Fundamental Matrix, Measure Resolvent Kernel and Stability Properties of Fractional Linear Delayed System with Discontinuous Initial Conditions

Abstract

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral system is introduced. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system, which allows us to prove that the corresponding resolvent system possesses a unique measure resolvent kernel. As a consequence, an integral representation of the solutions of the studied system is obtained. Then, using the obtained results, relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions are established.

1. Introduction

As an important tool for modelling many phenomena in various fields of science, fractional calculus and the fractional differential equations have been intensively investigated in the last decades. It seems that many natural systems can be represented more accurately via the memory data included through the fractional derivative formulation. For more details on fractional calculus theory and fractional differential equations, see the monographs of Kilbas et al. [1] and Podlubny [2]. For distributed order fractional differential equations, we refer Jiao at al. [3] and an application-oriented exposition is given in Diethelm [4]. The important case of impulsive differential and functional differential equations with fractional derivative and some applications are considered in the monograph of Stamova and Stamov [5].

From practical experience, it is well known that the stability of a process (in particular, of a stationary state) is the ability of the process to resist a priory unknown, small influences (disturbances). If such disturbances do not essentially change it, the process is said to be stable. It turns out that the investigation of stability is of utmost importance. So, the integral representations of the solutions of the studied models as main tools for investigation of different kinds of stability properties are an important theme for research. For deep information and a good historical overview of the stability properties for delayed and neutral systems with integer order derivatives we recommend the books by Hale and Lunel [6] and Kolmanovskii and Myshkis [7]. For the case of fractional-order derivatives, we prefer the surveys [8,9,10,11]. From the newest results, we note [12,13] (for fundamental theory) and [14,15,16] (for applications).

In the present work, we considered an initial (Cauchy) problem (IP) for a linear system with derivatives in Caputo’s sense of incommensurate order, distributed delays and discontinuous initial functions. The motivation to study distributed delay systems is because this type of delay includes as a special case, all types of delays (it follows from the Riesz’s theorem applied to the functional of Krasovskii), and in this sense, it is most appropriate to obtain results valid for all types of delays. To study the considered initial problem, we introduce a new approach via an equivalent Volterra–Stieltjes integral system, which allows us to obtain an integral representation of the solutions of the Cauchy problem for the studied class systems. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system, which allows us to prove straightforwardly that the corresponding resolvent system possesses a unique measure resolvent kernel. As a main result, we obtain a relation between the global fundamental matrix of the homogeneous system and the measure resolvent kernel, which is the unique solution of the resolvent system. Then, we establish two equivalent integral representations, where one of them is applicable even in cases when we interpret the space of the initial functions as state space. Then, using the obtained integral representations we study the relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions.

The paper is organised as follows. In Section 2, as usual we recall the most-used definitions of Riemann–Liouville and Caputo-type fractional derivatives and some of their properties. In the same section, we introduce the statement of the problem, as well as some necessary comments and auxiliary results used later. In Section 3, we introduce our approach for establishing of integral representation of the solutions of the initial problem for the studied systems with discontinuous initial functions. The obtained integral representation is applicable even in the case when the space of initial functions is treated as state space. In Section 4, using the results obtained in the previous section, we study the relations between the stability (uniform stability) of the zero solution of the system (3) and the different kinds of boundedness of its other solutions. Section 5 is devoted to some conclusions and comments.

2. Preliminaries and Problem Statement

We start with recall of the basic definitions of the fractional integral and derivative in the Riemann–Liouville sense, the Caputo-type fractional derivative as well as some of their properties. For additional details and other properties, we recommend [1,2].

Let $q\in (0,1]$ be an arbitrary number. With $L_1^{loc}(\mathbb{R},\mathbb{R})$ we will denote the linear space of all locally Lebesgue integrable functions $f:\mathbb{R}\to \mathbb{R}$ and by $B{L}_1^{loc}(\mathbb{R},\mathbb{R})\subset L_1^{loc}(\mathbb{R},\mathbb{R})$ its subspace of all locally bounded functions. Then, for arbitrary $a\in \mathbb{R}$, each $t>a$ and $f\in L_1^{loc}(\mathbb{R},\mathbb{R})$ the left-sided fractional integral operator and the Riemann–Liouville and Caputo-type left-sided fractional derivatives of order $\alpha \in (0,1)$ are defined via

$$I_{a+}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{a}{\overset{t}{\int }}{(t-s)}^{\alpha -1}f\left(s\right)ds,\hspace{1em}{}_{RL}D_{a+}^{\alpha }f\left(t\right)={\displaystyle \frac{d}{dt}}\left(I_{a+}^{1-\alpha }f\left(t\right)\right),$$

$${}_{C}D_{a+}^{\alpha }f\left(t\right){=}_{RL}{D}_{a+}^{\alpha }\left[f\left(s\right)-f\left(a\right)\right]\left(t\right),$$

and the following relations (see [1]) hold:

$$\left(a\right){}_{C}D_{a+}^{0}f\left(t\right)=f\left(t\right);\left(b\right){}_{C}D_{a+}^{\alpha }{I}_{a+}^{\alpha }f\left(t\right)=f\left(t\right);\left(c\right)I_{a+}^{\alpha }{}_{C}D_{a+}^{\alpha }f\left(t\right)=f\left(t\right)-f\left(a\right).$$

The following notations will be used, too: $J_a=[a,\infty )$, $q=(q_1,\dots ,q_n)$, $q_k\in (0,1)$, $k\in 〈n〉=\{1,2,\dots ,n\}$, ${\mathbb{R}}^{n\times n}$ denotes the real linear space of the square matrices with dimension $n\in \mathbb{N}$, $I,\Theta \in {\mathbb{R}}^{n\times n}$ are the identity and zero matrix, respectively, and $\mathbf{0}\in {\mathbb{R}}^n$ denotes the zero vector-column.

Everywhere below when we speak about the integer case we understand the same system in the form when $q_1=q_2=\dots =q_n=1$, i.e., the corresponding system with first-order derivatives.

Let $W:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$, $W(t,\theta )={\left\{w_{kj}(t,\theta )\right\}}_{k,j=1}^n$. With $B{V}^{loc}$ is denoted the linear space of the matrix valued functions $W(t,\theta )$ with bounded variation in $\theta$ on every compact interval $[c,d]\subset \mathbb{R}$ for every $t\in \mathbb{R}$, i.e., ${\displaystyle {\mathrm{Var}}_{[c,d]}W(t,·)=\sum _{k,j=1}^n{\mathrm{Var}}_{[c,d]}{w}_{kj}(t,·)}$ and $\left|W(t,\theta )\right|={\sum }_{k,j=1}^n\left|w_{kj}(t,\theta )\right|$. For $Y\left(t\right)={(y^1\left(t\right),\dots ,y^n\left(t\right))}^{\top }:J_a\to {\mathbb{R}}^n$, $\beta =({\beta }_1,\dots ,{\beta }_n)$, ${\beta }_k\in [-1,1]$, $k\in 〈n〉$ we use the notations $I_{\beta }\left(Y\left(t\right)\right)=\mathrm{diag}({\left(y_1\left(t\right)\right)}^{{\beta }_1},\dots ,{\left(y_n\left(t\right)\right)}^{{\beta }_n})$.

Let $h>0$ be arbitrary. With $\mathcal{BL}\left(\mathcal{PC}\right)$ we denote the Banach space of all vector functions

$$\varphi ={({\varphi }_1,\dots ,{\varphi }_n)}^{\top }:[-h,0]\to {\mathbb{R}}^n,$$

which are bounded and Lebesgue measurable (piecewise continuous) on the interval $[-h,0]$ with norm ${\displaystyle \parallel \varphi \parallel =\sum _{k=1}^n\underset{s\in [-h,0]}{sup}\left|{\varphi }_k\left(s\right)\right|<\infty }$. With $S^{\phi }$ we denote the set of all jump points of $\phi \in \mathcal{BL}$ and as usual $\mathcal{C}=C([-h,0],{\mathbb{R}}^n)$ is the subspace of all continuous functions, i.e., $\mathcal{C}\subset \mathcal{PC}\subset \mathcal{BL}$.

Let $a\in \mathbb{R}$ be arbitrary fixed and consider for $t>a$ the inhomogeneous fractional linear delayed system with incommensurate-type differential orders and distributed delays in the following general form:

$$D_{a+}^{q}x\left(t\right)={\int }_{-h}^0\left[d_{\theta }U(t,\theta )\right]x(t+\theta )+f\left(t\right),$$

(1)

with initial condition for each $\phi \in \mathcal{BL}$ as follows:

$$x_a\left(\theta \right)=x(a+\theta )=\phi \left(\theta \right)\mathrm{for}-h\le \theta \le 0,$$

(2)

where $D_{a+}^q=\mathrm{diag}{(}_C{D}_{a+}^{q_1},\dots {,}_C{D}_{a+}^{q_n})$, $D_{a+}^{q}x\left(t\right)={{(}_C{D}_{a+}^{q_1}{x}_1\left(t\right),\dots {,}_C{D}_{a+}^{q_n}{x}_n\left(t\right))}^{\top }$ (the symbol ^⊤ means transposition), $D_{a+}^{q_k}{x}_k\left(t\right){=}_C{D}_{a+}^{q_k}{x}_k\left(t\right)$, $k\in 〈n〉$; $x, f:J_a\to {\mathbb{R}}^n$;

$x\left(t\right)={(x_1\left(t\right),\dots ,x_n\left(t\right))}^{\top }$; $x_t\left(\theta \right)=x(t+\theta )$, $-h\le \theta \le 0$; $f\left(t\right)={(f_1\left(t\right),\dots ,f_n\left(t\right))}^{\top }$.

The Lebesgue decomposition in $\theta$ of $U(t,\theta )$ has the form

$$U(t,\theta )=U_j(t,\theta )+U_{ac}(t,\theta )+U_s(t,\theta )$$

with ${\displaystyle U_j(t,\theta )=\sum _{i\in {〈m〉}_0}{U}_j^i(t,\theta )}$, $i\in {〈m〉}_0=\{0,1,\dots ,m\}$, $m\in \mathbb{N}$, where $U_j(t,\theta )$, $U_{ac}(t,\theta )$, $U_s(t,\theta )$ are the jump, the absolutely continuous and the singular part, respectively, in the decomposition and the integral in (1) is understood in the Lebesgue–Stieltjes sense. The homogeneous system of (1) (i.e., $f\left(t\right)\equiv \mathbf{0}$, $t\in \mathbb{R}$) written in detail for any $k\in 〈n〉$ has the form:

$$D_{a+}^q{x}_k\left(t\right)=\sum _{j\in {〈m〉}_0}\left(\sum _{j\in 〈n〉}{\int }_{-h}^0{x}_l(t+\theta )d_{\theta }{u}_{k,j}^i(t,\theta )\right).$$

(3)

Definition 1

([6,7,17]). We say that for the kernel $U:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ the conditions (S) are fulfilled, if for each $i\in {〈m〉}_0$, $m\in \mathbb{N}$ and any $(t,\theta )\in J_a\times \mathbb{R}$ the following conditions hold:

(S1) 

The functions $U_j^i(t,\theta )$, $U_{ac}(t,\theta )$ and $U_s(t,\theta )$ are measurable in $(t,\theta )\in J_a\times \mathbb{R}$, continuous from left in θ on $(-h, 0)$, normalised so that $U_j^i(t,\theta )=U_{ac}(t,\theta )=U_s(t,\theta )=\Theta$ for $\theta \ge 0$, $U_j^i(t,\theta )=U_j^i(t,-{\sigma }_i)$ for $\theta \le -{\sigma }_i$, where ${\sigma }_i\left(t\right)\in C(\mathbb{R},[0,{\sigma }_i])$, ${\displaystyle {\sigma }_i=\underset{t\in \mathbb{R}}{max}{\sigma }_i\left(t\right)}$, $0={\sigma }_0<{\sigma }_1<{\sigma }_2<\dots <{\sigma }_m=h$, $U_{ac}(t,\theta )=U_{ac}(t,-h)$ and $U_s(t,\theta )=U_s(t,-h)$ when $\theta \le -h$.

(S2) 

The kernels $U_j^i(t,\theta )$, $U_{ac}(t,\theta ), U_s(t,\theta )\in B{V}^{loc}$, $u\left(t\right)=u_j\left(t\right)+u_{ac}(t,\theta )+u_s\left(t\right)$, where ${\displaystyle u_j\left(t\right)=\sum _{i\in 〈m〉}{u}_j^i\left(t\right)=\sum _{i\in 〈m〉}\underset{\eta \in [a,t]}{sup}\left|{Var}_{[-h,0]}{U}_j^i(t,·)\right|}$, ${\displaystyle u_{ac}\left(t\right)=\underset{\eta \in [a,t]}{sup}\left|{Var}_{[-h,0]}{U}_{ac}(t,·)\right|}$,

${\displaystyle u_s\left(t\right)=\underset{\eta \in [a,t]}{sup}\left|{Var}_{[-h,0]}{U}_s(t,·)\right|}$ and $u_j\left(t\right)$, $u_{ac}\left(t\right),u_s\left(t\right)\in B{L}_1^{loc}(J_a,[0,\infty ))$.

(S3) 

$U_j^i(t,\theta )={\left\{a_{kj}^i\left(t\right)H(\theta +{\sigma }_i\left(t\right))\right\}}_{k,j=1}^n$, $A^i\left(t\right)={\left\{a_{kj}^i\left(t\right)\right\}}_{k,j=1}^n\in B{L}_1^{loc}(J_a,{\mathbb{R}}^{n\times n})$, where $H\left(t\right)$ is the Heaviside function and the matrix ${\displaystyle \frac{\partial U_{ac}(·,\theta )}{\partial \theta }=\tilde{A}(·,\theta )\in B{L}_1^{loc}(J_a,{\mathbb{R}}^{n\times n})}$.

(S4) 

For each $t^{*}\in J_a$, the following relations hold:

$$\underset{t\to t^{*}}{lim}{\int }_{-h}^0|U^i(t,\theta )-U^i(t^{*},\theta )|\mathrm{d}\theta =0,\underset{t\to t^{*}}{lim}{\int }_{-h}^0|U_{ac}(t,\theta )-U_{ac}(t^{*},\theta )|\mathrm{d}\theta =0,$$

$$\underset{t\to t^{*}}{lim}{\int }_{-h}^0|U_s(t,\theta )-U_s(t^{*},\theta )|\mathrm{d}\theta =0.$$

For any $\phi \in \mathcal{BL}$, we denote the set of its all jump points by $S_{\phi }$ and assume that the set $S_{\phi }^i=\{t\in J_a|t-{\sigma }_i\left(t\right)\in S_{\phi }\}$ does not have limit points.

Remark 1.

As it is standard, the presented form of the initial condition (2) $x_a\left(\theta \right)=\phi \left(\theta \right)$, $-h\le \theta \le 0$ is chosen to consider the function $t\to x\left(t\right)$, $t\in J_a$ as a prolongation from right of the function $t\to \varphi (t-a)$ defined for $t\in [a-h,a]$. Note that some authors, instead the more weak condition (S4), assume that the kernels $U_j^i(t,\theta )$ are continuous in t, but this assumption excludes from considerations the important case of systems with variable concentrated delays (i.e., the system (1) can have only constant delays). We emphasise that for any $\phi \in \mathcal{PC}$ from Condition (S4) it follows that the set $S_{\phi }$ is at most finite.

Definition 2.

The vector function $x(t;a,\phi ,f)={(x_1\left(t\right),\dots ,x_n\left(t\right))}^{\top }$ is a solution of the IP (1), (2) in $J_a$ for some $\phi \in \mathcal{BL}$, if $x(t;a,\phi ,f)\in C(J_a,{\mathbb{R}}^n)$ satisfies the system (1) for all $t\in (a,\infty )$ and the initial condition (2) for each $t\in [a-h,a]$. (When misunderstanding is not possible, we will write only $x\left(t\right)$.)

Applying to both sides of (1) the left-sided fractional integral operator $I_{a+}^q$, we obtain the following auxiliary system

$$x\left(t\right)=\varphi \left(0\right)+I_{-1}(\Gamma \left(q\right))\left({\int }_a^t{I}_{q-1}(t-\eta ){\int }_{-h}^0\left[d_{\theta }U(\eta ,\theta )\right]x(\eta +\theta )\mathrm{d}\eta +{\int }_a^t{I}_{q-1}(t-s)f\left(\eta \right)\mathrm{d}\eta \right),$$

(4)

where $I_{-1}(\Gamma \left(q\right))=\mathrm{diag}({\Gamma }^{-1}\left(q_1\right),\dots ,{\Gamma }^{-1}\left(q_n\right))$.

Definition 3.

The vector function $X\left(t\right)={(x_1\left(t\right),\dots ,x_n\left(t\right))}^{\top }$ is a solution of the IP (4), (2) in $J_a$ if $x\left(t\right)\in C(J_a,{\mathbb{R}}^n)$ satisfies the system (4) for all $t\in (a,\infty )$ and the initial condition (2) for each $t\in [a-h,a]$.

In our exposition below, we need some auxiliary results for the case when the initial function $\phi \in \mathcal{BL}$ ($\phi$ is discontinuous).

Theorem 1

(see [18]). Let the following conditions hold:

1. 

Conditions (S) hold.

2. 

The function $f\in B{L}_1^{\mathrm{loc}}(J_a,{\mathbb{R}}^n)$.

Then, for arbitrary fixed initial point $a\in \mathbb{R}$ the following statements hold:

(i) 

Every solution $x(t;a,\phi ,f)$ of IP (1), (2) is a solution of the IP (4), (2) and vice versa.

(ii) 

For any initial function $\phi \in \mathcal{BL}$ and $f\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$, the IP (4), (2) has a unique solution in $J_a$.

For any $l\in 〈n〉$, define the following initial functions ${\phi }_0^l\in \mathcal{PC}$, ${\phi }_0^l\left(\theta \right):[-h,0]\to {\mathbb{R}}^n$ with ${\phi }_0^l\left(0\right)=I^l$ (where $I^l$ is the l-th column of the identity matrix $I\in {\mathbb{R}}^{n\times n}$ and ${\phi }_0^l\left(\theta \right)=\mathbf{0}$ for $\theta \in [-h,0)$.

Let $s\in \mathbb{R}$ be the low terminal and initial point. Consider for $t\ge s$ the IP (4), (2) in the homogeneous case of (4) (i.e., when $f\left(t\right)\equiv \mathbf{0}$, $t\in \mathbb{R}$) with initial function ${\phi }_0^l\left(\theta \right)={\phi }_0^l(t-s)\in \mathcal{PC}$, $\theta =t-s\in [-h,0]$, $l\in 〈n〉$ as follows:

$$\begin{array}{cc}\hfill C^l(t,s)& =C^l(t;s,{\phi }^l,\mathbf{0})=\hfill \\ & ={\phi }^l\left(0\right)+I_{-1}(\Gamma \left(q\right))\left({\int }_s^t{I}_{q-1}(t-\eta ){\int }_{-h}^0\left[d_{\theta }U(\eta ,\theta )\right]C^l(\eta +\theta ,s)\mathrm{d}\eta \right),t\in (s,\infty );\hfill \end{array}$$

(5)

$$C^l(t-s,s)=C^l(\theta ,s)={\phi }_0^l\left(\theta \right),\theta =t-s\in [-h,0].$$

(6)

Corollary 1.

Let the conditions (S) hold.

Then, the IP (5), (6) has a unique solution $C^l(t,s)$ for any $l\in 〈n〉$ and arbitrary fixed number $s\in \mathbb{R}$.

Proof. 

Since the statement of Theorem 1 holds for any fixed initial point $a\in \mathbb{R}$, then the assertion of Corollary 1 follows immediately from Theorem 1 for any $a\le s$. □

Definition 4.

The matrix-valued function

$$C(t,s)=(C^1(t,s),\dots ,C^n(t,s))=\left\{C_k^j\right\}(t,s){\}}_{k,j=1}^n:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n},$$

is called the fundamental (or Cauchy) matrix of the system (3) if for any $l\in 〈n〉$ its l-th column is a solution of the IP (5), (6).

As usual, everywhere below we will assume that $C(t,s)$ is prolonged as continuous in t function on $(-\infty ,s)$ for arbitrary fixed $s\in \mathbb{R}$.

Below we formulate as a corollary the important properties of the fundamental matrix $C(t,s)$ needed later in our exposition.

Corollary 2

(see [19,20]). Let the conditions (S) hold.

Then, the following statements hold:

(i) 

The system (3) has a unique fundamental matrix $C(t,s)$, which is absolutely continuous in t on every compact subinterval of $J_a$, i.e., for any fixed $s\in \mathbb{R}$, i.e., $C(·,s)\in A{C}_{loc}((a,\infty ),{\mathbb{R}}^{n\times n})$.

(ii) 

For arbitrary fixed initial point $s\in \mathbb{R}$ and each fixed $t>a$, $C(t,·):J_a\to {\mathbb{R}}^n$ is continuous at $s\in (a,t)\cup (t,\infty )$, right continuous at $s=a$, left continuous at $s=t$ and has jumps of the first kind for $s\in \{a,t\}$.

(iii) 

The fundamental matrix $C(t,s)$ has bounded variation in s on $[a,T]$ for arbitrary $T\in J_a$ and its total variation in s on $[a,t]$ is bounded in t, $t\in [a,T]$.

Theorem 2

([21], Corollary 2). Suppose that $\alpha \in (0,1)$ and the following conditions hold:

1. 

The functions $a\left(t\right),g\left(t\right),y\left(t\right)\in L_1^{loc}(J_a,{\mathbb{R}}_{+})$ and $a\left(t\right)$, $g\left(t\right)$ are nondecreasing.

2. 

The inequality $y\left(t\right)\le a\left(t\right)+g\left(t\right){\int }_a^t{(t-s)}^{\alpha -1}y\left(s\right)\mathrm{d}s$ holds for $t\in J_a$.

Then, for any $t\in J_a$ we have that $y\left(t\right)\le a\left(t\right)E_{\alpha }(g\left(t\right)\Gamma \left(\alpha \right)t^{\alpha })$.

Corollary 3.

Let the conditions (S) hold. Then, the fundamental matrix $C(t,s)$ has the following a priory estimate:

$$\left|C(t,s)\right|\le n{E}_{q_{*}}(n^2{\Gamma }^{-1}\left(q_M\right)u\left(t\right)\Gamma \left(q_{*}\right)t^{q_{*}})$$

(7)

Proof. 

Introduce the notations ${\displaystyle q_m=\underset{k\in 〈n〉}{min}{q}_k}$, ${\displaystyle q_M=\underset{k\in 〈n〉}{max}{q}_k}$ and since the function $\Gamma \left(z\right)$, $z\in {\mathbb{R}}_{+}$ has a local minimum at $z_{min}\approx 1.461$, where $\Gamma \left(z_{min}\right)\approx 0.885$ we obtain the following estimations:

$$\begin{array}{cc}\hfill & |I_{-1}(\Gamma \left(q\right))|=\sum _{k=1}^n{\Gamma }^{-1}\left(q_k\right)\le n{\Gamma }^{-1}\left(q_M\right),\hfill \\ \hfill & |I_{q-1}(t-s))|=\sum _{k=1}^n{(t-s)}^{q_k-1}\le n{(t-s)}^{q_{*}-1},\hfill \end{array}$$

(8)

where $q_{*}=q_M$ for $(t-s)\ge 1$ and $q_{*}=q_m$ when $(t-s)\le 1$.

From Corollary 1, it follows that the fundamental matrix $C(t,s)$ satisfies the following matrix system:

$$C(t,s)=I+I_{-1}(\Gamma \left(q\right))\left({\int }_s^t{I}_{q-1}(t-\eta ){\int }_{-h}^0\left[d_{\theta }U(\eta ,\theta )\right]C(\eta +\theta ,s)\right)\mathrm{d}\eta$$

(9)

and since

$$\left|{\int }_{-h}^0\left[d_{\theta }U(\eta ,\theta )\right]C(\eta +\theta ,s)\mathrm{d}\eta \right|\le u\left(t\right)\underset{\tau \in [s,\eta ]}{sup}\left|C(\tau ,s)\right|$$

then from (9), it follows that

$$\begin{array}{cc}\hfill \left|C\right(\tau ,s\left)\right|& \le n+n{\Gamma }^{-1}\left(q_M\right){\int }_s^{t}n{(t-\eta )}^{q_{*}-1}\left|{\int }_{-h}^0\left[d_{\theta }U(\eta ,\theta )\right]C(\eta +\theta ,s)\right|\mathrm{d}\eta \hfill \\ \hfill & \le n+n{\Gamma }^{-1}\left(q_M\right)u\left(t\right){\int }_s^{t}n{(t-\eta )}^{q_{*}-1}\underset{\tau \in [s,\eta ]}{sup}\left|C(\tau ,s)\right|\mathrm{d}\eta .\hfill \end{array}$$

(10)

Then, the estimation (7) follows from (10) and Theorem 2. □

3. Main Results

The main goal of this section is to introduce a new approach to study the IP (5), (6) via an equivalent Volterra–Stieltjes system. It is based on the existence and uniqueness of a global fundamental matrix of the system (3), which is a well known and often studied problem.

Consider for arbitrary initial function $\phi \in \mathcal{BL}$ and $f\in B{L}_1^{loc}([a,\infty ),{\mathbb{R}}^n)$ the Volterra–Stieltjes system

$$x\left(t\right)={\int }_a^t\left[d_{s}K(t,\theta )\right]x\left(s\right)+g\left(t\right),t\ge a,$$

(11)

where

$$\begin{array}{cc}\hfill g\left(t\right)=& \phi \left(0\right)+I_{-1}(\Gamma \left(q\right)){\int }_a^t{I}_{q-1}(t-\tau ){\int }_{-h}^{a-\tau }\left[d_{\theta }U(\tau ,\theta )\right]\phi (\tau +\theta )\mathrm{d}\tau \hfill \\ \hfill & +I_{-1}(\Gamma \left(q\right)){\int }_a^t{I}_{q-1}(t-\tau )f\left(\tau \right)\mathrm{d}\tau .\hfill \end{array}$$

(12)

Theorem 3.

Let the following conditions hold:

1. 

Conditions (S) hold.

2. 

The initial function $\phi \in \mathcal{BL}$ and the function $f\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$ are arbitrary.

Then, every solution $x(t;a,\phi ,f)$ of IP (1), (2) (IP (4), (2)) is a continuous solution of the system (11) for $t\ge a$ and vice versa.

Proof. 

Let $x\left(t\right)=x(t,a,\phi ,f)$ be the corresponding unique solution of the IP (1), (2) existing according Theorem 1 for the $f\left(t\right)\in B{L}_1^{loc}([a,\infty ),{\mathbb{R}}^n)$ and $\phi \in \mathcal{BL}$ mentioned in condition 2 of Theorem 3. According the first statement of Theorem 1, the solution $x\left(t\right)=x(t,a,\phi ,f)$ satisfies system (4) and then substituting this solution in (4) and splitting off in system (4) the parts that explicitly depend on the initial data, we obtain

$$\begin{array}{cc}\hfill x\left(t\right)=& \phi \left(0\right)+I_{-1}(\Gamma \left(q\right)){\int }_a^t{I}_{q-1}(t-\tau )\left({\int }_{a-\tau }^0\left[d_{\theta }U(\tau ,\theta )\right]x(\tau +\theta )\right)\mathrm{d}\tau \hfill \\ \hfill & +I_{-1}(\Gamma \left(q\right)){\int }_a^t{I}_{q-1}(t-\tau ){\int }_{-h}^{a-\tau }\left[d_{\theta }U(\tau ,\theta )\right]\phi (\tau +\theta )\mathrm{d}\tau \hfill \\ \hfill & +I_{-1}(\Gamma \left(q\right)){\int }_a^t{I}_{q-1}(t-\tau )f\left(\tau \right)\mathrm{d}\tau \hfill \\ \hfill & \stackrel{\tau +\theta =s}{=}{I}_{-1}(\Gamma \left(q\right)){\int }_a^t{I}_{q-1}(t-\tau )\left({\int }_a^s\left[d_{s}U(\tau ,s-\tau )\right]x\left(s\right)\right)\mathrm{d}\tau +g\left(t\right).\hfill \end{array}$$

(13)

Let us define the kernel $K(t,s):J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ via the equality

$$K(t,s)=I_{-1}(\Gamma \left(q\right)){\int }_s^{t}U(\tau ,s-\tau )I_{q-1}(t-\tau )\mathrm{d}\tau .$$

(14)

Applying Fubini’s theorem to the right side of (13), we have that

$$x\left(t\right)=I_{-1}(\Gamma \left(q\right)){\int }_a^t{d}_s\left[{\int }_s^{t}U(\tau ,s-\tau )I_{q-1}(t-\tau )\mathrm{d}\tau \right]x\left(s\right)+g\left(t\right)$$

(15)

and then using (14), system (15) can be rewritten as the Volterra–Stieltjes Equation (11).

The vice-versa statement can be proved in the reverse way. □

Definition 5

([22]). We say that $V(t,s):J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ is a Volterra measure kernel of type $B^{\infty }$ on $J_a$ (we denote $V(t,s)\in B^{\infty }$) if the following conditions hold:

(i) 

For any $t\in J_a$ the $V(t,·)\in {\mathbb{R}}^{n\times n}$ is a matrix of scalar real Borel measures on $\mathbb{R}$ with support in $J_a\cap (-\infty ,t]$;

(ii) 

${\displaystyle \underset{t\in J_a}{sup}\left|Va{r}_{s\in [a,t]}V(t,·)\right|<\infty }$;

(iii) 

For each Borel set $E\subset J_a$ the function $t\to K(t,E)$ is Borel measurable.

Definition 6.

We say that $W(t,s):J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ is a Volterra–Stieltjes measure kernel of type $VS$ on $J_a$ and denote $W(t,s)\in VS$, if $W(t,·)\in {\mathbb{R}}^{n\times n}$ is a matrix of scalar real Volterra–Stieltjes measures on $\mathbb{R}$ with support in $J_a\cap (-\infty ,t]$ and the following conditions hold:

(i) 

For the kernel $W:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ the conditions (S) hold;

(ii) 

${\displaystyle \underset{t\in J_a}{sup}\left|Va{r}_{s\in [a,t]}W(t,·)\right|<\infty }$.

We say that the function $V(t,s):J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ ($W(t,s):J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$) is a local kernel of type $B^{\infty }$ ($VS$) on $J_a$, and denote by $V(t,s)\in B_{loc}^{\infty }$ ($W(t,s)\in V{S}_{loc}$), if the conditions of Definition 5 (Definition 6) hold on any compact subsets of $J_a$.

Lemma 1.

Let $W(t,s)\in VS\left(V{S}_{loc}\right)$.

Then, we have that $W(t,s)\in B^{\infty }\left(B_{loc}^{\infty }\right)$.

Proof. 

Let $W(t,s)\in VS$. Then, the conditions (S) imply that the kernel $W(t,s)$ is a matrix with elements that are scalar Lebesgue–Stieltjes measures and hence they are also Borel measures. Then, from (S1) and (S2) it follows that the kernel $W(t,s)$ satisfies the conditions (i) and (iii) in Definition 5. Taking into account that the conditions (ii) in the Definition 5 and Definition 6 coincide, we can conclude that $W(t,s)\in B^{\infty }$. □

Lemma 2.

Let the following conditions hold:

1. 

$K(t,s)\in V{S}_{loc}$.

2. 

The functions $u_j\left(t\right),u_{ac}\left(t\right),u_s\left(t\right)$ are bounded.

Then, we have that $K(t,s)\in B^{\infty }$.

Proof. 

The statement follows immediately from Lemma 1. □

Lemma 3.

Let the following conditions hold:

1. 

The kernel $K(t,s)$ is defined via (14).

2. 

The kernel $U:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ appearing in (9) satisfies the conditions (S).

Then, $K(t,s)\in V{S}_{loc}$ and for every fixed $t\in J_a$ with $s\ge t$ we have that $K(t,s)=\Theta$.

Proof. 

Since $K(t,s)$ is defined via (14) and $U(t,s)$ satisfies the conditions (S), then from the conditions (S) it follows that that the kernel $K(t,s)$ is a matrix with elements that are scalar Lebesgue–Stieltjes measures and hence they are also Borel measures. Then, the assertion follows immediately from Lemma 1 and (14). □

The next simple corollary gives an explicit condition that guaranties that condition (ii) of Definition 6 holds.

Corollary 4.

Let the following conditions hold:

1. 

The conditions of Lemma 3 are fulfilled.

2. 

The functions $u_j\left(t\right),u_{ac}\left(t\right),u_s\left(t\right)$ are bounded.

Then, we have that $K(t,s)\in B^{\infty }$.

Proof. 

Condition 2 of the corollary implies that condition (ii) holds, i.e.,

$$\underset{t\in J_a}{sup}\left|Va{r}_{s\in [a,t]}K(t,·)\right|<\infty .$$

Then, applying Lemma 1 we obtain that the statement of Corollary 4 holds. □

The keystone of the approach introduced in the remarkable book [22] for solving linear systems having the same form as (6) with nonconvolution measure-valued Volterra kernels $K(t,s)\in B^{\infty }$, as well as obtaining an integral representation of their solutions, is based essentially on the possibility of solving the corresponding resolvent system for the kernel $K(t,s)$ in the form

$$\begin{array}{cc}\hfill & R(t,s)=-K(t,s)+{\int }_s^t{d}_{\eta }\left[K(t,\eta )\right]R(\eta ,s),\hfill \\ \hfill & R(t,s)=-K(t,s)-{\int }_s^t{d}_{\eta }\left[R(\eta ,s)\right]K(t,\eta ).\hfill \end{array}$$

(16)

The solution $R(t,s)\in B_{loc}^{\infty }$ of the system (16) if it exists is called measure resolvent for the kernel $K(t,s)\in B_{loc}^{\infty }$.

The proof in [22] of the problem of the existence of a measure resolvent $R(t,s)$ is based on non-trivial techniques using Banach algebras. Here, we introduce a new approach for establishing of this existence, without the necessity to use Banach algebra techniques. This approach is based on the existence and uniqueness of a global fundamental matrix for the system (3) (see Corollary 1 in [18]), which is a well known and often studied problem. The key result in our approach is to establish a relation between the existence of global fundamental matrix of the homogeneous system (3) and the existence of a measure resolvent kernel $R(t,s)\in B_{loc}^{\infty }$ for the measure kernel $K(t,s)\in B_{loc}^{\infty }$; this problem is treated in the next theorem.

Remark 2.

It must be noted that if $R(t,s)$ is a solution of system (16), then for every fixed $t\in J_a$ with $s\ge t$ we have that $R(t,s)=\Theta$.

Let $(t,s)\in J_a\times \mathbb{R}$ be arbitrary and $C(t,s)$ be the unique fundamental matrix of (3). Introduce the relation

$$R(t,s)=C(t,s)-{\mathbf{1}}_{s\in (-\infty ,t]}I.$$

(17)

where ${\mathbf{1}}_{s\in (-\infty ,t]}$ is the indicator function.

Theorem 4.

Let the following conditions hold:

1. 

The kernel $K(t,s)$ is defined via (14).

2. 

The kernel $U:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ appearing in (9) satisfies the conditions (S).

Then, the matrix $R(t,s)$ defined via the relation (17) is a solution of the resolvent Equation (16) if and only if, when the matrix $C(t,s)$ appearing in (17) is the unique fundamental matrix of (3).

Proof. 

Sufficiency.

Let $s\in J_a$ be an arbitrary fixed number. Then, for any $l\in 〈n〉$ the solution of the IP (5), (6) denoted by $x^l(t,s)$ exists according Corollary 1, i.e., for any $l\in 〈n〉$ we have that

$$x^l(t,s)=I^l+{\int }_a^t\left[d_{\eta }K(t,\eta )\right]x^l(\eta ,s)=I^l+{\int }_s^t\left[d_{\eta }K(t,\eta )\right]x^l(\eta ,s),$$

(18)

where $K(t,s)$ is defined via (14) and hence the fundamental matrix.

$C(t,s)=\{x^1(t,s),\dots ,x^n(t,s)\}$ is the unique solution of the following matrix system:

$$C(t,s)=I+{\int }_s^t\left[d_{\eta }K(t,\eta )\right]C(\eta ,s).$$

(19)

For the function $R(t,s)$ defined via (17), taking into account the properties of $C(t,s)$ mentioned in Corollary 2, we have that $R(·,s)\in A{C}_{loc}((a,\infty ),{\mathbb{R}}^{n\times n})$ for any fixed $s\in \mathbb{R}$, $R(t,·)\in B{L}_1^{loc}([a,t],{\mathbb{R}}^{n\times n})$.

Substituting (17) in (16) and integrating by parts for $s\le t$, we obtain that

$$\begin{array}{cc}\hfill R(t,s)& =I-I+{\int }_s^t\left[d_{\eta }K(t,\eta )\right](I+R(\eta ,s))={\int }_s^t\left[d_{\eta }K(t,\eta )\right](I+R(\eta ,s))\hfill \\ \hfill & =-K(t,s)+{\int }_s^t\left[d_{\eta }K(t,\eta )\right]R(\eta ,s)=-K(t,s)-{\int }_s^t\left[d_{\eta }R(\eta ,s)\right]K(t,\eta )\hfill \end{array}$$

(20)

and hence the function $R(t,s)$ defined via (17) is a solution of the resolvent Equation (16).

To complete the proof of the sufficiency, we must prove that $R(t,s)\in V{S}_{loc}$.

From (17), it follows that for any fixed $t\in J_a$ with $s\ge t$ we have that $R(t,s)=\Theta$, and since the kernel $K(t,·)\in {\mathbb{R}}^{n\times n}$ is with support in $J_a\cap (-\infty ,t]$ then the same is true for $R(t,s)$, too. According to Corollary 2, the fundamental matrix $C(t,s)$ has bounded variation in s on $[a,T]$ for any $T\in J_a$, and its total variation in s on $[a,t]$ is bounded in t, $t\in [a,T]$. Then, from (17) it follows that $R(t,s)$ has the same properties and hence $R(t,s)\in V{S}_{loc}$.

Necessity. Let $R(t,s)\in V{S}_{loc}$ be a solution of (16). Then, using (17) in a reverse way via (16) we obtain that $C(t,s)=I+R(t,s)$ satisfies (19) for $s\le t$. If we assume that (16) has two different solutions $R(t,s)\ne \overline{R}(t,s)$, then via (17) we can obtain that (19) has two different solutions $C(t,s)\ne \overline{C}(t,s)$, which is impossible and hence the system (16) has a unique solution. □

The next corollary is an adaptation of Theorem 2.5 in [22], Ch.10, in a form more convenient for practical application, in partial for our case too.

Corollary 5.

Let the following conditions hold:

1. 

The kernel $K(t,s)$ is defined via (14).

2. 

The kernel $U:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ appearing in (9) satisfies the conditions (S).

3. 

The function $f\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$.

Then, for any initial function $\phi \in \mathcal{BL}$ and each $t\in J_a$ the IP (6), (2) has a unique solution that possesses the following integral representation (variation of constants formula):

$$x\left(t\right)=g\left(t\right)-{\int }_a^t\left[d_{\eta }R(t,\eta )\right]g\left(\eta \right),$$

(21)

where the function $g\left(t\right)$ is defined via (12) and $R(t,s)\in V{S}_{loc}$ is the unique solution of (16) defined via (17).

Proof. 

From Lemma 1 and Lemma 3 it follows that the kernel $K(t,s)\in V{S}_{loc}$, and then applying Theorem 4 we obtain that the resolvent system (16) has a unique solution $R(t,s)\in V{S}_{loc}$. Then, the statement of Corollary 5 follows from Theorem 2.5 in [22], Ch.10. □

Theorem 5.

Let the conditions of Corollary 5 hold.

Then, for any initial function $\phi \in \mathcal{BL}$ and each $t\in J_a$ the unique solution of the IP (6), (2) has the following integral representation:

$$x\left(t\right)=C(t,a)\phi \left(0\right)+{\int }_a^{t}C(t,\eta )d_{\eta }g\left(\eta \right),$$

(22)

where the function $g\left(t\right)$ is defined via the Equality (12) and $C(t,s)$ is the fundamental matrix of (3).

Proof. 

Let the functions $\phi \in \mathcal{BL}$ and $f\left(t\right)\in B{L}_1^{loc}([a,\infty ),{\mathbb{R}}^n)$ be arbitrary. Then, according Theorems 1 and 3 the IP (11), (2) has a unique solution $x(t;a,\phi ,f)$ in $J_a$ and hence the system (3) has a unique fundamental matrix $C(t,s)$. Since the kernel $K(t,s)$ is defined via (9) and the kernel $U:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ appearing in (14) satisfies the conditions (S), then according to Lemma 3 we have that $K(t,s)\in V{S}_{loc}$. From Theorem 4, it follows that the kernel $R(t,s)$ defined via (17) is the unique solution of the resolvent Equation (16) and $R(t,s)\in V{S}_{loc}$. Applying Corollary 5, we obtain that for the unique solution of the IP (11), (2) $x(t;a,\phi ,f)$ the integral representation (21) holds. Integrating by parts the equality (21), we obtain that

$$\begin{array}{cc}\hfill x\left(t\right)& =g\left(t\right)-{\int }_a^t\left[d_{\eta }R(t,\eta )\right]g\left(\eta \right)=g\left(t\right)-{\int }_a^t\left[d_{\eta }C(t,\eta )\right]g\left(\eta \right)\hfill \\ \hfill & =g\left(t\right)-C(t,t)g\left(t\right)+C(t,0)g\left(0\right)+{\int }_a^{t}C(t,\eta )d_{\eta }g\left(\eta \right)\hfill \\ \hfill & =C(t,a)\phi \left(0\right)+{\int }_a^{t}C(t,\eta )d_{\eta }g\left(\eta \right),\hfill \end{array}$$

which completes the proof. □

Remark 3.

The obtained above results are a generalisation of the results in [23] obtained via another approach.

Now, we will establish an equivalent, but more applicable version of the integral representation (22), interpreting $\mathcal{PC}$ as state space to characterise the operators appearing in the obtained version. In our point of view, the new version of this formula will have great advantages in comparison with (22). The interpretation that uses $\mathcal{PC}$ as state space is widely accepted, and it has proven it is fruitful in many sections of the theory, especially sections related to asymptotic behaviour and periodicity of solutions.

Let $s\in \mathbb{R}$, $\mathcal{B}$ be an arbitrary real Banach space and $t,s\in J_a$ with $s\ge t$.

Definition 7

([24]). A two-parameter family of bounded linear operators $S(t,s):\mathcal{B}\to \mathcal{B}$, is called a forward evolutionary system on $\mathcal{B}$ whenever the following hold:

• $S(s,s)=I_{\mathcal{B}}$ for any $s\ge a$;
• $S(t,s)S(s,a)=S(t,a)$ for any $t\ge s\ge a$.

Everywhere below we will assume that the conditions of Theorem 1 hold. Then, for any $\phi \in \mathcal{PC}$ the IP (1), (2) has a forward non-continuable unique solution $x(t;a,\phi ,f)\in C(J_a,{\mathbb{R}}^n)$ and let for any $t\in J_a$ define the family of linear maps $S(t,s)\phi :\mathcal{PC}\to \mathcal{PC}$ via the relation:

$$S(t,a)\phi \left(\theta \right)=x_t(\theta ;a,\phi ,f).$$

(23)

Proposition 1.

Let the conditions of Theorem 1 hold and the kernel $K(t,s)$ is defined via (14).

Then, the operators $S(t,a)\phi :\mathcal{PC}\to \mathcal{PC}$ defined via (23) are continuous for any $t\in J_a$.

Proof. 

Let $\phi ,\tilde{\phi }\in \mathcal{PC}$ be arbitrary and $x\left(t\right)=(t;a,\phi ,f)$, $\tilde{x}\left(t\right)=(t;a,\tilde{\phi },f)\in C(J_a,{\mathbb{R}}^n)$ are the corresponding unique solutions of IP (4), (2). Using the same notations as in Corollary 3, we obtain that the estimations (8) hold and

$$\begin{array}{cc}\hfill & \left|\underset{-h}{\overset{0}{\int }}[d_{\theta }U(\eta ,\theta )(x(s+\theta )-\tilde{x}(s+\theta ))\right|\hfill \\ \hfill & \le u\left(t\right)\left(\underset{\tau \in [a-h,a]}{sup}|\phi (\tau -a)-\tilde{\phi }(\tau -a)|+\underset{\tau \in [a,s]}{sup}|x\left(\tau \right)-\tilde{x}\left(\tau \right)|\right)\hfill \\ \hfill & \le u\left(t\right)\left(\parallel \phi -\tilde{\phi }\parallel +\underset{\tau \in [a,s]}{sup}|x\left(\tau \right)-\tilde{x}\left(\tau \right)|\right).\hfill \end{array}$$

(24)

Substituting both solutions in (4), subtracting both equations and then from the estimations (8), (24) and Theorem 2 it follows that

$$\begin{array}{cc}\hfill & \underset{s\in [a,t]}{sup}|x\left(s\right)-\tilde{x}\left(s\right)|\hfill \\ \hfill & \le \parallel \phi -\tilde{\phi }\parallel +|I_{-1}(\Gamma \left(\alpha \right))|\left|{\int }_a^t{I}_{\alpha -1}(t-s){\int }_{-h}^0\left[d_{\theta }U(s,\theta )\right](x(s+\theta )-\tilde{x}(s+\theta ))\mathrm{d}s\right|\hfill \\ \hfill & \le \left(1+n{\Gamma }^{-1}\left(q_M\right)u\left(t\right)\right)\parallel \phi -\tilde{\phi }\parallel +n^2{\Gamma }^{-1}\left(q_M\right)u\left(t\right){\int }_a^t{(t-s)}^{q_{*}-1}\underset{\tau \in [a,s]}{sup}|x\left(\tau \right)-\tilde{x}\left(\tau \right)|\mathrm{d}s\hfill \\ \hfill & \le \left(1+n{\Gamma }^{-1}\left(q_M\right)u\left(t\right)\right)\parallel \phi -\tilde{\phi }\parallel E_{q_{*}}\left(n^2{\Gamma }^{-1}\left(q_M\right)\Gamma \left(q_{*}\right)t^{q_{*}}\right)\hfill \end{array}$$

holds for any fixed $t\in J_a$.

This completes the proof. □

Remark 4.

We note that if the initial function $\phi \in \mathcal{C}$, then $S(t,a)\phi :\mathcal{C}\to \mathcal{C}$ for any $t\in J_a$, but if $\phi \in \mathcal{PC}$, then $S(t,a)\phi :\mathcal{PC}\to \mathcal{C}$ when $t\ge a+h$. Moreover, Theorem 1 implies that the shifting operator (23) defines an evolutionary system on $\mathcal{PC}\left(\mathcal{C}\right)$ when the initial function $\phi \in \mathcal{PC}$ ($\phi \in \mathcal{C}$).

Define the matrix $C_0\left(\theta \right):[-h,0]\to {\mathbb{R}}^{n\times n}$ as $C_0\left(\theta \right)=({\phi }_0^1\left(\theta \right),\dots ,{\phi }_0^n\left(\theta \right))$, where ${\phi }^l\left(\theta \right)$, $l\in 〈n〉$ are the same functions as in the initial condition (6). It is clear that for $t=s$ we have $C_0(t-s)=C_0\left(0\right)=I$ and $C_0\left(\theta \right)=\Theta$ when $t<s$. Since each of its columns ${\phi }^l\left(\theta \right)\in \mathcal{PC}$, $l\in 〈n〉$, then from (23) it follows that for $t\ge s$

$$C_t(\theta ,s)=S(t,s)C_0\left(\theta \right).$$

(25)

As in the integer case, the map defined via (23) will be called a solution map or shifting operator (along the solutions trajectories) for the system (1). In the homogeneous case, i.e., when the forced term $f\left(t\right)\equiv \mathbf{0}$ in $J_a$, then $x(t;a,\phi ,\mathbf{0})$ denotes the corresponding unique solution of the IP (3), (2). It must be noted that the solution map has a key role in the abstract-type representations of the solutions in both cases (homogeneous and inhomogeneous systems).

According to (23), for any $\phi \in \mathcal{PC}$ the unique solution of the IP (3), (2) $x_t(\theta ;a,\phi ,\mathbf{0})$ has the form $x_t(\theta ;a,\phi ,\mathbf{0})=S(t,a)\phi \left(\theta \right)$ and the unique solution $x_t(\theta ;a,\mathbf{0},f)$ of the IP (1), (2) with initial function $\phi \left(\theta \right)\equiv \mathbf{0}$, $\theta \in [-h,0]$ (see [19,20]) has the form

$$x(t;a,\mathbf{0},f)={\int }_a^{t}C{(t,s)}^{RL}{D}^{1-q}f\left(s\right)\mathrm{d}s$$

for any $t\in J_a$.

Then, for $x_t(\theta ;a,\mathbf{0},f)$ we have the following representation

$$x_t(\theta ;a,\mathbf{0},f)={\int }_a^{t+\theta }{C}_t{(\theta ,s)}^{RL}{D}^{1-q}f\left(s\right)\mathrm{d}s={\int }_a^{t+\theta }C{(t+\theta ,s)}^{RL}{D}^{1-q}f\left(s\right)\mathrm{d}s.$$

Hence the superposition principle implies that for any fixed $t\in J_a$ we have that

$$x_t(\theta ;a,\phi ,f)=x(t+\theta ;a,\phi ,\mathbf{0})+{\int }_a^{t+\theta }C{(t+\theta ,s)}^{RL}{D}^{1-q}f\left(s\right)\mathrm{d}s,$$

(26)

and therefore from (23) and (26) it follows that for any fixed $t\in J_a$ we obtain

$$x_t(\theta ;a,\phi ,f)=S(t,a)\phi \left(\theta \right)+{\int }_a^{t+\theta }S(t,s)C_0{\left(\theta \right)}^{RL}{D}^{1-q}f\left(s\right)\mathrm{d}s,$$

(27)

where the integral in the right side of (27) is understood as a family of Euclidean space integrals parametrised via $\theta$.

Our aim is to give an interpretation of (27) when we use $\mathcal{PC}$ as state space. Note that this problem in the fractional case is more complicated in comparison with the integer case and cannot be realised via one family of compact linear operators parametrised via $t\in J_a$, allowing representation as vector-valued Lebesgue–Stieltjes integrals.

Integrating by parts the integral in the right side in (26), we have

$$\begin{array}{cc}\hfill & {\int }_a^{t+\theta }C{(t+\theta ,s)}^{RL}{D}^{1-q}g\left(s\right)\mathrm{d}s={\int }_a^{t+\theta }C(t+\theta ,s){\mathrm{d}}_{\mathrm{s}}{I}^{1-q}g\left(s\right)\hfill \\ & =I^{1-q}g(t+\theta )-{\int }_a^{t+\theta }{I}^{1-q}g\left(s\right){\mathrm{d}}_{\mathrm{s}}C(t+\theta ,s)\hfill \\ & =I_{-1}(\Gamma (1-q))\left({\int }_a^{t+\theta }{(t+\theta -\eta )}^{-q}g\left(\eta \right)\mathrm{d}\eta -{\int }_a^{t+\theta }\left[d_{s}C(t+\theta ,s)\right]\left({\int }_a^s{(s-\eta )}^{-q}g\left(\eta \right)\mathrm{d}\eta \right)\right),\hfill \end{array}$$

(28)

Define for arbitrary fixed $t\in J_a$, $g\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$ and $\theta \in [-h,0]$ the following two families of operators (parametrised via $t\in J_a$):

$$\begin{array}{cc}\hfill & \mathbf{I}\left(t\right)g\left(\theta \right)=\left(I^{1-q}g\right)(t+\theta )\hfill \\ \hfill & \mathbf{G}\left(t\right)g\left(\theta \right)=\left(G\left(t\right)g\right)(t+\theta )={\int }_a^{t+\theta }\left[d_{s}C(t+\theta ,s)\right]\left({\int }_a^s{(s-\eta )}^{-q}g\left(\eta \right)\mathrm{d}\eta \right)\hfill \end{array}$$

(29)

and hence for any fixed $t\in J_a$ the defined operators $\mathbf{I}\left(t\right),\mathbf{G}\left(t\right):B{L}_1^{loc}(J_a,{\mathbb{R}}^n)\to \mathcal{C}$ are linear. The equalities (29) and (28) show that in the fractional case the situation is essentially different compared with the integer case.

Theorem 6.

Let the conditions of Corollary 5 hold. Then, the operators defined via (29) of both families for any $t\in J_a$ are bounded and compact.

Proof. 

Let $t\in J_a$ be fixed, $g\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$ be arbitrary and $\theta \in [-h,0]$. Then, according Lemma 2.1 and (2.8) in [1], the operator $\mathbf{I}\left(t\right)$ is bounded and hence from (29) and Corollary 2, point 1, it follows that the operator $\mathbf{G}\left(t\right)$ is bounded too. Furthermore, from (29) it follows that if the operator $\mathbf{I}\left(t\right)$ is compact, then the operator $\mathbf{G}\left(t\right)$ is compact too.

Introduce for arbitrary fixed $t\in J_a$ the set

$$B_t^1=\{g\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n){|\parallel g\parallel }_t=\underset{\eta \in [a,t]}{sup}|g\left(\eta \right)\left|\right\}\le 1.$$

To apply the Arzela–Ascoli theorem, it is necessary to check that the set $\mathbf{I}\left(t\right)B_t^1$ is uniformly bounded and equicontinuous.

From (24) for arbitrary $g\in B_t^1$, we have that

$$\begin{array}{cc}\hfill \left|\mathbf{I}\right(t\left)g\right(\theta \left)\right|& =\underset{\theta \in [-h,0]}{sup}\left|\left(I^{1-q}g\right)(t+\theta )\right|\le I_{-1}(\Gamma (1-q))\underset{\theta \in [-h,0]}{sup}\left|{\int }_a^{t+\theta }{(t+\theta -\eta )}^{-q}g\left(\eta \right)\mathrm{d}\eta \right|\hfill \\ \hfill & \le I_{-1}(\Gamma (1-q))\underset{\eta \in [a,t]}{sup}\left|g\left(\eta \right)\right|\underset{\theta \in [-h,0]}{sup}\left|{\int }_a^{t+\theta }{(t+\theta -\eta )}^{-q}\mathrm{d}\eta \right|\hfill \\ \hfill & ={(1-q)}^{-1}{I}_{-1}(\Gamma (1-q))\underset{\eta \in [a,t]}{sup}\left|g\left(\eta \right)\right|\underset{\theta \in [-h,0]}{sup}{(t+\theta -\eta )}^{-q}\hfill \\ \hfill & \le {(1-q)}^{-1}{I}_{-1}(\Gamma (1-q))\underset{\eta \in [a,t]}{sup}\left|g\left(\eta \right)\right|\underset{\theta \in [-h,0]}{sup}{(t-a)}^{1-q}\hfill \\ \hfill & \le {(t-a)}^{1-q}{I}_{-1}(\Gamma (1-q))\underset{\eta \in [a,t]}{sup}\left|g\left(\eta \right)\right|\hfill \\ \hfill & \le {(t-a)}^{1-q}{I}_{-1}(\Gamma (2-q)),\hfill \end{array}$$

and hence the set $\mathbf{I}\left(t\right)B_t^1$ is uniformly bounded.

Since the function $h\left(\theta \right)={(t+\theta -a)}^{1-q}$ is uniformly continuous for $\theta \in [-h,0]$, then for any $\epsilon >0$ there exists $\delta \in (0,\epsilon )$ such that if $|{\theta }_1-{\theta }_2|<\delta$ then we have

${\displaystyle |h\left({\theta }_1\right)-h\left({\theta }_2\right)|<\frac{\epsilon \Gamma (2-q)}{3}}$ and hence

$$\begin{array}{cc}\hfill & |\mathbf{I}\left(t\right)\left({\theta }_1\right)-\mathbf{I}\left(t\right)\left({\theta }_2\right)|\hfill \\ \hfill & \le I_{-1}(\Gamma (1-q))\left|{\int }_a^{t+{\theta }_1}{(t+{\theta }_1-\eta )}^{-q}g\left(\eta \right)\mathrm{d}\eta -{\int }_a^{t+{\theta }_2}{(t+{\theta }_2-\eta )}^{-q}g\left(\eta \right)\mathrm{d}\eta \right|\hfill \\ \hfill & \le I_{-1}(\Gamma (1-q))\left|{\int }_a^{t+{\theta }_1}[{(t+{\theta }_1-\eta )}^{-q}-{(t+{\theta }_2-\eta )}^{-q}]g\left(\eta \right)\mathrm{d}\eta -{\int }_{t+{\theta }_1}^{t+{\theta }_2}{(t+{\theta }_2-\eta )}^{-q}g\left(\eta \right)\mathrm{d}\eta \right|\hfill \\ \hfill & \le I_{-1}(\Gamma (1-q)){\parallel g\parallel }_t\left|{\int }_a^{t+{\theta }_1}[{(t+{\theta }_2-\eta )}^{-q}-{(t+{\theta }_1-\eta )}^{-q}]\mathrm{d}\eta -{\int }_{t+{\theta }_1}^{t+{\theta }_2}{(t+{\theta }_2-\eta )}^{-q}\mathrm{d}\eta \right|\hfill \\ \hfill & \le I_{-1}(\Gamma (1-q)){\parallel g\parallel }_t\left|\left[{({\theta }_2-{\theta }_1)}^{1-q}-{(t+{\theta }_2-a)}^{1-q}+{(t+{\theta }_1-a)}^{1-q}\right]\right|+\left|{({\theta }_2-{\theta }_1)}^{1-q}\right|\hfill \\ \hfill & \le I_{-1}(\Gamma (1-q))\left(2{({\theta }_2-{\theta }_1)}^{1-q}+\left|{(t+{\theta }_2-a)}^{-q}-{(t+{\theta }_1-a)}^{-q}\right|\right)<\epsilon .\hfill \end{array}$$

Thus, the set $\mathbf{I}\left(t\right)B_t^1$ is equicontinuous and therefore the operator $\mathbf{I}\left(t\right)$ is compact for any $t\in J_a$. □

Finally, when we interpret $\mathcal{PC}$ as a state space, from (27), (28) and (29) we obtain the following representation:

$$x_t(\theta ;a,\phi ,f)=S(t,a)\phi \left(\theta \right)+I_{-1}(\Gamma (1-q))(\mathbf{I}\left(t\right)g\left(\theta \right)+\mathbf{G}\left(t\right)g\left(\theta \right)),$$

(30)

where, according to Theorem 6, the families of operators $\mathbf{I}\left(t\right),\mathbf{G}\left(t\right):B{L}_1^{loc}(J_a,{\mathbb{R}}^n)\to \mathcal{C}$ are compact for any $t\in J_a$.

4. Applications and Relations Between Boundedness and Stability

In this section, using the results obtained in the previous section, we study the relations between the stability of the zero solution of the system (3) and different kinds of boundedness of its other solutions.

It is well known (see [6,7]) that the study of the stability of any solution $x(t;a,\phi ,\mathbf{0})$ of (3), with initial function $\phi \in \mathcal{PC}$ can be reduced to the study of the stability of the zero solution, by changing the unknown and the initial functions. Thus, in the sequel we will have to study only the stability of the zero solution of (3) corresponding to the zero initial function.

Let us introduce the set ${\Phi }_{\delta }=\{\phi \in \mathcal{PC}|\parallel \phi \parallel <\delta \}$ and recall the stability definitions.

Definition 8

([7]). The zero solution of (3) is named based on the following definitions:

(A) 

Stable, for a given $a\in \mathbb{R}$ if for any $\epsilon >0$ there is $\delta (a,\epsilon )>0$ such that $\left|x\right(t;a,\phi ,\mathbf{0}\left)\right|\le \epsilon$ for each initial function $\phi \in {\Phi }_{\delta }$ and $t\in J_a$; in the opposite case the solution is called unstable;

(B) 

Uniformly stable, if for any $\epsilon >0$ there is $\delta \left(\epsilon \right)>0$ independent of a such that $\left|x\right(t;a,\phi ,\mathbf{0}\left)\right|\le \epsilon$ for any initial function $\phi \in {\Phi }_{\delta }$, $a\in \mathbb{R}$, $t\in J_a$;

(C) 

Asymptotically stable, if it is stable and for a given $a\in \mathbb{R}$ there is $\overline{\delta }\left(a\right)>0$, such that ${\displaystyle \underset{t\to \infty }{lim}x(t;a,\phi ,\mathbf{0})=\mathbf{0}}$ for any initial function $\phi \in {\Phi }_{\overline{\delta }}$, and the set $\Omega \left(a\right)\subset \mathcal{PC}$ of initial functions φ for which ${\displaystyle \underset{t\to \infty }{lim}x(t;a,\phi ,\mathbf{0})=\mathbf{0}}$ is called the attraction domain of the zero solution for the initial time a;

(D) 

Uniformly asymptotically stable, if it is uniformly stable and there is ${\delta }^{*}>0$ such that, for any $\gamma >0$ there is $t\left(\gamma \right)>0$ such that $\left|x\right(t;a,\phi ,\mathbf{0}\left)\right|\le \gamma$ for any $a\in \mathbb{R}$, $t\ge a+t\left(\gamma \right)$ and $\phi \in {\Phi }_{{\delta }^{*}}$. It clear that ${\Phi }_{{\delta }^{*}}\subseteq \Omega \left(a\right)$ for all $a\in \mathbb{R}$.

Theorem 7.

Let conditions 1 and 2 of Corollary 5 hold.

Then, for system (3) the following three statements are equivalent:

(i) 

The zero solution $x(t;a,\mathbf{0},\mathbf{0})=\mathbf{0}$ of the system (3) is stable;

(ii) 

All solutions of the system (3) are bounded;

(iii) 

For any $a\in \mathbb{R}$, there exists a constant $C\left(a\right)>0$ such that $\parallel S(t,a)\parallel \le C\left(a\right)$, $t\in J_a$.

Proof. 

(i) → (ii). Assume the contrary, that there exists an initial function $\tilde{\phi }\in \mathcal{PC}$ such that the corresponding solution $\tilde{x}(t;a,\tilde{\phi },\mathbf{0})$ of IP (3), (2) is unbounded and let $\epsilon >0$ be arbitrary. Then, there exists $\delta (a,\epsilon )>0$ such that for any $\phi \in \mathcal{PC}$ with $\parallel \phi \parallel <\delta$ we have that $\left|x\right(t;a,\phi ,\mathbf{0}\left)\right|\le \epsilon$ for $t\in J_a$, where $x(t;a,\phi ,\mathbf{0})$ is the corresponding solution of IP (3), (2). Introduce the initial function $\overline{\phi }\left(t\right)=\delta 2{\parallel \tilde{\phi }\parallel }^{-1}\tilde{\phi }\left(t\right)\in \mathcal{PC}$ and then we have that ${\displaystyle \parallel \overline{\phi }\left(t\right)\parallel =\delta 2\parallel \tilde{\phi }{\parallel }^{-1}\parallel \tilde{\phi }\left(t\right)\parallel \le \frac{\delta }{2}\le \delta }$. Since the system (3) is linear, then the function $\overline{x}(t;a,\overline{\phi },\mathbf{0})=\delta 2{\parallel \tilde{\phi }\parallel }^{-1}\tilde{x}(t;a,\tilde{\phi },\mathbf{0})$ is a solution of IP (3), (2) with initial function $\overline{\phi }\left(t\right)$. Then, we have that $|\overline{x}(t;a,\overline{\phi },\mathbf{0})|\le \epsilon$ for any $t\in J_a$ and hence we have $|\tilde{x}(t;a,\tilde{\phi },\mathbf{0})| =2{\delta }^{-1}\parallel \tilde{\phi }\parallel |\overline{x}(t;a,\overline{\phi },\mathbf{0})|\le 2{\delta }^{-1}\parallel \tilde{\phi }\parallel \epsilon$, which contradicts with our assumption.

(ii)→(iii). Since each solution of the system (3) is bounded, then for any $a\in \mathbb{R}$ and $\phi \in \mathcal{PC}$ there exists a constant $C(a,\phi )>0$, such that $\parallel S(t,a)\phi \parallel \le C(a,\phi )$ for any $t\in J_a$ and hence the Banach–Steinhaus theorem implies that there exists a constant $C\left(a\right)>0$ and $\parallel S(t,a)\parallel \le C\left(a\right)$ for any $t\in J_a$.

(iii) → (i). Let $\epsilon >0$ be arbitrary and then for any $\phi \in \mathcal{PC}$ we have that

$$\parallel S(t,a)\phi \parallel \le \parallel S(t,a)\parallel \parallel \phi \parallel \le C\left(a\right)\parallel \phi \parallel$$

for any $t\in J_a$. Choosing ${\displaystyle \delta =\frac{\epsilon }{C\left(a\right)}}$, we obtain that for each $t\in J_a$ and any $\parallel \phi \parallel <\delta$ we have that $\parallel S(t,a)\phi \parallel <\epsilon$ and hence the zero solution $x(t;a,\mathbf{0},\mathbf{0})$ of system (3) is stable. □

Corollary 6.

Let conditions 1 and 2 of Corollary 5 hold and the zero solution $x(t;a,\mathbf{0},\mathbf{0})$ of the system (3) be stable.

Then, there exists a constant $\overline{C}\left(s\right)>0$ for each $s\in \mathbb{R}$ such that $\left|C(t,s)\right|\le \overline{C}\left(s\right)$.

Proof. 

According Theorem 7, we have that (iii) holds. Then, from (iii) it follows that there exists $C\left(s\right)>0$ such that $\parallel S(t,s)\parallel \le C\left(s\right)$ for any $t\ge s$. Then, from (25) it follows that for $t\ge s$ the inequalities $|C_t(\theta ,s)|\le |C(t+\theta ,s)|\le \parallel S(t,s)\parallel \parallel C_0\left(\theta \right)\parallel \le n\parallel S(t,s)\parallel \le nC\left(s\right)$ hold. □

Remark 5.

We emphasise that in contrast with the integer case, the assertion of Corollary 6 is not equivalent with assertions (i)–(iii) of Theorem 7 and this is caused from the singular kernels of the fractional operators that impact on the boundedness in t of the operator $K(t,s)$ defined via (14). In the integer case when the operator $K(t,s)$ is bounded in t, the statement of Corollary 6 is equivalent to those of Theorem 7, but in the fractional case this is not enough and we must suppose that $u\left(t\right)=O\left(t^{-q_M}\right)$, which is illustrated via Example 1 in [25].

Theorem 8.

Let conditions 1 and 2 of Corollary 5 hold.

Then, for the system (3) the following three statements are equivalent:

(i) 

The zero solution $x(t;a,\mathbf{0},\mathbf{0})=\mathbf{0}$ of the system (3) is uniformly stable;

(ii) 

All solutions of the system (3) are uniformly bounded;

(iii) 

There exists a constant $C^{*}>0$ for any $a\in \mathbb{R}$, such that $\parallel S(t,a)\parallel \le C^{*}$, $t\in J_a$.

Corollary 7.

Let conditions 1 and 2 of Corollary 5 hold and the zero solution $x(t;a,\mathbf{0},\mathbf{0})$ of the system (3) be uniformly stable. Then, there exists a constant $\overline{C}>0$ for each $s\in \mathbb{R}$ such that $\left|C(t,s)\right|\le \overline{C}$.

The proofs of Theorem 8 and Corollary 7 are very similar to the corresponding proofs of Theorem 7 and Corollary 6 and that is why they will be omitted.

We will illustrate Theorem 7 with the following example:

Example 1.

Let $U_j^i(t,\theta )\equiv U_{ac}(t,\theta )\equiv U_s(t,\theta )\equiv \Theta$ for $i\ge 1$, $(t,\theta )\in [0,\infty )\times \mathbb{R}$, $h=1$, $A=\left(\begin{array}{cc}1& 0\\ 0& 2\end{array}\right)$, $U_j^0(t,\theta )=AH(\theta +1)$ and $q_1=\cdots =q_n=0,5\in (0,1]$. Then, system (3) obtains the form

$$\begin{array}{c}{D}_{0+}^{0,5}\left(\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 2\end{array}\right)\left(\begin{array}{c}{x}_1(t-1)\\ x_2(t-1)\end{array}\right)\mathrm{or}\left\{\begin{array}{cc} \hfill & D_{0+}^{0,5}{x}_1\left(t\right)=x_1(t-1)\hfill \\ & D_{0+}^{0,5}{x}_2\left(t\right)=2x_2(t-1)\hfill \end{array}\right.\end{array}$$

(31)

and the equation

$$det(z^{0,5}I-A{e}^{-z})=0$$

(32)

is the corresponding characteristic equation of the system (31). It is well known that the necessary and sufficient condition for the zero solution of system (31) to be stable (asymptotically stable) is all roots $z\in \mathbb{C}$ of (32) to have non-positive (negative) real parts. According to Corollary 3 in [26], if $|arg\left(z^{0,5}\right)| \ge {\displaystyle \frac{\pi }{4}}$ then all roots of (32) have non-positive real parts, which implies that the zero solution of system (31) is stable. Moreover, since (32) does not have pure imaginary roots and if $|arg\left(z^{0,5}\right)| >{\displaystyle \frac{\pi }{4}}$, then all roots $z\in \mathbb{C}$ of (32) have only negative real parts and hence the zero solution of system (31) will be asymptotically stable.

For any fixed $s\in [-1,0]$, the matrix equation

$$\begin{array}{c}{D}_{0+}^{0,5}Q(t,s)=\left(\begin{array}{cc}1& 0\\ 0& 2\end{array}\right)Q(t-1,s)\end{array}$$

with the initial condition $Q(t,s)\equiv \Theta ,t\in (-\infty ,s)$ and $Q(t,s)=I$ for any $t\in [s,0]$ possess a unique solution on the interval $t\in [0,\infty )$, which is a matrix-valued function $t\to Q(t,s):[0,\infty )\to {\mathbb{R}}^{2\times 2}$ (see [20] concerning the existence and other properties). Then, for any initial function $\phi \in \mathcal{PC}\cap BV([-1,0],{\mathbb{R}}^2)$ the corresponding solution $x\left(t\right)={(x_1(t,x_2\left(t\right))}^T$ of the IP (31), (2) has the representation $x\left(t\right)=\underset{-1}{\overset{0}{\int }}Q(t,s){\mathrm{d}}_s\phi \left(s\right)$ and ${\displaystyle \underset{t\in [0,\infty )}{sup}\left(\underset{s\in [-1,0]}{sup}Q(t,s)\right)<\infty }$ if the zero solution of system (31) is stable. Then, in the same way as in the proof of Theorem 7 we can establish that the statements of Theorem 7 hold for system (31).

5. Comments and Conclusions

In the present work a Cauchy (initial) problem for a fractional linear system with distributed delays, Caputo-type derivatives of incommensurate order and discontinuous initial functions was considered.

As a main result, first, a new straightforward approach was introduced to study the considered initial problem via an equivalent Volterra–Stieltjes integral system, which allowed us to obtain an integral representation of the solutions of the Cauchy problem for the studied class system. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system; this fact allowed us to prove that the corresponding resolvent system possesses a unique measure resolvent kernel. The proof in [22] of the problem of the existence of a measure resolvent $R(t,s)$ is based on non-trivial Banach algebra techniques. Here, we have introduced a new approach for establishing this existence, without the necessity to use these techniques. In our point of view, an approach based on the existence and uniqueness of a global fundamental matrix for the system (3) is not too complicated, since it is based on a well-known and often studied problem. The key result in our approach is the established relation between the global fundamental matrix of the homogeneous system (3) and the measure resolvent kernel $R(t,s)\in B_{loc}^{\infty }$ for the measure kernel $K(t,s)\in B_{loc}^{\infty }$.

As a consequence of this result, we have obtained an integral representation of the solutions of the studied systems. The integral representation of the solutions of the considered Cauchy problem obtained via the introduced approach is applicable even in the case when the initial functions are only locally bounded and Lebesgue measurable. With (30) we have established an equivalent, but more applicable version of the obtained integral representation (22) in cases when we interpret $\mathcal{PC}$ as state space. The interpretation that uses $\mathcal{PC}$ as state space is nowadays widely accepted and it has proven fruitful in many sections of the theory, especially sections related to asymptotic behaviour and periodicity of solutions. Analysing the operators appearing in the second addend in the right side in (30), we can conclude that in the fractional case the situation is essentially different compared with these in the integer case. More preciously speaking, it is clear that the fractional case is more complicated in comparison with the integer case and cannot be realised as in the integer case via the one-parameter family of compact linear operators parametrised via $t\in J_a$, each of them allowing representation as a vector-valued Lebesgue–Stieltjes integral.

Then, using the integral representations obtained in the previous section, we have obtained relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions. We emphasise that in contrast with the integer case, the assertion of Corollary 6 is not equivalent with assertions (i)–(iii) of Theorem 7 and this is caused from the singular kernels of the fractional operators that impact on the boundedness of the operator defined via (14). In the integer case, this operator is bounded on t and then the assertion of Corollary 6 is equivalent to those of Theorem 7, but in the fractional case this is not enough as is illustrated via the example in [25].