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

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.

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 Sections 2 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 𝐿^𝑝-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 by Ω⊂ℝ^𝑛, 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 from 1 to 𝑛 over repeating indices is assumed in all the present work, unless opposite is stated: 𝜕𝜕𝑥_ℎ((𝑎₀^ℎ𝑘_𝑖𝑗(𝑥,𝑡)+𝑎^ℎ𝑘_𝑖𝑗(𝑥)⋆)𝜕𝑢_𝑗(𝑥,𝑡)𝜕𝑥_𝑘)=−𝑓_𝑖0(𝑥,𝑡),𝑥∈Ω (2.1)𝑖,𝑗,ℎ,𝑘=1,2,…,𝑛, with boundary conditions: 𝑢_𝑖(𝑥,𝑡)=𝜓_𝑖(𝑥,𝑡),𝑥∈𝜕Ω_𝑢,((𝑎₀^ℎ𝑘_𝑖𝑗(𝑥,𝑡)+𝑎^ℎ𝑘_𝑖𝑗(𝑥)⋆)𝜕𝑢_𝑗(𝑥,𝑡)𝜕𝑥_𝑘)𝑛_ℎ(𝑥)=𝜙_𝑖(𝑥,𝑡),𝑥∈𝜕Ω_𝜎, (2.2)(2.3) holding for any 𝑡∈[0,𝑇]. Here (𝑎^ℎ𝑘_𝑖𝑗(𝑥)⋆𝑒^𝑗_𝑘)(𝑡)∶=^𝑡₀𝑎^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏)⋅𝑒^𝑗_𝑘(𝑥,𝜏)𝑑𝜏 (2.4) are Volterra integral operators with kernels 𝑎^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏); 𝑎₀^ℎ𝑘_𝑖𝑗(𝑥,𝑡) are instantaneous elastic coefficients (out-of-integral terms) and 𝑎^ℎ𝑘_𝑖𝑗(𝑥)∶=𝑎₀^ℎ𝑘_𝑖𝑗(𝑥,𝑡)+𝑎^ℎ𝑘_𝑖𝑗(𝑥)⋆; 𝑓_𝑖0 are components of a vector of external forces; 𝜙_𝑖(𝑥,𝑡) are components of a vector of boundary traction on the part 𝜕Ω_𝜎 of the external boundary; 𝜓(𝑥,𝑡) are components of the displacement vector on the rest part 𝜕Ω_𝑢 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 (𝑎^ℎ𝑘_𝑖𝑗(𝑥))^𝑛×𝑛_𝑛×𝑛 is assumed to be symmetric at each point 𝑥∈Ω: 𝑎^ℎ𝑘_𝑖𝑗(𝑥)=𝑎^𝑘ℎ_𝑗𝑖(𝑥)=𝑎^𝑖𝑘_ℎ𝑗(𝑥)=𝑎^ℎ𝑗_𝑖𝑘(𝑥). (2.5)

The tensor (𝑎₀^ℎ𝑘_𝑖𝑗(𝑥,𝑡))^𝑛×𝑛_𝑛×𝑛 is additionally positivedefinite, with elements bounded at each point 𝑥∈Ω𝑐₀𝜂^𝑗_𝑘𝜂^𝑗_𝑘≤𝑎₀^ℎ𝑘_𝑖𝑗(𝑥,𝑡)𝜂^𝑖_ℎ𝜂^𝑗_𝑘≤𝐶₀𝜂^𝑗_𝑘𝜂^𝑗_𝑘, (2.6)

for all 𝜂^𝑗_𝑘=𝜂^𝑘_𝑗∈ℝ and 𝑡∈[0,𝑇] where the constants 0<𝑐₀≤𝐶₀<∞ are independent of 𝑥 and 𝑡. For isotropic materials 𝑎^ℎ𝑘_𝑖𝑗=𝜆𝛿_ℎ𝑖𝛿_𝑘𝑗+𝜇𝛿_𝑖𝑗𝛿_ℎ𝑘+𝜇𝛿_𝑖𝑘𝛿_ℎ𝑗.

Example 2.1. (i) Often, the kernels 𝑎^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏) are of the convolution type and are taken in the exponential form 𝑎^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏)=^𝑚_𝑝=1𝛼_𝑝^ℎ𝑘_𝑖𝑗(𝑥)𝑒^{−𝛽𝑝(𝑥)(𝑡−𝜏)},if𝑡≥𝜏,0,if𝑡<𝜏, (2.7) where the 𝛽_𝑝 are piece-wise continuous functions, often just constants, and the 𝛼_𝑝^ℎ𝑘_𝑖𝑗(𝑥) are piece-wise continuous functions for 𝑥∈Ω.
(ii) The 𝑎^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏) may also be kernels of the Abel type (typical example for the relaxation kernels of concrete and cement): 𝑎^ℎ𝑘_𝑖𝑗𝐴(𝑥,𝑡,𝜏)=^{1ℎ𝑘𝑖𝑗}(𝑥,𝑡,𝜏)(𝑡−𝜏)^−𝛼+𝐴₂^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏)(𝜏)^−𝛽+𝐴₃^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏)𝑡^−𝛾,if𝑡≥𝜏,0,otherwise, (2.8) with 0≤𝛼,𝛽,𝛾<1. The 𝐴_𝑝^ℎ𝑘_𝑖𝑗, 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 𝑣_𝑖(𝑥)∈𝐻¹₀(Ω,𝜕Ω_𝑢), 𝑖=1,…,𝑛, where 𝐻¹₀(Ω,𝜕Ω_𝑢)∶={𝑣∈𝐻¹(Ω)∶𝑣(𝑥)=0,𝑥∈𝜕Ω_𝑢}, and integrate over the whole domain Ω. Integrating by parts and taking into account boundary condition (2.2), we obtain the following variational problem.

Find 𝑢_𝑗∈𝐻¹(Ω), 𝑗=1,…,𝑛, satisfying (2.2) and _Ω𝑎^ℎ𝑘_𝑖𝑗𝜕𝑢_𝑗𝜕𝑥_𝑘𝜕𝑣_𝑖𝜕𝑥_ℎ𝑑𝑥=𝑙(𝑣),𝑙(𝑣)∶=_Ω𝑓_𝑖0𝑣_𝑖𝑑𝑥+_𝜕Ω𝜎𝜙_𝑖𝑣_𝑖𝑑𝑠,∀𝑣_𝑖∈𝐻¹₀(Ω,𝜕Ω_𝑢),𝑖=1,…,𝑛. (3.1)

Definition 3.1 (General weak formulation). Consider the matrix of instantaneous elastic coefficients (𝑎₀^ℎ𝑘_𝑖𝑗)_𝑛×𝑛∈𝐶([0,𝑇];𝐋^∞(Ω)), the relaxation operators (𝑎^ℎ𝑘_𝑖𝑗⋆)_𝑛×𝑛, such that 𝑎^ℎ𝑘_𝑖𝑗(𝑡,𝜏)=0∀𝜏>𝑡, and 𝑎^ℎ𝑘_𝑖𝑗∈𝐶([0,𝑇];𝐿^∞(Ω)), and 𝑓₀∶=(𝑓_𝑖0)_𝑛∈𝐶([0,𝑇];H^-1(Ω)), the whole viscoelastic operator 𝑎^ℎ𝑘_𝑖𝑗=𝑎₀^ℎ𝑘_𝑖𝑗+𝑎^ℎ𝑘_𝑖𝑗⋆, the boundary tractions 𝜙∶=(𝜙_𝑖)_𝑛∈𝐶([0,𝑇];H^-1/2(𝜕Ω_𝜎)), and boundary displacements 𝜓∶=(𝜓_𝑖)_𝑛∈𝐶([0,𝑇];H^1/2(𝜕Ω_𝑢)).
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=𝑢+𝜓, where 𝜓∈𝐶([0,𝑇];H¹(Ω)) satisfies (𝜓|_𝜕Ω𝑢)=𝜓 and 𝑢_𝑖∈𝐶([0,𝑇];𝐻¹₀(Ω,𝜕Ω_𝑢)), 𝑖=1,…,𝑛, satisfies the integral identity [𝑎(𝑢,𝑣)](𝑡)∶=_Ω[𝑎^ℎ𝑘_𝑖𝑗𝜕𝑢_𝑗𝜕𝑥_𝑘](𝑡)𝜕𝑣_𝑖𝜕𝑥_ℎ𝑑𝑥=𝑙(̂𝑣)(𝑡)∀𝑡∈[0,𝑇], (3.2) for any 𝑣_𝑖∈𝐻¹₀(Ω,𝜕Ω_𝑢). The right hand-side of (3.2) is, for all 𝑡∈[0,𝑇], a linear functional on the H¹₀(Ω,𝜕Ω_𝑢)𝑙(̂𝑣)(𝑡)∶=_Ω(𝑓_𝑖0(𝑡)𝑣_𝑖−[𝑎^ℎ𝑘_𝑖𝑗𝜕𝜓_𝑗𝜕𝑥_𝑘](𝑡)𝜕𝑣_𝑖𝜕𝑥_ℎ)𝑑𝑥+_𝜕Ω𝜎𝜙_𝑖(𝑡)𝑣_𝑖𝑑𝑠. (3.3) The space of linear bounded functionals on 𝐇¹₀(Ω,𝜕Ω_𝑢) is denoted by 𝐻⁻¹(Ω).

We denote further𝑎₀(𝑢,𝑣)(𝑡)∶=∫_Ω𝑎₀^ℎ𝑘_𝑖𝑗(𝑥,𝑡)(𝜕𝑢_𝑗(𝑥,𝑡)/𝜕𝑥_𝑘)(𝜕𝑣_𝑖(𝑥)/𝜕𝑥_ℎ𝑑𝑥), 𝑎(𝑢,𝑣)(𝑡,𝜏)∶=∫_Ω𝑎^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏)(𝜕𝑢_𝑗(𝑥,𝜏)/𝜕𝑥_𝑘)(𝜕𝑣_𝑖(𝑥)/𝜕𝑥_ℎ)𝑑. Obviously, [𝑎(𝑢,𝑣)](𝑡)=𝑎₀(𝑢,𝑣)(𝑡)+∫^𝑡₀𝑎(𝑢,𝑣)(𝑡,𝜏)𝑑𝜏.

Note that 𝑎₀(𝑢,𝑣) and 𝑎(𝑢,𝑣) are bilinear forms on 𝐇¹₀(Ω,𝜕Ω_𝑢) for every 𝑡 and almost every 𝜏. We can rewrite the weak formulation (3.2) as follows: 𝑎₀(𝑢,𝑣)(𝑡)+^𝑡₀𝑎(𝑢,𝑣)(𝑡,𝜏)𝑑𝜏=𝑙(𝑣)(𝑡),𝑡∈[0,𝑇]. (3.4)

Let us finally rewrite (3.4) in the operator form. For this purpose we introduce the following notations: 𝐴_0𝑥𝑢∶=𝑎₀(𝑢,⋅),𝐴_𝑥𝑢∶=𝑎(𝑢,⋅),𝐿(𝑡)∶=𝑙(𝑣)(𝑡). (3.5)𝐴_0𝑥(𝑡),𝐴_𝑥(𝑡,𝜏)∶H¹₀(Ω,𝜕Ω_𝑢)→𝐻⁻¹(Ω) for all fixed 𝑡 and almost all 𝜏∈[0,𝑇]. Now we can represent (3.2) in the form 𝐴𝑢=𝐿(𝑡), where 𝐴⋅ is an infinite dimensional integro-differential linear operator 𝐴=𝐴_0𝑥⋅+𝐴_𝑥⋆ and the weak problem formulation (3.2) takes the form 𝐴_0𝑥(𝑡)𝑢(𝑡)+[𝐴_𝑥⋆𝑢](𝑡)=𝐿(𝑡). (3.6) 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). Let Ω⊂ℝ^𝑛 be a Lipschitz domain and 𝜕Ω_𝑢⊆𝜕Ω, let 𝐴_0𝑥∈𝐶([0,𝑇];ℒ(H¹₀(Ω,𝜕Ω_𝑢),𝐻⁻¹)), and let 𝐴_0𝑥(𝑡) be boundedly-invertible uniformly in [0,𝑇], 𝐴_𝑥(𝑡,𝜏)=0∀𝜏>𝑡, 𝐴_𝑥∈𝐶([0,𝑇];𝐿¹([0,𝑇],ℒ(H¹₀(Ω,𝜕Ω_𝑢),𝐻⁻¹))), and 𝑓∈𝐶([0,𝑇];𝐻⁻¹). Then there exists a unique global solution 𝑢 of the problem 𝐴_0𝑥(𝑡)𝑢(𝑡)+[𝐴_𝑥⋆𝑢](𝑡)=𝑓(𝑡) (3.7) in 𝐶([0,𝑇];H¹₀(Ω,𝜕Ω_𝑢)), which depends continuously on 𝑓, that is, ‖𝑢‖_{𝐶([0,𝑇];𝐻}¹_(Ω))≤𝐶₁‖𝑓‖_{𝐶([0,𝑇];𝐻−1),} (3.8) where the constant 𝐶₁ is independent of 𝑓, and if ‖𝐴⁻¹_0𝑥(𝑡)‖_{ℒ(𝐻−1,𝐻1)}≤1/𝑐₀, then 𝐶₁≤1𝐶(𝑐₀max_{𝑖,𝑗,ℎ,𝑘}‖𝑎^ℎ𝑘_𝑖𝑗(𝑡,𝜏)‖_{𝐶(([0,𝑇],𝐿1([0,𝑇],𝐿∞(Ω))))}), (3.9) where 𝐶 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 form 𝑢=𝑢+𝜓, where both 𝑢 and 𝜓 are in 𝐶([0,𝑇];𝐻¹(Ω)), then we can claim that our solution exists and is unique.

Lemma 3.3. Let Ω be a Lipschitz domain in ℝ^𝑛, the instantaneous elastic (out-of-integral) coefficients 𝑎₀^ℎ𝑘_𝑖𝑗∈𝐶([0,𝑇];𝐿^∞(Ω)) satisfy the positivity condition (the first of (2.6) with a constant 𝑐₀, and the relaxation kernels 𝑎^ℎ𝑘_𝑖𝑗∈𝐶(([0,𝑇],𝐿¹([0,𝑇],𝐿^∞(Ω)))). Then (i)𝐴_0𝑥 belongs to 𝐶([0,𝑇];ℒ(H¹₀(Ω,𝜕Ω_𝑢),𝐻⁻¹)), and 𝐴_0𝑥(𝑡) has an inverse operator 𝐴⁻¹_0𝑥(𝑡)  ∀𝑡∈[0,𝑇]. This inverse operator is uniformly bounded in [0,𝑇], that is, the following estimate: ‖𝐴⁻¹_0𝑥(𝑡)‖_{ℒ(𝐻−1,𝐻1)}≤1𝑐₀, (3.10) holds for any 𝑡∈[0,𝑇], and 𝑐₀ is independ on 𝑡;(ii)𝐴_𝑥(𝑡,𝜏) satisfies the following estimate: ‖𝐴_𝑥(𝑡,𝜏)‖_{ℒ(𝐻1(Ω),𝐻−1)}≤max_{𝑖,𝑗,ℎ,𝑘}‖𝑎^ℎ𝑘_𝑖𝑗(𝑡,𝜏)‖_𝐿∞(Ω), (3.11) for all 𝑡 and a.a 𝜏∈[0,𝑇]. Furthermore, the condition 𝑎^ℎ𝑘_𝑖𝑗∈𝐶(([0,𝑇],𝐿¹([0,𝑇],𝐿^∞(Ω))) implies that 𝐴_𝑥∈𝐶(([0,𝑇],𝐿¹([0,𝑇],ℒ(H¹₀(Ω,𝜕Ω_𝑢),𝐻⁻¹))).

Lemma 3.4. In both cases of Example 2.1, 𝑎^ℎ𝑘_𝑖𝑗∈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 function 𝑓(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 notation 𝒮(𝐽,𝑋), which should be interpreted as either 𝐶(𝐽,𝑋) or 𝐿^∞(𝐽,𝑋).

Corollary 4.2. Let 𝐽⊂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. Each 𝑔∈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 𝐵_𝑖,𝑛⊂𝐽 and finitely-valued functions 𝑔_𝑛(𝜏)∶=∑^𝑛_𝑖=1𝑔_𝑖,𝑛𝜒_{𝐵𝑖,𝑛}(𝜏),∀𝜏∈𝐽, such that lim_𝑛→∞‖𝑔_𝑛(𝜏)−𝑔(𝜏)‖_𝑋=0 a.e. on 𝐽. Here 𝜒_{𝐵𝑖,𝑛}(𝜏) is the characteristic function of 𝐵_𝑖,𝑛. Thus ‖𝐾(𝜏)𝑔_𝑛(𝜏)−𝐾(𝜏)𝑔(𝜏)‖_𝑍≤‖𝐾(𝜏)‖_{ℒ(𝑋,𝑍)}‖𝑔_𝑛(𝜏)−𝑔(𝜏)‖_𝑋→0as𝑛→∞ (4.1) for almost all 𝜏∈𝐽. 𝐾 is strongly measurable on 𝐽;𝐾(𝜏)𝑔_𝑛(𝜏)∶=∑^𝑛_𝑖=1𝐾(𝜏)𝑔_𝑖,𝑛𝜒_{𝐵𝑖,𝑛}(𝜏), ∀𝜏∈𝐽, where 𝐾(𝜏)𝑔_𝑖,𝑛 is, for all 𝑛, a strongly measurable 𝑍-valued function, owing to [12, Chapter 5, Corollary 2], and 𝜒_{𝐵𝑖,𝑛}∶𝐽→ℝ.  Owing to [13, the Appendix, Proposition (9a)], 𝐾(𝜏)𝑔_𝑛(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, _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<∞. (4.2)
Using Theorem 4.1 completes the proof.

Since for 𝐽=[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. Let 𝑋, 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 as 𝐽 is closed and bounded (i.e., compact), condition (ii) of Definition 4.3 implies that the following holds: |‖𝐾‖|_{𝑉(𝐶;𝐽;ℒ(𝑋,𝑍))}∶=sup_{𝑡∈𝐽𝐽}‖𝐾(𝑡,𝜏)‖_{ℒ(𝑋,𝑍)}𝑑𝜏<∞. (4.3) We will call in such a case |‖𝐾‖|_{𝑉(𝐶;𝐽;ℒ(𝑋,𝑍))} the kernel norm of the operator 𝐾⋆.

Corollary 4.5. Let 𝐽=[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 that 𝑓(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(ℝ^𝑛,𝑋)).

Corollary 4.6. Let 𝐽⊂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 ‖𝐾⋆𝑔‖_𝑍≤‖𝐾‖_{ℒ(𝑋,𝑍)}⋆‖𝑔‖_𝑋 (4.4) 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 Theorem 4.1 in application to Definition 4.3). If 𝐽 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 Definition 4.3, 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 the 𝐶(𝐽,𝑍)-norm follows from the continuity and the fact that 𝐽 is compact. And Theorem 4.1 delivers the continuity of the embedding ‖𝐾(𝑡,𝑠)⋆‖_{ℒ(𝒮(𝐽,𝑋),𝐶(𝐽,𝑍))}=sup_{‖𝑔‖𝒮(𝐽,𝑋)≤1𝐽}‖𝐾(𝑡,𝑠)𝑔(𝑠)𝑑𝑠‖_{𝐶(𝐽,𝑍)}≤sup_{‖𝑔‖𝒮(𝐽,𝑋)≤1}sup_{𝑡∈[0,𝑇]𝐽}‖𝐾(𝑡,𝑠)𝑔(𝑠)‖_𝑍𝑑𝑠≤sup_{‖𝑔‖𝒮(𝐽,𝑋)≤1}sup_{𝑡∈[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))}. (4.5)

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 spectrum 𝜎(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 𝑅(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. Let 𝒜 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. Let 𝑇 be a bounded linear operator over a Banach-space 𝑋, with lim_𝑛→∞sup‖𝑇^𝑛‖^1/𝑛_ℒ(𝑋)<1, (5.1) then (𝐼−𝑇)⁻¹∈ℒ(𝑋) can be represented as a series: (𝐼−𝑇)⁻¹=^∞_𝑗=0𝑇^𝑗. (5.2)

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. Let 𝐽⊂ℝ be a compact interval (𝐽=[0,𝑇]), and 𝑄⋆∈𝑉(𝐶;𝐽;ℂ). Then there exists a resolvent 𝑅(𝜆) of 𝑄⋆ in ℒ(𝒮([0,𝑇],ℂ)), given by the convergent series 𝑅(𝜆)=∑^∞_𝑗=0((𝑄⋆)^𝑗/𝜆^𝑗+1) for each 𝜆∈ℂ⧵{0}.

Proof. The class of kernels of Volterra operators from 𝑉(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: 𝑅+𝑄⋆𝑅=𝑅+𝑅⋆𝑄=𝑄, (5.3) such that 𝑅⋆∈𝑉(𝐶;[0,𝑇];ℂ).
Obviously, if 𝑄⋆∈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 𝜌(𝑄)=lim_𝑛→∞|‖(𝑄⋆)^𝑛‖|^1/𝑛 (5.4) is known from the theorem with the same title (see [14, Theorem 2, A.2.3] or [15, Theorem 18.9]). This means, lim_𝑛→∞|‖(𝑄⋆)^𝑛‖|^1/𝑛_{𝑉(𝐶;[0,𝑇];C)}=0for𝑄⋆∈𝑉(𝐶;[0,𝑇];ℂ). (5.5)
Let us give a justification of the assumptions of Theorem 5.3 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 ‖(𝑄⋆)^𝑛‖_{ℒ(𝐶[0,𝑇])}≤|‖(𝑄⋆)^𝑛‖|_{𝑉(𝐶;[0,𝑇];C)}. (5.6) 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 Theorem 5.3 are satisfied, for each 𝜆∈ℂ⧵{0} and 𝑄⋆∈𝑉(𝐶;[0,𝑇];ℂ), 𝑅(𝜆)≡(𝜆𝐼−𝑄⋆)⁻¹=^∞_𝑗=0(𝑄⋆)^𝑗𝜆^𝑗+1. (5.7)

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. Let 𝑋 be a Banach space, 𝐾⋆∈𝑉(𝐶;[0,𝑇];ℒ(𝑋)), 𝑛,𝑚=1,2,…. Then, for each 𝑛≥𝑚 one can estimate ‖^𝑛_𝑗=𝑚(−𝐾⋆)^𝑗‖_{ℒ(𝒮([0,𝑇],𝑋),𝐶([0,𝑇],𝑋))}≤‖^𝑛_𝑗=𝑚(‖𝐾‖_ℒ(𝑋)⋆)^𝑗‖_{ℒ(𝐶[0,𝑇])}, (6.1) where ℒ(𝐶[0,𝑇])≡ℒ(𝐶([0,𝑇],ℝ)).

Proof. Corollary 4.7 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 ‖^𝑛_𝑗=𝑚(−𝐾⋆)^𝑗‖_{ℒ(𝒮([0,𝑇],𝑋),𝐶([0,𝑇],𝑋))}=sup_{‖𝐹‖𝒮([0,𝑇],𝑋)≤1}‖^𝑛_𝑗=𝑚(−1)^𝑗[(𝐾⋆)^𝑗𝐹]‖_{𝐶([0,𝑇],𝑋)}≤sup_{‖𝐹‖𝒮([0,𝑇],𝑋)≤1}‖^𝑛_𝑗=𝑚‖[(𝐾⋆)^𝑗𝐹]‖𝑋‖_{𝐶([0,𝑇])}=sup_{‖𝐹‖𝒮([0,𝑇],𝑋)≤1}‖^𝑛_𝑗=𝑚‖[𝐾⋆[(𝐾⋆)^𝑗−1𝐹]]‖𝑋‖_{𝐶([0,𝑇])Corollary4.6}≤sup_{‖𝐹‖𝒮([0,𝑇],𝑋)≤1}‖^𝑛_𝑗=𝑚‖𝐾‖_ℒ(𝑋)⋆‖[(𝐾⋆)^𝑗−1𝐹]‖𝑋‖_{𝐶([0,𝑇])induction}≤sup_{‖𝐹‖𝒮([0,𝑇],𝑋)≤1}‖^𝑛_𝑗=𝑚(‖𝐾‖_ℒ(𝑋)⋆)^𝑗‖𝐹‖𝑋‖_{𝐶([0,𝑇])}≤sup_{‖𝐹‖𝒮([0,𝑇],𝑋)≤1}‖^𝑛_𝑗=𝑚(‖𝐾‖_ℒ(𝑋)⋆)^𝑗‖_{ℒ(𝐶[0,𝑇])}‖𝐹‖_{𝒮([0,𝑇],𝑋)},=‖^𝑛_𝑗=𝑚(‖𝐾‖_ℒ(𝑋)⋆)^𝑗‖_{ℒ(𝐶[0,𝑇]).} (6.2)

Theorem 6.2. Let 𝑋 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 𝑢(𝑡)+[𝐾⋆𝑢](𝑡)=𝐹(𝑡), (6.3) in 𝒮([0,𝑇];𝑋), which depends continuously on 𝐹, that is, ‖𝑢‖_{𝒮([0,𝑇],𝑋)}≤𝐶‖𝐹‖_{𝒮([0,𝑇],𝑋)}, (6.4) 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)⋆)≤𝐶(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): 𝑣(𝑡)∶=[𝑅𝐹](𝑡)≡^∞_𝑗=0(−1)^𝑗[(𝐾⋆)^𝑗𝐹](𝑡). (6.5)
According to Corollary 4.7, 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 𝑆_𝑛≡^𝑛_𝑗=0(−1)^𝑗(𝐾⋆)^𝑗 (6.6) is a Cauchy sequence, that is, that for any 𝜖 there exists a number 𝑁 such that ‖𝑆_𝑛−𝑆_𝑚‖_{ℒ(𝒮([0,𝑇],𝑋),𝐶([0,𝑇],𝑋))}<𝜖 for all 𝑛≥𝑚≥𝑁. We can estimate ‖𝑆_𝑛−𝑆_𝑚‖_{ℒ(𝒮([0,𝑇],𝑋),𝐶([0,𝑇],𝑋))}≡‖^𝑛_𝑗=𝑚(−1)^𝑗(𝐾⋆)^𝑗‖_{ℒ(𝒮([0,𝑇],𝑋),𝐶([0,𝑇],𝑋))Lemma6.1}≤‖^𝑛_𝑗=𝑚(‖𝐾‖_ℒ(𝑋)⋆)^𝑗‖_{ℒ(𝐶[0,𝑇])}. (6.7)
Note that, according to Corollary 4.6, ‖𝐾‖_ℒ(𝑋)⋆∈𝑉(𝐶;[0,𝑇];ℝ) since 𝐾⋆∈𝑉(𝐶;[0,𝑇];ℒ(𝑋,𝑋)). If the real-valued series 𝑅≡∑^∞_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,‖∑^𝑛_𝑗=𝑚(‖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 Lemma 5.4 for 𝑄≡−‖𝐾‖_ℒ(𝑋) and real 𝜆=−1. Then, owing to its statement, the real-valued series 𝑅 converges to 𝑅≡^∞_𝑗=0(‖𝐾‖_ℒ(𝑋)⋆)^𝑗=[𝐼−‖𝐾‖_ℒ(𝑋)⋆]⁻¹. (6.8) Owing to Lemma 5.4, the operator 𝑅 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, ‖𝑣‖_{𝒮([0,𝑇],𝑋)}≤‖^𝑛_𝑗=0(−𝐾⋆)^𝑗‖_{ℒ(𝒮([0,𝑇],𝑋),(𝒮([0,𝑇],𝑋))}‖𝐹‖_{𝒮([0,𝑇],𝑋)}≤‖𝑅‖_{ℒ(𝒮([0,𝑇])}‖𝐹‖_{𝒮([0,𝑇],𝑋)}. (6.9)
Furthermore, if there exists 𝑘⋆∈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, ‖𝑅‖_{ℒ(𝒮([0,𝑇])}≤‖^∞_𝑗=0(𝑘⋆)^𝑗‖_{ℒ(𝒮([0,𝑇])}=‖[𝐼−𝑘⋆]⁻¹‖_{ℒ(𝒮([0,𝑇])}<∞. (6.10) That is, ‖𝑅‖_{ℒ(𝒮([0,𝑇])}≤𝐶(𝑘⋆)<∞.
The fact that 𝑣(𝑡), given by convergent Neumann's series (6.5), is a solution of (6.3), can be checked by direct substitution: 𝑣(𝑡)+[𝐾⋆𝑣](𝑡)=^∞_𝑗=0(−1)^𝑗[(𝐾⋆)^𝑗𝐹](𝑡)+𝐾⋆^∞_𝑗=0(−1)^𝑗[(𝐾⋆)^𝑗=𝐹](𝑡)_∞𝑗=0(−1)^𝑗[(𝐾⋆)^𝑗𝐹](𝑡)+^∞_𝑗=1(−1)^𝑗−1[(𝐾⋆)^𝑗𝐹](𝑡)=𝐹(𝑡). (6.11) 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 𝑅 introduced in (6.5) to both sides of (6.3): 𝑅𝑢+𝑅𝐾⋆𝑢=𝑅𝐹,[𝑅𝑢](𝑡)+[𝑅𝐾⋆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). (6.12) Hence, 𝑢(𝑡)=[𝑅𝐹](𝑡), (6.13) that is, any solution is represented by Banach-valued Neumann's series. Since [𝑅𝐹] 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. Let 𝑋, 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 ‖𝐴₀(𝑡)⁻¹‖_{ℒ(𝑍,𝑌)}≤1𝑐₀∀𝑡∈[0,𝑇]. (6.14) Let 𝐴∈𝐶([0,𝑇];𝐿¹([0,𝑇],ℒ(𝑋,𝑍))). Then for 𝐾≡𝐴₀⁻¹𝐴 the following continuity property is valid: _𝐽‖𝐾(𝑡+Δ𝑡,𝜏)−𝐾(𝑡,𝜏)‖_{ℒ(𝑋,𝑌)}𝑑𝜏→0asΔ𝑡→0∀𝑡,𝑡+Δ𝑡∈𝐽. (6.15)

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)): _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)}≤1𝑑𝜏𝑐_0𝐽‖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)≤1𝑐_0𝐽‖𝐴(𝑡+Δ𝑡,𝜏)−𝐴(𝑡,𝜏)‖_{ℒ(𝑋,𝑍)}+1𝑑𝜏𝑐²₀|‖𝐴‖|_{𝑉(𝐶;[0,𝑇];ℒ(𝑋,𝑍))}‖𝐴₀(𝑡+Δ𝑡)−𝐴₀(𝑡)‖_{ℒ(𝑌,𝑍)}→0asΔ𝑡→0∀𝑡,𝑡+Δ𝑡∈𝐽. (6.16)(1)(2)(3)(4)(5)(6)(7)(8)(9)

Theorem 6.4. Let 𝑋, 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 𝐴₀(𝑡)𝑢(𝑡)+[𝐴⋆𝑢](𝑡)=𝑓(𝑡) (6.17) in 𝒮([0,𝑇];𝑋), which depends continuously on 𝑓, that is, ‖𝑢‖_{𝒮([0,𝑇],𝑋)}≤𝐶₁‖𝑓‖_{𝒮([0,𝑇],𝑍)}, (6.18) 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)𝐶(1/𝑐₀)𝑎⋆<∞.

Proof. Since, 𝐴₀(𝑡) is invertible, we can rewrite (6.17) as the following:  𝑢(𝑡)+𝐴₀(𝑡)⁻¹∫^𝑡₀𝐴(𝑡,𝜏)𝑢(𝜏)𝑑𝜏=𝐴₀(𝑡)⁻¹𝑓(𝑡). Observe that 𝐴₀(𝑡)⁻¹ does not depend on 𝜏, 𝑢(𝑡)+^𝑡₀𝐴₀(𝑡)⁻¹𝐴(𝑡,𝜏)𝑢(𝜏)𝑑𝜏=𝐴₀⁻¹(𝑡)𝑓(𝑡). (6.19) We denote 𝐾(𝑡,𝜏)≡𝐴₀(𝑡)⁻¹𝐴(𝑡,𝜏)and𝐹(𝑡)≡𝐴₀(𝑡)⁻¹𝑓(𝑡).
Let us show that the assumption of Theorem 6.2 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)Since 𝐴⋆∈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 Lemma 6.3 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.

Appendix

Proof of Lemma 3.4. Case (i). Each continuous Banach-valued function is strongly measurable and Bochner integrable. Hence, 𝑎^ℎ𝑘_𝑖𝑗∈𝐶([0,𝑇]²,𝐿^∞(Ω)) satisfies requirement (i) of Definition 4.3 in case (i) of Example 2.1. Requirement (ii) of Definition 4.3 is obviously valid for the 𝑎^ℎ𝑘_𝑖𝑗. Hence, (2.7) presents kernels from 𝑉(𝐶;[0,𝑇];𝐿^∞(Ω)).
Case (ii). Let us denote the weak singular multipliers, (𝑡−𝜏)^−𝛼, 𝜏^−𝛽 in case (ii) of Example 2.1, by 𝑣(𝑡,𝜏), in such a way that 𝑣(𝑡,𝜏)=(𝑡−𝜏)^−𝛼,if𝜏≤𝑡,0,if𝜏>𝑡, (A.1) or 𝜏𝑣(𝑡,𝜏)=^−𝛽,if𝜏≤𝑡,0,if𝜏>𝑡, (A.2)
The 𝑣(𝑡,𝜏) are Lebesgue-measurable on [0,𝑇]⊂ℝ and integrable w.r.t. 𝜏 for all 𝑡. Hence, 𝑣(𝑡,⋅)∈𝐿¹[0,𝑇]∀𝑡∈[0,𝑇]. According to [16], we can rewrite it as 𝑣(𝑡,⋅)∈𝐿¹([0,𝑇],ℝ))∀𝑡∈[0,𝑇].
Owing to the fact that 𝑣=0 for 𝜏>𝑡, and considering arbitrary 𝑡,𝑡+Δ𝑡∈[0,𝑇], let us perform now integration over 𝜏 and show that 𝑣∈𝐶([0,𝑇],𝐿¹([0,𝑇],ℝ)). For 𝑣 corresponding to the (𝑡−𝜏)^−𝛼, we get ∫^𝑇₀|𝑣(𝑡+Δ𝑡,𝜏)−𝑣(𝑡,𝜏)|𝑑𝜏=∫^𝑡₀(𝑡−𝜏)^−𝛼𝑑𝜏−∫^𝑡₀(𝑡+Δ𝑡−𝜏)^−𝛼𝑑𝜏+∫_𝑡^𝑡+Δ𝑡(𝑡+Δ𝑡−𝜏)^−𝛼𝑑𝜏=1/(1−𝛼)(−(𝑡−𝜏)^1−𝛼|^𝑡₀+(𝑡+Δ𝑡−𝜏)^1−𝛼|^𝑡₀−(𝑡+Δ𝑡−𝜏)^1−𝛼|_𝑡^𝑡+Δ𝑡)=1/(1−𝛼)(𝑡^1−𝛼+Δ𝑡^1−𝛼−(𝑡+Δ𝑡)^1−𝛼+Δ𝑡^1−𝛼)→0asΔ𝑡→0.
And for 𝑣 corresponding to the 𝜏^−𝛽, we calculate:∫^𝑇₀|𝑣(𝑡+Δ𝑡,𝜏)−𝑣(𝑡,𝜏)|𝑑𝜏=∫₀^𝑡+Δ𝑡(𝜏)^−𝛽𝑑𝜏−∫^𝑡₀(𝜏)^−𝛽𝑑𝜏=∫_𝑡^𝑡+Δ𝑡(𝜏)^−𝛽𝑑𝜏=1/(1−𝛽)(𝜏)^1−𝛽|_𝑡^𝑡+Δ𝑡=1/(1−𝛽)((𝑡+Δ𝑡)^1−𝛽−𝑡^1−𝛽)→0asΔ𝑡→0.  Owing to Definition 4.3, 𝑣⋆∈𝑉(𝐶;[0,𝑇];ℝ).
One can consider 𝑣 as a linear operator 𝑣(𝑡,𝜏)∶𝐿^∞(Ω)→𝐿^∞(Ω) defined for any function 𝑔∈𝐿^∞(Ω),on[0,𝑇]²⧵({𝑡=𝜏}∪{𝜏=0}), as following: 𝑔→𝑣(𝑡,𝜏)𝑔. Obviously,  𝑣∈𝐶([0,𝑇],𝐿¹([0,𝑇],ℒ(𝐿^∞(Ω),𝐿^∞(Ω)))), that is,  𝑣⋆∈𝑉(𝐶;[0,𝑇];ℒ(𝐿^∞(Ω),𝐿^∞(Ω))). The functions 𝐴_𝑙^ℎ𝑘_𝑖𝑗are in 𝐶([0,𝑇],𝐿^∞(Ω)),∀𝑡∈[0,𝑇]. Owing to Corollary 4.2, 𝑣(𝑡,⋅)𝐴_𝑙^ℎ𝑘_𝑖𝑗(𝑡,⋅)∈𝐿¹([0,𝑇],𝐿^∞(Ω))∀𝑡[0,𝑇]. Applying Lemma 6.3 one can varify that 𝑣𝐴_𝑙^ℎ𝑘_𝑖𝑗∈𝐶([0,𝑇],𝐿¹([0,𝑇],𝐿^∞(Ω)). According to Definition 4.3 , 𝑣𝐴_𝑙^ℎ𝑘_𝑖𝑗⋆∈𝑉(𝐶;[0,𝑇];ℒ(ℝ,𝐿^∞(Ω))), or, what is the same: 𝐴^ℎ𝑘_𝑖𝑗⋆∈𝑉(𝐶;[0,𝑇];𝐿^∞(Ω)).

Proof of Lemma 3.3. Let us start with the proof of (3.10). The bilinear form 𝑎₀(𝑢,𝑣) (introduced in Definition 3.1) is symmetric, owing to conditions (2.5), and continuous as follows from (2.6), that is, ∀𝑣∈H¹₀(Ω,𝜕Ω_𝑢), |𝑎₀(𝑢,𝑣)(𝑡)|∶=|_Ω𝑎^ℎ𝑘_𝑖𝑗0𝜕𝑢(𝑥,𝑡)^𝑗(𝑥,𝑡)𝜕𝑥_𝑘𝜕𝑣_𝑖(𝑥)𝜕𝑥_ℎ𝑑𝑥|≤max_{𝑖,𝑗,𝑘,ℎ}‖𝑎^ℎ𝑘_𝑖𝑗0(⋅,𝑡)‖_𝐿∞(Ω)‖𝑢(𝑡)‖_𝐻¹_(Ω)‖𝑣‖_𝐻¹_(Ω)(2.6)≤𝐶₀‖𝑢(𝑡)‖_𝐻¹_(Ω)‖𝑣‖_𝐻¹_(Ω),∀𝑡∈[0,𝑇]. (A.3)
Function 𝑢∶=𝑢−𝜓 must satisfy the integral identity (3.2) for any 𝑣∈H¹₀(Ω,𝜕Ω_𝑢). Due to symmetry (2.5), positivity (2.6) conditions, and to Korn's inequality (see [20, Chapter 1, Theorem 2.7]), the positivity condition required by the Lax-Milgram Theorem (see [16]) is satisfied: 𝑎₀(𝑢,𝑢)≡_Ω𝑎^ℎ𝑘_𝑖𝑗0𝜕𝑢_𝑗𝜕𝑥_𝑘𝜕𝑢_𝑖𝜕𝑥_ℎ𝑑𝑥_(2.5)=_Ω𝑎^ℎ𝑘_𝑖𝑗012(𝜕𝑢_𝑗𝜕𝑥_𝑘+𝜕𝑢_𝑘𝜕𝑥_𝑗)12(𝜕𝑢_𝑖𝜕𝑥_ℎ+𝜕𝑢_ℎ𝜕𝑥_𝑖)𝑑𝑥_(2.6)≥𝑐₀4_Ω(𝜕𝑢_𝑗𝜕𝑥_𝑘+𝜕𝑢_𝑘𝜕𝑥_𝑗𝜕𝑢)(^𝑗𝜕𝑥_𝑘+𝜕𝑢_𝑘𝜕𝑥_𝑗)𝑑𝑥_(Korn)≥𝑐₀‖𝑢‖²_𝐻1Ω. (A.4)
The assumptions of the Lax-Milgram Theorem are satisfied, and its application completes the proof for a fixed 𝑡∈[0,𝑇]. Owing to the definition of the norm of 𝐶([0,𝑇],𝐿^∞(Ω)) by sup_{𝑡∈[0,𝑇]}‖⋅‖_𝐿^∞_(Ω), 𝐴_0𝑥 is boundedly invertible uniformly in 𝑡 on [0,𝑇]. In order to check the continuity of 𝐴_0𝑥, that is, that 𝐴_0𝑥∈𝐶([0,𝑇];ℒ(H¹₀(Ω,𝜕Ω_𝑢),𝐻⁻¹)), let us consider 𝑡,𝑡+Δ𝑡∈[0,𝑇]. Owing to (A.3), ‖𝐴_0𝑥(𝑡+Δ𝑡)−𝐴_0𝑥(𝑡)‖_{ℒ(𝐻10(Ω),𝐻−1)}≤max_{𝑖,𝑗,𝑘,ℎ}‖𝑎₀^ℎ𝑘_𝑖𝑗(⋅,𝑡+Δ𝑡)−𝑎₀^ℎ𝑘_𝑖𝑗(⋅,𝑡)‖_𝐿∞(Ω)→0asΔ𝑡→0. (A.5)
In order to prove statement (ii) of Lemma 3.3, we first have to demonstrate that the linear operator 𝐴_𝑥(𝑡,𝜏)∶H¹₀(Ω,𝜕Ω_𝑢)→𝐻⁻¹(Ω) is bounded for each 𝑡 and almost all 𝜏∈[0,𝑇].
Let us consider the bilinear form 𝑎(𝑢,𝑣): |𝑎(𝑢,𝑣)(𝑡,𝜏)|≡|_Ω𝑎^ℎ𝑘_𝑖𝑗𝜕𝑢(𝑡,𝜏)^𝑗(𝜏)𝜕𝑥_𝑘𝜕𝑣_𝑖𝜕𝑥_ℎ𝑑𝑥|≤_Ω|𝑎^ℎ𝑘_𝑖𝑗𝜕𝑢(𝑡,𝜏)^𝑗(𝜏)𝜕𝑥_𝑘𝜕𝑣_𝑖𝜕𝑥_ℎ|𝑑𝑥≤max_{𝑖,𝑗,𝑘,ℎ}‖𝑎^ℎ𝑘_𝑖𝑗(𝑡,𝜏)‖_𝐿∞(Ω)‖𝑢(𝜏)‖_𝐻¹_(Ω)‖𝑣‖_𝐻¹_(Ω), (A.6)∀𝑡 and almost all 𝜏∈[0,𝑇]. Owing to estimate (A.6), the mapping 𝑣→𝑎(𝑢,𝑣), for each fixed 𝑢(⋅,𝑡)∈H¹₀(Ω,𝜕Ω_𝑢), is a linear continuous functional on H¹₀(Ω,𝜕Ω_𝑢), for all 𝑡 and almost all 𝜏∈[0,𝑇]. According to [21, Lemma  6.5.1], ‖𝐴_𝑥(𝑡,𝜏)ℒ‖_{(𝐻10(Ω,𝜕Ω𝑢),𝐻−1)}≤max_{𝑖,𝑗,𝑘,ℎ}‖𝑎^ℎ𝑘_𝑖𝑗(𝑡,𝜏)‖_𝐿∞(Ω)∀𝑡, a.a. 𝜏∈[0,𝑇]. At the following stage, it should be shown that 𝐴_𝑥(𝑡,𝜏) is strongly measurable and ‖𝐴_𝑥(𝑡,𝜏)‖_{ℒ(𝐻10(Ω,𝜕Ω𝑢),𝐻−1(Ω,𝜕Ω𝑢))} is integrable w.r.t. 𝜏 for all 𝑡 in [0,𝑇]. Since 𝑎^ℎ𝑘_𝑖𝑗(𝑡,⋅)∈𝐿¹([0,𝑇],𝐿^∞(Ω))∀𝑡∈[0,𝑇], it can be considered as a linear operator 𝑎^ℎ𝑘_𝑖𝑗(𝑡,𝜏)∶𝐿²(Ω)→𝐿²(Ω)  ∀𝑡 and a.a. 𝜏∈[0,𝑇]. That is, we can rewrite that 𝑎^ℎ𝑘_𝑖𝑗(𝑡,⋅)∈𝐿¹([0,𝑇],ℒ(𝐿²(Ω),𝐿²(Ω))∀𝑡∈[0,𝑇]. The operator 𝐴_𝑥(𝑡,𝜏) can be represented in the form 𝐴_𝑥(𝑡,𝜏)=[𝐾₂𝑓𝐾₁](𝑡,𝜏), where 𝐾₁∶=𝜕/𝜕𝑥_𝑘∈ℒ(𝐻¹₀(Ω,𝜕Ω_𝑢),𝐿²(Ω)), 𝑓∶=𝑎^ℎ𝑘_𝑖𝑗(𝑡,⋅)∈𝐿¹([0,𝑇],ℒ(𝐿²(Ω),𝐿²(Ω))∀𝑡∈[0,𝑇], and 𝐾₂∈ℒ(𝐿²(Ω),𝐻⁻¹(Ω,𝜕Ω_𝑢)) maps 𝑤_𝑖∈𝐿²(Ω) in the functionals ∫_Ω𝑤_𝑖(𝑥)(𝜕𝑣_𝑗/𝜕𝑥_ℎ)𝑑𝑥 defined on all 𝑣_𝑗∈𝐻¹₀(Ω,𝜕Ω_𝑢). Application of [12, Chapter 5, Section 5, Corollary 2] w.r.t. the operator 𝐾₂, acting on 𝑓𝐾₁, verifies that 𝐴_𝑥(𝑡,⋅)∈𝐿¹([0,𝑇],ℒ(H¹₀(Ω,𝜕Ω_𝑢),𝐻⁻¹(Ω,𝜕Ω_𝑢))),∀𝑡∈[0,𝑇].
Let us show that ‖𝐴_𝑥(𝑡,⋅)‖_{𝐿1([0,𝑇],ℒ(𝐻1(Ω),𝐻−1)} is continuous in the 𝑡-variable: ₀^𝑡+Δ𝑡‖𝐴_𝑥(𝑡+Δ𝑡,𝜏)−𝐴_𝑥(𝑡,𝜏)‖_{ℒ(𝐻10(Ω,𝜕Ω𝑢),𝐻−1)}≤𝑑𝜏_0𝑡+Δ𝑡max_{𝑖,𝑗,𝑘,ℎ}‖𝑎^ℎ𝑘_𝑖𝑗(⋅,𝑡+Δ𝑡,𝜏)−𝑎^ℎ𝑘_𝑖𝑗(⋅,𝑡,𝜏)‖_𝐿∞(Ω)𝑑𝜏→0asΔ𝑡→0 (A.7) for all 𝑡+Δ𝑡∈[0,𝑇], since 𝑎^ℎ𝑘_𝑖𝑗∈𝐶([0,𝑇],𝐿¹([0,𝑇];𝐿^∞(Ω))).

RemarkOwing to the estimate (3.11), the condition {𝑎^ℎ𝑘_𝑖𝑗(𝑥,𝑡,𝜏)=0𝑖𝑓𝜏>𝑡,∀𝑥∈Ω} additionally implies that 𝐴_𝑥⋆∈𝑉(𝐶;[0,𝑇];ℒ(H¹₀(Ω,𝜕Ω_𝑢),𝐻⁻¹(Ω,𝜕Ω_𝑢))), that is, 𝐴_𝑥⋆ is the Volterra integral operator.

Proof of Theorem 3.2. The statement of the theorem coincides with the one of Theorem 6.4 if it takes in its counterparts 𝒮∶=𝐶, 𝑋∶=H¹₀(Ω,𝜕Ω_𝑢) and 𝑍∶=𝐻⁻¹.

Acknowledgments

This research has been carried out in a framework of the author’s Ph.D. thesis finished in 2000. Main ideas presented in this work were particularly suggested by Professor S. E. Mikhailov from Glasgow Caledonian University. His help is highly acknowledged.