Mild solutions, variation of constants formula, and linearized stability for delay differential equations

The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a \textit{mild solution}, which is a solution under an initial condition having a discontinuous history function. Then the \textit{principal fundamental matrix solution} is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincar\'{e}-Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.


Introduction
Studies concerning with the variation of constants formula for delay differential equations (DDEs) have a long history of over fifty years. Nevertheless, the reason why we try to discuss the variation of constants formula in this paper is that such a consideration gives rise to a conceptual difficulty that is peculiar to the theory of DDEs. Specifically, it is usual to discuss DDEs within the scope of continuous history functions, but a class of discontinuous history functions emerges as initial conditions when we try to obtain the variation of constants formula. In connection with this, a matrix-valued solution having a certain discontinuous matrix-valued function as the initial condition is called the fundamental matrix solution. However, it is quite difficult to understand why the solution is called the "fundamental matrix solution" when compared with the theory of ordinary differential equations (ODEs).
This conceptual difficulty has arisen in the theoretical development about the variation of constants formula in the texts [18] and [19] by Jack Hale. In the revised edition [22], the theoretical development is rewritten based on the consideration in [34]. There also exist studies to understand the conceptual difficulty of the variation of constants formula for DDEs within the framework of Functional Analysis (e.g., see [7], [12], and [13]). In this framework, it is essential that the Banach space of continuous functions on closed and bounded interval endowed with the supremum norm is not reflexive, and the theory is constructed by using the so called "sun-star calculus". See [14] for the details. See also [36] for a survey article.
The idea of discussing the variation of constants formula for DDEs in this paper is to define a solution under an initial condition having a discontinuous history function as a mild solution. This concept comes from the analogy of the notion of mild solutions of abstract linear evolution equations, and its terminology also originates from this. It can be said that the notion of mild solutions is to elevate the technique to exchange the order of integration to a concept.
The dependence of the derivativeẋ(t) of an unknown function x on the past value of x is abstracted to the concept of retarded functional differential equations (RFDEs). In this paper, we consider an autonomous linear RFDĖ for a continuous linear map L : C([−r, 0], K n ) → K n . Here K = R or C, n ≥ 1 is an integer, and r > 0 is a constant, which are fixed throughout this paper. The derivative of x at 0 is interpreted as the right-hand derivative. We are using the following notations: • C([−r, 0], K n ) denotes the Banach space of all continuous functions from [−r, 0] to K n endowed with the supremum norm · . Here a norm |·| on K n , which is not necessarily the Euclidean norm, is fixed throughout this paper.
• For each t ≥ 0, x t : [−r, 0] → K n is a continuous function defined by In addition to the linear RFDE (1.1), we also consider a non-homogeneous linear RFDĖ x(t) = Lx t + g(t) (a.e. t ≥ 0) (1.2) for some g ∈ L 1 loc ([0, ∞), K n ). Here L 1 loc ([0, ∞), K n ) denotes the linear space of all locally Lebesgue integrable functions from [0, ∞) to K n defined almost everywhere. See also the notations given below. We refer the reader to [32] and [30] as references of the theory of Lebesgue integration for scalar-valued functions.
This paper is organized as follows: In Section 2, we introduce the notion of a history segment x t for a discontinuous function x : [−r, ∞) ⊃ dom(x) → K n . By using this, we also introduce the notion of a mild solution to the linear RFDE (1.1) under an initial condition ( In Section 3, we derive a differential equatioṅ satisfied by the principal fundamental matrix solution X L of (1.1). In the derivation, it is useful to use the notions of Volterra operator and Riemann-Stieltjes convolution. See Subsection 3.1 for the definitions and Subsection 3.3 for the fundamental properties. The above differential equation is the key to obtain a variation of constants formula. In Section 4, we consider the non-homogeneous linear RFDE (1.2). To study a mild solution of (1.2) under the initial condition (1.4), we also consider an integral equation for a continuous function G : [0, ∞) → K n with G(0) = 0. We show that the above integral equation has a unique solution x L (·; φ, G) under the initial condition (1.4).
In Section 5, we consider a non-homogeneous linear RFDĖ for a continuous function f : [0, ∞) → K n to motivate the use of the convolution for locally Riemann integrable functions. We show that the function x(·; f ) : [−r, ∞) → K n defined by x(·; f ) 0 = 0 and x(t; f ) := t 0 X L (t − u)f (u) du (1.8) for t ≥ 0 is a solution to Eq. (1.7) after developing the results of convolution for locally Riemann integrable functions. See Subsection 5.2 for the developments.
In Section 6, we study the non-homogeneous linear RFDE (1.2) under the initial condition (1.4) and find a variation of constants formula expressed by the principal fundamental matrix solution X L . For this purpose, we indeed consider the integral equation (1.6) for some continuous function G : [0, ∞) → K n with G(0) = 0. One of the main results of this paper is that the solution x L (·; φ, G) of (1.6) under the initial condition (1.4) satisfies for all t ≥ 0. HereẊ L (t) denotes the derivative of the locally absolutely continuous function X L | [0,∞) at t ≥ 0 (when it exists), and G L (·; φ) : [0, ∞) → K n is a function determined by the initial history function φ. See Subsection 6.2 for the detail of the derivation of the function G L (·; φ). We note that before we obtain the variation of constants formula (1.9), we show that holds for all t ≥ 0. Then the derivation of (1.9) is performed by defining a function for t ≥ 0 and showing that z := z L (·; φ) satisfies an integral equation because (1.12) shows that z L (·; φ) = x L (·; 0, G L (·; φ)) holds. Here we need to know the regularity of the function G L (·; φ), which is discussed in Subsection 6.3. In Section 7, we discuss the exponential stability of the principal fundamental matrix solution X L of the linear RFDE (1.1) and the uniform exponential stability of the C 0semigroup T L (t) t≥0 on the Banach space C([−r, 0], K n ) defined by We show that X L is α-exponentially stable if and only if T L (t) t≥0 is uniformly α-exponentially stable. See Theorems 7.3 and 7.4 for the details.
In Section 8, we apply the obtained variation of constants formulas to a proof of the stability part of the principle of linearized stability and Poincaré-Lyapunov theorem for RFDEs. This is indeed an appropriate modification of the proof for ODEs. However, the given proof makes clear the importance of the principal fundamental matrix solution. In the statement, we do not need to assume the uniqueness of a solution. Therefore, this should be compared with the proof relying on the nonlinear semigroup theory.
We have five appendices. In Appendix A, we collect results on Riemann-Stieltjes integrals for matrix-valued functions that are needed for this paper. In Appendix B, we give a proof of the representability of L by a Riemann-Stieltjes integral (1.3) because there does not seem to be any proof of the representability in the literature. In Appendix C, we discuss Gronwall's inequality and its variants used in the context of RFDEs. In Appendix D, we give lemmas that are used in the fixed point argument in this paper. In Appendix E, we continue to discuss the convolution. The contents of this appendix will not be used in this paper, but it will be useful to share the proofs of results on the convolution for matrix-valued locally Lebesgue integrable functions in the literature of RFDEs.

Notations
Throughout this paper, the following notations will be used.
• Let E = (E, · ) be a Banach space. For each subset I ⊂ R, let C(I, E) denote the linear space of all continuous functions from I to E. When the subset I is a closed and bounded interval, the linear space C(I, E) is considered as the Banach space of continuous functions endowed with the supremum norm · given by for f ∈ C(I, E).
• For each pair of Banach spaces E = (E, · ) and F = (F, · ), let B(E, F ) denote the linear space of all continuous linear maps (i.e., all bounded linear operators) from E to F . For each T ∈ B(E, F ), its operator norm is denoted by T . Then B(E, F ) is considered as the Banach space of continuous linear maps endowed with the operator norm. When F = E, B(E, F ) is also denoted by B(E).
• An n × n matrix A ∈ M n (K) is considered as a continuous linear map on the Banach space K n endowed with the given norm |·|. The operator norm of A is denoted by |A|. The linear space M n (K) of all n × n matrices is considered as the Banach space of matrices endowed with the operator norm.
• Let d ≥ 1 be an integer, X be a measurable set of R d , and Y = K n or M n (K).
-We say that a function f : it is measurable and is also measurable by the continuity of the norm |·|, and the above integral is the unsigned Lebesgue integral.
-Let L 1 (X, Y ) be the set of all Lebesgue integrable functions from X to Y defined almost everywhere. For f ∈ L 1 (X, Y ), let Then one can prove that holds.
• Let X be an interval of R and Y = K n or M n (K). Let L 1 loc (X, Y ) be the set of all has measure 0, and (iii) for each closed and bounded interval I contained in X, the restriction f | I : 2 Mild solutions and fundamental matrix solutions 2.1 Definitions

History segments and memory space
We first make clear the notion of history segments in our setting.
We call x t the history segment of x at t.
We note that dom(x t ) is expressed by In this paper, we need discontinuous initial history functions. For this purpose, we adopt the following space of history functions. Definition 2.2 (cf. [10]). We define a linear subspace and call it the memory space of L 1 -type. We consider M 1 ([−r, 0], K n ) as a seminormed space endowed with the seminorm · M 1 :  [10]. It is isomorphic to the product Banach space See also [3], [8], and references therein for the use of the product space.
For a continuous prolongation x of φ, holds.

Mild solutions
The following is the notion of a mild solution, whose introduction is a one of the contribution of this paper. We use the expression of L by the Riemann-Stieltjes integral (1.3) for ψ ∈ C([−r, 0], K n ).
Definition 2.5 (cf. [38]). Let φ ∈ M 1 ([−r, 0], K n ) be given. We say that a function x : [−r, ∞) ⊃ dom(x) → K n is a mild solution of the linear RFDE (1.1) under the initial condition x 0 = φ if the following conditions are satisfied: (i) x is a continuous prolongation of φ and (ii) for all t ≥ 0, the integrand in Eq. (2.1) is continuous with respect to θ ∈ [−r, 0]. Therefore, the integral in (2.1) is meaningful as a Riemann-Stieltjes integral. Eq. (2.1) is also expressed by where the third term of the right-hand side may depend on φ.
Remark 2.6. Eq. (2.1) appeared at [38, (5.19) in Corollary 5.13] after developing a nonlinear semigroup theory for some class of RFDEs. Compared with this approach, the method of this paper is considered to be taking the notion of mild solutions as a starting point.

Notation
t 0 x s ds For ease of notation, we introduce the following.
for some function space X .

2.2
t 0 x s ds and its properties We have the following lemma.
is continuous.
Proof. We define a function y : [−r, ∞) → K n by for t ≥ −r. Then y is continuous, and is continuous. Since the continuity of this function is ensured by the uniform continuity of y on any closed and bounded interval, the conclusion is obtained.
exists. See Graves [16,Section 2] for the definition of the Riemann integrability of functions on closed and bounded intervals taking values in normed spaces. We now show that when . Then x s ds holds for all t ≥ 0.
Proof. Let t > 0 be given. We consider a function f : Then the function f (·, s) is equal to x s . By applying Lemma 2.9 with this f , holds for all θ ∈ [−r, 0]. Since the right-hand side is equal to t 0 x s ds (θ), this shows the conclusion. We have the following corollary.
Lx s ds for all t ≥ 0. This shows that a mild solution coincides with a solution in the usual sense when the initial history function φ is continuous.

Existence and uniqueness of a mild solution
By using the contraction mapping principle with an a priori estimate, we will prove the unique existence of a mild solution of the linear RFDE (1.1) under an initial condition (1.4) We will use the following notation.
φ is a constant prolongation of φ.
In the following, we give a proof based on an a priori estimate. See Chicone [5, Subsection 2.1] for a similar argument.
Proof of Theorem 2.14. We divide the proof into the following steps.
Step 1: Reduction to a continuous unknown function and derivation of an a priori estimate. For a continuous prolongation x : [−r, ∞) ⊃ dom(x) → K n of φ, we consider the function y : [−r, ∞) → K n defined by Then y is a continuous function satisfying y 0 = 0. The problem of finding a mild solution holds for all t ≥ 0.
Step 2: Setting of function space. For each γ > L , Step 1 indicates that for a continuous function y : [−r, ∞) → K n satisfying y 0 = 0 and Eq. (2.6), we have Here the right-hand side converges to 0 as t → ∞. Therefore, holds (see Lemma D.1 for the detail). For each γ > L , let Y γ be the linear subspace of C([−r, ∞), K n ) given by which is considered as a normed space endowed with the norm · γ . Then Y γ is a Banach space (see Lemma D.2). We fix γ > L arbitrarily, and let Y := Y γ and · Y := · γ .
Step 3: Reduction to fixed point problem. We define a transformation T :  We now claim that T (Y ) ⊂ Y holds. Let y ∈ Y be given. In the same way as in Step 1, in order to obtain T y ∈ Y . By the assumption of y ∈ Y , y t ≤ y Y e γt holds for all t ≥ 0. Therefore, we have which implies sup t≥0 e −γt t 0 y s ds < ∞. Thus, T y ∈ Y is concluded.
Step 4: Application of contraction mapping principle. We now claim that the mapping T : Y → Y is a contraction. For any y 1 , y 2 ∈ Y , holds. Since we have y 1 s − y 2 s = e γs · e −γs (y 1 − y 2 ) s ≤ e γs y 1 − y 2 Y for the integrand in the right-hand side, is concluded. Therefore, T : Y → Y is a contraction. By applying the contraction mapping principle, there exists a unique y * ∈ Y such that T y * = y * .
The function is a mild solution of the linear RFDE (1.1) under the initial condition x 0 = φ. The uniqueness follows by the above discussion.
We hereafter use the following notation. We have the following corollary.
Proof. Let χ := αφ + βψ ∈ M 1 ([−r, 0], K n ) and x : [−r, ∞) ⊃ dom(x) → K n be the function defined by Since the map L and the Lebesgue integration are linear, x is a mild solution of the linear RFDE (1.1) under the initial condition x 0 = χ by the definition of mild solutions (see Definition 2.5). Therefore, (2.7) is a consequence of Theorem 2.14.

Fundamental matrix solutions
Since ODEs are special DDEs, it is natural to expect that the notions of fundamental systems of solutions and fundamental matrix solutions for linear ODEs are meaningful for DDEs in some way. However, the solution space of the linear RFDE (1.1) is infinitedimensional. Therefore, it is impossible to define these notions to (1.1) as a simple generalization.
A key to this consideration is to focus on a "finite-dimensionality". For this purpose, we consider an "instantaneous input" as an initial history function. We will use the following notation.
Sinceξ ∈ M 1 ([−r, 0], K n ) for each ξ ∈ K n , one can consider the mild solution x L ·;ξ : [−r, ∞) → K n of the linear RFDE (1.1) under the initial condition x 0 =ξ from Theorem 2.14. Then Corollary 2.15 yields that the subset S given by S := x L ·;ξ : [−r, ∞) → K n : ξ ∈ K n forms a linear space. We have the following lemma.
(b) The system of functions x 1 , . . . , x m is linearly independent.
Theorem 2.18. The linear space S is n-dimensional.
Proof. Let b 1 , . . . , b n be a basis of K n . From Lemma 2.17, the system of functions is linearly independent. Furthermore, for any x 1 , . . . , x n+1 ∈ S, the system of functions is linearly dependent from Lemma 2.17 because the system x 1 (0), . . . , x n+1 (0) ∈ K n of vectors is linearly dependent. Therefore, the statement holds.
Theorem 2.18 naturally leads us to the following definition.
Definition 2.19 (cf. [18], [19]). We call a basis of the n-dimensional linear space S a fundamental system of solutions to the linear RFDE (1.1). Equivalently, a fundamental system of solutions is the linear independent system x L ·;b 1 , . . . , x L ·;b n : [−r, ∞) → K n for some basis b 1 , . . . , b n of K n . We call a matrix-valued function having a fundamental system of solutions as its column vectors a fundamental matrix solution. In particular, we call the fundamental matrix solution satisfying X(0) = I the principal fundamental matrix solution. Here I denotes the identity matrix.
The above definition is considered as a natural generalization of the corresponding definition for linear ODEs (see [6,Definition 2.12 in Section 2.1 of Chapter 2]). See also [37,Definition 5.10] for a related definition.
Here the right-hand side is equal to x L (·; ξ 1ê1 + · · · + ξ nên ) from (2.7). Therefore, holds. Remark 2.21. We consider an autonomous linear ODĖ for some A ∈ M n (K). For a system of global solutions y 1 , . . . , y m : R → K n to the linear ODE (2.9), the following statements are equivalent: (a) For any t ∈ R, y 1 (t), . . . , y m (t) ∈ K n is linearly independent.
(c) The system of functions y 1 , . . . , y m is linearly independent.
The nontrivial part is (c) ⇒ (a), which is proved by the principle of superposition and by the unique existence of a solution of (2.9) under an initial condition Compared with this situation, the linear independence of vectors x 1 (t 0 ), . . . , x m (t 0 ) ∈ K n for each t 0 > 0 is not necessarily guaranteed for the functions x 1 , . . . , x m in Lemma 2.17 under the assumption that (a) or (b) in Lemma 2.17 holds. This should be compared with an example given by Popov [29], which is a three dimensional system of linear DDEs whose solution values are contained in a hyperplane of R 3 after a certain amount of time has elapsed. See also [19, Section 3.5] and [22, Section 3.5].

Consideration by Delfour
The definition of a mild solution in Definition 2.5 is also related to the consideration by Delfour [8]. In that paper, the author considered a continuous linear map . The author used the integral representation of L given by For the first term of the right-hand side of (2.10), we have under the exchange of order of integration. Here we have replaced φ with x s and have integrated from 0 to t with respect to s. In view of the above equality, it can be said that the concept of mild solutions in Definition 2.5 is also hidden in [8]. Theorem 2.14 and its proof should be compared with the existence and uniqueness result in [8].

Mild solutions for linear differential difference equations
We consider an autonomous linear differential difference equatioṅ for n × n matrices A, B 1 , . . . , B m ∈ M n (K) and τ 1 , . . . , τ m ∈ (0, r]. We refer the reader to [2] as a general reference of the theory of differential difference equations. The linear DDE (2.11) can be expressed in the form of the linear RFDE (1.1) by defining a continuous linear map L : , K n ) be given and x := x L (·; φ) for the above continuous linear map L. By the definition of mild solutions (see Definitions 2.5 and 2.7), x satisfies for all t ≥ 0. Since the last term is equal to x also satisfiesẋ by the Lebesgue differentiation theorem (see Subsection 3.2).
In this paper, a function defined on [0, ∞) is said to be of locally bounded variation if it is of bounded variation on any closed and bounded interval of [0, ∞). A function of locally bounded variation is also called a locally BV function. Then the above functionη is a function of locally bounded variation whose value is constant on [r, ∞). It is related to the reversal formula for Riemann-Stieltjes integrals (see Theorem A.9).
It will be turned out that the notions of Volterra operator and Riemann-Stieltjes convolution are useful to deduce a differential equation that is satisfied by the principal fundamental matrix solution X L : [−r, ∞) → M n (K) of the linear RFDE (1.1).

Definitions
Here the right-hand side is a Lebesgue integral. We call V the Volterra operator.
For details related to the Volterra operator as a linear operator on C([0, T ], K) for each T > 0, see [31]. By using the Lebesgue differentiation theorem (e.g., see [32, Theorem 1.3 in Section 1 of Chapter 3]) component-wise, it holds that V f is locally absolutely continuous (i.e., locally absolutely continuous on any closed and bounded interval of [0, ∞)), differentiable almost everywhere on [0, ∞), and holds for almost all t ∈ [0, ∞).
Here the right-hand side is a Riemann-Stieltjes integral. This function is called a Riemann-Stieltjes convolution.

Motivation
The following lemma motivates the use of Volterra operator and Riemann-Stieltjes convolution.
Proof. Let t > 0 be fixed. Since 0 θ x(s) ds = 0 by the assumption, we have We examine the right-hand side by dividing the consideration into the following cases: holds by the additivity of Riemann-Stieltjes integrals on sub-intervals.
Therefore, the expressions of L t 0 x s ds are obtained in combination with the reversal formula for Riemann-Stieltjes integrals (see Theorem A.9).

Properties of Volterra operator and Riemann-Stieltjes convolution
Throughout this subsection, let α : [0, ∞) → M n (K) be a function of locally bounded variation and f : [0, ∞) → M n (K) be a continuous function.

Continuity and local integrability
The following is a simple result about the continuity of Riemann-Stieltjes convolution.
given so that s < t. By the additivity of Riemann-Stieltjes integrals on sub-intervals, By combining this and the uniform continuity of f on closed and bounded intervals, the continuity of dα * f is obtained.
See [31,Lemma 10.4 in Section 10.3] for the corresponding result for scalar-valued functions. In this paper, we say that a function is locally Riemann integrable if it is Riemann integrable on any closed and bounded interval.
Theorem 3.5. dα * f is a sum of a continuous function and a function of locally bounded variation. Consequently, dα * f is locally Riemann integrable. Proof.
The first term in the right-hand side is continuous from Lemma 3.4. The second term is of locally bounded variation since holds for all t ≥ 0. Therefore, the conclusion holds.

Riemann-Stieltjes convolution under Volterra operator
The Riemann-Stieltjes convolution and Volterra operator are related in the following way.
holds. Consequently, dα * V f is locally absolutely continuous, differentiable almost everywhere, and satisfies holds for almost all t ≥ 0.
For the proof, we need the following theorem. It contains the result on iterated Riemann integrals for continuous functions on rectangles as a special case. holds.
See also [39,Theorem 15a in Section 15 of Chapter I]. We will give the proof in Appendix A.7.
Proof of Theorem 3.7. We extend the domain of definition of f to R by defining f (t) := f (0) for t ≤ 0. By the proof of Lemma 3.4, we have by using the Volterra operator and the Riemann-Stieltjes convolution. Since holds by the integration by parts formula for Riemann-Stieltjes integrals.
The following is a corollary of Theorem 3.7. It will not be used in the sequel.
Consequently, dα * f is of locally bounded variation, differentiable almost everywhere, and satisfies for almost all t ≥ 0.
Proof. By the fundamental theorem of calculus, f = f (0) + V f ′ holds. By combining this and (3.3), the expression of dα * f is obtained. Since V (dα * f ′ ) is locally absolutely continuous, it is also of locally bounded variation. Therefore, the expression of dα * f yields that dα * f is of locally bounded variation. The remaining properties are consequences of the fact that matrix-valued functions of bounded variation are differentiable almost everywhere. This is obtained by applying the corresponding result 1 for real-valued functions component-wise.

Differential equation and principal fundamental matrix solution
As an application of Theorem 3.7, one can derive a differential equation that is satisfied by x ·;ξ for each ξ ∈ K n .
Proof. Eq. (3.5) is a consequence of the definition of mild solutions and Lemma 3.3. Since x| [0,∞) is continuous, Theorem 3.7 and Eq. (3.5) yield that holds for all t ≥ 0. In combination with Theorem 3.5, it holds that x| [0,∞) is locally absolutely continuous, differentiable almost everywhere, and satisfieṡ for almost all t ≥ 0. The remaining expression in Eq. (3.6) is a consequence of the reversal formula for Riemann-Stieltjes integrals.
We obtain the following result as a direct consequence of Theorem 3.10 and (2.8). We omit the proof.

Non-homogeneous linear RFDEs
In this section, we study a non-homogeneous linear RFDE (1.2)

Non-homogeneous linear RFDE and mild solutions
It is natural to define the notion of mild solutions to Eq. (1.2) in the following way.
We note that holds for a mild solution of Eq. (1.2) under the initial condition x t 0 = φ.
for almost all t ≥ t 0 .
Proof. By the translation, we may assume t 0 = 0. Since L : C([−r, 0], K n ) → K n is a bounded linear operator, holds for all t ≥ 0 from Corollary 2.12. Then the fundamental theorem of calculus and the Lebesgue differentiation theorem yield that holds for almost all t ≥ 0.
We assume that dom(g) = [0, ∞) and consider the function is a solution (in the Carathéodory sense).

Integral equation with a general forcing term
More generally, for a given t 0 ≥ 0 and a given continuous function G : [t 0 , ∞) → K n with G(t 0 ) = 0, we can discuss a solution of the following integral equation Here the assumption G(t 0 ) = 0 is natural because the right-hand side of (4.1) is equal to The notion of a solution of (4.1) can be defined in the similar way as in Definition 4.1. The following theorem holds. The following proof should be compared with the proof of Theorem 2.14.
Proof of Theorem 4.4. By the translation, it is sufficient to consider the case t 0 = 0. We will solve the integral equation locally and will connect the obtained local solutions. For this purpose, we need to consider an integral equation under the initial condition x σ = ψ for each σ ≥ 0 and each ψ ∈ M 1 ([−r, 0], K n ). Here an appropriate forcing term is given by for t ≥ σ. Then we are going to consider an integral equation under the initial condition x σ = ψ. The remainder of the proof is divided into the following steps.
Step 1: Existence and uniqueness of a local solution. We fix the above σ and ψ.
By defining a continuous function y : [−r, ∞) → K n by which is an integral equation under the initial condition y 0 = 0. We choose a constant a > 0 so that L a < 1 and consider a closed subset Y of the Banach space C([−r, a], K n ) given by Then it holds that T is contractive, and the application of the contraction mapping principle yields the unique existence of a fixed point y * of T . By defining a function it is concluded that x * is a solution of Eq. (4.2). We note that such a local solution is unique by the choice of the above a.
Step 2: Existence and uniqueness of a (global) solution. We note that the time a > 0 of existence of a local solution to Eq. (4.2) in Step 1 does not depend on the considered integral equation (4.2) and the specified initial condition x σ = ψ. In this step, we will show that by connecting these local solutions, we obtain a global solution. For this purpose, for each σ ≥ 0 and each ψ ∈ M 1 ([−r, 0], K n ), let x(·; σ, ψ) : [σ − r, σ + a] → K n be the obtained unique solution of Eq. (4.2) under an initial condition x σ = ψ. We fix σ and ψ. Let x := x(·; σ, ψ) and y := x(·; σ + a, x σ+a ).
We now claim that the function z : is a solution to Eq. (4.2). We note that this definition makes sense because y σ+a = x σ+a . To show the claim, it is sufficient to consider the case t ∈ [σ + a, σ + 2a]. In this case, we have where In the above equations, one can replace y s and x s with z s . Therefore, in view of it holds that z is a solution of Eq. (4.2) under the initial condition x t 0 = φ. By repeating the above procedure, a global solution of the original integral equation (4.1) is obtained. By the uniqueness of each local solution, such a global solution is unique.
Remark 4.5. Let K = R. In [21], Hale and Meyer studied the following equation are linear for each t ∈ R, and h : R → R n is a locally Lebesgue integrable function. In We hereafter use the following notation.
We obtain the following corollary. It will be a basics to consider a variation of constants formula for Eq. (1.6).
Corollary 4.6. For any φ ∈ M 1 ([−r, 0], K n ) and any continuous function G : for all t ≥ 0. Since the last term is equal to by the linearity of L, Theorem 4.4 yields x = x L (·; φ, G).
In the same way as in Theorem 3.10, we obtain the following theorem. The proof can be omitted.

Variation of constants formula for non-homogeneous linear ODEs
As a motivation to introduce convolution for locally Riemann integrable functions on [0, ∞), we first recall the variation of constants formula for a non-homogeneous linear for an n × n matrix A ∈ M n (K) and a continuous function f : with the matrix exponential. This is the variation of constants formula for (5.1), which is obtained by finding an equation of y = y(t) under the change of variable x(t) = e tA y(t). Indeed, the function y must satisfy an initial condition y(t 0 ) = e −t 0 A ξ anḋ This procedure to derive the formula (5.2) corresponds to replacing a constant vector v ∈ K n in the general solution for the linear ODE (2.9) with a vector-valued function y = y(t). This is the reason for the terminology of the variation of constants formula. The above method to derive (5.2) should be called the method of variation of constants. which is equivalent to (5.2) is helpful. Here the initial time t 0 is set to 0, and it has been omitted in x A (t; ξ, f ). The first term of the right-hand side of (5.3) is the solution of the linear ODE (2.9) under the initial condition x(0) = ξ. Therefore, the second term of the right-hand side of (5.3) is the solution of (5.1) under the initial condition x(0) = 0. This can be checked directly by differentiating the second term as We note that this gives another proof of (5.3).

Convolution and non-homogeneous linear RFDEs
For a continuous linear map L : C([−r, 0], K n ) → K n and a continuous function f : [0, ∞) → K n , we consider the non-homogeneous linear RFDE (1.7) Since R ∋ t → e tA ∈ M n (K) is the principal fundamental matrix solution of the linear ODE (2.9) in the sense that it is a matrix solution to (2.9) and e tA | t=0 is the identity matrix, it is natural to ask whether the function x(·; f ) : [−r, ∞) → K n defined by x(·; f ) 0 = 0 and (1.8) x(t; f ) := x(t) = Lx t (t ≥ 0).
In Theorem 3.11, we obtained the differential equation that is satisfied by X L . However, it is not direct to prove that the function x(·; f ) is a solution to (1.7) by differentiating the right-hand side of (1.8) as in the case of the non-homogeneous linear ODE (5.1) because one cannot take the term X L (t) out of the integral. This comes from the property that initial value problems of RFDEs cannot be solved backward in general. Therefore, one needs to treat the integral of the right-hand side of (1.8) as it is.
Such an integral is a convolution for locally (Riemann) integrable functions, which should be distinguished from the convolution for integrable functions. The convolution for locally integrable functions has been used in the literature of DDEs. For example, see [2,Chapter 1] with the context of the Laplace transform. The convolution is also used in [34] and [14], however, the detail has been omitted there.

Convolution and Riemann-Stieltjes convolution
In this subsection, we study a convolution of the following type. See [31,Section 5.3] for the convolution of continuous functions. We note that when f is a constant function, then holds for all t ≥ 0. In the same way, g * f = g(0)V f holds when g is constant.
Lemma 5.2 (cf. [31]). Let f, g : [0, ∞) → M n (K) be locally Riemann integrable functions. If f is continuous, then g * f is a sum of a continuous function and a locally absolutely continuous function.
Proof. By using (5.4), holds. Therefore, the conclusion is obtained by showing that g * f is continuous when be given so that s < t. By the same reasoning as in the proof of Lemma 3.4, we have By combining this and the uniform continuity of f on closed and bounded intervals, the continuity of g * f is obtained.

Convolution of locally BV functions and continuous functions
By using Theorem 3.7, one can obtain the following result on the regularity of convolution.
The above result is considered as the finite-dimensional version of [33, Theorem 3.2] (i.e., the case that the Banach space X in [33, Theorem 3.2] is finite-dimensional) except the equality In the following, we give a simpler proof of Theorem 5.3 based on Theorem 3.7.
Proof of Theorem 5.3. Since V f is continuously differentiable and (V f )(0) = O, holds for all t ≥ 0 by the integration by parts formula for Riemann-Stieltjes integrals and from Theorem A. 19. By combining the obtained equality and Theorem 3.7, the equality (5.5) is obtained. for from Theorems 3.7 and 5.3. This formula is easy to remember. We note that the above holds for all t ≥ 0.
We have the following corollaries. Proof. From Theorems 5.3 and 3.7, we have where V 2 f := V (V f ). Since the right-hand side is equal to g * (V f ) from Theorem 5.3, the equality is obtained. (g * f ) ′ = g(0)f + dg * f holds.
2. If g is locally absolutely continuous, then g * f is continuously differentiable and holds.
Here g ′ * f : [0, ∞) → M n (K) be the function defined by for t ≥ 0, where the integrals are Lebesgue integrals.
Proof. 1. Under the assumption, dg * f is continuous from Lemma 3.4 and Remark 3.6. Therefore, the conclusion follows by the formula (5.5).
2. The continuous differentiability of g * f follows by the statement 1. When g is locally absolutely continuous, holds from Theorem A.20.

Associativity of Riemann-Stieltjes convolution
For the proof of Theorem 5.9 below, we need the following result.

A formula for non-homogeneous equation with trivial initial history
Let L : C([−r, 0], K n ) → K n be a continuous linear map. We recall that for a continuous map G : [0, ∞) → K n with G(0) = 0, the function x L (·; 0, G) : [−r, ∞) → K n denotes the unique solution of an integral equation under the initial condition x 0 = 0. In this subsection, as an application of the results in Subsection 5.2, we show that the function x(·; f ) : [−r, ∞) → K n defined by x(·; f ) 0 = 0 and (1.8) is a solution to the non-homogeneous linear RFDE (1.7). Theorem 5.9 (cf. [35]). Let f : [0, ∞) → K n be a continuous function. Then holds for all t ≥ 0.
The above proof of Theorem 5.9 is different from the proofs in the literature (e.g., see [35,Section 4]).

Variation of constants formula
In this section, we obtain a "variation of constants formula" for the non-homogeneous linear RFDE (1.2)ẋ (t) = Lx t + g(t) (a.e. t ≥ 0) for some g ∈ L 1 loc ([0, ∞), K n ) expressed by X L . In view of Corollary 4.6, we will divide our consideration into the following steps: Then the full formula for the mild solution of (1.2) under the initial condition x 0 = φ ∈ M 1 ([−r, 0], K n ) is obtained by combining the above formulas. In Step 1, for a given continuous function G : [0, ∞) → K n with G(0) = 0, we indeed consider the integral equation (5.7) under the initial condition x 0 = 0 and try to find a formula for the solution x L (·; 0, G) expressed by X L .
Remark 6.1. Since x 0 = 0 =0, Eq. (5.7) is equivalent to The following is the main result of this section.
We will call the formula (1.9) the variation of constants formula for Eq. (1.6). The definition of the function G L (·; φ) : [0, ∞) → K n for φ ∈ M 1 ([−r, 0], K n ) will be given later. For this definition, the expression of L by the Riemann-Stieltjes integral (1.3) for ψ ∈ C([−r, 0], K n ) is a key tool.

Motivation: Naito's consideration
We first concentrate our consideration to the case that g ∈ C([0, ∞), K n ) and φ ∈ C([−r, 0], K n ). From Theorem 5.9, we only need to find a formula for x L (·; φ, 0) in this case.
Naito [26,Theorem 6.5] has discussed an expression of the form In the above formula, x : [−r, ∞) → K n is the solution of the linear RFDE (1.1) under the initial condition x 0 = φ ∈ C([−r, 0], K n ), andφ : [−r, ∞) → K n is the function defined bȳ See also Notation 1. Although the study of [26] is in the setting of infinite retardation, we are now interpreting this in the setting of finite retardation (i.e., the history function space is C([−r, 0], K n )). We note that the matrix-valued function X : [0, ∞) → M n (K) is defined by using the inverse Laplace transform. See [26] for the detail. See also [27], where an interpretation of the matrix-valued function X is given. In our setting, a formula expressed by the principal fundamental matrix solution X L is true. To see this, let y(t) := x L (t; φ, 0) −φ(t) for t ∈ [−r, ∞). Then the function y : [−r, ∞) → K n satisfies y 0 = 0 anḋ See also the proof of Theorem 2.14. Since the function [0, ∞) ∋ t → Lφ t ∈ K n is continuous, we obtain by applying Theorem 5.9.

Derivation of a general forcing term
The formula (6.1) is not sufficient for the application to the linearized stability. See Section 8 for the detail of the application of the variation of constants formula to the linearized stability. We now introduce the following function. for t ≥ 0.
From the expression (2.2) for a mild solution, the function z L (·; φ) satisfies for all t ≥ 0. The second term of the right-hand side is further calculated as follows: • When t ∈ [0, r), θ ∈ [−r, 0] satisfies t + θ ≥ 0 if and only if θ ∈ [−t, 0]. Since by the additivity of Riemann-Stieltjes integrals on sub-intervals.
• When t ≥ r, the second term is equal to This leads to the following definition.
By definition, G L (0; φ) = 0 holds. Summarizing the above discussion, we obtain the following lemma.
under the initial condition z 0 = 0.

Regularity of the general forcing term
To study Eq. (1.12), it is important to reveal the regularity of the function G L (·; φ) for each φ ∈ M 1 ([−r, 0], K n ).

Forcing terms for continuous initial histories
Before we tackle this problem, we find a differential equation satisfied by z := z L (·; φ) for φ ∈ C([−r, 0], K n ). It should be noted that this is not straightforward because (1.11) is only valid for t ≥ 0. Let x := x L (·; φ, 0) andx := x L ·; φ(0), 0 . In view of for each t ≥ 0, we express the linear RFDE (1.1) aṡ by using the additivity of Riemann-Stieltjes integrals on sub-intervals. Here we are interpreting that the second term of the right-hand side is equal to 0 when t ≥ r. More precisely, we introduce the following.
We note that since g L (·; φ) is not necessarily continuous, Theorem 5.9 is not sufficient to obtain an expression of z = z L (·; φ) by X L .

Relationship with the forcing terms
Comparing (1.12) and (6.2), it is natural to expect that holds for all t ≥ 0 when φ ∈ C([−r, 0], K n ). We now justify this relationship.
Proof. When t ≥ r, holds. Therefore, the right-hand side of (6.4) is equal to 0 for all t ≥ r. We next consider the case t ∈ [0, r). In this case, we have by the additivity of Riemann-Stieltjes integrals on sub-intervals. Since the expression (6.4) is obtained. Since [0, ∞) ∋ t → Lφ t ∈ K n is continuous and [0, ∞) ∋ t → η(−t)φ(0) is of locally bounded variation, the local Riemann integrability of g L (·; φ) follows by the expression (6.4).
The following theorem reveals a connection between G L (·; φ) and g L (·; φ).
Proof. For the first term of the definition of G L (t; φ), we have  Here the second term of the right-hand side is equal to including the case t = r. The proof is complete in view of where the integration by parts formula for Riemann-Stieltjes integrals is used.
Case 2: t ∈ (r, ∞). Since we have shown that (6.5) holds for t = r, which also implies that the right-hand side of (6.5) is equal to for all t ≥ r.
Remark 6.11. G L (t; φ) is also expressed as In this paper, we do not need the above expression. By combining the obtained results, we obtain the following result on the regularity of G L (·; φ). See also [34, Remark 2.10(ii) in Chapter 2].

Formulas for trivial initial histories
Since X L | [0,∞) is locally absolutely continuous (see Theorem 3.11), by the integration by parts formula for matrix-valued absolutely continuous functions 2 , holds for any g ∈ L 1 loc ([0, ∞), K n ). Here X L (0) = I and (V g)(0) = 0 are also used. The following theorem is motivated by this. Theorem 6.13. Let G : [0, ∞) → K n be a continuous function with G(0) = 0. Then (1.10) holds for all t ≥ 0.
Proof. Let X := X L | [0,∞) . We define a function x : [−r, ∞) → K n by x 0 = 0 and for t ≥ 0. By applying Corollary 5.6, we have V x| [0,∞) = X * G in combination with the fundamental theorem of calculus. Furthermore, we have from Theorems 3.11 and 5.7. Therefore, x satisfies for all t ≥ 0. This implies that (1.10) holds by applying Theorems 4.7 and 4.4.

Formulas for homogeneous equations
We next find an expression of x L (·; φ, 0) by X L as an application of Theorem 6.13.
holds for all t ≥ 0.
Proof. From Lemma 6.5 and Theorem 6.13 together with Theorem 6.12, holds for all t ≥ 0. Then the formula (6.7) is obtained in view of for t ≥ 0. We have the following corollary.
Proof. From Theorem 6.12, holds. Therefore, the formula (6.8) is obtained from (6.7) by using the integration by parts formula for matrix-valued absolutely continuous functions.
Corollary 6.17 should be compared with [25,Theorem 1.11], where the inverse Laplace transform is used to obtain a formula.

Derivation of the main result of this section
Theorem 6.2 is a combination of Theorems 6.13 and 6.15 in view of Corollary 4.6. Therefore, the proof can be omitted.
The following is a corollary of Theorem 6.2, which is a combination of Corollaries 6.14 and 6.17 in view of Corollary 4.6. The proof can be omitted. Corollary 6.18. If φ ∈ C([−r, 0], K n ) and G = V g for some g ∈ L 1 loc ([0, ∞), K n ), then holds for all t ≥ 0.

Variation of constants formula for linear differential difference equations
We apply Theorem 6.15 to an autonomous linear differential difference equation (2.11) for n × n matrices A, B 1 , . . . , B m ∈ M n (K) and τ 1 , . . . , τ m ∈ (0, r]. We recall that the linear DDE (2.11) can be expressed in the form of the linear RFDE (1.1) by defining a continuous linear map L : C([−r, 0], K n ) → K n by (2.12) for ψ ∈ C([−r, 0], K n ). For the above mentioned application, we need to calculate the function G L (·; φ) for each φ ∈ M 1 ([−r, 0], K n ) based on Definition 6.4. By the linearity of L → G L (·; φ), this can be reduced to the calculation of G L k (·; φ) for each k ∈ {0, . . . , m}, where L k : C([−r, 0], K n ) → K n is the continuous linear map given by for k ∈ {1, . . . , m}. We have the following lemma. Lemma 6.19. Let φ ∈ M 1 ([−r, 0], K n ) be given. Then the following statements hold: holds.
By combining the above expressions, the conclusion is obtained.
Proof. Let φ ∈ M 1 ([−r, 0], K n ) be given. From Lemma 6.19, is locally absolutely continuous. Therefore, Theorem 6.15 and the integration by parts formula for absolutely continuous functions yield that holds for all t ≥ 0. We now fix k ∈ {1, . . . , m} and find an expression of the integral Lemma 6.19 shows thatĠ L k (t; φ) = B k φ(t − τ k ) holds for almost all t ∈ [0, τ k ], anḋ G L k (t; φ) = 0 holds for all t ∈ (τ k , ∞). Then the integral is expressed as follows: • When t ∈ [0, τ k ], the integral becomes • When t ∈ (τ k , ∞), the integral becomes This completes the proof.
Furthermore, by the formal exchange of order of integration, the function X is interpreted as a "matrix-valued solution" to the linear RFDE (1.1). Indeed, Hale argued that X satisfies (i) X 0 =Î, (ii) X| [0,∞) is locally absolutely continuous, and (iii) X satisfieṡ However, the above integral does not make sense in general because X is not continuous.

Volterra convolution integral equations and fundamental matrix solutions
Let x := x L ·;ξ | [0,∞) for some ξ ∈ K n and suppose η(0) = O. By using the integration by parts formula for Riemann-Stieltjes integrals and Theorem A.19 in (3.5) we have for all t ≥ 0. Here (V x)(0) = 0 is also used. The above calculation shows that the function

Exponential stability of principal fundamental matrix solution
For a continuous linear map L : C([−r, 0], K n ) → K n , we consider a linear RFDE (1.1) Let X L : [−r, ∞) → M n (K) be the principal fundamental matrix solution. We use the following terminology.
Definition 7.1. We say that the principal fundamental matrix solution X L is exponentially stable if there exist constants M ≥ 1 and α > 0 such that holds for all t ≥ 0. We also say that X L is α-exponentially stable.
In the following calculations, it is useful to extend the domain of definition of X L to R by letting X L (t) := O for t ∈ (−∞, −r).
holds for all t ∈ R.
Proof. By the assumption, one can choose a constant M 0 ≥ 1 so that holds for all t ≥ 0. Since the statement is trivial when t ≤ 0, we only have to consider the case t > 0. Let θ ∈ [−r, 0]. When t + θ ≥ 0, we have The above estimate also holds when t + θ < 0 because X L (t + θ) = O in this case. Therefore, the conclusion is obtained. [19], [22]). If X L is α-exponentially stable for some α > 0, then the holds for all t ∈ R. Since the statement is trivial when t = 0, we only have to consider the case t > 0. Let θ ∈ [−r, 0] and φ ∈ C([−r, 0], K n ) be given. Then holds from Corollary 6.17 (see Definition 6.6 for the definition of g L (t; φ)). We divide the consideration into the following cases.
Case 1: t + θ ≥ 0. For the first term of the right-hand side, holds. For the second term, holds from Lemma 7.2. Since holds for all t ∈ [0, r) (see Lemma A.4) and g L (t; φ) = 0 for all t ≥ r, we have We note that r 0 e αu du = 1 α (e αr − 1) holds.
Case 2: t + θ < 0. In this case, we have By combining the estimates obtained in Cases 1 and 2, holds for some M ≥ 1. Therefore, the conclusion is obtained.
The converse of Theorem 7.3 also holds.
Proof. By the assumption, we choose a constant M 0 ≥ 1 so that holds for all t ≥ 0. We fix ξ ∈ K n and let Then the map K n ∋ ξ → φ ξ ∈ C([−r, 0], K n ) is linear from Corollary 2.7. Since X L (·)ξ = x L ·;ξ , we have This yields that the linear operator We now show that X L is α-exponentially stable by dividing the following cases.
Case 2: t ∈ [0, r]. In this case, X L (t) is estimated by Here 1 = e −αt e αt is used.
By combining the above estimates, the conclusion is obtained.  In this section, we consider a non-autonomous RFDĖ For the continuous map h, we assume that h(t, φ) = o( φ ) holds as φ → 0 uniformly in t. This means that for every ε > 0, there exists a δ > 0 such that for all (t, φ) It follows that h(t, 0) = 0 for all t ∈ R, and the RFDE (8.1) has the zero solution. (8.1) is considered as a perturbation of the linear RFDE (1.1) Let X L : [−r, ∞) → M n (R) be the principal fundamental matrix solution and T L (t) t≥0 be the C 0 -semigroup on C([−r, 0], R n ) generated by (1.1). We also consider a non-autonomous RFDĖ for a map N : R × C([−r, 0], R n ) → R n with the following properties: • For each t ∈ R, the map N (t) : C([−r, 0], R n ) → R n defined by for φ ∈ C([−r, 0], R n ) is a bounded linear operator.
See also [22, Subsection 6.6.3] for a related discussion.
. This yields the conclusion.

Variation of constants formula and non-linear equations
In this subsection, we consider a non-autonomous RFDĖ for some continuous map For each φ ∈ C([−r, 0], R n ) and each T > 0, a continuous function x : [t 0 − r, t 0 + T ] → R n is called a solution of the RFDE (8.3) under an initial condition x t 0 = φ if the following conditions are satisfied: (i) Here the derivative of x at t 0 and t 0 + T are understood as the right-hand derivative at t 0 and the left-hand derivative at t 0 + T , respectively.
, t 0 ∈ R, and T > 0 be given. Then for a continuous function x : x is a solution of the RFDE (8.3) under the initial condition x t 0 = φ if and only if x satisfies We note that the above statement is not a simple application of Corollaries 4.6 and 6.14 because there is no method of variation of constants for RFDEs (see Subsection 5.1).
from Theorem 5.9 or Corollary 6.14. Therefore, the expression of x is obtained by coming back to the condition on x.

Stability part of principle of linearized stability
The statement in the following theorem is the stability part of the principle of linearized stability for RFDEs. Theorem 8.3 (cf. [14]). If X L is exponentially stable, then there exist M ≥ 1, β > 0, and a neighborhood U of 0 in C([−r, 0], R n ) such that for every t 0 ∈ R, every φ ∈ U , and every non-continuable solution x of the RFDE (8.1) under the initial condition x t 0 = φ, x is defined for all t ≥ t 0 and satisfies Remark 8.4. See [2,Chapter 11] for the corresponding result for differential difference equations. See [11] for the general result of the principle of linearized stability in the context of nonlinear semigroups. See also [14, Chapter VII] for a general treatment of the principle of linearized stability and its application to RFDEs under the local Lipschitz continuity of h.
In the proof of Theorem 8.3 given below, the Peano existence theorem and the continuation of solutions for RFDEs play key roles. See [19,Chapter 2] and [22,Chapter 2] for the fundamental theory of RFDEs.
Proof of Theorem 8.3. We divide the proof into the following steps.
Step 1: Choice of a neighborhood of 0 and a non-continuable solution. From Lemma 7.2 and Theorem 7.3, we choose constants M ≥ 1 and α > 0 so that and T L (t) ≤ M e −αt (t ≥ 0) hold. We also choose an ε > 0 so that Since h(t, φ) = o( φ ) as φ → 0 uniformly in t, there exists a δ > 0 for this ε > 0 with the following properties: Then holds. From now on, we fix t 0 ∈ R and φ ∈ U and proceed with the discussion. By applying the Peano existence theorem for RFDEs, the RFDĖ has a solution under the initial condition x t 0 = φ. Let x be a non-continuable solution of the RFDE (8.4) under this initial condition. Then its domain of definition is written as [t 0 − r, t 0 + T ) for some T ∈ (0, ∞]. Step 2: Estimate by Gronwall's inequality. Let t ∈ [t 0 , t 0 + T ) and θ ∈ [−r, 0]. By applying Theorem 8.2, When t + θ < t 0 , the estimate M e −αt e αu ε x u du also holds in view of These estimates yield by applying Gronwall's inequality (see Lemma C.1). This means that holds for all t ∈ [t 0 , t 0 + T ).
Step 3: Proof by contradiction. We next show that T is equal to ∞, i.e., the noncontinuable solution x is defined on [t 0 − r, ∞). We suppose T < ∞ and derive a contradiction. Since x t < δ holds for all t ∈ [t 0 − r, t 0 + T ), we have This shows that x| [t 0 ,t 0 +T ) is Lipschitz continuous. In particular, x| [t 0 ,t 0 +T ) is uniformly continuous, and therefore, the limit lim t↑t 0 +T x(t) exists. Since this yields the existence of the limit lim i.e., ψ ∈ U , by taking the limit as t ↑ t 0 + T in the inequality (8.5). Then the RFDE (8.4) has a solution under the initial condition x t 0 +T = ψ by the Peano existence theorem for RFDEs, and one can construct a continuation of x. It contradicts the property that x is non-continuable. Therefore, T should be infinity.
The above steps yield the conclusion.
The above proof of Theorem 8.3 is an appropriate modification of the stability part of the principle of linearized stability for ODEs (e.g., see [6,Section 2.3]). It also should be compared with [35, Theorem 2 and its proof]. We note that the continuity of the higher-order term h in the RFDE (8.1) is sufficient for the proof.

Poincaré-Lyapunov theorem for RFDEs
The Poincaré-Lyapunov theorem is also extended to RFDEs as follows. See [6, Exercise 2.79] for the theorem for ODEs.
Theorem 8.5. Let σ ∈ R be given. If X L is exponentially stable, then there exist M ≥ 1, β > 0, and a neighborhood U of 0 in C([−r, 0], R n ) such that for every t 0 ≥ σ, every φ ∈ U , and every non-continuable solution x of the RFDE (8.2) under the initial condition x t 0 = φ, x is defined for all t ≥ t 0 and satisfies Proof. From Lemma 7.2 and Theorem 7.3, we choose constants M 0 ≥ 1 and α > 0 so that and hold. We also choose an ε > 0 so that Since N (t) → 0 as t → ∞, there is an a ∈ R for this ε > 0 such that N (t) < ε holds for all t ≥ a. There also exists a δ > 0 with the following properties: (i) For all φ ∈ C([−r, 0], R n ), φ < δ implies φ ∈ U 0 .
We first consider the case (I) σ ≥ a. Since holds for all t ≥ σ and all φ < δ, the conclusion is obtained in the same way as in the proof of Theorem 8.3. We next consider the case (II) σ < a. We divide the proof into the following steps.
Step 1: Choice of a neighborhood of 0 and a non-continuable solution. We choose an R > ε so that R > sup t∈ [σ,a] N (t) . We define open sets U and U by holds. We now fix t 0 ≥ σ and φ ∈ U , and let x : [t 0 − r, t 0 + T ) → R n be a non-continuable solution of an RFDEẋ under the initial condition x t 0 = φ.
Step 2: Estimate by Gronwall's inequality. Let t ∈ [t 0 , t 0 + T ) and θ ∈ [−r, 0]. By applying Theorem 8.2, holds for the non-continuable solution x : [t 0 − r, t 0 + T ) → R n . When t + θ ≥ t 0 , we have When t + θ ≤ 0, the estimate M 0 e −αt e αu ( N (u) + ε) x u du also holds in view of These estimates yield and we obtain by Gronwall's inequality (see Lemma C.1). This means that holds for all t ∈ [t 0 , t 0 + T ). The remaining consideration is further divided into the following cases: • Case: t 0 + T < a. Let t ∈ [t 0 , t 0 + T ). Since t < a, the exponential term in (8.6) is estimated from above by e [M 0 (R+ε)−α](t−t 0 ) , which is equal to by the choice of β. In view of σ ≤ t 0 ≤ t < a, the above is also estimated from above by e M 0 (R−ε)(a−σ) e −β(t−t 0 ) , and therefore, inequality (8.5) holds with M = M 0 e M 0 (R−ε)(a−σ) .
The remainder of the proof is same as in the proof of Theorem 8.3. This completes the proof.

Acknowledgment
This work was supported by JSPS Grant-in-Aid for Young Scientists Grant Number JP19K14565.
A Riemann-Stieltjes integrals with respect to matrix-valued functions Throughout this appendix, let K = R or C, n ≥ 1 be an integer, and [a, b] be a closed and bounded interval of R. In this appendix, we study Riemann-Stieltjes integrals with respect to matrix-valued functions. We refer the reader to [

A.1 Definitions
Definition A.1. Let P : a = x 0 < x 1 < · · · < x m = b be a partition of [a, b] for some integer m ≥ 1. For a finite sequence ξ := (ξ k ) m k=1 satisfying which is called the norm of (P, ξ).
The above terminology comes from [15]. We call S(f ; α, (P, ξ)) the Riemann-Stieltjes sum of f with respect to α under the tagged partition (P, ξ).
We note that such a J is unique if it exists. It is called the Riemann-Stieltjes integral of f with respect to α and is denoted by b a dα(x) f (x).

A.1.1 Remarks
Remark A.4. One can also consider a sum which is different from S(f ; α, (P, ξ)) in general. If a limit of the above sum as |(P, ξ)| → 0 exists in the sense of Definition A.3, we will write the limit as In this case, the sum S(f ; α, (P, ξ)) and the integral b a dα(x) f (x) belong to K n .

A.2 Reduction to scalar-valued case
Since the linear space M n (K) is finite-dimensional, the operator norm |·| on M n (K) is equivalent to the norm |·| 2 on M n (K) defined by where a i,j is the (i, j)-component of the matrix A ∈ M n (K). This means that the notion of convergence in M n (K) can be treated component-wise.
Lemma A.6. Let f, α : [a, b] → M n (K) be functions. Then the following properties are equivalent: (a) f is Riemann-Stieltjes integrable with respect to α. The proof is based on the definition of the matrix product and on the property that the operator norm |·| is equivalent to the norm |·| 2 given in (A.1). Therefore, we omit the proof.
Proof. By the definition of the product of a matrix and a vector, the i-th component of S(f ; α, (P, ξ)) ∈ K n is equal to n j=1 S(f j ; α i,j , (P, ξ)).
Therefore, the conclusion is obtained by the triangle inequality.
The converse of Lemma A.7 does not necessarily hold as the following example shows. . This means that f is Riemann-Stieltjes integrable with respect to α for any pair (g, β) of functions.
In view of the above example, the Riemann-Stieltjes integration of vector-valued functions with respect to matrix-valued functions is not completely reduced to that for scalarvalued functions. However, it is often useful to reduce the integration to scalar-valued case in view of Lemma A.7.

A.3 Fundamental results
The following are fundamental results on Riemann-Stieltjes integrals for matrix-valued functions. holds.
We call Eq. (A.2) the reversal formula for Riemann-Stieltjes integrals. The proof is obtained by returning to the definition of Riemann-Stieltjes integrals. Therefore, it can be omitted.

A.3.2 Integration by parts formula
The following is the integration by parts formula for Riemann-Stieltjes integrals with respect to matrix-valued functions.
The proof is basically same as the proof for the case n = 1 (i.e., the scalar-valued case). See [31,Proposition D.3] for the proof of this case. See also [39, Theorems 4a and 4b in Chapter 1].

A.4.1 Matrix-valued functions of bounded variation
We first recall the definition of matrix-valued functions of bounded variation. holds.
Proof. Let (P, ξ) be a tagged partition of [a, b] given in Definition A.1. Since |AB| ≤ |A||B| holds for any A, B ∈ M n (K), we have |f (x)|.
Then the remaining proof is essentially same as the scalar-valued case.
Remark A.14. In the completely similar way, (A.4) also holds for any continuous function f : [a, b] → K n . This can also be seen from Lemma A.13 because for any A ∈ M n (K) of the form A = (a 0 · · · 0) (a ∈ K n , 0 ∈ K n ), |A| = |a| holds.

A.4.2 Integrability of matrix-valued functions
The following is a fundamental theorem on the Riemann-Stieltjes integrability for scalarvalued functions. The following is the result on additivity of Riemann-Stieltjes integrals with respect to matrix-valued functions on sub-intervals. The proof is same as that for the case n = 1. See [31,Proposition D.2] for the proof. We note that the statement can be proved by considering partitions of [a, b] with c ∈ (a, b) as an intermediate point.
Remark A. 18. In Theorem A. 17, the assumptions that f is continuous and α is of bounded variation are essential because these assumptions ensure the existence of three integrals (see (A.3) and Theorem A.16). Without these assumptions, the integral in the left-hand side does not necessarily exist even if the integrals in the right-hand side exist. Such a situation will occur when the functions f and α share a discontinuity at c. See [39, Section 5 in Chapter I] for the detail.

A.5 Integration with respect to continuously differentiable functions
The following theorem shows a relationship between Riemann-Stieltjes integrals and Riemann integrals.
holds. Here the right-hand side is a Riemann integral.
Since the above statement is not mentioned in [39] and [31] even for the case n = 1, we now give an outline of the proof.
Outline of the proof of Theorem A. 19. Let (P, ξ) be a tagged partition of [a, b] given in Definition A.1. Let holds for each k ∈ {1, . . . , m} by the fundamental theorem of calculus, we have From this, we also have By combining this and the uniform continuity of α ′ , one can obtain the conclusion.
When n = 1 and K = R, one can use the mean value theorem for the proof of Theorem A.19.

A.6 Integration with respect to absolutely continuous functions
The following theorem should be compared with Theorem A. 19.
holds. Here the right-hand side is a Lebesgue integral.
See [39, Theorem 6a in Chapter I] for the proof of the scalar-valued case. We note that the existence of the Riemann-Stieltjes integral in the left-hand side is ensured by Theorem A.16 because the absolutely continuous function α is of bounded variation. We also note that the function Since it is interesting to compare the proof of Theorem A.19 and the proof of Theorem A.20, we now give an outline of the proof.
Outline of the proof of Theorem A.20. Let (P, ξ) be a tagged partition of [a, b] given in In combination with the uniform continuity of f , the conclusion is obtained by taking the limit as |(P, ξ)| → 0.

A.7 Proof of the theorem on iterated integrals
In this subsection, we give a proof of Theorem 3.8.
Proof of Theorem 3.8. We define a bounded linear operator T : C([a, b], M n (K)) → M n (K) by for g ∈ C([a, b], M n (K)). From Lemma 2.9, the left-hand side of (3.4) is equal to T d c f (·, y) dy, which is also equal to d c T f (·, y) dy since T is a bounded linear operator. By the definition of T , this integral is equal to the right-hand side of (3.4). This completes the proof.

B Riesz representation theorem
Throughout this appendix, let K = R or C and let [a, b] be a closed and bounded interval of R.
The following is the cerebrated Riesz representation theorem.
holds for all f ∈ C([a, b], K).
In a proof of Theorem B.1 (e.g., see discussions on [31, Chapter 9]), we construct such a function α by using a continuous linear extension holds for all f ∈ C([a, b], K n ). Corollary B.3 has been used in the literature of RFDEs (e.g., see [18], [19], [22], and [14]). We now give the proof of Corollary B.3 because it is not given in these references. Here y i denotes the i-th component of y ∈ K n . Since A i,j is a continuous linear functional, one can choose a function α i,j : [a, b] → K of bounded variation so that holds for all g ∈ C([a, b], K) from Theorem B.1. By using f = n j=1 f j e j for f = (f 1 , . . . , f n ), we have From Lemma A.7, this yields that holds for all f ∈ C([a, b], K n ) by defining a matrix-valued function α : [a, b] → M n (K) of bounded variation by α := (α i,j ) i,j . This completes the proof.

C Variants of Gronwall's inequality
Throughout this appendix, let [a, b] be a closed and bounded interval of R.

C.1 Gronwall's inequality and its generalization
The following is known as Gronwall's inequality. Outline of the proof. To use a technique for scalar homogeneous linear ODEs, let v(t) := t a β(s)u(s) ds. Then the given inequality becomeṡ where the non-negativity of β is used. Since the left-hand side is the derivative of the function t → v(t) + α, it is natural to consider the derivative of Then it holds that this function is strictly monotonically decreasing, which yields the conclusion.
The following is a generalized version of Gronwall's inequality.
Lemma C.2 (refs. [19], [20], [22]  By letting v(t) := t a β(s)u(s) ds, one can obtain Then the first inequality is obtained by integrating both sides in combination with

C.2 Gronwall's inequality and RFDEs
In this subsection, let r > 0 and E = (E, · ) be a normed space. It holds that the function [a, b] ∋ t → u t ∈ C([−r, 0], E) is continuous.
In the context of RFDEs, it is often convenient to use the following result rather than to use Gronwall's inequality directly. Proof. Let t ∈ [a, b] be fixed and θ ∈ [−r, 0] be given. When t + θ ≥ a, we have Here the property that α is monotonically increasing and the non-negativity of β are used. When t + θ ≤ a, we have u(t + θ) ≤ u a .
By combining the above inequalities, we obtain u t ≤ max{ u a , α(t)} + t a β(s) u s ds.
For the notation y t , see Subsection C.2. holds.
Lemma D.2. If E is a Banach space, then Y γ is also a Banach space.
Proof. Let (y k ) ∞ k=1 be a Cauchy sequence in Y γ . We choose ε > 0. Then for all sufficiently large k, ℓ ≥ 1, we have y k − y ℓ γ ≤ ε. From Lemma D.1, this means that for all sufficiently large k, ℓ ≥ 1, y k (t) − y ℓ (t) ≤ εe γt holds for all t ≥ 0. This implies that (y k (t)) ∞ k=1 is a Cauchy sequence for each t ≥ 0, and therefore, (y k ) ∞ k=1 has the limit function y : [−r, ∞) → E with y 0 = 0. Since the above relation shows that the convergence of (y k ) ∞ k=1 to y is uniform on each closed and bounded interval of R by taking the limit as ℓ → ∞, the limit function y is continuous. Then it is concluded that y k − y γ ≤ ε holds for all sufficiently large k ≥ 1, which implies that (y k ) ∞ k=1 converges to y in Y γ .

E Convolution continued
In this appendix, we discuss the convolution for functions in L 1 loc ([0, ∞), M n (K)). The purpose here is to share results on the convolution and their proofs in the literature of RFDEs. The results discussed here extend the results in Subsection 5.2, but they will not be used in this paper. See also [25,Proposition A.4, Theorems A.5, A.6, A.7 in Appendix A].

E.1 Convolution for locally essentially bounded functions and locally Lebesgue integrable functions
We first recall that a function g ∈ L 1 loc ([0, ∞), M n (K)) is said to be locally essentially

E.2 Convolution for locally Lebesgue integrable functions
The notion of convolution in Definition E.1 is not satisfactory in the sense that the condition on f and g is not symmetry. To introduce the notion of convolution for functions in L 1 loc ([0, ∞), M n (K)), we need the following.
Theorem E.3. Let f, g ∈ L 1 loc ([0, ∞), M n (K)) be given. Then the following statements hold: 1. For almost all t > 0, u → g(t − u)f (u) belongs to L 1 loc ([0, t], M n (K)). In the following, we give a direct proof of Theorem E.3 by using Fubini's theorem and Tonelli's theorem for functions on the Euclidean space R d . See [32, Theorems 3.1 and 3.2 in Section 3 of Chapter 2] for these statements and their proofs.
In the same way, we define the functionḡ : R → M n (K). Thenf ,ḡ : R → M n (K) are locally Lebesgue integrable functions. Let T > 0 be fixed. The remainder of the proof is divided into the following steps.
Step 1: Setting of triangle region and function. We consider a closed set A of R 2 given by A := (t, u) ∈ R 2 : t ∈ [0, T ], u ∈ [0, t] .
See Fig. 1 for the picture of A. Then the characteristic function 1 A is measurable and holds for all (t, u) ∈ R 2 . We define a function h : R 2 → M n (K) by h(t, u) := 1 A (t, u)ḡ(t − u)f (u).
• The functions are measurable functions defined almost everywhere.
• We have including the possibility that all the unsigned Lebesgue integrals are ∞.
Step 3: Application of Fubini's theorem. By Step 2, we have Since the last term is finite, it holds that h is integrable. By applying Fubini's theorem component-wise, the following statements hold: • For almost all u ∈ R, the function R ∋ t → h(t, u) ∈ M n (K) is Lebesgue integrable.
• For almost all t ∈ R, the function R ∋ u → h(t, u) ∈ M n (K) is Lebesgue integrable.
• The functions u → R h(t, u) dt ∈ M n (K), t → R h(t, u) du ∈ M n (K) belong to L 1 (R, M n (K)).
• The equalities hold.
Step 4 In the same way, we define the functiong : R → M n (K). Sincef ,g : R → M n (K) are Lebesgue integrable functions, one can prove the following statements as in the scalarvalued case: 4 1 ′ . For almost all t ∈ R, the function u →g(t − u)f (u) is a Lebesgue integrable function defined almost everywhere.

E.3 Convolution under Volterra operator
The convolution for functions in L 1 loc ([0, ∞), M n (K)) and the Volterra operator are related in the following way.
The above theorem is an extension of Corollary 5.5.
Proof of Theorem E.5. For each t > 0, holds for any f ∈ L 1 loc ([0, ∞), M n (K)). Here I : [0, ∞) → M n (K) denote the constant function whose value is equal to the identity matrix.
The following is a result on the regularity of convolution. It should be compared with Theorem 5.3.