# A Quasistatic Contact Problem for Viscoelastic Materials with Slip-Dependent Friction and Time Delay

Authors: Si-sheng Yao; Nan-jing Huang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

A mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation is introduced and studied, in which the contact is bilateral, the friction is modeled with Tresca’s friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. The variational formulation of the mathematical model is given as a quasistatic integro-differential variational inequality system. Based on arguments of the time-dependent variational inequality and Banach's fixed point theorem, an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system is proved under some suitable conditions. Furthermore, the behavior of the solution with respect to perturbations of time-delay term is considered and a convergence
result is also given.


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## Body

## 1. Introduction

The phenomena of contact between deformable bodies or between deformable and rigid bodies are abound in industry and daily life. Contact of braking pads with wheels and that of tires with roads are just a few simple examples [1]. Because of the importance of contact processes in structural and mechanical systems, a considerable effort has been made in their modeling and numerical simulations (see [1–4] and the references therein). What is worth to be taken particularly is some engineering papers that discussed the developed mathematical modeling to a practically interesting problem [5, 6]. Owing to their inherent complexity, contact phenomena are modeled by nonlinear evolutionary problems that are difficult to analyze (see [1]). The first work concerned with the study of frictional contact problems within the framework of variational inequalities was made in [7]. Comprehensive references for the analysis and numerical approximation of contact problems include [1, 8]. Mathematical, mechanical, and numerical state-of-the-art on contact mechanics can be found in the proceedings [9, 10] and in the special issue [11].When a viscoelastic material undergoes a small deformation gradient with a relatively slow force applied, the process of the relative contact problem can be modeled by a quasistatic system. At a relatively short duration, the effect of temperature changes caused by energy dissipation on the deformation of the material is usually negligible. Rigorous mathematical treatment of quasistatic problems is a test of recent years. The reason lies in the considerable difficulties that the process of frictional contact presents in the modeling and analysis because of the complicated surface phenomena involved (see [12]). By employing Banach's fixed point theorem, Chau et al. [13] got some existence and uniqueness results for two quasistatic problems which describe the frictional contact between a deformable body and an obstacle. They also proved that the solution of the viscoelastic problem converges to the solution of the corresponding elastic problem. By using arguments for time-dependent elliptic variational inequalities and Banach's fixed point theorem, Rodriguez-Aros et al. [10] dealt with the existence of a unique solution to an evolutionary variational inequality with Volterra-type integral term. Delost and Fabre [14] presented a valid approximation method for a quasistatic abstract variational inequality with time-independent constraint and applied its results to the approximation of the quasistatic evolution of an elastic body in bilateral contact with a rigid foundation. Very recently, Vollebregt and Schuttelaars [15] studied the quasistatic analysis of a contact problem with slip-velocity-dependent friction. For more works concerned with the quasistatic contact problems, we refer to [1, 16] and the references therein.Quasistatic contact problems for viscoelastic or other materials with explicit time-dependent operators were investigated in a large number of papers. Applying the theory of evolutionary hemivariational inequality, Migórski et al. [17] proved the existence and the regularity properties of the unique weak solution to a nonlinear explicit time-dependent elastic-viscoplastic frictional contact problem with multivalued subdifferential boundary conditions. In [18], Migórski et al. considered a class of quasistatic contact problems for explicit time-dependent viscoelastic materials with subdifferential frictional contact conditions. Based on the fixed point theorem for multivalued mappings and variational-hemivariational inequality theory, Costea and Matei [19] proved the existence of weak solution for the general and unified framework contact models. They also discussed the uniqueness, the boundedness, and the stability of the weak solution under some suitable conditions.It is well known that time-delay phenomena are frequently encountered in various technical systems, such as electric, pneumatic and hydraulic networks, and chemical processes. For example, regarding polymer under the action of alternative stress, the stress will lag behind the strain, which is just a time delay phenomenon. It gives us a mechanism of the time-delay phenomena appeared in contact problems. Comincioli [20] proved the existence and uniqueness for a kind of variational inequality with time-delay. For general results of variational inequalities with time delay, we refer to [21–23]. However, to the best of our knowledge, there is no papers to study contact problems for viscoelastic materials with time-delay.Motivated and inspired by the work mentioned above, in this paper, we introduce and study a mathematical model which describes an explicit time-dependent quasistatic frictional contact problem between a deformable body and a foundation, in which the contact is bilateral, the friction is modeled with Tresca's friction law with the friction bound depending on the total slip, and the behavior of the material is described with a viscoelastic constitutive law with time delay. We give the variational formulation of the mathematical model as a quasistatic integro-differential variational inequality system. By using the arguments of time-dependent variational inequality and Banach's fixed point theorem, we prove an existence and uniqueness of the solution for the quasistatic integro-differential variational inequality system under some suitable conditions. Furthermore, we consider the behavior of the solution with respect to perturbations of time-delay term and show a convergence result. The results presented in this paper generalize and improve some known results of [1, 24].The paper is structured as follows. In Section2, we list the necessary assumptions on the data and derive the variational formulation for the problem. In this part, an example which is assumed to the Kelvin-Voigt viscoelastic constitutive law with long memory is given, which represents a constitutive equation of the form (2.19). In Section 3, we prove the existence and uniqueness of the solution to the quasistatic integro-differential variational inequality system. In Section 4, we study the behavior of the solution with respect to perturbations of time-delay term and derive the convergence result.


## 2. Preliminaries

Letℛd be a d-dimensional Euclidean space and 𝒮d the space of second order symmetric tensors on ℛd. Let Ω⊂ℛd be open, connected, and bounded with a Lipschitz boundary Γ that is divided into three disjoint measurable parts Γ1, Γ2, and Γ3 such that meas(Γ1)>0. Let L2(Ω) be the Lebesgue space of 2-integrable functions and Wk,p(Ω) the Sobolev space of functions whose weak derivatives of orders less than or equal to k are p-integrable on Ω. Let Hk(Ω)=Wk,2(Ω).Since the boundary is Lipschitz continuous, the outward unit normal which is denoted byν exists a.e. on Γ. For T>0, and let I¯=˙[0,T] be the bounded time interval of interest. Let ℜ(u) be the range of displacement u. Since the body is clamped on Γ1, the displacement field vanishes there. Surface traction of density f2 acts on Γ2 and a body force of density f0 is applied in Ω. The contact is bilateral, that is, the normal displacement uν vanishes on Γ3 at any time.The canonical inner products and corresponding norms onℛd and 𝒮d are defined as follows:
(2.1)u·v=uivi,∥v∥=(v·v)1/2,∀u,v∈ℛd,σ·τ=σijτij,∥τ∥=(τ·τ)1/2,∀σ,τ∈𝒮d.
Everywhere in the sequel the index i and j run between 1 and d and the summation convention over repeated indices is implied.In the following we denote(2.2)H={v=(v1,v2,…,vd)T∣vi∈L2(Ω),1≤i≤d}=L2(Ω)d,(u,v)H=∫Ω‍ui(x)vi(x)dx,Q={τ=(τij)∣τij=τji∈L2(Ω),1≤i,j≤d}=L2(Ω)sd×d,(σ,τ)Q=∫Ω‍σi,j(x)τi,j(x)dx,Q1=Ω×I,H1={v=(v1,v2,…,vd)T∣vi∈H1(Ω),1≤i≤d}=H1(Ω)d,V={v∈H1∣v=0onΓ1},V1={v∈V∣vν=0onΓ3},
where H and Q are Hilbert spaces with the canonical inner products. The associated norms on the spaces will be denoted by ∥·∥H and ∥·∥Q, respectively.Define(2.3)(u,v)H1=(u,v)H+(ε(u),ε(v))Q,∥v∥H1=(v,v)H1,∀u,v∈H1.
It is easy to verify that (H1,∥·∥H1) is a real Hilbert space. Since V is a closed subspace of the space H1 and meas(Γ1)>0, the following Korn's inequality holds:
(2.4)∥ε(v)∥Q≥ι∥v∥H1,∀v∈V,
where ι denotes a positive constant depending only on Ω and Γ1. We define the inner product (·,·)V and the norm ∥·∥V on V by
(2.5)(u,v)V=(ε(u),ε(v))Q,∥v∥V=∥ε(v)∥Q,∀u,v∈V.
It follows that ∥·∥H1 and ∥·∥V are equivalent norms on V. Thus, (V,∥·∥V) is a real Hilbert space and V1 is also a real Hilbert space under the inner product of the space V given by (2.5).For every elementv∈H1, we also use the notation v for the trace of v on Γ and we denote by vν and vτ the normal and the tangential components of v on Γ given by
(2.6)vν=v·ν,vτ=v-vνν.
We also denote by σν and στ the normal and the tangential traces of a function σ∈Q, and we recall that when σ is a regular function, that is, σ∈C1(Ω¯)sd×d, then
(2.7)σν=(σν)·ν,στ=σν-σνν,
and the following Green's formula holds:
(2.8)(σ,ε(v))Q+(Divσ,v)H=∫Γ‍σν·vda,∀v∈H1.We model the friction with Tresca's friction law, where the friction boundg is assumed to depend on the accumulated slip of the surface. In this model we try to incorporate changes in the contact surface structure resulted from sliding. Therefore, g=g(t,St(u˙)) on Γ3×I with St(u˙)(x) being the accumulated slip at the point x on Γ3 over the time period I¯ as
(2.9)St(u˙)=∫0t‍∥u˙τ(s)∥ds,t∈I¯.
It follows that ∥στ∥≤g(St(u˙)) on Γ3. When the strict inequality holds, the material point is in the stick zone: u˙τ=0, while when the equality holds, ∥στ∥=g(St(u˙)), the material point is in the slip zone: στ=-λu˙τ for some λ>0.Letr be a constant satisfying 0<r<T and set Q-r=Ω×(-r,0). Let 𝔅 be the Borel σ-algebra of the interval [-r,0] and μ(·) be a given finite signed measure defined on ([-r,0],𝔅). Zhu [22] defined the time-delay operator G as follows: for any h∈L2(Ω×(-r,∞))𝒮d×d,
(2.10)(Gh)(t,x)=˙∫-r0‍h(t+θ,x)μ(dθ).
In order to make the above integral coherent, we always take the integrand to be a Borel correction of h (by which we mean a Borel measurable function that is equal to h almost everywhere).Some special cases of the operatorG are as follows:  (i) LetΩ1={ω1,ω2,…,ωn,…}, ℒ1=2Ω1, and
(2.11)μ1(A)=∑ωi∈A‍μ1(ωi)=∑ωi∈A‍pi,A∈ℒ1,whereμ1(ωi)=pi,i=1,2,…,n,…,pi∈ℛ+ and μ1(∅)=0. Then it is easy to see that
(2.12)(Gh)(t,x)=∫-r0‍h(t+θ,x)μ1(dθ)=∑ωi∈(-r,0)‍pih(t+ωi,x),which can be used to describe the countably many discrete delays.(ii) LetΩ2=ℛ, ℒ2 be a σ-algebra of Ω2, m a Lebesgue measure, f a Lebesgue measurable function, and
(2.13)μ2(A)=∫A‍f(x)dm,A∈ℒ2.Then(2.14)(Gh)(t,x)=∫-r0‍h(t+θ,x)μ2(dθ)=∫-r0‍h(t+θ,x)∫dθ‍f(x)dm.Moreover, lettingf≡1, then
(2.15)(Gh)(t,x)=∫-r0‍h(t+θ,x)μ2(dθ)=∫-r0‍h(t+θ,x)dθ.Remark 2.1.
It is easy to see that, for anyh∈L2(Ω×(-r,∞))𝒮d×d, Gh as an element in L2(Ω×(0,∞))𝒮d×d is independent of the choices of Borel corrections for h.Remark 2.2.
Sinceμ is a very general regular measure, (2.10) can be used in many cases such as finitely many and countably many discrete delays. At this stage, we note that (2.10) contains a very wide class of time-delay operators.The following lemma is a fundamental result for operatorG.Lemma 2.3 (see [22]).
Forh∈L2((-r,∞)×Ω;ℛd), we have Gh∈L2((0,∞)×Ω;ℛd). Furthermore, for any g∈L2((0,∞)×Ω;ℛd),0≤s≤+∞ and 0≤s0≤r, the following inequality holds:
(2.16)|∫Ω‍dx∫0s+s0‍(Gh)·gdt|≤12∫Ω‍dx∫0s+s0‍∥g∥2dt+12|μ|([-r,0])2∫Ω‍dx∫-rs‍∥h∥2dt+12|μ|([-r,0])|μ|([-s0,0])∫Ω‍dx∫-rs+s0‍∥h∥2dt.Now we consider the contact problem. For anyu~∈ℜ(u), based on (2.10), we derive the time-delay operator 𝒢 of the form
(2.17)(𝒢h)(t,ε(u~))=˙∫-r0‍h(t+θ,ε(u~))μ(dθ).Remark 2.4.
Replacingx with ε(u~) in (2.10) and letting s0=0 and g=Gh in Lemma 2.3, it is easy to know that
(2.18)∥𝒢∥≤|μ|([-r,0]).Under the previous assumptions, the classical formulation of the frictional contact problem with total slip dependent friction bound and the time-delay is as follows. For anyu~∈ℜ(u), find a displacement field u:Ω×I¯→ℛd and a stress field σ:Ω×I¯→𝒮d such that
(2.19)σ(t)=𝒜(t,ε(u˙(t)))+ℬ(t,ε(u(t)))+𝒢h(t,ε(u~))inΩ×I¯,(2.20)Divσ(t)+f0(t)=0inΩ×I¯,(2.21)u(t)=0onΓ1×I¯,(2.22)σ(t)ν=f2(t)onΓ2×I¯,(2.23)uν(t)=0,∥στ(t)∥≤g(t,St(u˙(t)))onΓ3×I¯,(2.24)u(0)=u0inΩ.We present a short description of the equations and conditions in Problems (2.19)–(2.24). For more details and mechanical interpretation, we refer to [1, 16]. Here (2.19) represents the viscoelastic constitutive law in which 𝒜, ℬ, and 𝒢 are given nonlinear operators, called the viscosity operator, elasticity operator, and time-delay operator, respectively. The prime represents the derivative with respect to the time variable, and therefore u˙ represents the velocity field. Note that the explicit dependence of the viscosity, elasticity, and time-delay operators 𝒜, ℬ, and 𝒢 with respect to the time variable means that the model involve the situations when the properties of the material depend on the temperature, that is, its evolution in time is prescribed. Equality (2.20) represents the equilibrium equation where Divσ=(σij,j) represents the divergence of stress. Conditions (2.21) and (2.22) are the displacement and traction boundary conditions, respectively. Equation (2.23) represents the frictional contact conditions and (2.24) is the initial condition in which the function u0 denotes the initial displacement field.In the study of mechanical problems (2.19)–(2.24), we assume that 𝒜, ℬ, g, and h satisfy the following conditions.H(𝒜): 𝒜:Q1×𝒮d→𝒮d is an operator such that  (i)
∥𝒜(x,t1,ε1)-𝒜(x,t2,ε2)∥Q≤L(|t1-t2|+∥ε1-ε2∥Q), for all ε1,ε2∈𝒮d,t1,t2∈I¯, a.e. x∈Ω with L>0;(ii)
((𝒜(x,t,ε1)-𝒜(x,t,ε2)),(ε1-ε2))Q≥M∥ε1-ε2∥Q2, for all ε1,ε2∈𝒮d, a.e. (x,t)∈Q1 with M>0;(iii)
for anyε∈𝒮d,(x,t)↦𝒜(x,t,ε) is measurable on Q1;(iv)
the mapping(x,t)↦𝒜(x,t,0)∈L2(Q1)d×d.H(ℬ): ℬ:Q1×𝒮d→𝒮d is an operator such that  (i)
∥ℬ(x,t,ε1)-ℬ(x,t,ε2)∥Q≤L1∥ε1-ε2∥Q, for all ε1,ε2∈𝒮d, a.e. (x,t)∈Q1 with L1>0;(ii)
for anyε∈𝒮d,(x,t)↦ℬ(x,t,ε) is measurable on Q;(iii)
the mapping(x,t)↦ℬ(x,t,0)∈L2(Q1)d×d.H(g): g:Γ3×I×ℛ→ℛ+ is an operator such that  (i)
there existsL2>0 such that for all t1,t2∈I¯, u1,u2∈ℛ,x∈Ω, |g(x,t1,u1)-g(x,t2,u2)|≤L2(|t1-t2|+|u1-u2|);(ii)
for anyu∈ℛ,(x,t)↦g(x,t,u) is measurable;(iii)
the mapping(x,t)↦g(x,t,0)∈L2(Γ3×I);(iv)
∥στ∥<g(t,St(u˙(t)))⇒u˙τ=0,∥στ∥=g(t,St(u˙(t)))⇒ exists λ≥0 such that στ=-λu˙τ.H(h): h:Q1×𝒮d→𝒮d is an operator such that  (i)
∥h(x,t,ε1)-h(x,t,ε2)∥Q≤L3∥ε1-ε2∥Q, for all ε1,ε2∈𝒮d, a.e. (x,t)∈Q1 with L3>0;(ii)
for anyε∈𝒮d,(x,t)↦h(x,t,ε) is measurable on Q1;(iii)
(h(x,t,ε1)-h(x,t,ε2),(ε1-ε2))Q≥M'∥ε1-ε2∥Q2, for all ε1,ε2∈𝒮d, a.e. (x,t)∈Q1 with M'>0;(iv)
the mapping(x,t)↦h(x,t,0)∈L2(Q1)d×d.In the following, we provide an elementary example of the mechanical problem which hold the constitutive law equation (2.19).Example 2.5.
Let𝒜 and ℬ be nonlinear operators which describe the viscous and the elastic properties of the material and satisfy the conditions H(𝒜) and H(ℬ), respectively, while 𝒞 is the linear relaxation operator. The following example is assumed to be the Kelvin-Voigt viscoelastic constitutive law with long memory of the form
(2.25)σ(t)=𝒜(t,ε(u˙(t)))+ℬ(t,ε(u(t)))+∫0t‍𝒞(t-s)ε(u(s))ds,
which represents a constitutive equation of the form (2.19).Contact problems involving viscoelastic materials with long memory have been studied in [25, 26]. For more detail on the long memory models, we refer to [27, 28].The famous time-temperature superposition principle tells us that when materials are applied with the alternating stress, the reaction time is an inverse proportion to the effect of the frequency. Hence, the influence of increasing the time (or reducing the frequency) and elevating temperature to materials is equivalent.The sinusoidally driven indentation test was shown to be effective for viability characterization of articular cartilage. Based on the viscoelastic correspondence principle, Argatov [5] described the mechanical response of the articular cartilage layer in the framework of viscoelastic model. Using the asymptotic modeling approach, Argatov analyzed and interpreted the results of the indentation test. Now, deriving from the (30) and (115) in [5], and noting the relationship between time and frequency, we write the viscoelastic constitutive law in the following form:
(2.26)σ(t)=a1t2a2+t2ε(u˙(t))+b1+b2t2b3+t2ε(u(t)),
where a1,a2 and b1,b2,b3 are some parameters which rely on the characteristic relaxation time of strain under an applied step in stress, the equilibrium elastic modulus, and the glass elastic modulus.It is easy to verify that𝒜(t,ε(u˙(t)))=˙(a1t2/(a2+t2))ε(u˙(t)) and ℬ(t,ε(u(t)))=˙((b1+b2t2)/(b3+t2))ε(u(t)) satisfy the assumption H(𝒜) and H(ℬ), respectively.Next, we denote byf(t) the element of V1 given by
(2.27)(f(t),v)V=(f0(t),v)H+(f2(t),v)L2(Γ2)d,∀v∈V1,a.e.t∈I¯.When we assume that the body force and surface traction satisfyf0∈C(I¯;H) and f2∈C(I¯;L2(Γ2)), we can get
(2.28)f∈C(I¯;V1).Letj:I×L2(Γ3)×V1→ℛ be the functional defined as follows:
(2.29)j(t,v;w)=∫Γ3‍g(t,v)∥wτ∥da,∀v∈L2(Γ3),w∈V1.We notice that, by the assumption H(g), the integral in (2.29) is well defined.Lemma 2.6 (Gronwall's inequality).
Assume thatf,g∈C[a,b] satisfy
(2.30)f(t)≤g(t)+c∫at‍f(s)ds,t∈[a,b],
where c>0 is a constant. Then
(2.31)f(t)≤g(t)+c∫at‍g(s)ec(t-s)ds,t∈[a,b].
Moreover, if g is nondecreasing, then
(2.32)f(t)≤g(t)ec(t-a),t∈[a,b].Proceeding in a standard way with these notations, we combine (2.8)–(2.24) to obtain the following variational formulation.Problem 1.
Find a displacementu:I¯→V1 such that (2.24) holds and
(2.33)(𝒜(t,ε(u˙(t))),ε(v-u˙(t)))Q+(ℬ(t,ε(u(t))),ε(v-u˙(t)))Q+(𝒢h(t,ε(u~)),ε(v-u˙(t)))Q+j(t,St(u˙);v)-j(t,St(u˙);u˙(t))≥(f(t),v-u˙(t))V,∀v∈V1,u~∈ℜ(u),a.e.t∈I¯.
We first introduce the following problem.Problem 2.
Find a displacementu:I¯→V1 such that (2.24) holds and
(2.34)(𝒜(t,ε(u˙(t))),ε(v-u˙(t)))Q+(𝒢h(t,ε(u~)),ε(v-u˙(t)))Q+j(t,St(u˙);v)-j(t,St(u˙);u˙(t))≥(f(t),v-u˙(t))V,∀v∈V1,u~∈ℜ(u),a.e.t∈I¯.For solving the above problems, we derive some results for an elliptic variational inequality of the second kind: Givenf∈X, find u∈V such that
(2.35)(A(t,u),v-u)V+j(v)+j(u)≥(f,v-u)V,∀v∈V,a.e.t∈I¯.Lemma 2.7 (see [1]).
Letj:V→ℛ¯ be a proper, convex, and lower semicontinuous functional. Then for any f∈V, there exists a unique element u:=Proxj(f) such that
(2.36)u∈V,(u,v-u)V+j(v)-j(u)≥(f,v-u)V,∀v∈V,a.e.t∈I¯.Lemma 2.8.
LetV be a Hilbert space. Assume that H(𝒜) holds and j:V→ℛ¯ is a proper, convex, lower semicontinuous functional. Then for any f∈V, variational inequality (2.35) has a unique solution.Proof.
For anyf∈V, let ρ>0 be a parameter to be chosen later. Since ρj:V→ℛ¯ is again a proper, convex, and lower semicontinuous functional, we can define an operator T:I¯×V→V by
(2.37)T(t,v)=Proxρj(ρf-ρA(t,v)+v),∀v∈V,a.e.t∈I¯,
where (A(t,u),v)V=˙(𝒜(x,t,ε(u)),ε(v))Q (see (2.5)). We will show that with a suitable choice of ρ the operator T is a contractive mapping on I¯×V. To this end, let u,v∈V. Since Prox is a nonexpansive mapping, it follows from (2.37) that
(2.38)∥T(t,u)-T(t,v)∥V2≤∥u-v-ρ(A(t,u)-A(t,v))∥V2=∥u-v∥V2-2ρ(A(t,u)-A(t,v),u-v)V+ρ2∥A(t,u)-A(t,v)∥V2.
Using the assumption H(𝒜) and (2.5), we obtain
(2.39)∥T(t,u)-T(t,v)∥V2≤(1-2ρM+ρ2L2)∥u-v∥V2.
If 0≤ρ≤2M/L2, then
(2.40)0<1-2ρM+ρ2L2<1.
Taking
(2.41)α=(1-2ρM+ρ2L2)1/2,
we deduce that α∈(0,1) and
(2.42)∥T(t,u)-T(t,v)∥V2≤α∥u-v∥V2,
which shows that T:I¯×V→V is a contractive mapping. Therefore, T has a fixed point u, that is,
(2.43)u=Proxρj(ρf-ρA(t,u)+u),a.e.t∈I¯,
which implies
(2.44)(u,v-u)V+ρj(v)-ρj(u)≥(ρf-ρA(t,u)+u,v-u)V,∀v∈V,a.e.t∈I¯.
It follows that
(2.45)ρ[(A(t,u),v-u)V+j(v)-j(u)]≥ρ(f,v-u)V,∀v∈V,a.e.t∈I¯.
Since ρ>0, we deduce from the above inequality that u is a solution of variational inequality (2.35).
To show the uniqueness, we assume that there exist two solutionsu1,u2∈V of variational inequality (2.35). Then for any v∈V and a.e.t∈I¯, we have
(2.46)(A(t,u1),v-u1)V+j(v)+j(u1)≥(f,v-u1)V,(A(t,u2),v-u2)V+j(v)+j(u2)≥(f,v-u2)V.
Since j is proper, we know that j(u1)<∞ and j(u2)<∞. Taking v=u2 in the first inequality and v=u1 in the second one and adding the corresponding inequalities, we get
(2.47)(A(t,u1)-A(t,u2),u1-u2)V≤0.
Using (2.5) and H(𝒜), we obtain that u1=u2, which completes the proof of Lemma 2.8.Remark 2.9.
Lemma2.8 is a generalization of Theorem 4.1 of [1].


## 3. Main Results

In this section, we present an existence and uniqueness result concerned with the solution of Problem1. Throughout this section, we assume that H(𝒜), H(ℬ), H(g), H(h), and (2.28) hold.Theorem 3.1.
Problem2 has a unique solution u∈C1(I¯;V1).The proof of Theorem3.1 is based on fixed point arguments and is established in several steps. Let η∈C(I¯;Q) and ξ∈C(I¯;V1) be arbitrarily given. We consider the following auxiliary variational problem.Problem 3.
Findwηξ:I¯→V1 such that for any v∈V1(3.1)(𝒜(t,ε(wηξ(t))),ε(v-wηξ(t)))Q+(η(t),ε(v-wηξ(t)))Q+j(t,St(ξ);v)-j(t,St(ξ);wηξ(t))≥(f(t),v-wηξ(t))V,a.e.t∈I¯.Lemma 3.2.
There exists a unique solutionwηξ∈C(I¯;V1) to Problem 3.Proof.
For each fixedt∈I¯, in terms of hypotheses (2.9), H(𝒜), H(g), and (2.29), Problem 3 is an elliptic variational inequality on Q. It follows from Lemma 2.8 that Problem 3 is uniquely solvable. Let wηξ(t)∈V1 be the unique solution of Problem 3. Now we show that wηξ(t)∈C(I¯,V1).
Suppose thatt1,t2∈I¯. For simplicity we write wηξ(ti)=wi, η(ti)=ηi and f(ti)=fi with i=1,2. Using (3.1) for t=t1,t2, we have
(3.2)(𝒜(t1,ε(w1)),ε(v-w1))Q+(η1,ε(v-w1))Q+j(t1,St1(ξ);v)-j(t1,St1(ξ);w1)≥(f1,v-w1)V,(3.3)(𝒜(t2,ε(w2)),ε(v-w2))Q+(η2,ε(v-w2))Q+j(t2,St2(ξ);v)-j(t2,St2(ξ);w2)≥(f2,v-w2)V.
By adding two inequalities with v=w2 in (3.2) and v=w1 in (3.3), we get (3.4)(𝒜(t1,ε(w1))-𝒜(t2,ε(w2)),ε(w1-w2))Q≤(f1-f2,w1-w2)V+(η1-η2,ε(w1-w2))Q+D(t1,t2,ξ,w1,w2),
where
(3.5)D(t1,t2,ξ,w1,w2)=j(t1,St1(ξ);w2)-j(t1,St1(ξ);w1)+j(t2,St2(ξ);w1)-j(t2,St2(ξ);w2).
It implies that
(3.6)(𝒜(t1,ε(w1))-𝒜(t1,ε(w2)),ε(w1-w2))Q≤(𝒜(t1,ε(w2))-𝒜(t2,ε(w2)),ε(w1-w2))Q+D(t1,t2,ξ,w1,w2)+(η1-η2,ε(w1-w2))Q+(f1-f2,w1-w2)V.
By H(𝒜), we get
(3.7)(𝒜(t1,ε(w1))-𝒜(t1,ε(w2)),ε(w1-w2))Q≥M∥w1-w2∥V2,∥𝒜(t1,ε(w2))-𝒜(t2,ε(w2))∥Q≤L|t1-t2|.
Constituting a trace operator γ:V→L2(Γ3) that γv=v∣Γ3, since γ is a linear continuous operator, it implies that there exists a constant c>0 such that
(3.8)∥v∥L2(Γ3)≤c∥v∥V.
It follows from (2.9), (2.29), (3.8), and H(g) that
(3.9)D(t1,t2,ξ,w1,w2)≤L2(1+∥ξ∥C(I¯;V))|t1-t2|·∥w1-w2∥.
By (3.6)-(3.7) and (3.9), we have
(3.10)∥w1-w2∥V≤1M(∥f1-f2∥V+∥η1-η2∥Q+|t1-t2|(L+L2(1+∥ξ∥C(I¯;V)))),
which implies that wηξ∈C(I¯;V1). This completes the proof of Lemma 3.2.In order to get the unique solution of Problem2, we derive the following operator Λη:C(I¯;V1)→C(I¯;V1) defined by
(3.11)Ληξ=wηξ,∀ξ∈C(I¯;V1).Lemma 3.3.
For anyη∈C(I¯;Q), the operator Λη has a unique fixed point ξη∈C(I¯;V1).Proof.
Letη∈C(I¯;Q) and ξ1,ξ2∈C(I¯;V1). We denote by wi the solution of Problem 3 with ξ=ξi for i=1,2. By an argument similar to that used in obtaining (3.6), we get
(3.12)(𝒜(t,ε(w1))-𝒜(t,ε(w2)),ε(w1-w2))Q≤D(t,ξ1,ξ2,w1,w2),
where
(3.13)D(t,ξ1,ξ2,w1,w2)=j(t,St(ξ1);w2)-j(t,St(ξ1);w1)+j(t,St(ξ2);w1)-j(t,St(ξ2);w2).
Using (2.29), (3.8), (2.9), and H(g), we deduce that, for any t∈I¯,
(3.14)D(t,ξ1,ξ2,w1,w2)≤c1∫0t‍∥ξ1(s)-ξ2(s)∥Vds∥w1(t)-w2(t)∥V,
where c1=cL2. By using the similar method in obtaining (3.10), we have
(3.15)∥w1(t)-w2(t)∥V≤c1M∫0t‍∥ξ1(s)-ξ2(s)∥Vds,a.e.t∈I¯.
Since wi=wηξi=Ληξi, we rewrite the above inequality as
(3.16)∥Ληξ1(t)-Ληξ2(t)∥V≤c1M∫0t‍∥ξ1(s)-ξ2(s)∥Vds,a.e.t∈I¯.
For w∈C(I¯;V1), let
(3.17)∥w∥β=maxt∈Ie-βt∥w(t)∥V,
where β>0 is a constant which will be chosen later. Clearly, ∥·∥β defines a norm on the space C(I¯;V1) and
(3.18)e-βt∥Ληξ1(t)-Ληξ2(t)∥V≤c1e-βtM∫0t‍eβse-βs∥ξ1(s)-ξ2(s)∥Vds≤c1e-βtM∫0t‍eβs∥ξ1-ξ2∥βds≤c1Mβ∥ξ1-ξ2∥β,a.e.t∈I¯.
Thus,
(3.19)∥Ληξ1-Ληξ2∥β≤c1Mβ∥ξ1-ξ2∥β
and so the operator Λη is a contraction on the space C(I¯;V1) endowed with the equivalent norm ∥·∥β if we choose β such that Mβ>c1. Therefore, the operator Λη has a unique fixed point ξη∈C(I¯;V1), which completes the proof of Lemma 3.3.In what follows, for anyη∈C(I¯;Q), we write
(3.20)wη=wηξη.
By Ληξη=ξη, (3.11), and (3.20), we have
(3.21)wη=ξη.
Taking ξ=ξη in (3.1) and using (3.20) and (3.21), we deduce that, for any v∈V1,
(3.22)(𝒜(t,ε(wη(t))),ε(v-wη(t)))Q+(η,ε(v-wη(t)))Q+j(t,St(wη);v)-j(t,St(wη);wη(t))≥(f(t),v-wη(t))V,a.e.t∈I¯.
Let wη:I¯→V1 be the function given by
(3.23)uη(t)=∫0t‍wη(x)ds+u0,a.e.t∈I¯.
In addition, we define the operator Λ:C(I¯;Q)→C(I¯;Q) by
(3.24)Λη=∫-r0‍h(t+θ,ε(u~η))μ(dθ),∀η∈C(I¯;Q),a.e.t∈I¯.Lemma 3.4.
The operatorΛ has a unique fixed point η*∈C(I¯;Q).Proof.
For anyη1,η2∈C(I¯;Q), let ui=uηi, u~i=u~ηi and wi=wηi with i=1,2. Using (3.22) and arguments similar to those used in the proof of Lemma 3.3, we obtain
(3.25)∥w1(t)-w2(t)∥V≤c2(∥η1(t)-η2(t)∥Q+∫0t‍∥w1(s)-w2(s)∥Vds),a.e.t∈I¯,
where
(3.26)c2=max{1M,cM}.
An application of the Gronwall inequality yields
(3.27)∥w1(t)-w2(t)∥V≤c3(∥η1(t)-η2(t)∥Q+∫0t‍∥η1(s)-η2(s)∥Qds),a.e.t∈I¯,
where
(3.28)c3=max{c2,c22ec2T}.
Thus,
(3.29)∫0t‍∥w1(s)-w2(s)∥Vds≤c4∫0t‍∥η1(s)-η2(s)∥Qds,a.e.t∈I¯,
where c4=c3(1+T). For the operator Λ defined in (3.24), by u~∈ℜ(u), (2.5), Remark 2.4, and H(h), we obtain
(3.30)∥Λη1(t)-Λη2(t)∥Q=∥∫-r0‍[h(t+θ,ε(u~1))-h(t+θ,ε(u~2))]μ(dθ)∥Q≤|μ|([-r,0])∥h(t+θ,ε(u~1))-h(t+θ,ε(u~2))∥Q≤L3|μ|([-r,0])∥u1-u2∥V≤c5∫0t‍∥η1-η2∥Vds,a.e.t∈I¯,
where c5=L3|μ|([-r,0])c4. By using (3.30) and the similar proof of Lemma 3.3, we get the result of Lemma 3.4. This completes the proof.Now we prove Theorem3.1.Proof of Theorem3.1.
Letη*∈C(I¯;Q) be the fixed point of Λ and let uη*∈C1(I¯;Q) be the function defined by (3.24) for η=η*. For any v∈V1 and a.e. t∈I¯, it follows from u˙η*=wη* and (3.22) that
(3.31)(𝒜(t,ε(u˙η*(t))),ε(v-u˙η*(t)))Q+(η*,ε(v-u˙η*(t)))Q+j(t,St(u˙η*);v)-j(t,St(u˙η*);u˙η*(t))≥(f(t),v-u˙η*(t))V.
Now inequality (2.34) follows from (3.24) and (3.30). Moreover, since (3.23) implies uη*(0)=u0, we conclude that uη* is a solution of Problem 2.
Letu1,u2∈C(I¯;V1) be two solutions to Problem 2 and let wi=u˙i for i=1,2. Then we have
(3.32)ui(t)=∫0t‍wi(s)ds+u0,a.e.t∈I¯.
For a.e. t∈I¯, by the similar argument used in obtaining (3.6), we have
(3.33)(𝒜(t,ε(w1(t)))-𝒜(t,ε(w2(t))),ε(w1(t)-w2(t)))Q≤(Λη1(t)-Λη2(t),ε(w1(t)-w2(t)))Q+D(t,w1,w2),
where
(3.34)D(t,w1,w2)=j(t,St(w1);w2(t))-j(t,St(w1);w1(t))+j(t,St(w2);w1(t))-j(t,St(w2);w2(t)).
Using (2.29), (3.8), (2.9), and H(g), we deduce that
(3.35)D(t,w1,w2)≤c1∫0t‍∥w1(s)-w2(s)∥Vds∥w1(t)-w2(t)∥V,a.e.t∈I¯.
From the assumption H(𝒜) and relations (3.30)–(3.35), we know that, for a.e. t∈I¯,
(3.36)∥w1(t)-w2(t)∥V≤c6(∥u1(t)-u2(t)∥V+∫0t‍∥w1(s)-w2(s)∥Vds),
where
(3.37)c6=max{L3|μ|([-r,0])M,c1M}.
An application of the Gronwall inequality yields
(3.38)∥w1(t)-w2(t)∥V≤c7(∥u1(t)-u2(t)∥V+∫0t‍∥u1(s)-u2(s)∥Vds),a.e.t∈I¯,
where
(3.39)c7=max{c6,c62ec6T}.
Recalling the definition (3.32) of u1 and u2, and letting c8=c7(1+T), we obtain
(3.40)∥w1(t)-w2(t)∥V≤c8∫0t‍∥w1(s)-w2(s)∥Vds,a.e.t∈I¯
and so the Gronwall inequality implies that w1=w2. By definition (3.32), we see that u1=u2, which completes the proof of Theorem 3.1.Theorem 3.5.
Problem1 has a unique solution u∈C1(I¯;V1).Proof.
Letζ∈C(I¯;V1) and denote by uζ∈C(I¯;V1) the solution of the following problem:
(3.41)(𝒜(t,ε(u˙ζ(t))),ε(v-u˙ζ(t)))Q+(𝒢h(t,ε(u~ζ)),ε(v-u˙ζ(t)))Q+j(t,St(u˙ζ);v)-j(t,St(u˙ζ);u˙ζ(t))≥(f(t)-ζ(t),v-u˙ζ(t))V,∀v,u~ζ∈V1,a.e.t∈I¯,
where
(3.42)(ℬ(t,ε(uζ(t))),ε(v-u˙ζ(t))Q=(ζ(t),v-u˙ζ(t))V.
From Theorem 3.1, we know that there exists a unique solution uζ for problem (3.41).
Consider the operatorΥ:C(I¯;V*)→C(I¯;V*) defined by
(3.43)Υζ(t)=ℬ(t,ε(uζ(t))),∀ζ∈C(I¯;V1),a.e.t∈I¯.
Now we show that the operator Υ has a unique fixed point. In fact, for any ζ1,ζ2∈C(I¯;V1), let u1=uζ1 and u2=uζ2 be the corresponding solutions to (3.41). Then it is easy to see that u1,u2∈C(I¯;V1). For any u~1∈ℜ(u1) and u~2∈ℜ(u2), by the similar argument used in obtaining (3.33), we have
(3.44)(ζ2(t)-ζ1(t),w1(t)-w2(t))V+D(t,w1,w2)≥(𝒜(t,ε(w1(t)))-𝒜(t,ε(w2(t))),ε(w1(t)-w2(t)))Q+(𝒢h(t,ε(u~1))-𝒢h(t,ε(u~2)),ε(w1(t)-w2(t)))Q,a.e.t∈I¯.
By H(𝒜), H(h), (2.5), (3.30), (3.32), and (3.35), we get
(3.45)∥w1(t)-w2(t)∥V≤c9(∥ζ1(t)-ζ2(t)∥V+∫0t‍∥w1(s)-w2(s)∥Vds),a.e.t∈I¯,
where
(3.46)c9=max{1M+M′,1+cM+M′}.
An application of the Gronwall inequality yields
(3.47)∫0t‍∥w1(s)-w2(s)∥Vds≤c10∫0t‍∥ζ1(s)-ζ2(s)∥Vds,a.e.t∈I¯,
where c10=(1+T)max{c9,c92ec9T}. From H(ℬ), (3.43), and (3.47), we have
(3.48)∥Υζ1(t)-Υζ2(t)∥V2≤c11∫0t‍∥ζ1(s)-ζ2(s)∥V2ds,a.e.t∈I¯,
where c11=Tc102L12. Iterating the last inequality p times, we obtain
(3.49)∥Υpζ1(t)-Υpζ2(t)∥V2≤c11ptp-1(p-1)!∫0t‍∥ζ1(s)-ζ2(s)∥V2ds,a.e.t∈I¯,
which leads to
(3.50)∥Υpζ1-Υpζ2∥L2(I¯;V)≤(c11pTp(p-1)!)1/2∥ζ1-ζ2∥L2(I¯;V),a.e.t∈I¯.
Since limp→∞(c11pTp/(p-1)!)1/2=0, the previous inequality implies that, for p large enough, a power Υp of Υ is a contraction. It follows that there exists a unique element ζ*∈V1 such that Υpζ*=ζ*. Moreover, since
(3.51)Υp(Υζ*)=Υ(Υpζ*)=Υζ*,
we deduce that Υζ* is also a fixed point of the operator Υp. By the uniqueness of the fixed point that Υζ*=ζ*, we know that ζ* is a fixed point of Υ. The uniqueness of the fixed point of Υ results straightforward from the uniqueness of the fixed point of Υp. This implies that uζ* is the unique solution of Problem 1, which completes the proof of Theorem 3.5.Remark 3.6.
When𝒢=0 and all the viscosity and elasticity operators 𝒜 and ℬ are explicitly time dependent, Theorem 3.5 reduces to Theorem 10.2 of [1]. Furthermore, Theorem 3.5 is also a generalization of Theorem 2.1 of [24].


## 4. A Convergence Result

In this section, we study the dependence of the solution to Problem1 with respect to perturbations of the operator h. We assume that H(𝒜), H(ℬ), H(g), and H(h) hold and, for any β>0, let hβ be a perturbation of the operator h.We consider the following problem.Problem 4.
Finduβ:I¯→V1 such that
(4.1)(𝒜(t,ε(u˙β(t))),ε(v-u˙β(t)))Q+(ℬ(t,ε(uβ(t))),ε(v-u˙β(t)))Q+(𝒢hβ(t,ε(u~β)),ε(v-u˙β(t)))Q+j(t,St(u˙β);v)-j(t,St(u˙β);u˙β(t))≥(f(t),v-u˙β(t))V,∀v∈V1,u~β∈ℜ(uβ),(4.2)uβ(0)=u0.
It follows from Theorem 3.5 that, for each β>0, Problem 4 has a unique solution denoted by uβ∈C(I¯;V1).
In order to get the convergence result, we need the following assumption:(4.3)limβ→0∥hβ(x,t,ε)-h(x,t,ε)∥Q=0.
Now we give the convergence result.Theorem 4.1.
Assume thatH(𝒜), H(ℬ), H(g), H(h), and (2.28) hold. Then the solution uβ of Problem 4 converges to the solution u of Problem 1, that is,
(4.4)limβ→0∥uβ-u∥C(I¯;V1)=0.Proof.
For anyβ>0 and a.e. t∈I¯, let
(4.5)ηβ=∫-r0‍hβ(t+θ,ε(u~β(θ)))μ(dθ),η=∫-r0‍h(t+θ,ε(u~(θ)))μ(dθ).
Keeping in mind (4.1) and (2.33), and using H(𝒜) and H(ℬ), by the similar argument used in obtaining (3.36), we have
(4.6)∥wβ(t)-w(t)∥V≤c12(∥ηβ(t)-η(t)∥V+∫0t‍∥wβ(s)-w(s)∥Vds),a.e.t∈I¯,
where
(4.7)c12=max{1M,c+L1M}.
Hence,
(4.8)∫0t‍∥wβ(s)-w(s)∥Vds≤c13∫0t‍∥ηβ(s)-η(s)∥Vds,a.e.t∈I¯,
where
(4.9)c13=(1+T)max{c12,c122ec12T}.
For a.e. t∈I¯, it follows from (4.5), (2.5), and Remark 2.4 that
(4.10)∥ηβ(t)-η(t)∥Q≤|μ|([-r,0])∥hβ(t+θ,ε(u~β(θ)))-h(t+θ,ε(u~(θ)))∥Q≤|μ|([-r,0])[∥hβ(t+θ,ε(u~β(θ)))-hβ(t+θ,ε(u~(θ)))∥Q+∥hβ(t+θ,ε(u~(θ)))-h(t+θ,ε(u~(θ)))∥Q].
Noting u~β∈ℜ(uβ), from the last inequality, (4.3), (4.8), and H(h), we have
(4.11)∥wβ(t)-w(t)∥V≤c13|μ|([-r,0])TL3∫0t‍∥wβ(s)-w(s)∥Vds,a.e.t∈I¯.
It follows from the Gronwall inequality that wβ=w and so the convergence result (4.4) is a consequence of (3.32). This completes the proof.



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