(t) \quad \forall t \in \mathbb{R}_{+} \tag{20}\\ & (\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(t)), \boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}+(\mathcal{S} \boldsymbol{u}(t), \boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q} \tag{21}\\ & \quad+\left(\mathcal{B} \boldsymbol{u}(t), v_{\nu}^{+}-u_{\nu}^{+}(t)\right)_{L^{2}\left(\Gamma_{3}\right)}+(P \boldsymbol{u}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V} \\ & \quad \geq(\boldsymbol{f}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V} \quad \forall \boldsymbol{v} \in U, \forall t \in \mathbb{R}_{+} \end{align*}(20)σ(t)=Eε(u(t))+Su(t)∀t∈R+(21)(Eε(u(t)),ε(v)−ε(u(t)))Q+(Su(t),ε(v)−ε(u(t)))Q+(Bu(t),vν+−uν+(t))L2(Γ3)+(Pu(t),v−u(t))V≥(f(t),v−u(t))V∀v∈U,∀t∈R+
(iii) Let A : V → V A : V → V A:V rarr VA: V \rightarrow VA:V→V and φ : Q × L 2 ( Γ 3 ) × V → R φ : Q × L 2 Γ 3 × V → R varphi:Q xxL^(2)(Gamma_(3))xx V rarrR\varphi: Q \times L^{2}\left(\Gamma_{3}\right) \times V \rightarrow \mathbb{R}φ:Q×L2(Γ3)×V→R be defined by equalities
( A u , v ) V = ( E ε ( u ) , ε ( v ) ) Q + ( P u , v ) V φ ( x , v ) = ( σ , ε ( v ) ) Q + ( ξ , v ν + ) L 2 ( Γ 3 ) ( A u , v ) V = ( E ε ( u ) , ε ( v ) ) Q + ( P u , v ) V φ ( x , v ) = ( σ , ε ( v ) ) Q + ξ , v ν + L 2 Γ 3 {:[(Au","v)_(V)=(Eepsi(u)","epsi(v))_(Q)+(Pu","v)_(V)],[varphi(x","v)=(sigma","epsi(v))_(Q)+(xi,v_(nu)^(+))_(L^(2)(Gamma_(3)))]:}\begin{aligned} & (A \boldsymbol{u}, \boldsymbol{v})_{V}=(\mathcal{E} \varepsilon(\boldsymbol{u}), \boldsymbol{\varepsilon}(\boldsymbol{v}))_{Q}+(P \boldsymbol{u}, \boldsymbol{v})_{V} \\ & \varphi(x, \boldsymbol{v})=(\boldsymbol{\sigma}, \boldsymbol{\varepsilon}(\boldsymbol{v}))_{Q}+\left(\xi, v_{\nu}^{+}\right)_{L^{2}\left(\Gamma_{3}\right)} \end{aligned}(Au,v)V=(Eε(u),ε(v))Q+(Pu,v)Vφ(x,v)=(σ,ε(v))Q+(ξ,vν+)L2(Γ3)
for all u , v ∈ V , x = ( σ , ξ ) ∈ Q × L 2 ( Γ 3 ) u , v ∈ V , x = ( σ , ξ ) ∈ Q × L 2 Γ 3 u,v in V,x=(sigma,xi)in Q xxL^(2)(Gamma_(3))\boldsymbol{u}, \boldsymbol{v} \in V, x=(\boldsymbol{\sigma}, \xi) \in Q \times L^{2}\left(\Gamma_{3}\right)u,v∈V,x=(σ,ξ)∈Q×L2(Γ3). We prove that A : V → V A : V → V A:V rarr VA: V \rightarrow VA:V→V is a strongly monotone and Lipschitz continuous operator and there exists β ≥ 0 β ≥ 0 beta >= 0\beta \geq 0β≥0 such that
φ ( x 1 , u 2 ) − φ ( x 1 , u 1 ) + φ ( x 2 , u 1 ) − φ ( x 2 , u 2 ) ≤ β ‖ x 1 − x 2 ‖ Q × L 2 ( Γ 3 ) ‖ u 1 − u 2 ‖ V ∀ x 1 , x 2 ∈ Q × L 2 ( Γ 3 ) , u 1 , u 2 ∈ V . φ x 1 , u 2 − φ x 1 , u 1 + φ x 2 , u 1 − φ x 2 , u 2 ≤ β x 1 − x 2 Q × L 2 Γ 3 u 1 − u 2 V ∀ x 1 , x 2 ∈ Q × L 2 Γ 3 , u 1 , u 2 ∈ V . {:[varphi(x_(1),u_(2))-varphi(x_(1),u_(1))+varphi(x_(2),u_(1))-varphi(x_(2),u_(2))],[ <= beta||x_(1)-x_(2)||_(Q xxL^(2)(Gamma_(3)))||u_(1)-u_(2)||_(V)quad AAx_(1)","x_(2)in Q xxL^(2)(Gamma_(3))","u_(1)","u_(2)in V.]:}\begin{aligned} & \varphi\left(x_{1}, \boldsymbol{u}_{2}\right)-\varphi\left(x_{1}, \boldsymbol{u}_{1}\right)+\varphi\left(x_{2}, \boldsymbol{u}_{1}\right)-\varphi\left(x_{2}, \boldsymbol{u}_{2}\right) \\ & \leq \beta\left\|x_{1}-x_{2}\right\|_{Q \times L^{2}\left(\Gamma_{3}\right)}\left\|\boldsymbol{u}_{1}-\boldsymbol{u}_{2}\right\|_{V} \quad \forall x_{1}, x_{2} \in Q \times L^{2}\left(\Gamma_{3}\right), \boldsymbol{u}_{1}, \boldsymbol{u}_{2} \in V . \end{aligned}φ(x1,u2)−φ(x1,u1)+φ(x2,u1)−φ(x2,u2)≤β‖x1−x2‖Q×L2(Γ3)‖u1−u2‖V∀x1,x2∈Q×L2(Γ3),u1,u2∈V.
Moreover, we prove that for every n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N there exists s n > 0 s n > 0 s_(n) > 0s_{n}>0sn>0 such that
‖ S u 1 ( t ) − S u 2 ( t ) ‖ Q + ‖ B u 1 ( t ) − B u 2 ( t ) ‖ L 2 ( Γ 3 ) ≤ s n ∫ 0 t ‖ u 1 ( s ) − u 2 ( s ) ‖ V d s ∀ u 1 , u 2 ∈ C ( R + ; V ) , ∀ t ∈ [ 0 , n ] S u 1 ( t ) − S u 2 ( t ) Q + B u 1 ( t ) − B u 2 ( t ) L 2 Γ 3 ≤ s n ∫ 0 t   u 1 ( s ) − u 2 ( s ) V d s ∀ u 1 , u 2 ∈ C R + ; V , ∀ t ∈ [ 0 , n ] {:[||Su_(1)(t)-Su_(2)(t)||_(Q)+||Bu_(1)(t)-Bu_(2)(t)||_(L^(2)(Gamma_(3)))],[quad <= s_(n)int_(0)^(t)||u_(1)(s)-u_(2)(s)||_(V)ds quad AAu_(1)","u_(2)in C(R_(+);V)","AA t in[0","n]]:}\begin{aligned} & \left\|\mathcal{S} \boldsymbol{u}_{1}(t)-\mathcal{S} \boldsymbol{u}_{2}(t)\right\|_{Q}+\left\|\mathcal{B} \boldsymbol{u}_{1}(t)-\mathcal{B} \boldsymbol{u}_{2}(t)\right\|_{L^{2}\left(\Gamma_{3}\right)} \\ & \quad \leq s_{n} \int_{0}^{t}\left\|\boldsymbol{u}_{1}(s)-\boldsymbol{u}_{2}(s)\right\|_{V} d s \quad \forall \boldsymbol{u}_{1}, \boldsymbol{u}_{2} \in C\left(\mathbb{R}_{+} ; V\right), \forall t \in[0, n] \end{aligned}‖Su1(t)−Su2(t)‖Q+‖Bu1(t)−Bu2(t)‖L2(Γ3)≤sn∫0t‖u1(s)−u2(s)‖Vds∀u1,u2∈C(R+;V),∀t∈[0,n]
These properties allow to use Theorem 2 in [3]. In this way we prove the existence of a unique function u ∈ C ( R + ; U ) u ∈ C R + ; U u in C(R_(+);U)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; U\right)u∈C(R+;U) which satisfies the historydependent variational inequality (21), for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+.
(iv) Let σ σ sigma\boldsymbol{\sigma}σ be the function given by (20); then, the couple ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) satisfies (20)-(21) for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+and, moreover, it has the regularity u ∈ C ( R + ; U ) u ∈ C R + ; U u in C(R_(+);U)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; U\right)u∈C(R+;U), σ ∈ C ( R + ; Q ) σ ∈ C R + ; Q sigma in C(R_(+);Q)\boldsymbol{\sigma} \in C\left(\mathbb{R}_{+} ; Q\right)σ∈C(R+;Q). This concludes the existence part in Theorem 1. The uniqueness part follows from the uniqueness of the solution of the inequality (21), guaranteed by Theorem 2 in [3].

Acknowledgement

The work of the first two authors was supported within the Sectoral Operational Programme for Human Resources Development 2007-2013, co-financed by the European Social Fund, under the projects POSDRU/107/1.5/S/76841 and POSDRU/88/1.5/S/60185, respectively, entitled Modern Doctoral Studies: Internationalization and Interdisciplinarity, at University Babeş-Bolyai, Cluj-Napoca, Romania.

References

[1] J. Jarušek and M. Sofonea. On the solvability of dynamic elastic-viscoplastic contact problems. Zeitschrift für Angewandte Matematik und Mechanik (ZAMM). 88:3-22, 2008.
[2] M. Shillor, M. Sofonea, J.J. Telega. Models and Analysis of Quasistatic Contact. Lect. Notes Phys. Springer, Berlin Heidelberg, 2004.
[3] M. Sofonea and A. Matei. History-dependent quasivariational inequalities arising in Contact Mechanics. European Journal of Applied Mathematics. 22:471-491, 2011.

4 Proofs

Variational formulation. Assume in what follows that ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) are sufficiently regular functions which satisfy (11)-(77) and let v ∈ U v ∈ U v in U\boldsymbol{v} \in Uv∈U and t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+be given. We integrate equation (1) with the initial conditions (7) to obtain
(22) σ ( t ) = E ε ( u ( t ) ) + ∫ 0 t G ( σ ( s ) , ε ( u ( s ) ) ) d s + σ 0 − E ε ( u 0 ) . (22) σ ( t ) = E ε ( u ( t ) ) + ∫ 0 t   G ( σ ( s ) , ε ( u ( s ) ) ) d s + σ 0 − E ε u 0 . {:(22)sigma(t)=Eepsi(u(t))+int_(0)^(t)G(sigma(s)","epsi(u(s)))ds+sigma_(0)-Eepsi(u_(0)).:}\begin{equation*} \boldsymbol{\sigma}(t)=\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(t))+\int_{0}^{t} \mathcal{G}(\boldsymbol{\sigma}(s), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))) d s+\boldsymbol{\sigma}_{0}-\mathcal{E} \boldsymbol{\varepsilon}\left(\boldsymbol{u}_{0}\right) . \tag{22} \end{equation*}(22)σ(t)=Eε(u(t))+∫0tG(σ(s),ε(u(s)))ds+σ0−Eε(u0).
Next, we use Green formula and the equilibrium equation (2) to see that
∫ Ω σ ⋅ ( ε ( v ) − ε ( u ( t ) ) ) d x = ∫ Ω f 0 ( t ) ⋅ ( v − u ( t ) ) d x + ∫ Γ σ ν ⋅ ( v − u ( t ) ) d a ∫ Ω   σ ⋅ ( ε ( v ) − ε ( u ( t ) ) ) d x = ∫ Ω   f 0 ( t ) ⋅ ( v − u ( t ) ) d x + ∫ Γ   σ ν ⋅ ( v − u ( t ) ) d a int_(Omega)sigma*(epsi(v)-epsi(u(t)))dx=int_(Omega)f_(0)(t)*(v-u(t))dx+int_(Gamma)sigma nu*(v-u(t))da\int_{\Omega} \boldsymbol{\sigma} \cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))) d x=\int_{\Omega} \boldsymbol{f}_{0}(t) \cdot(\boldsymbol{v}-\boldsymbol{u}(t)) d x+\int_{\Gamma} \boldsymbol{\sigma} \boldsymbol{\nu} \cdot(\boldsymbol{v}-\boldsymbol{u}(t)) d a∫Ωσ⋅(ε(v)−ε(u(t)))dx=∫Ωf0(t)⋅(v−u(t))dx+∫Γσν⋅(v−u(t))da.
We split the boundary integral over Γ 1 , Γ 2 Γ 1 , Γ 2 Gamma_(1),Gamma_(2)\Gamma_{1}, \Gamma_{2}Γ1,Γ2 and Γ 3 Γ 3 Gamma_(3)\Gamma_{3}Γ3 and, since v − u ( t ) = 0 v − u ( t ) = 0 v-u(t)=0\boldsymbol{v}-\boldsymbol{u}(t)=\mathbf{0}v−u(t)=0 a.e. on Γ 1 Γ 1 Gamma_(1)\Gamma_{1}Γ1, and σ ν = f 2 ( t ) σ ν = f 2 ( t ) sigma nu=f_(2)(t)\boldsymbol{\sigma} \boldsymbol{\nu}=\boldsymbol{f}_{2}(t)σν=f2(t) a.e. on Γ 2 Γ 2 Gamma_(2)\Gamma_{2}Γ2 we deduce that
∫ Ω σ ⋅ ( ε ( v ) − ε ( u ( t ) ) ) d x = ∫ Ω f 0 ( t ) ⋅ ( v − u ( t ) ) d x + ∫ Γ 2 f 2 ( t ) ⋅ ( v − u ( t ) ) d a + ∫ Γ 3 σ ν ⋅ ( v − u ( t ) ) d a ∫ Ω   σ ⋅ ( ε ( v ) − ε ( u ( t ) ) ) d x = ∫ Ω   f 0 ( t ) ⋅ ( v − u ( t ) ) d x + ∫ Γ 2   f 2 ( t ) ⋅ ( v − u ( t ) ) d a + ∫ Γ 3   σ ν ⋅ ( v − u ( t ) ) d a {:[int_(Omega)sigma*(epsi(v)-epsi(u(t)))dx=int_(Omega)f_(0)(t)*(v-u(t))dx],[quad+int_(Gamma_(2))f_(2)(t)*(v-u(t))da+int_(Gamma_(3))sigma nu*(v-u(t))da]:}\begin{aligned} \int_{\Omega} \boldsymbol{\sigma} \cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))) d x=\int_{\Omega} \boldsymbol{f}_{0}(t) \cdot(\boldsymbol{v}-\boldsymbol{u}(t)) d x \\ \quad+\int_{\Gamma_{2}} \boldsymbol{f}_{2}(t) \cdot(\boldsymbol{v}-\boldsymbol{u}(t)) d a+\int_{\Gamma_{3}} \boldsymbol{\sigma} \boldsymbol{\nu} \cdot(\boldsymbol{v}-\boldsymbol{u}(t)) d a \end{aligned}∫Ωσ⋅(ε(v)−ε(u(t)))dx=∫Ωf0(t)⋅(v−u(t))dx+∫Γ2f2(t)⋅(v−u(t))da+∫Γ3σν⋅(v−u(t))da
Moreover, since
σ ν ⋅ ( v − u ( t ) ) = σ ν ( v ν − u ν ( t ) ) + σ τ ⋅ ( v τ − u τ ( t ) ) a.e. on Γ 3 , σ ν ⋅ ( v − u ( t ) ) = σ ν v ν − u ν ( t ) + σ τ ⋅ v τ − u τ ( t )  a.e. on  Γ 3 , sigma nu*(v-u(t))=sigma_(nu)(v_(nu)-u_(nu)(t))+sigma_(tau)*(v_(tau)-u_(tau)(t))quad" a.e. on "Gamma_(3),\boldsymbol{\sigma} \boldsymbol{\nu} \cdot(\boldsymbol{v}-\boldsymbol{u}(t))=\sigma_{\nu}\left(v_{\nu}-u_{\nu}(t)\right)+\boldsymbol{\sigma}_{\tau} \cdot\left(\boldsymbol{v}_{\tau}-\boldsymbol{u}_{\tau}(t)\right) \quad \text { a.e. on } \Gamma_{3},σν⋅(v−u(t))=σν(vν−uν(t))+στ⋅(vτ−uτ(t)) a.e. on Γ3,
taking into account (6) we obtain
(23) ∫ Ω σ ⋅ ( ε ( v ) − ε ( u ( t ) ) ) d x = ∫ Ω f 0 ( t ) ⋅ ( v − u ( t ) ) d x + ∫ Γ 2 f 2 ( t ) ⋅ ( v − u ( t ) ) d a + ∫ Γ 3 σ ν ( v ν − u ν ( t ) ) d a (23) ∫ Ω   σ ⋅ ( ε ( v ) − ε ( u ( t ) ) ) d x = ∫ Ω   f 0 ( t ) ⋅ ( v − u ( t ) ) d x + ∫ Γ 2   f 2 ( t ) ⋅ ( v − u ( t ) ) d a + ∫ Γ 3   σ ν v ν − u ν ( t ) d a {:[(23)int_(Omega)sigma*(epsi(v)-epsi(u(t)))dx=int_(Omega)f_(0)(t)*(v-u(t))dx],[quad+int_(Gamma_(2))f_(2)(t)*(v-u(t))da+int_(Gamma_(3))sigma_(nu)(v_(nu)-u_(nu)(t))da]:}\begin{align*} \int_{\Omega} \boldsymbol{\sigma} \cdot(\boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t))) d x=\int_{\Omega} \boldsymbol{f}_{0}(t) \cdot(\boldsymbol{v}-\boldsymbol{u}(t)) d x \tag{23}\\ \quad+\int_{\Gamma_{2}} \boldsymbol{f}_{2}(t) \cdot(\boldsymbol{v}-\boldsymbol{u}(t)) d a+\int_{\Gamma_{3}} \sigma_{\nu}\left(v_{\nu}-u_{\nu}(t)\right) d a \end{align*}(23)∫Ωσ⋅(ε(v)−ε(u(t)))dx=∫Ωf0(t)⋅(v−u(t))dx+∫Γ2f2(t)⋅(v−u(t))da+∫Γ3σν(vν−uν(t))da
We write now
(24) σ ν ( t ) ( v ν − u ν ( t ) ) = ( σ ν ( t ) + p ( u ν ( t ) ) + ξ ( t ) ) ( v ν − g ) + ( σ ν ( t ) + p ( u ν ( t ) ) + ξ ( t ) ) ( g − u ν ( t ) ) − ( p ( u ν ( t ) ) + ξ ( t ) ) ( v ν − u ν ( t ) ) a.e. on Γ 3 (24) σ ν ( t ) v ν − u ν ( t ) = σ ν ( t ) + p u ν ( t ) + ξ ( t ) v ν − g + σ ν ( t ) + p u ν ( t ) + ξ ( t ) g − u ν ( t ) − p u ν ( t ) + ξ ( t ) v ν − u ν ( t )  a.e. on  Γ 3 {:[(24)sigma_(nu)(t)(v_(nu)-u_(nu)(t))=(sigma_(nu)(t)+p(u_(nu)(t))+xi(t))(v_(nu)-g)],[quad+(sigma_(nu)(t)+p(u_(nu)(t))+xi(t))(g-u_(nu)(t))],[quad-(p(u_(nu)(t))+xi(t))(v_(nu)-u_(nu)(t))quad" a.e. on "Gamma_(3)]:}\begin{align*} & \sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right)=\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(v_{\nu}-g\right) \tag{24}\\ & \quad+\left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(g-u_{\nu}(t)\right) \\ & \quad-\left(p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(v_{\nu}-u_{\nu}(t)\right) \quad \text { a.e. on } \Gamma_{3} \end{align*}(24)σν(t)(vν−uν(t))=(σν(t)+p(uν(t))+ξ(t))(vν−g)+(σν(t)+p(uν(t))+ξ(t))(g−uν(t))−(p(uν(t))+ξ(t))(vν−uν(t)) a.e. on Γ3
then we use the contact conditions (5) and the definition (14) of the set U U UUU to see that
(25) ( σ ν ( t ) + p ( u ν ( t ) ) + ξ ( t ) ) ( v ν − g ) ≥ 0 (26) ( σ ν ( t ) + p ( u ν ( t ) ) + ξ ( t ) ) ( g − u ν ( t ) ) = 0 (27) ξ ( t ) ( v ν − u ν ( t ) ) ≤ ( ∫ 0 t b ( t − s ) u ν + ( s ) d s ) ( v ν + − u ν + ( t ) ) (25) σ ν ( t ) + p u ν ( t ) + ξ ( t ) v ν − g ≥ 0 (26) σ ν ( t ) + p u ν ( t ) + ξ ( t ) g − u ν ( t ) = 0 (27) ξ ( t ) v ν − u ν ( t ) ≤ ∫ 0 t   b ( t − s ) u ν + ( s ) d s v ν + − u ν + ( t ) {:[(25)(sigma_(nu)(t)+p(u_(nu)(t))+xi(t))(v_(nu)-g) >= 0],[(26)(sigma_(nu)(t)+p(u_(nu)(t))+xi(t))(g-u_(nu)(t))=0],[(27)xi(t)(v_(nu)-u_(nu)(t)) <= (int_(0)^(t)b(t-s)u_(nu)^(+)(s)ds)(v_(nu)^(+)-u_(nu)^(+)(t))]:}\begin{align*} & \left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(v_{\nu}-g\right) \geq 0 \tag{25}\\ & \left(\sigma_{\nu}(t)+p\left(u_{\nu}(t)\right)+\xi(t)\right)\left(g-u_{\nu}(t)\right)=0 \tag{26}\\ & \xi(t)\left(v_{\nu}-u_{\nu}(t)\right) \leq\left(\int_{0}^{t} b(t-s) u_{\nu}^{+}(s) d s\right)\left(v_{\nu}^{+}-u_{\nu}^{+}(t)\right) \tag{27} \end{align*}(25)(σν(t)+p(uν(t))+ξ(t))(vν−g)≥0(26)(σν(t)+p(uν(t))+ξ(t))(g−uν(t))=0(27)ξ(t)(vν−uν(t))≤(∫0tb(t−s)uν+(s)ds)(vν+−uν+(t))
a.e. on Γ 3 Γ 3 Gamma_(3)\Gamma_{3}Γ3. We combine (24)-(27) to deduce that
σ ν ( t ) ( v ν − u ν ( t ) ) ≥ − p ( u ν ( t ) ) ( v ν − u ν ( t ) ) − ( ∫ 0 t b ( t − s ) u ν + ( s ) d s ) ( v ν + − u ν + ( t ) ) a.e. on Γ 3 σ ν ( t ) v ν − u ν ( t ) ≥ − p u ν ( t ) v ν − u ν ( t ) − ∫ 0 t   b ( t − s ) u ν + ( s ) d s v ν + − u ν + ( t )  a.e. on  Γ 3 {:[sigma_(nu)(t)(v_(nu)-u_(nu)(t)) >= -p(u_(nu)(t))(v_(nu)-u_(nu)(t))],[quad-(int_(0)^(t)b(t-s)u_(nu)^(+)(s)ds)(v_(nu)^(+)-u_(nu)^(+)(t))quad" a.e. on "Gamma_(3)]:}\begin{aligned} & \sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right) \geq-p\left(u_{\nu}(t)\right)\left(v_{\nu}-u_{\nu}(t)\right) \\ & \quad-\left(\int_{0}^{t} b(t-s) u_{\nu}^{+}(s) d s\right)\left(v_{\nu}^{+}-u_{\nu}^{+}(t)\right) \quad \text { a.e. on } \Gamma_{3} \end{aligned}σν(t)(vν−uν(t))≥−p(uν(t))(vν−uν(t))−(∫0tb(t−s)uν+(s)ds)(vν+−uν+(t)) a.e. on Γ3
and, therefore,
(28) ∫ Γ 3 σ ν ( t ) ( v ν − u ν ( t ) ) d a ≥ − ∫ Γ 3 p ( u ν ( t ) ) ( v ν − u ν ( t ) ) d a − ∫ Γ 3 ( ∫ 0 t b ( t − s ) u ν + ( s ) d s ) ( v ν + − u ν + ( t ) ) d a (28) ∫ Γ 3   σ ν ( t ) v ν − u ν ( t ) d a ≥ − ∫ Γ 3   p u ν ( t ) v ν − u ν ( t ) d a − ∫ Γ 3   ∫ 0 t   b ( t − s ) u ν + ( s ) d s v ν + − u ν + ( t ) d a {:[(28)int_(Gamma_(3))sigma_(nu)(t)(v_(nu)-u_(nu)(t))da >= -int_(Gamma_(3))p(u_(nu)(t))(v_(nu)-u_(nu)(t))da],[-int_(Gamma_(3))(int_(0)^(t)b(t-s)u_(nu)^(+)(s)ds)(v_(nu)^(+)-u_(nu)^(+)(t))da]:}\begin{gather*} \int_{\Gamma_{3}} \sigma_{\nu}(t)\left(v_{\nu}-u_{\nu}(t)\right) d a \geq-\int_{\Gamma_{3}} p\left(u_{\nu}(t)\right)\left(v_{\nu}-u_{\nu}(t)\right) d a \tag{28}\\ -\int_{\Gamma_{3}}\left(\int_{0}^{t} b(t-s) u_{\nu}^{+}(s) d s\right)\left(v_{\nu}^{+}-u_{\nu}^{+}(t)\right) d a \end{gather*}(28)∫Γ3σν(t)(vν−uν(t))da≥−∫Γ3p(uν(t))(vν−uν(t))da−∫Γ3(∫0tb(t−s)uν+(s)ds)(vν+−uν+(t))da
Then, combining (23) and (28) and using notation (15)-(17) we obtain
(29) ( σ ( t ) , ε ( v ) − ε ( u ( t ) ) ) Q + ( P u ( t ) , v − u ( t ) ) V + ( B u ( t ) , v ν + − u ν + ( t ) ) L 2 ( Γ 3 ) ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ U (29) ( σ ( t ) , ε ( v ) − ε ( u ( t ) ) ) Q + ( P u ( t ) , v − u ( t ) ) V + B u ( t ) , v ν + − u ν + ( t ) L 2 Γ 3 ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ U {:[(29)(sigma(t)","epsi(v)-epsi(u(t)))_(Q)+(Pu(t)","v-u(t))_(V)],[quad+(Bu(t),v_(nu)^(+)-u_(nu)^(+)(t))_(L^(2)(Gamma_(3))) >= (f(t)","v-u(t))_(V)quad AA v in U]:}\begin{align*} & (\boldsymbol{\sigma}(t), \boldsymbol{\varepsilon}(\boldsymbol{v})-\boldsymbol{\varepsilon}(\boldsymbol{u}(t)))_{Q}+(P \boldsymbol{u}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V} \tag{29}\\ & \quad+\left(\mathcal{B} \boldsymbol{u}(t), v_{\nu}^{+}-u_{\nu}^{+}(t)\right)_{L^{2}\left(\Gamma_{3}\right)} \geq(\boldsymbol{f}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V} \quad \forall \boldsymbol{v} \in U \end{align*}(29)(σ(t),ε(v)−ε(u(t)))Q+(Pu(t),v−u(t))V+(Bu(t),vν+−uν+(t))L2(Γ3)≥(f(t),v−u(t))V∀v∈U
The variational formulation P V P V P_(V)\mathcal{P}_{V}PV is now a consequence of (22) and (29).
Proof of Theorem 1, We need some preliminary results.
Theorem 2 Let ( X , ‖ ⋅ ‖ X ) X , ‖ ⋅ ‖ X (X,||*||_(X))\left(X,\|\cdot\|_{X}\right)(X,‖⋅‖X) be a real Banach space and let Λ : C ( R + ; X ) → C ( R + ; X ) Λ : C R + ; X → C R + ; X Lambda:C(R_(+);X)rarr C(R_(+);X)\Lambda: C\left(\mathbb{R}_{+} ; X\right) \rightarrow C\left(\mathbb{R}_{+} ; X\right)Λ:C(R+;X)→C(R+;X) be a nonlinear operator with the following property: there exists c > 0 c > 0 c > 0c>0c>0 such that
(30) ‖ Λ u ( t ) − Λ v ( t ) ‖ X ≤ c ∫ 0 t ‖ u ( s ) − v ( s ) ‖ X d s (30) ‖ Λ u ( t ) − Λ v ( t ) ‖ X ≤ c ∫ 0 t   ‖ u ( s ) − v ( s ) ‖ X d s {:(30)||Lambda u(t)-Lambda v(t)||_(X) <= cint_(0)^(t)||u(s)-v(s)||_(X)ds:}\begin{equation*} \|\Lambda u(t)-\Lambda v(t)\|_{X} \leq c \int_{0}^{t}\|u(s)-v(s)\|_{X} d s \tag{30} \end{equation*}(30)‖Λu(t)−Λv(t)‖X≤c∫0t‖u(s)−v(s)‖Xds
for all u , v ∈ C ( R + ; X ) u , v ∈ C R + ; X u,v in C(R_(+);X)u, v \in C\left(\mathbb{R}_{+} ; X\right)u,v∈C(R+;X) and for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+. Then the operator Λ Λ Lambda\LambdaΛ has a unique fixed point η ∗ ∈ C ( R + ; X ) η ∗ ∈ C R + ; X eta^(**)in C(R_(+);X)\eta^{*} \in C\left(\mathbb{R}_{+} ; X\right)η∗∈C(R+;X).
Theorem 2 represents a simplified version of Corollary 2.5 in the paper
M. Sofonea, C. Avramescu, A. Matei, A Fixed point result with applications in the study of viscoplastic frictionless contact problems, Communications on pure and Applied Analysis 7 (2008), 645-658.
Note that in (30) and below, the notation Λ η ( t ) Λ η ( t ) Lambda eta(t)\Lambda \eta(t)Λη(t) represents the value of the function Λ η Λ η Lambda eta\Lambda \etaΛη at the point t t ttt, i.e. Λ η ( t ) = ( Λ η ) ( t ) Λ η ( t ) = ( Λ η ) ( t ) Lambda eta(t)=(Lambda eta)(t)\Lambda \eta(t)=(\Lambda \eta)(t)Λη(t)=(Λη)(t).
Let X X XXX be a real Hilbert space with inner product ( ⋅ , ⋅ ) X ( ⋅ , ⋅ ) X (*,*)_(X)(\cdot, \cdot)_{X}(⋅,⋅)X and associated norm ‖ ⋅ ‖ X ‖ ⋅ ‖ X ||*||_(X)\|\cdot\|_{X}‖⋅‖X and let Y Y YYY be a normed space with the norm ‖ ⋅ ‖ Y ‖ ⋅ ‖ Y ||*||_(Y)\|\cdot\|_{Y}‖⋅‖Y. Assume given a set K ⊂ X K ⊂ X K sub XK \subset XK⊂X, the operators A : X → X , R : C ( R + ; X ) → C ( R + ; Y ) A : X → X , R : C R + ; X → C R + ; Y A:X rarr X,R:C(R_(+);X)rarr C(R_(+);Y)A: X \rightarrow X, \mathcal{R}: C\left(\mathbb{R}_{+} ; X\right) \rightarrow C\left(\mathbb{R}_{+} ; Y\right)A:X→X,R:C(R+;X)→C(R+;Y), the functional φ : Y × X → R φ : Y × X → R varphi:Y xx X rarrR\varphi: Y \times X \rightarrow \mathbb{R}φ:Y×X→R and a function f : R + → X f : R + → X f:R_(+)rarr Xf: \mathbb{R}_{+} \rightarrow Xf:R+→X such that:
K K KKK is a closed, convex, nonempty subset of X X XXX.
(32) { (a) There exists m > 0 such that ( A u 1 − A u 2 , u 1 − u 2 ) X ≥ m ‖ u 1 − u 2 ‖ X 2 ∀ u 1 , u 2 ∈ X . (b) There exists L > 0 such that ‖ A u 1 − A u 2 ‖ X ≤ L ‖ u 1 − u 2 ‖ X ∀ u 1 , u 2 ∈ X . (33) { (a) For all y ∈ Y , φ ( y , ⋅ ) : X → R is convex and lsc. (b) There exists α > 0 such that φ ( y 1 , u 2 ) − φ ( y 1 , u 1 ) + φ ( y 2 , u 1 ) − φ ( y 2 , u 2 ) ≤ α ‖ y 1 − y 2 ‖ Y ‖ u 1 − u 2 ‖ X ∀ y 1 , y 2 ∈ Y , ∀ u 1 , u 2 ∈ X . ‖ R u 1 ( t ) − R u 2 ( t ) ‖ Y ≤ r n ∫ 0 t ‖ u 1 ( s ) − u 2 ( s ) ‖ X d s ∀ u 1 , u 2 ∈ C ( R + ; X ) , ∀ t ∈ [ 0 , n ] . f ∈ C ( R + ; X ) . (32)  (a) There exists  m > 0  such that  A u 1 − A u 2 , u 1 − u 2 X ≥ m u 1 − u 2 X 2 ∀ u 1 , u 2 ∈ X .  (b)   There exists  L > 0  such that  A u 1 − A u 2 X ≤ L u 1 − u 2 X ∀ u 1 , u 2 ∈ X . (33)  (a) For all  y ∈ Y , φ ( y , ⋅ ) : X → R  is convex and lsc.   (b) There exists  α > 0  such that  φ y 1 , u 2 − φ y 1 , u 1 + φ y 2 , u 1 − φ y 2 , u 2 ≤ α y 1 − y 2 Y u 1 − u 2 X ∀ y 1 , y 2 ∈ Y , ∀ u 1 , u 2 ∈ X . R u 1 ( t ) − R u 2 ( t ) Y ≤ r n ∫ 0 t   u 1 ( s ) − u 2 ( s ) X d s ∀ u 1 , u 2 ∈ C R + ; X , ∀ t ∈ [ 0 , n ] . f ∈ C R + ; X . {:[(32){[" (a) There exists "m > 0" such that "],[quad(Au_(1)-Au_(2),u_(1)-u_(2))_(X) >= m||u_(1)-u_(2)||_(X)^(2)quad AAu_(1)","u_(2)in X.],[" (b) "" There exists "L > 0" such that "],[||Au_(1)-Au_(2)||_(X) <= L||u_(1)-u_(2)||_(X)quad AAu_(1)","u_(2)in X.]:}],[(33){[" (a) For all "y in Y","varphi(y","*):X rarrR" is convex and lsc. "],[" (b) There exists "alpha > 0" such that "],[varphi(y_(1),u_(2))-varphi(y_(1),u_(1))+varphi(y_(2),u_(1))-varphi(y_(2),u_(2))],[ <= alpha||y_(1)-y_(2)||_(Y)||u_(1)-u_(2)||_(X)quad AAy_(1)","y_(2)in Y","AAu_(1)","u_(2)in X.],[quad||Ru_(1)(t)-Ru_(2)(t)||_(Y) <= r_(n)int_(0)^(t)||u_(1)(s)-u_(2)(s)||_(X)ds],[AAu_(1)","u_(2)in C(R_(+);X)","AA t in[0","n].],[f in C(R_(+);X).]:}]:}\begin{align*} & \left\{\begin{array}{l} \text { (a) There exists } m>0 \text { such that } \\ \quad\left(A u_{1}-A u_{2}, u_{1}-u_{2}\right)_{X} \geq m\left\|u_{1}-u_{2}\right\|_{X}^{2} \quad \forall u_{1}, u_{2} \in X . \\ \text { (b) } \text { There exists } L>0 \text { such that } \\ \left\|A u_{1}-A u_{2}\right\|_{X} \leq L\left\|u_{1}-u_{2}\right\|_{X} \quad \forall u_{1}, u_{2} \in X . \end{array}\right. \tag{32}\\ & \left\{\begin{array}{l} \text { (a) For all } y \in Y, \varphi(y, \cdot): X \rightarrow \mathbb{R} \text { is convex and lsc. } \\ \text { (b) There exists } \alpha>0 \text { such that } \\ \varphi\left(y_{1}, u_{2}\right)-\varphi\left(y_{1}, u_{1}\right)+\varphi\left(y_{2}, u_{1}\right)-\varphi\left(y_{2}, u_{2}\right) \\ \leq \alpha\left\|y_{1}-y_{2}\right\|_{Y}\left\|u_{1}-u_{2}\right\|_{X} \quad \forall y_{1}, y_{2} \in Y, \forall u_{1}, u_{2} \in X . \\ \quad\left\|\mathcal{R} u_{1}(t)-\mathcal{R} u_{2}(t)\right\|_{Y} \leq r_{n} \int_{0}^{t}\left\|u_{1}(s)-u_{2}(s)\right\|_{X} d s \\ \forall u_{1}, u_{2} \in C\left(\mathbb{R}_{+} ; X\right), \forall t \in[0, n] . \\ f \in C\left(\mathbb{R}_{+} ; X\right) . \end{array}\right. \tag{33} \end{align*}(32){ (a) There exists m>0 such that (Au1−Au2,u1−u2)X≥m‖u1−u2‖X2∀u1,u2∈X. (b)  There exists L>0 such that ‖Au1−Au2‖X≤L‖u1−u2‖X∀u1,u2∈X.(33){ (a) For all y∈Y,φ(y,⋅):X→R is convex and lsc.  (b) There exists α>0 such that φ(y1,u2)−φ(y1,u1)+φ(y2,u1)−φ(y2,u2)≤α‖y1−y2‖Y‖u1−u2‖X∀y1,y2∈Y,∀u1,u2∈X.‖Ru1(t)−Ru2(t)‖Y≤rn∫0t‖u1(s)−u2(s)‖Xds∀u1,u2∈C(R+;X),∀t∈[0,n].f∈C(R+;X).
We have the following result.
Theorem 3 Assume that (31) -(35) hold. Then there exists a unique function u ∈ C ( R + ; K ) u ∈ C R + ; K u in C(R_(+);K)u \in C\left(\mathbb{R}_{+} ; K\right)u∈C(R+;K) such that, for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+, the inequality below holds:
(36) ( A u ( t ) , v − u ( t ) ) X + φ ( R u ( t ) , v ) − φ ( R u ( t ) , u ( t ) ) ≥ ( f ( t ) , v − u ( t ) ) X ∀ v ∈ K (36) ( A u ( t ) , v − u ( t ) ) X + φ ( R u ( t ) , v ) − φ ( R u ( t ) , u ( t ) ) ≥ ( f ( t ) , v − u ( t ) ) X ∀ v ∈ K {:[(36)(Au(t)","v-u(t))_(X)+varphi(Ru(t)","v)-varphi(Ru(t)","u(t))],[ >= (f(t)","v-u(t))_(X)quad AA v in K]:}\begin{gather*} (A u(t), v-u(t))_{X}+\varphi(\mathcal{R} u(t), v)-\varphi(\mathcal{R} u(t), u(t)) \tag{36}\\ \geq(f(t), v-u(t))_{X} \quad \forall v \in K \end{gather*}(36)(Au(t),v−u(t))X+φ(Ru(t),v)−φ(Ru(t),u(t))≥(f(t),v−u(t))X∀v∈K
Theorem 3 represents a simplified version of Theorem 2 in 3 .
We turn to the main steps of the proof of Theorem 1.
Lemma 1 For each function u ∈ C ( R + ; V ) u ∈ C R + ; V u in C(R_(+);V)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right)u∈C(R+;V) there exists a unique function S u ∈ C ( R + ; Q ) S u ∈ C R + ; Q Su in C(R_(+);Q)\mathcal{S} \boldsymbol{u} \in C\left(\mathbb{R}_{+} ; Q\right)Su∈C(R+;Q) such that
(37) S u ( t ) = ∫ 0 t G ( S u ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε ( u 0 ) ∀ t ∈ R + (37) S u ( t ) = ∫ 0 t   G ( S u ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε u 0 ∀ t ∈ R + {:(37)Su(t)=int_(0)^(t)G(Su(s)+Eepsi(u(s))","epsi(u(s)))ds+sigma_(0)-Eepsi(u_(0))quad AA t inR_(+):}\begin{equation*} \mathcal{S} \boldsymbol{u}(t)=\int_{0}^{t} \mathcal{G}(\mathcal{S} \boldsymbol{u}(s)+\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(s)), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))) d s+\boldsymbol{\sigma}_{0}-\mathcal{E} \boldsymbol{\varepsilon}\left(\boldsymbol{u}_{0}\right) \quad \forall t \in \mathbb{R}_{+} \tag{37} \end{equation*}(37)Su(t)=∫0tG(Su(s)+Eε(u(s)),ε(u(s)))ds+σ0−Eε(u0)∀t∈R+
Moreover, the operator S : C ( R + ; V ) → C ( R + ; Q ) S : C R + ; V → C R + ; Q S:C(R_(+);V)rarr C(R_(+);Q)\mathcal{S}: C\left(\mathbb{R}_{+} ; V\right) \rightarrow C\left(\mathbb{R}_{+} ; Q\right)S:C(R+;V)→C(R+;Q) satisfies the following property: for every n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N there exists s n > 0 s n > 0 s_(n) > 0s_{n}>0sn>0 such that
(38) ‖ S u 1 ( t ) − S u 2 ( t ) ‖ Q ≤ s n ∫ 0 t ‖ u 1 ( s ) − u 2 ( s ) ‖ V d s ∀ u 1 , u 2 ∈ C ( R + ; V ) , ∀ t ∈ [ 0 , n ] (38) S u 1 ( t ) − S u 2 ( t ) Q ≤ s n ∫ 0 t   u 1 ( s ) − u 2 ( s ) V d s ∀ u 1 , u 2 ∈ C R + ; V , ∀ t ∈ [ 0 , n ] {:[(38)||Su_(1)(t)-Su_(2)(t)||_(Q) <= s_(n)int_(0)^(t)||u_(1)(s)-u_(2)(s)||_(V)ds],[AAu_(1)","u_(2)in C(R_(+);V)","AA t in[0","n]]:}\begin{gather*} \left\|\mathcal{S} \boldsymbol{u}_{1}(t)-\mathcal{S} \boldsymbol{u}_{2}(t)\right\|_{Q} \leq s_{n} \int_{0}^{t}\left\|\boldsymbol{u}_{1}(s)-\boldsymbol{u}_{2}(s)\right\|_{V} d s \tag{38}\\ \forall \boldsymbol{u}_{1}, \boldsymbol{u}_{2} \in C\left(\mathbb{R}_{+} ; V\right), \forall t \in[0, n] \end{gather*}(38)‖Su1(t)−Su2(t)‖Q≤sn∫0t‖u1(s)−u2(s)‖Vds∀u1,u2∈C(R+;V),∀t∈[0,n]
Proof. Let u ∈ C ( R + ; V ) u ∈ C R + ; V u in C(R_(+);V)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right)u∈C(R+;V) and consider the operator Λ : C ( R + ; Q ) → C ( R + ; Q ) Λ : C R + ; Q → C R + ; Q Lambda:C(R_(+);Q)rarr C(R_(+);Q)\Lambda: C\left(\mathbb{R}_{+} ; Q\right) \rightarrow C\left(\mathbb{R}_{+} ; Q\right)Λ:C(R+;Q)→C(R+;Q) defined as follows
(39) Λ τ ( t ) = ∫ 0 t G ( τ ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε ( u 0 ) ∀ τ ∈ C ( R + ; Q ) , t ∈ R + (39) Λ τ ( t ) = ∫ 0 t   G ( τ ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε u 0 ∀ τ ∈ C R + ; Q , t ∈ R + {:[(39)Lambda tau(t)=int_(0)^(t)G(tau(s)+Eepsi(u(s))","epsi(u(s)))ds+sigma_(0)-Eepsi(u_(0))],[AA tau in C(R_(+);Q)","t inR_(+)]:}\begin{align*} \Lambda \boldsymbol{\tau}(t)= & \int_{0}^{t} \mathcal{G}(\boldsymbol{\tau}(s)+\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(s)), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))) d s+\boldsymbol{\sigma}_{0}-\mathcal{E} \boldsymbol{\varepsilon}\left(\boldsymbol{u}_{0}\right) \tag{39}\\ & \forall \boldsymbol{\tau} \in C\left(\mathbb{R}_{+} ; Q\right), t \in \mathbb{R}_{+} \end{align*}(39)Λτ(t)=∫0tG(τ(s)+Eε(u(s)),ε(u(s)))ds+σ0−Eε(u0)∀τ∈C(R+;Q),t∈R+
The operator Λ Λ Lambda\LambdaΛ depends on u u u\boldsymbol{u}u but, for simplicity, we do not indicate explicitly this dependence.
Let τ 1 , τ 2 ∈ C ( R + ; Q ) τ 1 , τ 2 ∈ C R + ; Q tau_(1),tau_(2)in C(R_(+);Q)\boldsymbol{\tau}_{1}, \boldsymbol{\tau}_{2} \in C\left(\mathbb{R}_{+} ; Q\right)τ1,τ2∈C(R+;Q) and let t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+. Then, using (39) and (9) we have
‖ Λ τ 1 ( t ) − Λ τ 2 ( t ) ‖ Q = ‖ ∫ 0 t G ( τ 1 ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s − ∫ 0 t G ( τ 2 ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s ‖ Q ≤ L G ∫ 0 t ‖ τ 1 ( s ) − τ 2 ( s ) ‖ Q d s Λ τ 1 ( t ) − Λ τ 2 ( t ) Q = ∫ 0 t   G τ 1 ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) d s − ∫ 0 t   G τ 2 ( s ) + E ε ( u ( s ) ) , ε ( u ( s ) ) d s Q ≤ L G ∫ 0 t   τ 1 ( s ) − τ 2 ( s ) Q d s {:[||Lambdatau_(1)(t)-Lambdatau_(2)(t)||_(Q)],[=||int_(0)^(t)G(tau_(1)(s)+Eepsi(u(s)),epsi(u(s)))ds-int_(0)^(t)G(tau_(2)(s)+Eepsi(u(s)),epsi(u(s)))ds||_(Q)],[quad <= L_(G)int_(0)^(t)||tau_(1)(s)-tau_(2)(s)||_(Q)ds]:}\begin{aligned} & \left\|\Lambda \boldsymbol{\tau}_{1}(t)-\Lambda \boldsymbol{\tau}_{2}(t)\right\|_{Q} \\ & =\left\|\int_{0}^{t} \mathcal{G}\left(\boldsymbol{\tau}_{1}(s)+\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(s)), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))\right) d s-\int_{0}^{t} \mathcal{G}\left(\boldsymbol{\tau}_{2}(s)+\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(s)), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))\right) d s\right\|_{Q} \\ & \quad \leq L_{\mathcal{G}} \int_{0}^{t}\left\|\boldsymbol{\tau}_{1}(s)-\boldsymbol{\tau}_{2}(s)\right\|_{Q} d s \end{aligned}‖Λτ1(t)−Λτ2(t)‖Q=‖∫0tG(τ1(s)+Eε(u(s)),ε(u(s)))ds−∫0tG(τ2(s)+Eε(u(s)),ε(u(s)))ds‖Q≤LG∫0t‖τ1(s)−τ2(s)‖Qds
Next, we use Theorem 2 to see that Λ Λ Lambda\LambdaΛ has a unique fixed point in C ( R + ; Q ) C R + ; Q C(R_(+);Q)C\left(\mathbb{R}_{+} ; Q\right)C(R+;Q), denoted S u S u Su\mathcal{S} \boldsymbol{u}Su. And, finally, we combine (39) with equality Λ ( S u ) = S u Λ ( S u ) = S u Lambda(Su)=Su\Lambda(\mathcal{S} \boldsymbol{u})=\mathcal{S} \boldsymbol{u}Λ(Su)=Su to see that (37) holds.
To proceed, let u 1 , u 2 ∈ C ( R + ; V ) , n ∈ N u 1 , u 2 ∈ C R + ; V , n ∈ N u_(1),u_(2)in C(R_(+);V),n inN\boldsymbol{u}_{1}, \boldsymbol{u}_{2} \in C\left(\mathbb{R}_{+} ; V\right), n \in \mathbb{N}u1,u2∈C(R+;V),n∈N and let t ∈ [ 0 , n ] t ∈ [ 0 , n ] t in[0,n]t \in[0, n]t∈[0,n]. Then, using (37) and taking into account (8), (9) we write
‖ S u 1 ( t ) − S u 2 ( t ) ‖ Q ≤ K ( ∫ 0 t ‖ ε ( u 1 ( s ) ) − ε ( u 2 ( s ) ) ‖ Q d s + ∫ 0 t ‖ S u 1 ( s ) − S u 2 ( s ) ‖ Q d s ) = K ( ∫ 0 t ‖ u 1 ( s ) − u 2 ( s ) ‖ V d s + ∫ 0 t ‖ S u 1 ( s ) − S u 2 ( s ) ‖ Q d s ) S u 1 ( t ) − S u 2 ( t ) Q ≤ K ∫ 0 t   ε u 1 ( s ) − ε u 2 ( s ) Q d s + ∫ 0 t   S u 1 ( s ) − S u 2 ( s ) Q d s = K ∫ 0 t   u 1 ( s ) − u 2 ( s ) V d s + ∫ 0 t   S u 1 ( s ) − S u 2 ( s ) Q d s {:[||Su_(1)(t)-Su_(2)(t)||_(Q)],[ <= K(int_(0)^(t)||epsi(u_(1)(s))-epsi(u_(2)(s))||_(Q)ds+int_(0)^(t)||Su_(1)(s)-Su_(2)(s)||_(Q)ds)],[quad=K(int_(0)^(t)||u_(1)(s)-u_(2)(s)||_(V)ds+int_(0)^(t)||Su_(1)(s)-Su_(2)(s)||_(Q)ds)]:}\begin{aligned} & \left\|\mathcal{S} \boldsymbol{u}_{1}(t)-\mathcal{S} \boldsymbol{u}_{2}(t)\right\|_{Q} \\ & \leq \mathcal{K}\left(\int_{0}^{t}\left\|\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{1}(s)\right)-\boldsymbol{\varepsilon}\left(\boldsymbol{u}_{2}(s)\right)\right\|_{Q} d s+\int_{0}^{t}\left\|\mathcal{S} \boldsymbol{u}_{1}(s)-\mathcal{S} \boldsymbol{u}_{2}(s)\right\|_{Q} d s\right) \\ & \quad=\mathcal{K}\left(\int_{0}^{t}\left\|\boldsymbol{u}_{1}(s)-\boldsymbol{u}_{2}(s)\right\|_{V} d s+\int_{0}^{t}\left\|\mathcal{S} \boldsymbol{u}_{1}(s)-\mathcal{S} \boldsymbol{u}_{2}(s)\right\|_{Q} d s\right) \end{aligned}‖Su1(t)−Su2(t)‖Q≤K(∫0t‖ε(u1(s))−ε(u2(s))‖Qds+∫0t‖Su1(s)−Su2(s)‖Qds)=K(∫0t‖u1(s)−u2(s)‖Vds+∫0t‖Su1(s)−Su2(s)‖Qds)
where K K K\mathcal{K}K is a positive constant which depends on G G G\mathcal{G}G and E E E\mathcal{E}E. Using now a Gronwall argument we deduce that
‖ S u 1 ( t ) − S u 2 ( t ) ‖ Q ≤ K e n K ∫ 0 t ‖ u 1 ( s ) − u 2 ( s ) ‖ V d s S u 1 ( t ) − S u 2 ( t ) Q ≤ K e n K ∫ 0 t   u 1 ( s ) − u 2 ( s ) V d s ||Su_(1)(t)-Su_(2)(t)||_(Q) <= Ke^(nK)int_(0)^(t)||u_(1)(s)-u_(2)(s)||_(V)ds\left\|\mathcal{S} \boldsymbol{u}_{1}(t)-\mathcal{S} \boldsymbol{u}_{2}(t)\right\|_{Q} \leq \mathcal{K} e^{n \mathcal{K}} \int_{0}^{t}\left\|\boldsymbol{u}_{1}(s)-\boldsymbol{u}_{2}(s)\right\|_{V} d s‖Su1(t)−Su2(t)‖Q≤KenK∫0t‖u1(s)−u2(s)‖Vds
This inequality shows that (38) holds with s n = K e n K s n = K e n K s_(n)=Ke^(nK)s_{n}=\mathcal{K} e^{n \mathcal{K}}sn=KenK.
Next, we use the operator S : C ( R + ; V ) → C ( R + ; Q ) S : C R + ; V → C R + ; Q S:C(R_(+);V)rarr C(R_(+);Q)\mathcal{S}: C\left(\mathbb{R}_{+} ; V\right) \rightarrow C\left(\mathbb{R}_{+} ; Q\right)S:C(R+;V)→C(R+;Q) defined in Lemma 1 to obtain the following equivalence result.
Lemma 2 Let ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) be a couple of functions such that u ∈ C ( R + ; V ) u ∈ C R + ; V u in C(R_(+);V)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right)u∈C(R+;V), σ ∈ C ( R + ; Q ) σ ∈ C R + ; Q sigma in C(R_(+);Q)\boldsymbol{\sigma} \in C\left(\mathbb{R}_{+} ; Q\right)σ∈C(R+;Q). Then, ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) is a solution of Problem P V P V P_(V)\mathcal{P}_{V}PV if and only if (20) and (21) hold.
Proof. Assume that ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) is a solution of Problem P V P V P_(V)\mathcal{P}_{V}PV and let t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+. Using (18) we have
σ ( t ) − E ε ( u ( t ) ) = ∫ 0 t G ( σ ( s ) − E ε ( u ( s ) ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε ( u 0 ) σ ( t ) − E ε ( u ( t ) ) = ∫ 0 t   G ( σ ( s ) − E ε ( u ( s ) ) + E ε ( u ( s ) ) , ε ( u ( s ) ) ) d s + σ 0 − E ε u 0 sigma(t)-Eepsi(u(t))=int_(0)^(t)G(sigma(s)-Eepsi(u(s))+Eepsi(u(s)),epsi(u(s)))ds+sigma_(0)-Eepsi(u_(0))\boldsymbol{\sigma}(t)-\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(t))=\int_{0}^{t} \mathcal{G}(\boldsymbol{\sigma}(s)-\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(s))+\mathcal{E} \boldsymbol{\varepsilon}(\boldsymbol{u}(s)), \boldsymbol{\varepsilon}(\boldsymbol{u}(s))) d s+\boldsymbol{\sigma}_{0}-\mathcal{E} \boldsymbol{\varepsilon}\left(\boldsymbol{u}_{0}\right)σ(t)−Eε(u(t))=∫0tG(σ(s)−Eε(u(s))+Eε(u(s)),ε(u(s)))ds+σ0−Eε(u0)
and, using the definition (37) of the operator S S S\mathcal{S}S, we obtain (20). Moreover, combining (19) and (20) we deduce (21).
Conversely, assume that ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) satisfies (20) and (21). Then by (20) and the definition (37) of the operator S S S\mathcal{S}S we obtain (18). Moreover, using (20) and (21) we find (19) which concludes the proof.
We are now in position to provide the proof of Theorem 1.
Proof. We define the operator A : V → V A : V → V A:V rarr VA: V \rightarrow VA:V→V and the form φ : Q × L 2 ( Γ 3 ) × V → R φ : Q × L 2 Γ 3 × V → R varphi:Q xxL^(2)(Gamma_(3))xx V rarrR\varphi: Q \times L^{2}\left(\Gamma_{3}\right) \times V \rightarrow \mathbb{R}φ:Q×L2(Γ3)×V→R by equalities
(40) ( A u , v ) V = ( E ε ( u ) , ε ( v ) ) Q + ( P u , v ) V (41) φ ( x , v ) = ( σ , ε ( v ) ) Q + ( ξ , v ν + ) L 2 ( Γ 3 ) (40) ( A u , v ) V = ( E ε ( u ) , ε ( v ) ) Q + ( P u , v ) V (41) φ ( x , v ) = ( σ , ε ( v ) ) Q + ξ , v ν + L 2 Γ 3 {:[(40)(Au","v)_(V)=(Eepsi(u)","epsi(v))_(Q)+(Pu","v)_(V)],[(41)varphi(x","v)=(sigma","epsi(v))_(Q)+(xi,v_(nu)^(+))_(L^(2)(Gamma_(3)))]:}\begin{align*} & (A \boldsymbol{u}, \boldsymbol{v})_{V}=(\mathcal{E} \varepsilon(\boldsymbol{u}), \varepsilon(\boldsymbol{v}))_{Q}+(P \boldsymbol{u}, \boldsymbol{v})_{V} \tag{40}\\ & \varphi(x, \boldsymbol{v})=(\boldsymbol{\sigma}, \boldsymbol{\varepsilon}(\boldsymbol{v}))_{Q}+\left(\xi, v_{\nu}^{+}\right)_{L^{2}\left(\Gamma_{3}\right)} \tag{41} \end{align*}(40)(Au,v)V=(Eε(u),ε(v))Q+(Pu,v)V(41)φ(x,v)=(σ,ε(v))Q+(ξ,vν+)L2(Γ3)
for all u , v ∈ V , x = ( σ , ξ ) ∈ Q × L 2 ( Γ 3 ) u , v ∈ V , x = ( σ , ξ ) ∈ Q × L 2 Γ 3 u,v in V,x=(sigma,xi)in Q xxL^(2)(Gamma_(3))\boldsymbol{u}, \boldsymbol{v} \in V, x=(\boldsymbol{\sigma}, \xi) \in Q \times L^{2}\left(\Gamma_{3}\right)u,v∈V,x=(σ,ξ)∈Q×L2(Γ3). We also consider the operator R : C ( R + ; V ) → C ( R + ; Q × L 2 ( Γ 3 ) ) R : C R + ; V → C R + ; Q × L 2 Γ 3 R:C(R_(+);V)rarr C(R_(+);Q xxL^(2)(Gamma_(3)))\mathcal{R}: C\left(\mathbb{R}_{+} ; V\right) \rightarrow C\left(\mathbb{R}_{+} ; Q \times L^{2}\left(\Gamma_{3}\right)\right)R:C(R+;V)→C(R+;Q×L2(Γ3)) defined by
(42) R u ( t ) = ( S u ( t ) , B u ( t ) ) ∀ u ∈ C ( R + ; V ) (42) R u ( t ) = ( S u ( t ) , B u ( t ) ) ∀ u ∈ C R + ; V {:(42)Ru(t)=(Su(t)","Bu(t))quad AA u in C(R_(+);V):}\begin{equation*} \mathcal{R} \boldsymbol{u}(t)=(\mathcal{S} \boldsymbol{u}(t), \mathcal{B} \boldsymbol{u}(t)) \quad \forall \boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right) \tag{42} \end{equation*}(42)Ru(t)=(Su(t),Bu(t))∀u∈C(R+;V)
where, recall, S S S\mathcal{S}S and B B B\mathcal{B}B are the operators given in (37) and (16), respectively.
With this notation we consider the problem of finding a function u u u\boldsymbol{u}u : R + → V R + → V R_(+)rarr V\mathbb{R}_{+} \rightarrow VR+→V such that, for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+, the following inequality holds:
(43) ( A u ( t ) , v − u ( t ) ) V + φ ( R u ( t ) , v ) − φ ( R u ( t ) , u ( t ) ) ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ V . (43) ( A u ( t ) , v − u ( t ) ) V + φ ( R u ( t ) , v ) − φ ( R u ( t ) , u ( t ) ) ≥ ( f ( t ) , v − u ( t ) ) V ∀ v ∈ V . {:[(43)(Au(t)","v-u(t))_(V)+varphi(Ru(t)","v)-varphi(Ru(t)","u(t))],[ >= (f(t)","v-u(t))_(V)quad AA v in V.]:}\begin{gather*} (A \boldsymbol{u}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V}+\varphi(\mathcal{R} \boldsymbol{u}(t), \boldsymbol{v})-\varphi(\mathcal{R} \boldsymbol{u}(t), \boldsymbol{u}(t)) \tag{43}\\ \geq(\boldsymbol{f}(t), \boldsymbol{v}-\boldsymbol{u}(t))_{V} \quad \forall \boldsymbol{v} \in V . \end{gather*}(43)(Au(t),v−u(t))V+φ(Ru(t),v)−φ(Ru(t),u(t))≥(f(t),v−u(t))V∀v∈V.
To solve (43) we employ Theorem 3 with X = V , K = U X = V , K = U X=V,K=UX=V, K=UX=V,K=U and Y = Q × L 2 ( Γ 3 ) Y = Q × L 2 Γ 3 Y=Q xxL^(2)(Gamma_(3))Y=Q \times L^{2}\left(\Gamma_{3}\right)Y=Q×L2(Γ3). It is clear that (31) holds. Next, we use (8), (10) and the Sobolev
trace theorem to see that the operator A A AAA is strongly monotone and Lipschitz continuous, i.e. it verifies condition (32). In addition, we note that the functional φ φ varphi\varphiφ satisfies condition (33). We also use (38), assumption (12) and definition (42) to see that (34) holds, too. Finally, using (11) and (17) we deduce that f f fff has the regularity expressed in (35). It follows now from Theorem 3 that there exists a unique function u ∈ C ( R + ; V ) u ∈ C R + ; V u in C(R_(+);V)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right)u∈C(R+;V) which solves the inequality (433), for any t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+.
Based on the results above we deduce the existence of a unique function u ∈ C ( R + ; V ) u ∈ C R + ; V u in C(R_(+);V)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; V\right)u∈C(R+;V) which satisfies (21). Let σ σ sigma\boldsymbol{\sigma}σ be the function given by (20); then, the couple ( u , σ u , σ u,sigma\boldsymbol{u}, \boldsymbol{\sigma}u,σ ) satisfies (20)-(21) for all t ∈ R + t ∈ R + t inR_(+)t \in \mathbb{R}_{+}t∈R+and, moreover, it has the regularity u ∈ C ( R + ; U ) , σ ∈ C ( R + ; Q ) u ∈ C R + ; U , σ ∈ C R + ; Q u in C(R_(+);U),sigma in C(R_(+);Q)\boldsymbol{u} \in C\left(\mathbb{R}_{+} ; U\right), \boldsymbol{\sigma} \in C\left(\mathbb{R}_{+} ; Q\right)u∈C(R+;U),σ∈C(R+;Q). This concludes the existence part in Theorem 1. The uniqueness part follows from the uniqueness of the solution of the inequality (21), guaranteed by Theorem 3.
  1. *anca.farcas@ubbcluj.ro, University Babeş-Bolyai, 400110 Cluj-Napoca, Romania
    † † ^(†){ }^{\dagger}† fpatrulescu@ictp.acad.ro, Tiberiu Popoviciu Institute of Numerical Analysis P.O. Box 68-1 and University Babeş-Bolyai, 400110 Cluj-Napoca, Romania
    ‡ ‡ ^(‡){ }^{\ddagger}‡ sofonea@univ-perp.fr, Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France
2012

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