# Existence and Stability Estimate for the Solution of the Ageing Hereditary Linear Viscoelasticity Problem

Authors: Julia Orlik
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

The paper is concerned with the existence and stability of weak (variational) solutions
for the problem of the quasistatic evolution of a viscoelastic material under
mixed inhomogenous Dirichlet-Neumann boundary conditions. The main novelty
of the paper relies in dealing with continuous-in-time weak solutions and allowing nonconvolution and weak-singular Volterra's relaxation kernels.


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## Body

## 1. Introduction

Our work deals with mathematical problems arising from considering the behavior of hereditary viscoelastic solids. These result in the system of elliptic partial differential equations in space variables, whose coefficients are Volterra integral operators of the second kind in time, which allow for weak-singular kernels. In Sections2 and 3 a general mathematical model of the boundary value problem of the inhomogeneous hereditary ageing viscoelasticity is given in the classical and weak formulations. The main result of this section, Theorem 3.2, proves  the existence and uniqueness of the solution to the general problem of the hereditary ageing viscoelasticity with mixed boundary conditions. The proof of the main results is shifted to the end of the paper, since it is based on the main result of Section 4 applied to the class of continuous Banach-valued functions with values in the Sobolev spaces. Section 4 can be considered separately of the mechanical background of the problem and can be interesting from the point of view of "Linear Volterra integral operators in Banach spaces". It includes definition of classes of Volterra operators with operator-valued kernels acting in the space of continuous Banach-valued functions. We also present some well-known results from the theory of Bochner's integral and their extension to the introduced classes. Lemma 5.4 delivers statements about the spectral radius and representation as a series for scalar Volterra operators. Theorem 6.2 delivers the main result of this section. It gives a statement about  the solvability of Volterra's integral equation in the space of continuous functions with values in Banach spaces. The idea of its proof is the reduction of a statement for Banach-valued functions to the corresponding statement for real-valued functions. Similar ideas were proposed in [1] for Volterra operators with kernels depending on a parameter. The main principle for obtaining this purpose is given by Theorem 4.1. The question of the solvability of nonconvolutional Volterra integral equations in Banach spaces was considered in [2–10], but under somewhat stricter assumptions for the kernel classes and almost all the proofs were given by using semigroup theory or fixed-point arguments. That is, in [2] the same problem is considered but the solvability proof restricts on a special kind of the kernel's weak singularity and is based on the fixed-point argument. The work in [6] allows for weak singular kernels, but again of the same as in [2] special type, (t-τ)-α. This allows to get a simple estimate for the partial sum of the Neumann series in the form of an explicit formula, which shows its convergence. In the work [8], the kernels are supposed to be continuous on the whole time interval. References [3, 5] consider evolutional integrodifferential equations with weak singular kernels, but such equations can be transformed by partial integration to the Volterra integral equation with bounded kernels. While we are looking for the solution in the space of continuous Banach-valued functions, [7] proceeds in Lp-spaces, p<∞. Papers [4, 9, 10] first assume the existence of the resolvent of the operator kernel part, that is, that its spectral radius is zero, and second do not permit the dependence of the operator kernel part on both time-variables separately.


## 2. Definition of the Problem

We consider a linear viscoelastic and aging (of the non-convolutional integral type) body, which is subjected to some external loading. We denote the domain occupied by the body byΩ⊂ℝn, which is a Lipschitz domain.We are going to consider the equilibrium equations for such a solid. We would like to recall that viscoelastic solid is still a solid, its deformation is slow and we restrict ourselves to the quasi-static, that is, classical for the solid mechanics statement of problem, without the inertial term. A summation from1 to n over repeating indices is assumed in all the present work, unless opposite is stated:
(2.1)∂∂xh((aijhk0(x,t)+aijhk(x)⋆)∂uj(x,t)∂xk)=-fi0(x,t),x∈Ωi,j,h,k=1,2,…,n, with boundary conditions:
(2.2)ui(x,t)=ψi(x,t),x∈∂Ωu,(2.3)((aijhk0(x,t)+aijhk(x)⋆)∂uj(x,t)∂xk)nh(x)=ϕi(x,t),x∈∂Ωσ,
holding for any t∈[0,T]. Here
(2.4)(aijhk(x)⋆ekj)(t):=∫0taijhk(x,t,τ)·ekj(x,τ)dτ
are Volterra integral operators with kernels aijhk(x,t,τ); aijhk0(x,t) are instantaneous elastic coefficients (out-of-integral terms) and a̲ijhk(x):=aijhk0(x,t)+aijhk(x)⋆; fi0 are components of a vector of external forces; ϕi(x,t) are components of a vector of boundary traction on the part ∂Ωσ of the external boundary; ψ(x,t) are components of the displacement vector on the rest part ∂Ωu of the boundary. All functions are supposed to be continuous w.r.t. t∈[0,T] and sufficiently smooth w.r.t. x in domain Ω (for performing a partial integration). The whole viscoelastic operator tensor (a̲ijhk(x))n×nn×n is assumed to be symmetric at each point x∈Ω:
(2.5)a̲ijhk(x)=a̲jikh(x)=a̲hjik(x)=a̲ikhj(x).The tensor(aijhk0(x,t))n×nn×n  is additionally positivedefinite, with elements bounded at each point x∈Ω(2.6)c0ηkjηkj≤aijhk0(x,t)ηhiηkj≤C0ηkjηkj,for allηkj=ηjk∈ℝ and t∈[0,T] where the constants 0<c0≤C0<∞ are independent of x and t. For isotropic materials a̲ijhk=λ̲δhiδkj+μ̲δijδhk+μ̲δikδhj.Example 2.1.
(i) Often, the kernelsaijhk(x,t,τ) are of the convolution type and are taken in the exponential form
(2.7)aijhk(x,t,τ)={∑p=1mαijhkp(x)e-βp(x)(t-τ),ift≥τ,0,ift<τ,
where the βp are piece-wise continuous functions, often just constants, and the αijhkp(x) are piece-wise continuous functions for x∈Ω.
(ii) Theaijhk(x,t,τ) may also be kernels of the Abel type (typical example for the relaxation kernels of concrete and cement):
(2.8)aijhk(x,t,τ)={Aijhk1(x,t,τ)(t-τ)-α+Aijhk2(x,t,τ)(τ)-β+Aijhk3(x,t,τ)t-γ,ift≥τ,0,otherwise,
with 0≤α,β,γ<1. The Aijhkp, p=1,2,3, are continuous in t and τ, and piece-wise continuous in x∈Ω.


## 3. Weak Problem Formulation and Main Results

In order to obtain the variational formulation, we multiply (2.1) by test functions vi(x)∈H01(Ω,∂Ωu), i=1,…,n, where H01(Ω,∂Ωu):={v∈H1(Ω):v(x)=0,x∈∂Ωu}, and integrate over the whole domain Ω. Integrating by parts and taking into account boundary condition (2.2), we obtain the following variational problem.Finduj∈H1(Ω), j=1,…,n, satisfying (2.2) and
(3.1)∫Ωa̲ijhk∂uj∂xk∂vi∂xhdx=l(v),l(v):=∫Ωfi0vidx+∫∂Ωσϕivids,∀vi∈H01(Ω,∂Ωu),i=1,…,n.Definition 3.1 (General weak formulation).
Consider the matrix of instantaneous elastic coefficients(aijhk0)n×n∈C([0,T];L∞(Ω)), the relaxation operators (aijhk⋆)n×n, such that aijhk(t,τ)=0∀τ>t, and aijhk∈C([0,T];L∞(Ω)), and f0:=(fi0)n∈C([0,T];H-1(Ω)), the whole viscoelastic operator a̲ijhk=aijhk0+aijhk⋆, the boundary tractions ϕ:=(ϕi)n∈C([0,T];H-1/2(∂Ωσ)), and boundary displacements ψ:=(ψi)n∈C([0,T];H1/2(∂Ωu)).
One defines a weak solution of problem (2.1)–(2.3) as a vector-valued function u∈C([0,T];H1(Ω)), which can be represented in the form u=û+ψ̃, where ψ̃∈C([0,T];H1(Ω)) satisfies (ψ̃|∂Ωu)=ψ and ûi∈C([0,T];H01(Ω,∂Ωu)), i=1,…,n, satisfies the integral identity
(3.2)[a̲(û,v)](t):=∫Ω[a̲ijhk∂ûj∂xk](t)∂vi∂xhdx=l̂(v)(t)∀t∈[0,T],
for any vi∈H01(Ω,∂Ωu). The right hand-side of (3.2) is, for all t∈[0,T], a linear functional on the H01(Ω,∂Ωu)(3.3)l̂(v)(t):=∫Ω(fi0(t)vi-[a̲ijhk∂ψ̃j∂xk](t)∂vi∂xh)dx+∫∂Ωσϕi(t)vids.
The space of linear bounded functionals on H01(Ω,∂Ωu) is denoted by H-1(Ω).We denote furthera0(û,v)(t):=∫Ωaijhk0(x,t)(∂ûj(x,t)/∂xk)(∂vi(x)/∂xhdx), a(û,v)(t,τ):=∫Ωaijhk(x,t,τ)(∂ûj(x,τ)/∂xk)(∂vi(x)/∂xh)d. Obviously, [a̲(û,v)](t)=a0(û,v)(t)+∫0ta(û,v)(t,τ)dτ.Note thata0(û,v) and a(û,v) are bilinear forms on H01(Ω,∂Ωu) for every t and almost every τ. We can rewrite the weak formulation (3.2) as follows:
(3.4)a0(û,v)(t)+∫0ta(û,v)(t,τ)dτ=l(v)(t),t∈[0,T].Let us finally rewrite (3.4) in the operator form. For this purpose we introduce the following notations:
(3.5)A0xû:=a0(û,·),Axû:=a(û,·),L(t):=l(v)(t).A0x(t),Ax(t,τ):H01(Ω,∂Ωu)→H-1(Ω) for all fixed t and almost all τ∈[0,T]. Now we can represent (3.2) in the form A̲û=L(t), where A̲· is an infinite dimensional integro-differential linear operator A̲=A0x·+Ax⋆ and the weak problem formulation (3.2) takes the form
(3.6)A0x(t)û(t)+[Ax⋆û](t)=L(t).
Equation (3.6) provides the most general form of the time-space integro-differential dependencies of the considered problem. The following theorem is used as an auxiliary result for showing the solvability of such equation.Theorem 3.2 (Data stability).
LetΩ⊂ℝn be a Lipschitz domain and ∂Ωu⊆∂Ω, let A0x∈C([0,T];ℒ(H01(Ω,∂Ωu),H-1)), and let A0x(t) be boundedly-invertible uniformly in [0,T], Ax(t,τ)=0∀τ>t, Ax∈C([0,T];L1([0,T],ℒ(H01(Ω,∂Ωu),H-1))), and f∈C([0,T];H-1). Then there exists a unique global solution u of the problem
(3.7)A0x(t)u(t)+[Ax⋆u](t)=f(t)
in C([0,T];H01(Ω,∂Ωu)), which depends continuously on f, that is,
(3.8)∥u∥C([0,T];H1(Ω))≤C1∥f∥C([0,T];H-1),
where the constant C1 is independent of f, and if ∥A0x-1(t)∥ℒ(H-1,H1)≤1/c0, then
(3.9)C1≤C̃(1c0maxi,j,h,k∥aijhk(t,τ)∥C(([0,T],L1([0,T],L∞(Ω))))),
where C̃ is some real-valued function, independent of f.The rest of the paper is aimed to prove this theorem. The final proof of this theorem as well as of two following lemmas can  be found in the  Appendix . Since we are searching for the weak solution in the formu=û+ψ̂, where both û and ψ̂ are in C([0,T];H1(Ω)), then we can claim that our solution exists and is unique.Lemma 3.3.
LetΩ be a Lipschitz domain in ℝn, the instantaneous elastic (out-of-integral) coefficients aijhk0∈C([0,T];L∞(Ω)) satisfy the positivity condition (the first of (2.6) with a constant c0, and the relaxation kernels aijhk∈C(([0,T],L1([0,T],L∞(Ω)))). Then (i)
A0x belongs to C([0,T];ℒ(H01(Ω,∂Ωu),H-1)), and A0x(t) has an inverse operator A0x-1(t)  ∀t∈[0,T]. This inverse operator is uniformly bounded in [0,T], that is, the following estimate:
(3.10)∥A0x-1(t)∥ℒ(H-1,H1)≤1c0,
holds for any t∈[0,T], and c0 is independ on t;(ii)
Ax(t,τ) satisfies the following estimate:
(3.11)∥Ax(t,τ)∥ℒ(H1(Ω),H-1)≤maxi,j,h,k∥aijhk(t,τ)∥L∞(Ω),
for all t and a.a τ∈[0,T]. Furthermore, the condition aijhk∈C(([0,T],L1([0,T],L∞(Ω))) implies that Ax∈C(([0,T],L1([0,T],ℒ(H01(Ω,∂Ωu),H-1))).Lemma 3.4.
In both cases of Example2.1, aijhk∈C(([0,T],L1([0,T],L∞(Ω)))), that is, are the Volterra kernels.


## 4. Volterra Integral Operators in Banach Spaces

The following theorem (which can be found, e.g., in [11, section 7.2]) delivers the tool, which we will multiply use for obtaining the main results of this section.Theorem 4.1.
A strongly-measurable Banach-valued functionf(s) is Bochner-integrable, if its norm in the corresponding Banach-space is Lebesgue integrable, and it follows (see [12, page 133]), ∥∫Jf(s)λ(ds)∥X≤∫J∥f(s)∥Xλ(ds).We introduce the notation𝒮(J,X), which should be interpreted as either C(J,X) or L∞(J,X).Corollary 4.2.
LetJ⊂ℝ be a finite segment (J=[0,T]), X and Z be Banach-spaces, and K∈L1(J,ℒ(X,Z)). Then K(·)g(·)∈L1(J,Z) for all g∈𝒮(J,X).Proof.
Eachg∈C(J,X) is Bochner integrable, hence, strongly measurable, and each g∈L∞(J,X) is strongly measurable, according to definition of the space L∞(J;X). Thus we can find disjoint sets Bi,n⊂J and finitely-valued functions gn(τ):=∑i=1ngi,nχBi,n(τ),∀τ∈J, such that limn→∞∥gn(τ)-g(τ)∥X=0 a.e. on J.  Here χBi,n(τ) is the characteristic function of Bi,n. Thus
(4.1)∥K(τ)gn(τ)-K(τ)g(τ)∥Z≤∥K(τ)∥ℒ(X,Z)∥gn(τ)-g(τ)∥X→0asn→∞
for almost all τ∈J. K is strongly measurable on J;K(τ)gn(τ):=∑i=1nK(τ)gi,nχBi,n(τ), ∀τ∈J, where K(τ)gi,n is, for all n, a strongly measurable Z-valued function, owing to [12, Chapter 5, Corollary 2], and χBi,n:J→ℝ.  Owing to [13, the Appendix, Proposition (9a)], K(τ)gn(τ) is strongly measurable on J. Then, according to the convergence theorem of Egorov, in a view of (4.1) (see [13, page 1013]), K(τ)g(τ) is also strongly measurable on J.
Owing to the fact that∥K(·)ℒ∥(X,Z)∈L1(J), we have,
(4.2)∫J∥K(τ)g(τ)∥Zdτ≤∫J∥K(τ)∥ℒ(X,Z)∥g(τ)∥Xdτ≤∥g∥𝒮(J,X)∫J∥K(τ)∥ℒ(X,Z)dτ<∞.
Using Theorem4.1 completes the proof.Since forJ=[0,T], C(J,X)↪L∞(J,X), it would be enough to consider only L∞ instead of 𝒮 in Corollary 4.2 and also in further considerations. Nevertheless, we often will need this theory specifically for continuous Banach-valued functions in the main part of this work. Therefore it is reasonable to keep 𝒮.Now we define a class of Banach-valued Volterra operators.Definition 4.3.
LetX, Z be Banach spaces and J⊂ℝ be a finite segment [0,T], 0≤T<∞, or the half-infinite interval [0,∞). Suppose an operator kernel K is such that K(t,τ)=0∀τ>t, K∈C(J,L1(J,ℒ(X,Z))), that is,(i)
K:(t,τ)→ℒ(X,Z), ∀t∈J and almost all τ∈J, K(t,τ) is strongly measurable w.r.t. τ, for all t∈J, and ∥K(t,τ)∥ℒ(X,Z) is integrable w.r.t. τ for all t in J, that is, ∫J∥K(t,τ)∥ℒ(X,Z)dτ<∞∀t∈J;(ii)
∫J∥K(t+Δt,τ)-K(t,τ)∥ℒ(X,Z)dτ→asΔt→0∀t,t+Δt∈J.Call K⋆ defined by g↦(K⋆g)(t):=∫JK(t,τ)g(τ)dτ, for g∈𝒮(J,X) a Volterra integral operator. The set of all Volterra integral operators is denoted by V(C;J;ℒ(X,Z)).Remark 4.4.
Observe that, as soon asJ is closed and bounded (i.e., compact), condition (ii) of Definition 4.3 implies that the following holds:
(4.3)|∥K∥|V(C;J;ℒ(X,Z)):=supt∈J∫J∥K(t,τ)∥ℒ(X,Z)dτ<∞.
We will call in such a case |∥K∥|V(C;J;ℒ(X,Z)) the kernel norm of the operator K⋆.Corollary 4.5.
LetJ=[0,T], and let X, Z be Banach spaces. If K⋆∈V(C;J;ℒ(X,Z)), g∈𝒮(J,X), one can view f(t,τ):=K(t,τ)g(τ) also as an element of C(J,L1(J,Z)).Proof.
The fact thatf(t,·)∈L1(J,Z) follows from Corollary 4.2. Let us show the continuity: ∥K(t+Δt,·)g(·)-K(t,·)g(·)∥L1(J,Z)=∫J∥K(t+Δt,τ)g(τ)-K(t,τ)g(τ)∥Zdτ≤∫J∥(K(t+Δt,τ)-K(t,τ))∥ℒ(X,Z)∥g(τ)∥Xdτ≤∥g∥𝒮(J,X)∫J∥K(t+Δt)-K(t)∥ℒ(X,Z)dτ→asΔt→0∀t,t+Δt∈J. For abbreviation, we will set V(C;J;X):=V(C;J;ℒ(ℝn,X)).Corollary 4.6.
LetJ⊂ℝ be a segment, X and Z be Banach-spaces, K⋆∈V(C;J;ℒ(X,Z)). Then ∥K∥ℒ(X,Z)⋆∈V(C;J;ℝ) and the following estimate
(4.4)∥K⋆g∥Z≤∥K∥ℒ(X,Z)⋆∥g∥X
holds for all g∈𝒮(J,X) and t∈J.Proof.
An operator(∥K∥ℒ(X,Z)⋆u)(t)≡∫J∥K∥ℒ(X,Z)(t,τ)u(τ)dτ,∀u∈𝒮(J) has the scalar kernel ∥K(t,τ)∥ℒ(X,Z):J2→ℝ, which obviously satisfies all requirements of Definition 4.3, since K⋆∈V(C;J;ℒ(X,Z)). The estimate (4.4) follows directly from Theorem 4.1.Corollary 4.7 (from Theorem4.1 in application to Definition 4.3).
IfJ is a finite segment and X, Z are Banach spaces, then V(C;J;ℒ(X,Z))↪ℒ(𝒮(J,X),C(J,Z)), that is any operator K⋆∈V(C;J;ℒ(X,Z)) can be identified with a bounded linear mapping from 𝒮(J,X) to C(J,Z).Proof.
The continuity is implied from condition (ii) from Definition4.3, the compactness of the interval and the boundedness of ∥g(τ)∥X by applying Theorem 4.1 (see proof to Corollary 4.5).
The boundedness in theC(J,Z)-norm follows from the continuity and the fact that J is compact. And Theorem 4.1 delivers the continuity of the embedding
(4.5)∥K(t,s)⋆∥ℒ(𝒮(J,X),C(J,Z))=sup∥g∥𝒮(J,X)≤1∫J∥K(t,s)g(s)ds∥C(J,Z)≤sup∥g∥𝒮(J,X)≤1supt∈[0,T]∫J∥K(t,s)g(s)∥Zds≤sup∥g∥𝒮(J,X)≤1supt∈[0,T]∫J∥K(t,s)∥ℒ(X,Z)∥g(s)∥Xds≤sup∥g∥𝒮(J,X)≤1(|∥K∥|V(C;J;ℒ(X,Z))∥g∥𝒮(J,X))≤∥K∥V(C;J;ℒ(X,Z)).


## 5. Some Auxiliary Results from the Spectral Theory

In this section we give definitions of the spectrum and of the resolvent for an element of a Banach algebra with a unit used in [14,  A.2.1].Definition 5.1.
The  spectrumσ(Q) of Q from a Banach-algebra with a unit I is the set of all λ∈ℂ for which (λI-Q)-1 does not exist or is not bounded. Furthermore, for λ∈ℂ∖σ(Q), (λI-Q) is invertible and the inverse R̲̃(λ)≡(λI-Q)-1 is called the  resolvent of Q, and belongs to the same algebra.The following definition is known from in [14,  A.2.1] or  in [15,  18.8].Definition 5.2.
Let𝒜 be a Banach-algebra with unit, Q∈𝒜 and σ(Q) be the spectrum of Q. Then one calls the number ρ(Q)=sup{|λ|∣λ∈σ(Q)} the  spectral radius of Q.In [16, Theorem 3.6] about Neumann's series the following is proven.Theorem 5.3.
LetT be a bounded linear operator over a Banach-space X, with
(5.1)limn→∞sup∥Tn∥ℒ(X)1/n<1,
then (I-T)-1∈ℒ(X) can be represented as a series:
(5.2)(I-T)-1=∑j=0∞Tj.Now, we formulate and prove a lemma about the spectral radius and representation of a resolvent for scalar Volterra operators as Neuman's series. It prepares the basis for the main result of this section.Lemma 5.4.
LetJ⊂ℝ be a compact interval (J=[0,T]), and Q⋆∈V(C;J;ℂ). Then there exists a resolvent R̲̃(λ) of Q⋆ in ℒ(𝒮([0,T],ℂ)), given by the convergent series R̲̃(λ)=∑j=0∞((Q⋆)j/λj+1) for each λ∈ℂ∖{0}.Proof.
The class of kernels of Volterra operators fromV(C;J;ℂ), on the compact interval forms a Banach algebra with product ⋆. This follows from [17, Theorem 9.5.3(ii)]. Such an algebra, owing to [18, Theorem 2, Section 20, Chapter 4], does not have a unit. According to [19, page 228], we can add an artificial unit to our algebra 𝒜 kernels from V(C;[0,T];ℂ). Owing to [17, Theorem 9.5.5(ii)], every kernel Q, Q⋆∈V(C;[0,T];ℂ) on some bounded interval [0,T]⊂ℝ has an element R (called there a resolvent of Q), which satisfies the following equality:
(5.3)R+Q⋆R=R+R⋆Q=Q,
such that R⋆∈V(C;[0,T];ℂ).
Obviously, ifQ⋆∈V(C;[0,T];ℂ), then, for all λ∈ℂ∖{0}, P⋆≡-(1/λ)Q⋆ belongs also to V(C;[0,T];ℂ), and, hence, has a resolvent (in the sense of [17]) in V(C;[0,T];ℂ), satisfying R+P⋆R=R+R⋆P=P.
Considering separately each of two last equalities and adding-I on both sides, we varify that (I+P)-1=I-R. Consequently, I-(1/λ)Q is invertible and, hence, all nonzero λ∈ℂ, are outside σ(Q) for Q⋆∈V(C;[0,T];ℂ).
In [14, Theorem 1.2.,  A.2.2] or [15,  18.6] it claims that the spectrum σ(Q) of an element Q∈𝒜 is nonempty. Hence, only λ=0 belongs to the spectrum, and this implies that the spectral radius ρ(Q) for kernels from V(C;[0,T];ℂ) is zero.
Furthermore, the following spectral radius formula(5.4)ρ(Q)=limn→∞|∥(Q⋆)n∥|1/n
is known from the theorem with the same title (see [14, Theorem 2,  A.2.3] or [15, Theorem 18.9]). This means,
(5.5)limn→∞|∥(Q⋆)n∥|V(C;[0,T];C)1/n=0forQ⋆∈V(C;[0,T];ℂ).
Let us give a justification of the assumptions of Theorem5.3 on Q⋆∈V(C;[0,T];ℂ). Since V(C;[0,T];ℂ) is an algebra, (Q⋆)n∈V(C;[0,T];ℂ), for any n=2,3,…. According to Corollary 4.7, V(C;[0,T];ℂ)↪ℒ(C([0,T],ℂ)), and we can estimate
(5.6)∥(Q⋆)n∥ℒ(C[0,T])≤|∥(Q⋆)n∥|V(C;[0,T];C).
According to (5.5), condition (5.1) is satisfied for V(C;[0,T];ℂ), completed by the unit from ℒ(𝒮[0,T]).
Since the assumptions of Theorem5.3 are satisfied, for each λ∈ℂ∖{0} and Q⋆∈V(C;[0,T];ℂ),
(5.7)R̲̃(λ)≡(λI-Q⋆)-1=∑j=0∞(Q⋆)jλj+1.


## 6. Solvability of the Volterra Integral Equations in Banach Spaces

We will use an idea of [1, Lemma 2.7], in which a class of Volterra operators depending on a complex parameter on a strip is considered, and which claims that their resolvents on a finite segment exist and belong to the same operator class. The idea is to reduce the proof of the Neumann series convergence for such Volterra operators to the convergence for corresponding scalar Volterra operators by taking supremum from kernels w.r.t. the complex parameter.Lemma 6.1.
LetX be a Banach space, K⋆∈V(C;[0,T];ℒ(X)), n,m=1,2,…. Then, for each n≥m one can estimate
(6.1)∥∑j=mn(-K⋆)j∥ℒ(𝒮([0,T],X),C([0,T],X))≤∥∑j=mn(∥K∥ℒ(X)⋆)j∥ℒ(C[0,T]),
where ℒ(C[0,T])≡ℒ(C([0,T],ℝ)).Proof.
Corollary4.7 claims that, since K⋆∈V(C;[0,T];ℒ(X)) and F∈𝒮([0,T],X), K⋆F∈C([0,T],X). We can estimate
(6.2)∥∑j=mn(-K⋆)j∥ℒ(𝒮([0,T],X),C([0,T],X))=sup∥F∥𝒮([0,T],X)≤1∥∑j=mn(-1)j[(K⋆)jF]∥C([0,T],X)≤sup∥F∥𝒮([0,T],X)≤1∥∑j=mn∥[(K⋆)jF]∥X∥C([0,T])=sup∥F∥𝒮([0,T],X)≤1∥∑j=mn∥[K⋆[(K⋆)j-1F]]∥X∥C([0,T])≤Corollary 4.6sup∥F∥𝒮([0,T],X)≤1∥∑j=mn∥K∥ℒ(X)⋆∥[(K⋆)j-1F]∥X∥C([0,T])≤inductionsup∥F∥𝒮([0,T],X)≤1∥∑j=mn(∥K∥ℒ(X)⋆)j∥F∥X∥C([0,T])≤sup∥F∥𝒮([0,T],X)≤1∥∑j=mn(∥K∥ℒ(X)⋆)j∥ℒ(C[0,T])∥F∥𝒮([0,T],X),=∥∑j=mn(∥K∥ℒ(X)⋆)j∥ℒ(C[0,T]).Theorem 6.2.
LetX be a Banach space, K⋆∈V(C;[0,T];ℒ(X)) and F∈𝒮([0,T],X). Then there exists a unique global solution of the problem
(6.3)u(t)+[K⋆u](t)=F(t),
in 𝒮([0,T];X), which depends continuously on F, that is,
(6.4)∥u∥𝒮([0,T],X)≤C∥F∥𝒮([0,T],X),
where the constant C=C(∥K∥ℒ(X)⋆) is independent of F. Morover, if ∥K(t,τ)∥ℒ(X)≤k(t,τ)∀t and a.a. τ∈[0,T], where k⋆∈V(C;[0,T];ℝ), then C(∥K∥ℒ(X)⋆)≤C̃(k⋆)<∞.Proof.
We will look for a solution to (6.3) in the form of the Neumann series applied to the right handside function F(t):
(6.5)v(t):=[R̲F](t)≡∑j=0∞(-1)j[(K⋆)jF](t).
According to Corollary4.7, K⋆F∈C([0,T],X), since K⋆∈V(C;[0,T];ℒ(X)) and F∈𝒮([0,T],X). Hence, all terms of series (6.5), except of the zero-term, belong to C([0,T],X). As for the zero-term, (K⋆)0=I, where I is an identity operator over 𝒮[0,T]. And IF∈𝒮([0,T],X), since F∈𝒮([0,T],X). Our aim is to show, that the Banach-valued Neumann's series ∑j=1∞(-1)j(K⋆)j(t) converges w.r.t. the norm in ℒ(𝒮([0,T],X),C([0,T],X)). The spaces L∞([0,T],X) and C([0,T],X) are complete as Banach spacesm , that is, each Cauchy sequence converges. It is known, that the space of bounded linear operators mapping from a Banach space into a Banach space is also Banach space. Hence, it is enough to show that
(6.6)Sn≡∑j=0n(-1)j(K⋆)j
is a Cauchy sequence, that is, that for any ϵ there exists a number N such that ∥Sn-Sm∥ℒ(𝒮([0,T],X),C([0,T],X))<ϵ for all n≥m≥N. We can estimate
(6.7)∥Sn-Sm∥ℒ(𝒮([0,T],X),C([0,T],X))≡∥∑j=mn(-1)j(K⋆)j∥ℒ(𝒮([0,T],X),C([0,T],X))≤Lemma 6.1∥∑j=mn(∥K∥ℒ(X)⋆)j∥ℒ(C[0,T]).
Note that, according to Corollary4.6, ∥K∥ℒ(X)⋆∈V(C;[0,T];ℝ) since K⋆∈V(C;[0,T];ℒ(X,X)). If the real-valued series R̲̃≡∑j=0∞(∥K∥ℒ(X)⋆)j is convergent and bounded in ℒ(C[0,T]), there must exist a number N such that ∀n,m≥N,∥∑j=mn(∥K∥ℒ(X)⋆)j∥ℒ(C[0,T])<ϵ. The sequence of the partial sums (6.6) will be fundamental in ℒ(𝒮([0,T],X),C([0,T],X)).
Let us recall Lemma5.4 for Q≡-∥K∥ℒ(X) and real λ=-1. Then, owing to its statement, the real-valued series R̲̃ converges to
(6.8)R̲̃≡∑j=0∞(∥K∥ℒ(X)⋆)j=[I-∥K∥ℒ(X)⋆]-1.
Owing to Lemma 5.4, the operator R̲̃ is bounded in ℒ(C[0,T]).
Owing to (6.7), this implies that the sequence of the partial sums (6.6) of the series (6.5) is fundamental in ℒ(𝒮([0,T],X),C([0,T],X)). Consequently, we get that v(t)∈𝒮([0,T],X). Note that we excluded m=0 in Lemma 6.1, since, for 𝒮=L∞, (K⋆)0∉ℒ(𝒮([0,T],X),C([0,T],X)), but belongs to ℒ(𝒮([0,T],X),𝒮([0,T],X)). Applying an estimate similar to (6.1) for m=0,
(6.9)∥v∥𝒮([0,T],X)≤∥∑j=0n(-K⋆)j∥ℒ(𝒮([0,T],X),(𝒮([0,T],X))∥F∥𝒮([0,T],X)≤∥R̲̃∥ℒ(𝒮([0,T])∥F∥𝒮([0,T],X).
Furthermore, if there existsk⋆∈V(C;[0,T];ℝ) such that ∥K(t,τ)∥ℒ(X)≤k(t,τ)  ∀t and a.a. τ∈[0,T], then, according to (6.8) and Lemma 5.4,
(6.10)∥R̲̃∥ℒ(𝒮([0,T])≤∥∑j=0∞(k⋆)j∥ℒ(𝒮([0,T])=∥[I-k⋆]-1∥ℒ(𝒮([0,T])<∞.
That is, ∥R̲̃∥ℒ(𝒮([0,T])≤C̃(k⋆)<∞.
The fact thatv(t), given by convergent Neumann's series (6.5), is a solution of (6.3), can be checked by direct substitution:
(6.11)v(t)+[K⋆v](t)=∑j=0∞(-1)j[(K⋆)jF](t)+K⋆∑j=0∞(-1)j[(K⋆)jF](t)=∑j=0∞(-1)j[(K⋆)jF](t)+∑j=1∞(-1)j-1[(K⋆)jF](t)=F(t).
It remains only to prove the uniqueness of the solution to (6.3).
Suppose,u is any solution to (6.3). We apply from the left the operator R̲ introduced in (6.5) to both sides of (6.3):
(6.12)R̲u+R̲K⋆u=R̲F,[R̲u](t)+[R̲K⋆u](t)=∑j=0∞(-1)j[(K⋆)ju](t)+∑j=0∞(-1)j(K⋆)j[K⋆u](t)=∑j=0∞(-1)j[(K⋆)ju](t)+∑j=1∞(-1)j-1[(K⋆)ju](t)=u(t).
Hence,
(6.13)u(t)=[R̲F](t),
that is, any solution is represented by Banach-valued Neumann's series. Since [R̲F] belongs to 𝒮([0,T],X), solution (6.5) is unique in 𝒮([0,T],X). Now, the theorem is proven completely.Lemma 6.3.
LetX, Y, Z be Banach spaces, A0∈C([0,T];ℒ(Y,Z)), and let A0(t) be boundedly invertible uniformly in [0,T], that is, there exists constant c0>0 such that
(6.14)∥A0(t)-1∥ℒ(Z,Y)≤1c0∀t∈[0,T].
Let A∈C([0,T];L1([0,T],ℒ(X,Z))). Then for K≡A0-1A the following continuity property is valid:
(6.15)∫J∥K(t+Δt,τ)-K(t,τ)∥ℒ(X,Y)dτ→0asΔt→0∀t,t+Δt∈J.Proof.
We use estimates (6.14), and the properties that A0∈C([0,T];ℒ(Y,Z)) and A⋆ belongs to V(C;[0,T];ℒ(X,Z)):
(6.16)∫J∥K(t+Δt,τ)-K(t,τ)∥ℒ(X,Y)dτ=∫J∥A0-1(t+Δt)A(t+Δt,τ)-A0-1(t)A(t,τ)∥ℒ(X,Y)dτ=∫J∥A0-1(t+Δt)(A(t+Δt,τ)-A(t,τ))+(A0-1(t+Δt)-A0-1(t))A(t,τ)∥ℒ(X,Y)dτ≤∫J∥A0-1(t+Δt)∥ℒ(Z,Y)∥(A(t+Δt,τ)-A(t,τ))∥ℒ(X,Z)dτ+∫J∥A0-1(t+Δt)-A0-1(t)∥ℒ(Z,Y)∥A(t,τ)∥ℒ(X,Z)dτ≤1c0∫J∥A(t+Δt,τ)-A(t,τ)∥ℒ(X,Z)dτ+|∥A∥|V(C;[0,T];ℒ(X,Z))∥A0-1(t)(A0(t)-A0(t+Δt))A0-1(t+Δt)∥ℒ(Y)≤1c0∫J∥A(t+Δt,τ)-A(t,τ)∥ℒ(X,Z)dτ+1c02|∥A∥|V(C;[0,T];ℒ(X,Z))∥A0(t+Δt)-A0(t)∥ℒ(Y,Z)→0asΔt→0∀t,t+Δt∈J.Theorem 6.4.
LetX, Z be Banach spaces, A0∈C([0,T];ℒ(X,Z)), and let A0(t) be boundedly invertible uniformly in [0,T], A⋆∈V(C;[0,T];ℒ(X,Z)), and f∈𝒮([0,T];Z). Then there exists a unique global solution u of the problem
(6.17)A0(t)u(t)+[A⋆u](t)=f(t)
in 𝒮([0,T];X), which depends continuously on f, that is,
(6.18)∥u∥𝒮([0,T],X)≤C1∥f∥𝒮([0,T],Z),
where the constant C1=C1(∥A0-1(t)∥ℒ(Z,X),∥A0-1(t)A∥ℒ(X)⋆) is independent of f.
Morover, if∥A0-1(t)∥ℒ(Z,X)≤1/c0 and ∥A(t,τ)∥ℒ(X,Z)≤a(t,τ)  ∀t and a.a. τ∈[0,T], where c0 is a constant and a⋆∈V(C;[0,T];ℝ), then C1≤(1/c0)C̃(1/c0)a⋆<∞.Proof.
Since,A0(t) is invertible, we can rewrite (6.17) as the following:  u(t)+A0(t)-1∫0tA(t,τ)u(τ)dτ=A0(t)-1f(t). Observe that A0(t)-1 does not depend on τ,
(6.19)u(t)+∫0tA0(t)-1A(t,τ)u(τ)dτ=A0-1(t)f(t).
We denote K(t,τ)≡A0(t)-1A(t,τ)andF(t)≡A0(t)-1f(t).
Let us show that the assumption of Theorem6.2 are satisfied for (6.19), that is, that K⋆∈V(C;[0,T];ℒ(X)) and F(t)∈𝒮([0,T],X). In order to prove that K⋆∈V(C;[0,T];ℒ(X)) we should show the justification of all the requirements of Definition 4.3.
(i)
SinceA⋆∈V(C;[0,T];ℒ(X,Z)), A(t,·)∈L1([0,T],ℒ(X,Z))  ∀t∈[0,T] (see Definition 4.3). According to [12, Chapter 5, Corollary 2], K(t,·)∈L1([0,T],ℒ(X))  ∀t∈[0,T], since A0-1(t) does not depend on τ and belongs to ℒ(X,Z)  ∀t∈[0,T]. Hence, owing  Theorem 4.1
, condition (i) of Definition 4.3 is satisfied.(ii)
Applying Lemma6.3 for A0∈C([0,T];ℒ(X,Z)) and A⋆∈V(C;[0,T];ℒ(X,Z)), in order to justify the continuity, completes the proof that K⋆∈V(C;[0,T];ℒ(X)). It remains to check, if F∈𝒮([0,T],X). According to [16, Theorem 3.3], F(t) belongs to X for all or almost all t∈[0,T]. Continuity of F w.r.t. t for continuous f(t), can be obtained in a similar way as in Lemma 6.3, taking into account the continuity of A0(t). Using Theorem 6.2 completes the proof.



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*Source: 828315-2009-07-12.xml*