The Perron Condition for Delayed Systems with Caputo Fractional Derivatives

Abstract

In the present work, we study a class of nonhomogeneous linear systems with fractional derivatives in Caputo’s sense of incommensurate order and distributed delays, satisfying the Perron condition. More precisely we study the impact of the Perron condition on the boundedness of the fundamental and extended fundamental matrices of the corresponding homogeneous system. We prove that if the nonhomogeneous system satisfies the Perron condition, then the fundamental matrix $C(t,s)$ and the extended fundamental matrix $Q(t,s)$ of the homogeneous system are uniformly bounded. As a consequence we obtain that the boundedness of the matrix $Q(t,s)$ is a necessary and sufficient condition for the uniform stability of the zero solution of the homogeneous system. Furthermore, we also prove that the extended fundamental matrix $Q(t,s)$ tends to zero when $t\to \infty$, which is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution of the homogeneous system under study.

1. Introduction

Analysis of practical experience in modelling many real-world phenomena shows that many of them can be represented more accurately by taking into account memory data incorporated through the nonlocal nature of the fractional derivatives. Detailed information on the theory of fractional calculus and fractional differential equations can be found in the monographs of Kilbas et al. [1], Podlubny [2], and Hilfer [3]. Fractional differential equations of distributed order are well considered in the book of Jiao et al. [4]. A good overview of the topic of fractional impulsive differential equations and their applications can be found in the monograph of Stamova and Stamov [5].

In 1930, in ref. [6], O. Perron introduced his famous condition for inhomogeneous linear systems with first-order derivatives of the form ${\displaystyle \frac{dx}{dt}=A\left(t\right)x\left(t\right)+f\left(t\right)}$, where the elements of the $n\times n$, $n\in \mathbb{N}$ dimensional matrix $A\left(t\right)$ are real and measurable functions for $t\ge 0$. The condition states that if for any continuous and bounded forcing term $f\left(t\right)$ the corresponding solution of the Cauchy problem with $x\left(0\right)=0$ is also a bounded function for $t\ge 0$, then if the matrix $A\left(t\right)$ is a bounded function for $t\ge 0$, then this condition is equivalent to the uniform asymptotic stability of the zero solution of the homogeneous system.

The Perron condition has also been widely studied for the case of linear functional-differential systems with first-order derivatives (see Halanay [7], and Kolmanovskii et al. [8]). To our knowledge, the influence of the Perron condition on the stability properties even for a class of linear fractional systems (equations) with delay has not been studied. This article is devoted to filling this gap.

The survey [9] and the references cited therein provide a good overview of contemporary research in the field of stability properties for the fractional delayed equations and systems. From the recent works devoted to this topic we refer to the articles [10,11,12] devoted to stability results that are important from our point of view.

In the present work, we consider a nonhomogeneous linear system with incommensurate fractional derivatives in the sense of Caputo and distributed delays. For this system we introduce the Perron condition and study its influence on the asymptotic properties of the corresponding homogeneous system.

The paper is organized as follows. As is standard in Section 2, we present an overview of the definitions of the left Riemann–Liouville fractional integral, Riemann–Liouville and Caputo derivatives, and some of their basic properties. In addition, a statement of the problem under study is given, together with some necessary comments and results used later. In Section 3, our main results are presented. We study the influence of the Perron condition on the boundedness behaviour of the fundamental matrix $C(t,s)$ and the extended fundamental matrix $Q(t,s)$ for the homogeneous system, when the corresponding nonhomogeneous delayed system satisfies the Perron condition. We first establish that the extended fundamental matrix $Q(t,s)$ is uniformly bounded, and then that it tends uniformly to zero when $t\to \infty$. It is also proved (under some natural conditions, as in the classical case) that this result is a necessary and sufficient condition for the zero solution of the homogeneous system under study to be uniformly asymptotically stable. As usual, the last section, Section 4, is dedicated to some comments and conclusions.

2. Preliminaries and Problem Statement

In this section, for readers’ convenience, following the monographs [1,2], we present the notations used and recall the definitions of Riemann–Liouville and Caputo fractional derivatives, with some of their needed properties. Moreover, we also expose some auxiliary results that are used in the exposition below.

With $L_1^{loc}(\mathbb{R},{\mathbb{R}}^n)$ we denote the linear spaces of all locally Lebesgue integrable functions $f:\mathbb{R}\to \mathbb{R}$ and by $B{L}_1(\mathbb{R},\mathbb{R})\subset B{L}_1^{loc}(\mathbb{R},\mathbb{R})\subset L_1^{loc}(\mathbb{R},\mathbb{R})$ are denoted the subspaces of all bounded and all locally bounded functions.

For arbitrary $a\in \mathbb{R}$ and any $g\in L_1^{loc}(\mathbb{R},\mathbb{R})$, we define the left-sided fractional integral operator, Riemann–Liouville and Caputo fractional derivatives of order $q\in (0,1]$ via the equalities

$$\left(I_{a+}^{q}g\right)\left(t\right)={\displaystyle \frac{1}{\Gamma \left(q\right)}}{\displaystyle \underset{a}{\overset{t}{\int }}}{(t-\nu )}^{q-1}g\left(\nu \right)d\nu ,{}^{RL}D_{a+}^{q}g\left(t\right)={\displaystyle \frac{d}{dt}}\left(I_{a+}^{1-q}g\right)\left(t\right),$$

$${}^{C}D_{a+}^{q}g\left(t\right)={}^{RL}D_{a+}^q\left[g\left(\nu \right)-g\left(a\right)\right]\left(t\right),t>a$$

$$D_{a+}^{0}g\left(t\right)=g\left(t\right).$$

The notations ${\mathbb{R}}_{+}=(0,\infty )$ ${\overline{\mathbb{R}}}_{+}=[0,\infty )$, $J_a=[a,\infty )$, $a\in \mathbb{R}$, $q=(q_1,\dots ,q_n)$, $q_k\in (0,1]$ with $k\in ⟨n⟩=\{1,\dots ,n\}$, $q_M\in \underset{k\in ⟨n⟩}{max}{q}_k$, will be used too; the real linear space of the square matrices with dimension $n\in \mathbb{N}$ is denoted by ${\mathbb{R}}^{n\times n}$, where $\mathbf{I},\Theta \in {\mathbb{R}}^{n\times n}$ are the identity and zero matrices, respectively, and $\mathbf{0}\in {\mathbb{R}}^n$ denotes the zero vector column.

For $W(t,\theta )={\left\{w_{kj}(t,\theta )\right\}}_{k,j=1}^n:\times \to {\mathbb{R}}^{n\times n}$ for any fixed $(t,\theta )\in {\mathbb{R}}^2$ we denote by $\left|W(t,\theta )\right|={\displaystyle \sum _{k,j=1}^n}\left|w_{kj}(t,\theta )\right|$ the following matrix norm in ${\mathbb{R}}^{n\times n}$ and by $B{V}^{loc}(\mathbb{R}\times \mathbb{R},{\mathbb{R}}^{n\times n})=B{V}^{loc}$ 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 any $t\in \mathbb{R}$, i.e., $Va{r}_{[c,d]}W(t,·)={\displaystyle \sum _{k,j=1}^n}Va{r}_{[c,d]}{w}_{kj}(t,·)$. By $BC(\mathbb{R},{\mathbb{R}}^n)$ ($BAC(J_a,{\mathbb{R}}^n)$) we denote the linear space of the bounded continuous (bounded absolutely continuous) vector-valued functions and for any vector $y\left(t\right)={(y^1\left(t\right),\dots ,y^n\left(t\right))}^T:\mathbb{R}\to {\mathbb{R}}^n$ we use the notation $I_{\beta }\left(y\left(t\right)\right)=diag({\left(y_1\left(t\right)\right)}^{{\beta }_1},\dots ,{\left(y_n\left(t\right)\right)}^{{\beta }_n})$, $\beta =({\beta }_1,\dots ,{\beta }_n)$, ${\beta }_k\in [-1,1]$, $k\in ⟨n⟩$.

Let $a\in \mathbb{R}$, $h>0$ be arbitrary fixed numbers and consider for $t>a$ the inhomogeneous linear system with Caputo-type fractional derivatives of incommensurate orders and distributed delays

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

(1)

where $D_{a+}^q=diag({}_{C}D_{a+}^{q_1},\dots ,{}_{C}D_{a+}^{q_n})$, $D_{a+}^{q_k}{x}_k\left(t\right)={}^{C}D_{a+}^{{\alpha }_k}{x}_k\left(t\right),f\left(t\right)={(f_1\left(t\right),\dots ,f_n\left(t\right))}^T$ and $x\left(t\right)={(x_1\left(t\right),\dots ,x_n\left(t\right))}^T:\mathbb{R}\to {\mathbb{R}}^n$ (the symbol T used throughout the text means transposition), $x_t\left(\theta \right)=x(t+\theta )$, $-h\le \theta \le 0$, $U:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$, $k\in ⟨n⟩=\{1,\dots ,n\}$ and $f\in L_1^{loc}(\mathbb{R},{\mathbb{R}}^n)$.

The Banach space $\mathbf{BL}$ of initial vector functions

$$\phi ={\left({\phi }_1,\dots ,{\phi }_n\right)}^T:[-h,0]\to {\mathbb{R}}^n$$

consists of all locally bounded and Lebesgue measurable functions on the interval $[-h,0]$ vector functions, with norm $\left|\right|\phi \left|\right|={\displaystyle \sum _{k=1}^n}\underset{s\in [-h,0]}{sup}\left|{\phi }_k\left(s\right)\right|<\infty$. The real linear space $P{C}_{loc}$ ($PA{C}_{loc}$) consists of all vector functions $\phi ={\left({\phi }_1,\dots ,{\phi }_n\right)}^T:\mathbb{R}\to {\mathbb{R}}^n$ which are piecewise continuous (piecewise absolutely continuous) over any compact interval $[a,b]\subset \mathbb{R}$. This means that any function can be broken down into a finite number of separate continuous (absolutely continuous) pieces. In addition, $\mathbf{PC}=P{C}_{loc}([-h,0],{\mathbb{R}}^n)$ and $\mathbf{PAC}=PA{C}_{loc}([-h,0],{\mathbb{R}}^n)$.

For any fixed $a\in \mathbb{R}$ we introduce an initial condition for each $\phi \in \mathbf{BL}$ as follows:

$$x_t\left(\theta \right):=x(t+\theta ):=\phi \left(\theta \right)\mathrm{for}t+\theta \le a,\mathrm{where}-h\le \theta \le 0.$$

(2)

The detailed view of the corresponding homogeneous system of (1) (i.e., $f\left(t\right)\equiv \mathbf{0},t\in \mathbb{R}$) has the form

$$D_{a+}^q{x}_k\left(t\right)=\sum _{i\in {⟨m⟩}_0}\left(\sum _{j\in ⟨n⟩}{\displaystyle \underset{-h}{\overset{0}{\int }}}{x}_j(t+\theta )d_{\theta }{u}_{k,j}^i(t,\theta )\right),k\in ⟨n⟩.$$

(3)

In our exposition below we will suppose that for any fixed $t\in \mathbb{R}$ the kernel $U(t,·)\in B{V}^{loc}$ in $\theta$ and its Lebesgue decomposition has the form

$$U(t,\theta )=U^{ju}(t,\theta )+U^{ac}(t,\theta )+U^s(t,\theta ),$$

where $U^{ju}(t,\theta ),U^{ac}(t,\theta ),U^s(t,\theta )$ are the jump, absolutely continuous, and singular parts in the decomposition, respectively, with $U^{ju}(t,\theta )={\displaystyle \sum _{i\in {⟨m⟩}_0}}{U}_i^{ju}(t,\theta )$, $i\in {⟨m⟩}_0=\left\{0,1,\dots ,m\right\},m\in \mathbb{N}$.

Remark 1. 

The approach based on the Lebesgue decomposition of the kernel was introduced in the middle of the twentieth century form the leading specialists in the area of functional-differential equations, A. Myshkis [13] and J. Hale [14]. It is accepted worldwide from the specialists in this area, especially for systems with distributed delay and integer (mainly first order) derivatives, as well as for systems with distributed delays and fractional derivatives. This approach allows us to formulate more precisely the needed conditions for the investigated systems with different types of delays, and helps to compare the new conditions with those used for the proof of the classical results for first-order derivatives.

The conditions (S) given in the next definition explicitly illustrate and clarify the importance and the applicability of the approach based on the Lebesgue decomposition of the kernels.

Definition 1 

([8]). We say that the conditions (S) for the kernel $U:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ hold when for each $(t,\theta )\in \mathbb{R}\times \mathbb{R}={\mathbb{R}}^2$ and for any partition $0={\sigma }_0<{\sigma }_1<{\sigma }_2<\dots <{\sigma }_m=h$ of $[0,h]$ the following conditions are fulfilled:

(S1). 

The functions $U_i^{ju}(t,\theta ),U^{ac}(t,\theta ),U^s(t,\theta )$, $i\in {⟨m⟩}_0,m\in \mathbb{N}$ are measurable in $\mathbb{R}\times \mathbb{R}$, left continuous in θ on $(-h,0)$ and normalized so, that $U_i^{ju}(t,\theta )=U^{ac}(t,\theta )=U^s(t,\theta )=\Theta$ for $\theta \ge 0$, $U_i^{ju}(t,\theta )=U_i^{ju}(t,-{\sigma }_i)$ for $\theta \le -{\sigma }_i$, $U^{ac}(t,\theta )=U^{ac}(t,-h)$ and $U^s(t,\theta )=U^s(t,-h)$ for $\theta \le -h$.

(S2). 

The functions $u^{ju}\left(t\right),u^{ac}\left(t\right),u^s\left(t\right)\in B{L}_1(J_a,{\overline{\mathbb{R}}}_{+})$, where

$$u^{ju}\left(t\right)=\sum _{i\in ⟨m⟩}{u}_i^{ju}\left(t\right)=\sum _{i\in {⟨m⟩}_0}\underset{\eta \in [a,t]}{sup}\left|Va{r}_{[-h,0]}{U}_i^{ju}(\eta ,·)\right|,$$

$$u^{ac}\left(t\right)=\underset{\eta \in [a,t]}{sup}\left|Va{r}_{[-h,0]}{U}^{ac}(\eta ,·)\right|,u^s\left(t\right)=\underset{\eta \in [a,t]}{sup}\left|Va{r}_{[-h,0]}{U}^s(\eta ,·)\right|$$

and $u\left(t\right)=u^{ju}\left(t\right)+u^{ac}\left(t\right)+u^s\left(t\right)$.

(S3). 

$U_i^{ju}(t,\theta )={\left\{a_i^{kj}\left(t\right)H(\theta +{\sigma }_i)\right\}}_{k,j=1}^n$, $A_i\left(t\right)={\left\{a_i^{kj}\left(t\right)\right\}}_{k,j=1}^n\in B{L}_1(\mathbb{R},{\mathbb{R}}^{n\times n})$, $H\left(t\right)$ is the Heaviside function and the matrix ${\displaystyle \frac{\partial U^{ac}(·,\theta )}{\partial \theta }}=\partial U^{ac}(·,\theta )=A_{*}(·,\theta )\in B{L}_1^{loc}(\mathbb{R},{\mathbb{R}}^{n\times n}).$

(S4). 

The following relations hold [13]:

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

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

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

for each $t^{*}\in \mathbb{R}$.

For any $\phi \in \mathbf{BL}$ we denote the set of all its jump points by $S_{\phi }$. It is clear that for any $\phi \in \mathbf{PC}\left(\mathbf{PAC}\right)$ the set $S_{\phi }$ is at most finite.

Remark 2. 

The integral in (1) is understood in the Lebesgue–Stieltjes sense and then for any $\phi \in \mathbf{BL}$ used in the initial condition (2) we mean that the function $t\to x\left(t\right)$, $t\in J_a$ is considered as a prolongation of the function $t\to \phi (t-a)$, for $t\in J_a$. Moreover, it is clear that for the initial problem (IP) (1), (2) with arbitrary initial point $a\in \mathbb{R}$ it is necessary that the conditions (S) must be fulfilled only when $t\in J_a$.

Definition 2. 

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

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

$$x\left(t\right)=\phi \left(0\right)+I_{-1}(\Gamma \left(q\right))({\displaystyle \underset{a}{\overset{t}{\int }}}{I}_{q-1}(t-s){\displaystyle \underset{-h}{\overset{0}{\int }}}\left[d_{\theta }U(s,\theta )\right]x(s+\theta ))ds+{\displaystyle \underset{a}{\overset{t}{\int }}}{I}_{q-1}(t-s)f\left(s\right)ds),$$

(4)

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

Definition 3. 

The vector function $x\left(t\right)={(x_1\left(t\right),...,x_n\left(t\right))}^T$ 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 when the initial function $\phi$ is discontinuous, i.e., $\phi \in \mathbf{BL}$ stated in the next theorem and its corollaries below:

Theorem 1 

([15]). Let $a\in \mathbb{R}$ be an arbitrary fixed initial point and the following conditions are fulfilled:

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

Then, the following statements hold:

• Every solution $x(t;a,\phi ,f)$ of IP (1), (2) is a solution of the IP (4), (2) and vice versa.
• For any initial function $\phi \in \mathbf{BL}$ and $f\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$ the IP (4), (2) has a unique solution in $J_a$.

Consider the following two matrix initial problems (IP):

(C) Let ${\phi }^C\left(\theta \right)={\left\{{\phi }_k^j\left(\theta \right)\right\}}_{k,j\in ⟨n⟩}\in PAC([-h,0],{\mathbb{R}}^{n\times n})$ with ${\phi }^C\left(0\right)=\mathrm{I},{\phi }^C\left(\theta \right)=\Theta ,\theta \in (-\infty ,0)$ and ${\phi }^{C_k}\left(\theta \right)$ denotes the k-th column of ${\phi }^C\left(\theta \right)$. For any fixed $s\in \mathbb{R}$ with $s\ge a$ introduce for $t>s$ the matrix system

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

(5)

with the initial condition

$$C_t(\theta ,s):=C(t+\theta ,s):={\phi }^C\left(\theta \right),t+\theta \le s,\theta \in [-h,0]$$

(6)

(Q) For any fixed $s\in [a-h,a]$ consider the initial function with ${\phi }_s^Q\left(\theta \right)=\mathrm{I},\theta \in [s,0],{\phi }_s^Q\left(\theta \right)=\Theta ,\theta \in (-\infty ,s)$, where ${\phi }_s^{Q_k}\left(\theta \right)$ denotes the k-th column of ${\phi }_s^Q\left(\theta \right)$ and introduce for $t>a$ the matrix system as follows:

$$\begin{array}{cc}\hfill Q(t,s)& =Q(t,s,{\phi }_s^Q,\mathbf{0})\hfill \\ \hfill & ={\mathbf{1}}_{\left\{s\le t\le a\right\}}\mathbf{I}+I_{-1}(\Gamma \left(q\right)){\displaystyle \underset{a}{\overset{t}{\int }}}{I}_{q-1}(t-\tau ){\displaystyle \underset{-h}{\overset{0}{\int }}}\left[d_{\theta }U(\tau ,\theta )\right]Q(\tau +\theta ,s)d\tau ,t\in (a,\infty ),\hfill \end{array}$$

(7)

with the initial condition

$$Q_t(\theta ,s):=Q(t+\theta ,s):={\phi }_s^Q\left(\theta \right),t+\theta \le a,\theta \in [-h,0].$$

(8)

Remark 3. 

Note that $C(t,s)$ is a unique solution of the IP (5), (6) for any fixed $s\ge a$ as an initial point, if and only if its columns $C^j(t,s)$, $j\in ⟨n⟩$ are the unique solutions of IP (3), (2) with initial condition $C^j(s,s)=I^j$ and $C^j(t,s)=\mathbf{0}$ for $t<s$, where $I^j$ denote the j-th column of $\mathrm{I}$.

Analogically $Q(t,s)$ is a unique solution of the IP (7), (8) for any fixed $s\in [a-h,a]$ if and only if its columns $Q^j(t,s)$, $j\in ⟨n⟩$ are the unique solutions of IP (3), (2) with initial point $a\in \mathbb{R}$ and initial condition $Q^j(t,s)=I^j$ for $a\ge t\ge s$ and $Q^j(t,s)=\mathbf{0}$ for $t\in (s,\infty )$.

Definition 4 

([16]). If for any fixed $s\in \mathbb{R}$ ($s\in [a-h,a]$) the matrix function $C(t,s)$ ($Q(t,s)$) is the unique solution of the IP (5), (6) (IP (7), (8)) then it will be called a fundamental (extended fundamental) matrix for the system (3). Everywhere below we will suppose that $C(t,s)\equiv \Theta$ when $t\in (-\infty ,s)$. It is clear that when $s=a$ we have that $Q(t,a)\equiv C(t,a)$ for $t\in J_a$.

Below as a corollary we summarize the important properties (for details, see [16,17,18]) of these matrices, which will be used in our exposition.

Corollary 1 

([17]). Let the conditions (S) be fulfilled.

Then the following statements hold:

• $C(·,s)\in AC(J_s,{\mathbb{R}}^{n\times n})$ for any fixed $s\ge a\in \mathbb{R}$ as an arbitrary initial point and has a jump of the first kind when $t=s$.
• $Q(·,s)\in AC(J_a,{\mathbb{R}}^{n\times n})$ for each fixed $s\in [a-h,a]$, when $t\in [a-h,a]$ and has a jump of the first kind for $t=s$.
• $C(t,·):J_a\to {\mathbb{R}}^{n\times n}$ is continuous at $s\in (a,t)\cup (t,\infty )$, right continuous at $s=a$ and left continuous at $s=t$. In addition, $C(t,·)\in B{V}^{loc}(\mathbb{R},{\mathbb{R}}^{n\times n})$ (i.e., has bounded variation in s on every compact interval $[a,T]$, $T\in {\mathbb{R}}_{+}$).
• $Q(t,·):[a-h,a]\to {\mathbb{R}}^{n\times n}$ is locally bounded and Lebesgue measurable in s.

Remark 4. 

In addition we will use the results in [16,17,18] from which it follows that for any initial function $\phi \in \mathbf{BL}$ with $\phi \in BV([-h,0],{\mathbb{R}}^n)$ or with $\phi \in \mathbf{PAC}=PAC([-h,0],{\mathbb{R}}^n)$ the IP (3), (2) has a unique solution $x(t;a,\phi ,\mathbf{0})\in B{V}^{loc}(J_a,{\mathbb{R}}^n)$ or respectively $x(t;a,\phi ,\mathbf{0})\in A{C}^{loc}(J_a,{\mathbb{R}}^n)$.

For the reader’s convenience, we add as corollary the statement of Corollary 2 in [19] which is an important generalization of the well-known Bellman–Gronwall’s lemma for the case of fractional derivatives.

Corollary 2 

([19]). Suppose that $\beta \in (0,1)$ and the following conditions hold:

(i) 

The functions $\alpha \left(t\right),g\left(t\right),y\left(t\right)\in L_1^{loc}(J_a,{\mathbb{R}}_{+})$

(ii) 

The functions $\alpha \left(t\right),g\left(t\right)$ are nondecreasing and $g\left(t\right)$ is bounded on $J_a$.

(iii) 

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

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

At the end of this section, for the reader’s convenience, we will also recall the definitions of the different types of stability.

Definition 5 

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

(A) 

Stable, if for a given $a\in \mathbb{R}$ and for any $\epsilon >0$, there is $\delta \left(\epsilon \right)>0$ such that $\left|x\left(t\right)\right|=\left|x(t,a,\phi ,\mathbf{0})\right|\le \epsilon$ for each initial function $\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n)$ 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 $\left|x\left(t\right)\right|=\left|x(t,a,\phi ,\mathbf{0})\right|\le \epsilon$ for any initial function $\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n)$, $a\in \mathbb{R}$, $t\in J_a$;

(C) 

Asymptotically stable, if it is stable and for a given $a\in \mathbb{R}$ there is $\delta \left(a\right)>0$, such that $\underset{t\to \infty }{lim}\left|x(t,a,\phi ,\mathbf{0})\right|=0$ for any initial function $\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n)$;

(D) 

Uniformly asymptotically stable, if it is uniformly stable and $\underset{t\to \infty }{lim}\left|x(t,a,\phi ,\mathbf{0})\right|=0$.

3. Main Results

The main goal in this section is to study the behaviours of the homogeneous linear delayed system (3) when the corresponding inhomogeneous delayed system (1) satisfies the Perron condition. We will study in more detail the influence of the Perron condition on the boundedness of the fundamental and extended fundamental matrix of (3). First we will establish that the Perron condition implies uniform boundedness of $C(t,s)$. Then we will prove that the uniform boundedness of the fundamental matrix $C(t,s)$ implies that the extended fundamental matrix $Q(t,s)$ is also uniformly bounded. As a consequence it will be proven that the uniform boundedness of $Q(t,s)$ is a necessary and sufficient condition for the stability of the zero solution of the studied homogeneous systems. Furthermore we will establish that a necessary and sufficient condition for uniform asymptotic stability of the zero solution of the studied homogeneous system is $Q(t,s)$ to tend to zero uniformly in s when $t\to \infty$.

We start our study with establishing under which conditions some well-known classical results in the case of delayed systems with first-order derivatives stay preserved in the fractional case too.

In view on the applications everywhere below we will assume that the kernel does not have singular part, i.e., $U^s(t,\theta )\equiv \Theta$.

Remark 5. 

There are two reasons to introduce the condition $U^s(t,\theta )\equiv \Theta$:

1. 

As far as we know, no satisfactory physical or practical interpretation has been established for the singular part of the Lebesgue decomposition.

This conclusion can be confirmed by analyzing both models with fractional derivatives and models with integer derivatives.

2. 

From a technical point of view, it is more convenient without the singular part since its partial derivative in θ generally is not Lebesgue integrable even on compact subintervals of $\mathbb{R}$. This leads to great difficulty in introducing additional conditions on the singular part that have scientific meaning in practical aspects.

Without the singular part, the kernel $U(t,\theta )$ can be rewritten as

$$U(t,\theta )=\sum _{i\in ⟨m_0⟩}{U}_i^{ju}(t,\theta )+U^{ac}(t,\theta )=\sum _{i\in ⟨m_0⟩}{A}_i\left(t\right)H(\theta +{\sigma }_i)+{\displaystyle \underset{0}{\overset{\theta }{\int }}}{A}_{*}(t,\xi )d\xi$$

(9)

Thus, for $t\in J_a$ taking into account (9), the system (1) obtains the form

$$\begin{array}{ccc}\hfill D_{a+}^{q}x\left(t\right)& & =\sum _{i\in ⟨m_0⟩}{A}_i\left(t\right)x(t-{\sigma }_i)+{\displaystyle \underset{-h}{\overset{0}{\int }}}{A}_{*}(t,\theta )x(t+\theta )d\theta +f\left(t\right)\hfill \\ \hfill & & \stackrel{t+\theta =\alpha }{=}\sum _{i\in ⟨m_0⟩}{A}_i\left(t\right)x(t-{\sigma }_i)+{\displaystyle \underset{t-h}{\overset{t}{\int }}}{A}_{*}(t,\alpha -t)x\left(\alpha \right)d\theta +f\left(t\right).\hfill \end{array}$$

(10)

Denoting by $B{C}_a(J_a,{\mathbb{R}}^n)$ ($BA{C}_a(J_a,{\mathbb{R}}^n)$) the linear space of all $f\left(t\right)\in BC(J_a,{\mathbb{R}}^n)$ ($f\left(t\right)\in BA{C}_a(J_a,{\mathbb{R}}^n)$) with $f\left(a\right)=\mathbf{0}$, it is clear that $BA{C}_a(J_a,{\mathbb{R}}^n)\subset B{C}_a(J_a,{\mathbb{R}}^n)$.

Definition 6 

([6]). We will say that the system (3) satisfies the condition of Perron if for every vector function $f\in BC(J_a,{\mathbb{R}}^n)$, the corresponding unique solution $x(t;a,\mathbf{0},f)$ of the IP (10), (3) with zero initial function is also bounded.

Theorem 2. 

Let the following conditions be fulfilled:

1. 

For the kernel $U(t,\theta )$ appearing in (1) and (3) the conditions (S) hold and $\underset{t\in J_a}{sup}\left|u\left(t\right)\right|=U<\infty$.

2. 

The system (1) satisfies the condition of Perron.

Then there exists a constant $C>0$ such that ${\displaystyle \underset{a}{\overset{t}{\int }}}\left|C(t,s)\right|ds\le C$ for $t\in J_a$.

Proof. 

In virtue of Theorem 4.3 in [16], we have that the IP (1), (2) with initial function $\phi \left(\theta \right)\equiv \mathbf{0},-h\le \theta \le 0$ has a unique solution that possesses an integral representation via the fundamental matrix $C(t,s)$ of the system (3), as follows

$$x_f\left(t\right)={\displaystyle \underset{a}{\overset{t}{\int }}}C(t,s){}^{RL}D_{a+}^{1-q}f\left(s\right)ds$$

(11)

for arbitrary $f\left(t\right)\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$. Note that since all columns $C(t,s)$ are solutions of the homogeneous system (3), then $C(t,s)$ obviously does not depend on the choice of the forcing term $f\left(t\right)$. Then from (11) and Corollary 1 it follows that

$$x_f\left(t\right)={\displaystyle \underset{a}{\overset{t}{\int }}}C(t,s){}^{RL}D_{a+}^{1-q}f\left(s\right)ds={\displaystyle \underset{a}{\overset{t}{\int }}}C(t,s)D_{a+}^{1-q}f\left(s\right)ds\in BA{C}_a(J_a,{\mathbb{R}}^n).$$

(12)

Introduce the norm ${∥·∥}_{[a,t]}=\underset{s\in [a,t]}{sup}\left|·\right]$ in $B{C}_a([a,t],{\mathbb{R}}^n)$ and the same norm in the subspace $BA{C}_a([a,t],{\mathbb{R}}^n)$ too.

Let $f\in BA{C}_a([a,t],{\mathbb{R}}^n)\subset B{C}_a([a,t],{\mathbb{R}}^n)$ be arbitrary and then from Theorem 2 in page 92, Formula (2.4.17) of ref. [1] for any $t\in J_a$ we have that

$$D_{a+}^{1-q}f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle \underset{a}{\overset{t}{\int }}}{(t-s)}^{q-1}{f}^{\prime }\left(s\right)ds:=\left(I_{a+}^q{f}^{\prime }\right)\left(t\right).$$

(13)

Using (13) we introduce the operator ${\Im }_t$ for every $t\in J_a$ as follows

$${\Im }_t\left(f\right)=\left(I_{a+}^q{f}^{\prime }\right)\left(t\right):BA{C}_a([a,t],{\mathbb{R}}^n)\to B{C}_a([a,t],{\mathbb{R}}^n),$$

(14)

and so, we define a collection of linear operators ${\left\{{\Im }_t\right\}}_{t\in J_a}$. Then, according to Lemma 2.1 in page 72 of ref. [1], we obtain that every operator ${\Im }_t\in {\left\{{\Im }_t\right\}}_{t\in J_a}$ defined via (14) is bounded with respect to the norm ${∥·∥}_{[a,t]}$. Since $BA{C}_a([a,t],{\mathbb{R}}^n)\subset B{C}_a([a,t],{\mathbb{R}}^n)$ is a subspace, according to the Hahn–Banach extension theorem, for any operator ${\Im }_t\in {\left\{{\Im }_t\right\}}_{t\in J_a}$ there exists a linear and bounded extension ${\tilde{\Im }}_t:B{C}_a([a,t],{\mathbb{R}}^n)\to B{C}_a([a,t],{\mathbb{R}}^n)$ of the operator ${\Im }_t$ with ${\tilde{\Im }}_t\left(f\right)={\Im }_t\left(f\right)$ for any $f\in BA{C}_a([a,t],{\mathbb{R}}^n)$ and ${∥{\tilde{\Im }}_t∥}_{[a,t]}={∥{\Im }_t∥}_{[a,t]}$.

For any fixed $t\in J_a$ and $f\in B{C}_a([a,t],{\mathbb{R}}^n)$ we define the operator

$${\Re }_t:B{C}_a([a,t],{\mathbb{R}}^n)\to B{C}_a([a,t],{\mathbb{R}}^n)$$

via the equality (11) as follows

$$\left({\Re }_{t}f\right)\left(t\right)=x_f\left(t\right)={\displaystyle \underset{a}{\overset{t}{\int }}}C(t,s)\left({\tilde{\Im }}_{t}f\right)\left(s\right)ds.$$

(15)

Since $C(t,s)$ is the fundamental matrix of (3) and $C(t,s)=\Theta$ for $s>t$, for any fixed $t\in J_a$ there exists a constant $C_t>0$ such that $\underset{s\in [a,t]}{sup}\left|C(t,s)\right|\le C_t$. Taking into account that ${\tilde{\Im }}_t$ is linear and bounded, from (15) for every $f\in B{C}_a([a,t],{\mathbb{R}}^n)$ it follows that

$$\left|\left({\Re }_{t}f\right)\left(t\right)\right|\le {\displaystyle \underset{a}{\overset{t}{\int }}}\left|C(t,s)\right|\left|\left({\tilde{\Im }}_{t}f\right)\left(s\right)\right|ds\le (t-a)C_t∥{\tilde{\Im }}_t∥{∥f∥}_{[a,t]}.$$

(16)

Thus, from (16) it follows that the Volterra-type operator ${\Re }_t$ is linear and bounded for any fixed $t\in J_a$. Then, from Tietze extension theorem, for any fixed $t\in J_a$ it follows that there exists a linear and bounded extension ${\tilde{\Re }}_t:B{C}_a([a,t],{\mathbb{R}}^n)\to B{C}_a([a,t],{\mathbb{R}}^n)$ of the operator ${\Re }_t$ with ${\tilde{\Re }}_t{|}_{B{C}_a([a,t],{\mathbb{R}}^n)}={\Re }_t$, ${∥{\tilde{\Re }}_t∥}_{J_a}={∥{\Re }_t∥}_{[a,t]}$ and for the obtained collection of operators ${\left\{{\tilde{\mathbb{R}}}_t\right\}}_{t\in J_a}$ we have that ${∥{\tilde{\Re }}_t\left(f\right)∥}_{J_a}\le {∥{\tilde{\Re }}_t∥}_{J_a}{∥f∥}_{J_a}$ for any $f\in B{C}_a([a,t],{\mathbb{R}}^n)$.

According to the Peron condition, for any forcing term $f\in B{C}_a([a,t],{\mathbb{R}}^n)$ with ${∥f∥}_{J_a}<\infty$ the corresponding solution $x_f\left(t\right)$ of IP (1), (2) with zero initial function is bounded, i.e., we have that ${∥x_f\left(t\right)∥}_{J_a}<\infty$ and hence $x_f\in B{C}_a([a,t],{\mathbb{R}}^n)$. Then in virtue of the Banach–Steinhaus theorem, there exists a constant $C_{*}>0$ with ${\displaystyle \underset{t\in J_a}{sup}{∥{\tilde{\Re }}_t∥}_{J_a}\le C_{*}}$, and hence for any $f\in B{C}_a([a,t],{\mathbb{R}}^n)$ we have that ${∥{\tilde{\Re }}_t\left(f\right)∥}_{J_a}\le C_{*}{∥f∥}_{J_a}$ for arbitrary ${\tilde{\Re }}_t\in {\left\{{\tilde{\Re }}_t\right\}}_{t\in J_a}$. Thus, from (16) we obtain

$$\left|\left({\Re }_{t}f\right)\left(t\right)\right|=\left|x_f\left(t\right)\right|=\left|{\displaystyle \underset{a}{\overset{t}{\int }}}C(t,s)\left({\tilde{\Im }}_{t}f\right)\left(s\right)ds\right|\le {∥{\tilde{\Re }}_t∥}_{J_a}{∥f∥}_{J_a}\le C_{*}{∥f∥}_{J_a}.$$

(17)

Let $t\in J_a$ be arbitrary fixed and $C(t,s)={\left\{c_{k,j}(t,s)\right\}}_{k,j\in ⟨n⟩}$ be the fundamental matrix. Following [7] introduce the matrix $G\left(s\right)={\left\{g_{j,k}\left(s\right)\right\}}_{j,k\in ⟨n⟩}$ with $g_{j,k}\left(s\right)=sign{c}_{k,j}(t,s)$ for any $k,j\in ⟨n⟩$. For any $k\in ⟨n⟩$ we have

$$\sum _{j\in ⟨n⟩}{c}_{k,j}(t,s)sign{c}_{k,j}(t,s)=\sum _{j\in ⟨n⟩}{c}_{k,j}(t,s)g_{j,k}\left(s\right)=\sum _{j\in ⟨n⟩}\left|c_{k,j}(t,s)\right|$$

and let denote $g_k\left(s\right)={(g_{1,k}\left(s\right),\dots ,g_{n,k}\left(s\right))}^T$.

For any $k\in ⟨n⟩$ there exists a bounded sequence ${\left\{g_k^l\left(s\right)\right\}}_{l\in \mathbb{N}}\subset B{C}_a(J_a,{\mathbb{R}}^n)$ with ${∥g_k^l∥}_{J_a}\le \overline{C}$, $l\in \mathbb{N}$, $\overline{C}>0$, such that $\underset{l\to \infty }{lim}{∥{\tilde{\Im }}_t{g}_k^l-g_k∥}_{[a,t]}=\mathbf{0}$. Then from (17) it follows that for any $k\in ⟨n⟩$

$$\left|{\displaystyle \underset{a}{\overset{t}{\int }}}C(t,s)\left({\tilde{\Im }}_t{g}_k^l\right)\left(s\right)ds\right|\le {∥{\tilde{\Re }}_t∥}_{J_a}{∥g_k^l∥}_{J_a}\le C_{*}{∥g_k^l∥}_{J_a}\le \overline{C}{C}_{*}$$

(18)

and hence ${\displaystyle \underset{a}{\overset{t}{\int }}}\left|C(t,s)\right|ds=\left|{\displaystyle \underset{a}{\overset{t}{\int }}}C(t,s)g_k\left(s\right)ds\right|\le \overline{C}{C}_{*}$ which completes the proof. □

Remark 6. 

The Perron condition is assumed for all bounded continuous functions ($f\in BC(\mathbb{R},\mathbb{R})$). Note that from the zero initial condition ($x\left(t\right)\equiv \mathbf{0},t\in [a-h,h]$) it follows that $f\left(a\right)=0$ and hence we can use instead $BC(\mathbb{R},\mathbb{R})$ the subspace $B{C}_a(\mathbb{R},\mathbb{R})=\{f\in BC(\mathbb{R},\mathbb{R})|f\left(a\right)=0\}$. This fact brings one additional advantage, since for any $f\in B{C}_a(\mathbb{R},\mathbb{R})$ we have that $D_{a+}^{q}f\left(t\right)={}^{RL}D_{a+}^{q}f\left(t\right)$ for each $t\ge a$ (see ref. [1], Formula (2.4.12)). That is why we start our considerations with $f\in BA{C}_a(\mathbb{R},\mathbb{R})$ since in this case we obtain for $t\ge a$ the following equalities (see ref. [1], Formula (2.4.17)):

$${}^{C}D_{a+}^{q}f\left(t\right)={}^{RL}D_{a+}^{q}f\left(t\right)={\displaystyle \frac{d}{dt}}\left(\left(I_{a+}^{1-q}f\left(\nu \right)\right)\left(t\right)\right)=\left(I_{a+}^{1-q}\left({\displaystyle \frac{d}{d\nu }}f\left(\nu \right)\right)\left(t\right)\right)={\int }_a^t{(t-\nu )}^{-q}{f}^{\prime }\left(\nu \right)d\nu .$$

Then we extend the considered operator to the whole of $B{C}_a(\mathbb{R},\mathbb{R})$, a technical tool that is not needed in systems with first-order derivatives.

Corollary 3. 

Let the conditions of Theorem 2 hold.

Then there exists a constant $C_{\infty }>0$ such that $\left|C(t,s)\right|<C_{\infty }$ for any $t\in J_a$.

Proof. 

From Theorem 2 it follows that there exists a constant $C>0$ such that ${\displaystyle \underset{a}{\overset{t}{\int }}}\left|C(t,s)\right|ds\le C$ for $t\in J_a$.

Assume that there exists a sequence ${\left\{(t_l,s_l)\right\}}_{l\in \mathbb{N}}$ such that for any $l\in \mathbb{N}$, $a\le s_l\le t_l$, with $t_l\in J_a$, such that $\left|C(t_l,s_l)\right|\ge l$. Suppose that $\underset{l\in \mathbb{N}}{sup}\left|t_l\right|\le T<\infty$. Then since $a\le s_l\le t_l$ for any $l\in \mathbb{N}$, we obtain that ${\left\{(t_l,s_l)\right\}}_{l\in \mathbb{N}}\subset [a,T]\times [a,T]$ with $s\le t$. Then according to Theorem 6 in ref. [18] we have that the fundamental matrix $C(t,s)$ is continuous for $(t,s)\in [a,T]\times [a,T]$ with $t\ne s$ and has a first kind jump for $t=s$. Hence $C(t,s)$ is bounded, which contradicts our assumption that $|C(t_l,s_l)|\ge l$, $l\in \mathbb{N}$. So, the sequence ${\left\{t_l\right\}}_{l\in \mathbb{N}}$ is unbounded. It is clear that without loss of generality we can suppose that $\left|C(t_l,s_l)\right|=\underset{s\in [a,t_l]}{max}\left|C(t_l,s)\right|$ for any $l\in \mathbb{N}$.

Let $s,s_0\in [a,t)$ with $s\le s_0$ be arbitrary. Then from Corollary 1 it follows that $C(t,s)=C(t,s_0)C(s_0,s)$ and hence

$$|C(t,s)-C(t,s_0)|\le \left|C(t,s_0)\right|\left|C(s_0,s)-\mathbf{I}\right|.$$

(19)

Since $C(t,s)$ is uniformly continuous for $s\in [a,\overline{t}]$ with $\overline{t}\in (s_0,t)$ then for any $\epsilon \in (0,{\displaystyle \frac{1}{2}})$ there exists $|s_0-s|<\delta <\overline{t}-s_0$ such that $\left|C\right(s_0,s)-\mathbf{I}|<\epsilon$ for any $|s_0-s|<\delta$ and then from (19) it follows that $|C(t,s)-C(t,s_0)|\le {\displaystyle \frac{1}{2}}\left|C(t,s_0)\right|$ or equivalently ${\displaystyle \frac{1}{2}}\left|C(t,s_0)\right|\le \left|C(t,s)\right|\le {\displaystyle \frac{3}{2}}\left|C(t,s_0)\right|$ for any $\left|s_0-s\right|<\delta$ too.

Consider the case when $\underset{l\in \mathbb{N}}{sup}\left|s_l\right|<\infty$ and then we have

$$C>{\displaystyle \underset{a}{\overset{t_l}{\int }}}\left|C(t_l,s_l)\right|ds>{\displaystyle \underset{s_l}{\overset{s_l+\delta }{\int }}}\left|C(t_l,s)\right|ds\ge {\displaystyle \frac{\delta }{2}}\left|C(t_l,s_l)\right|\ge {\displaystyle \frac{\delta }{2}}l$$

which contradicts the assumption that $\underset{l\in \mathbb{N}}{sup}\left|s_l\right|<\infty$. If ${\left\{s_l\right\}}_{l\in \mathbb{N}}$ is unbounded then we obtain

$$C>{\displaystyle \underset{a}{\overset{t_l}{\int }}}\left|C(t_l,s_l)\right|ds>{\displaystyle \underset{s_l-\delta }{\overset{s_l}{\int }}}\left|C(t_l,s)\right|ds\ge {\displaystyle \frac{\delta }{2}}\left|C(t_l,s_l)\right|,$$

which contradicts the assumption that the sequence ${\left\{\left|C(t_l,s_l)\right|\right\}}_{l\in \mathbb{N}}$ is unbounded. □

Theorem 3. 

Let the conditions of Theorem 2 hold.

Then there exists a constant $Q_{\infty }^{*}>0$ such that $\underset{t\in J_a}{sup}\left|Q(t,s)\right|\le Q_{\infty }^{*}$ for arbitrary $s\in [a-h,a]$.

Proof. 

Let $s\in [a-h,a]$ be arbitrary, $Q(\tau ,s)$ be the unique matrix solution of the IP (7), (8) with initial function ${\phi }_s^Q\left(\theta \right),\theta \in [-h,0]$ and $Q_k(\tau ,s)$ be the k-th column of the matrix $Q(\tau ,s)$. Then from (7) for arbitrary $s\in [a-h,a]$ and any $t\in J_a$ it follows that

$$\begin{array}{c}\hfill \begin{array}{c}\left|Q_k(t,s)\right|\le {\mathbf{1}}_{\left\{s\le t\le a\right\}}+\left|{\displaystyle \frac{1}{q_k\Gamma \left(q_k\right)}}\right|{\displaystyle \underset{a}{\overset{t}{\int }}}\left|{\displaystyle \underset{-h}{\overset{0}{\int }}}\left[d_{\theta }U(\tau ,\theta )\right]Q_k(\tau +\theta ,s)\right|d{(t-\tau )}^{q_k}\hfill \\ \le {\mathbf{1}}_{\left\{s\le t\le a\right\}}+\left|{\displaystyle \frac{U}{q_k\Gamma \left(q_k\right)}}\right|{\displaystyle \underset{a}{\overset{t}{\int }}}\left|\underset{\eta \in [a-h,\tau ]}{sup}{Q}_k(\eta ,s)\right|d{(t-\tau )}^{q_k}\hfill \\ \le {\mathbf{1}}_{\left\{s\le t\le a\right\}}+{\displaystyle \frac{U}{q_k\Gamma \left(q_k\right)}}{\displaystyle \underset{a}{\overset{t}{\int }}}\left|\underset{\eta \in [a-h,a]}{sup}{Q}_k(\eta ,s)\right|d{(t-\tau )}^{q_k}+{\displaystyle \frac{U_t}{q_k\Gamma \left(q_k\right)}}{\displaystyle \underset{a}{\overset{t}{\int }}}\left|\underset{\eta \in [a,\tau ]}{sup}{Q}_k(\eta ,s)\right|d{(t-\tau )}^{q_k}\hfill \\ \le \underset{=C_t}{\underbrace{{\mathbf{1}}_{\left\{s\le t\le a\right\}}+{\displaystyle \frac{U{(t-a)}^{q_k}}{\Gamma (1+q_k)}}}}+\underset{=g}{\underbrace{{\displaystyle \frac{U}{\Gamma \left(q_k\right)}}}}{\displaystyle \underset{a}{\overset{t}{\int }}}{(t-\tau )}^{q_k-1}\underset{\eta \in [a,\tau ]}{sup}\left|Q_k(\eta ,s)\right|d\tau .\hfill \end{array}\end{array}$$

(20)

According Corollary 2 from (20) we obtain

$$\begin{array}{cc}\hfill \underset{\tau \in [a-h,t]}{sup}\left|Q_k(\tau ,s)\right|& \le \underset{=C_t}{\underbrace{{\mathbf{1}}_{\left\{s\le t\le a\right\}}+{\Gamma }_{max}U{(t-a)}^{q_{max}}}}\hfill \\ \hfill & +\underset{=g}{\underbrace{{\displaystyle \frac{U}{\Gamma \left(q_k\right)}}}}{\displaystyle \underset{a}{\overset{t}{\int }}}{(t-\tau )}^{q_{max}-1}\underset{\eta \in [a,\tau ]}{sup}\left|Q_k(\eta ,s)\right|d\tau \hfill \\ \hfill & \le C_t{E}_{q_{max}}\left(gU{t}^{q_{max}}\right)\hfill \end{array}$$

(21)

and hence for any initial function ${\phi }_s^Q\left(\theta \right)$ the corresponding solution $\left|Q\right(t,s\left)\right|$ of the IP (7), (8) for any fixed $t\in J_a$ is uniformly bounded in s, $s\in [a-h,a]$.

Since the system (3) is linear and homogeneous then the matrix $\tilde{Q}(t,s):=Q(t,s)-Q(t,a)$ is the unique matrix solution of the IP (7), (8) with initial function ${\phi }_s^Q\left(\theta \right)-{\phi }_a^Q\left(\theta \right),\theta \in [-h,0]$. Then as above we obtain that

$$\begin{array}{cc}\hfill & \left|Q_k(t,s)-Q_k(t,a)\right|\hfill \\ \hfill & \le {\mathbf{1}}_{\left\{s\le t<a\right\}}+\left|{\displaystyle \frac{1}{q_k\Gamma \left(q_k\right)}}\right|{\displaystyle \underset{a}{\overset{t}{\int }}}\left|{\displaystyle \underset{-h}{\overset{0}{\int }}}\left[d_{\theta }U(\tau ,\theta )\right]\left(Q_k(\tau +\theta ,s)-Q_k(t,a)\right)\right|d{(t-\tau )}^{q_k}\hfill \\ \hfill & \le {\mathbf{1}}_{\left\{s\le t<a\right\}}+{\displaystyle \frac{U}{q_k\Gamma \left(q_k\right)}}{\displaystyle \underset{a}{\overset{t}{\int }}}\left|\underset{\eta \in [a-h,\tau ]}{sup}{Q}_k(\eta ,s)-Q_k(\eta ,a)\right|d{(t-\tau )}^{q_k}\hfill \\ \hfill & \le {\mathbf{1}}_{\left\{s\le t<a\right\}}+{\displaystyle \frac{U}{q_k\Gamma \left(q_k\right)}}{\displaystyle \underset{a}{\overset{t}{\int }}}\left|\underset{\eta \in [a-h,a]}{sup}{Q}_k(\eta ,s)-Q_k(\eta ,a)\right|d{(t-\tau )}^{q_k}\hfill \\ \hfill & +{\displaystyle \frac{U}{q_k\Gamma \left(q_k\right)}}{\displaystyle \underset{a}{\overset{t}{\int }}}\left|\underset{\eta \in [a,\tau ]}{sup}{Q}_k(\eta ,s)-Q_k(\eta ,a)\right|d{(t-\tau )}^{q_k}\hfill \\ \hfill & \le \underset{C_t}{\underbrace{{\mathbf{1}}_{\left\{s\le t<a\right\}}+{\Gamma }_{max}U{(t-a)}^{q_{max}}}}+\underset{=g}{\underbrace{{\displaystyle \frac{U}{\Gamma (1+q_k)}}}}{\displaystyle \underset{a}{\overset{t}{\int }}}{(t-\tau )}^{q_k-1}\underset{\eta \in [a,\tau ]}{sup}\left|Q_k(\eta ,s)-Q_k(\eta ,a)\right|d\tau \hfill \\ \hfill & \le C_t{E}_{q_{max}}\left(gU{t}^{q_{max}}\right)\hfill \end{array}$$

(22)

and thus as above according Corollary 2 we have that $|\tilde{Q}(t,s)|=|Q(t,s)-Q(t,a)|$ is uniformly bounded in s for $s\in [a-h,a]$ for any fixed $t\in J_a$ too.

Define the family of linear functionals indexed via $t\in J_a$ as follows:

$$\left({\mathbf{Q}}_t\phi \right):={\displaystyle \underset{a-h}{\overset{a}{\int }}}\tilde{Q}(t,s)d\phi \left(s\right),\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n).$$

(23)

Since the right side of (22) does not depend from s as above, it follows, that for arbitrary $s\in [a-h,a]$, $\underset{\tau \in [a-h,t]}{sup}\left|Q(\tau ,s)-Q(\tau ,a)\right|\le n{C}_t{E}_{q_{max}}\left(gU{t}^{q_{max}}\right)={\tilde{Q}}_t$ and hence we obtain that $\left|\left({\mathbf{Q}}_t\phi \right)\right|\le {\tilde{Q}}_t\underset{s\in [a-h,a]}{Var}\phi \left(s\right)$. Thus the functionals ${\mathbf{Q}}_t$ are bounded for any $t\in J_a$.

Then in virtue of Banach–Steinhaus theorem there exist a constant $Q_{\infty }>0$ such that ${\tilde{Q}}_t\le Q_{\infty }$ for any $t\in J_a$ and then we have that $\underset{t\in J_a}{sup}\left|\tilde{Q}(\tau ,s)\right|=\underset{t\in J_a}{sup}\left|Q(\tau ,s)-Q(\tau ,a)\right|\le Q_{\infty }$, i.e., $\left|\tilde{Q}(\tau ,s)\right|$ is uniformly bounded in s for $s\in [a-h,a]$. According Corollary 3 there exists a constant $C_{\infty }>0$ such that $\left|C(t,s)\right|<C_{\infty }$ for any $t\in J_a$ and taking into account that $\underset{t\in J_a}{sup}\left|Q(t,a)\right|=\underset{t\in J_a}{sup}\left|C(t,a)\right|\le C_{\infty }$ and $\underset{t\in J_a}{sup}\left|Q(t,s)-Q(t,a)\right|<Q_{\infty }$ for any $t\in J_a$ we obtain that $\underset{t\in J_a}{sup}\left|Q(t,s)\right|\le Q_{\infty }^{*}=C_{\infty }+Q_{\infty }$ for arbitrary $s\in [a-h,a]$. □

Corollary 4. 

Let the conditions of Theorem 2 hold.

Then the uniform boundedness of the extended fundamental matrix $Q(t,s)$ is a necessary and sufficient condition for the uniform stability of the zero solution of (3).

Proof. 

Necessity:

Let the zero solution of (3) be uniform stable. Then for arbitrary initial function $\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n)$ in virtue of Theorem 7 in [18] we have that the corresponding solution is bounded, i.e., there exists a constant $\underset{t\in J_a}{sup}\left|x\left(t\right)\right|\le C_{\phi }$. Then since for any $s\in [a-h,a]$ the functions ${\phi }_s^Q\in BV([a-h,a],{\mathbb{R}}^n)$ and hence all columns $Q_k(\tau ,s)$, $k\in ⟨n⟩$ are uniformly bounded functions. Thus $Q(t,s)$ is uniformly bounded too.

Sufficiency:

Let $\underset{t\in J_a}{sup}\left|Q(t,s)\right|\le Q_{\infty }^{*}$ for any $s\in [a-h,a]$ and $\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n)$ be arbitrary. Then according Theorem 4.4 in [16] and [17] the corresponding unique solution $x\left(t\right)=x(t,a,\phi ,\mathbf{0})$ of IP (3), (2) has the integral representation

$$x\left(t\right)={\displaystyle \underset{a-h}{\overset{a}{\int }}}Q(t,s)d\overline{\phi }\left(s\right),$$

(24)

where $\overline{\phi }\left(s\right)=\phi (s-a),s\in (a-h,a]$ and $\overline{\phi }(a-h)=\mathbf{0}$.

From (24) it follows that

$$\left|x(t,a,\phi ,\mathbf{0})\right|=\underset{t\in J_a}{sup}\left|Q(t,s)\right|\le Q_{\infty }^{*}Va{r}_{s\in [a-h,a]}\overline{\phi }\left(s\right)\le Q_{\infty }^{*}Va{r}_{s\in [a-h,a]}\phi \left(s\right)$$

and hence $\left|x(t,a,\phi ,\mathbf{0})\right|$ is a bounded solution. Thus, the statement follows form Theorem 7 in [18]. □

Theorem 4. 

Let the conditions of Theorem 2 hold.

Then $\underset{t\to \infty }{lim}\left|Q(t,s)\right|=0$ uniformly for any $s\in [a-h,a]$.

Proof. 

Let $x\left(t\right)=x(t,a,\phi ,\mathbf{0})$ be the unique solution of the IP (3), (2) for any $\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n)$ and hence since $\underset{t\in J_a}{sup}\left|Q(t,s)\right|\le Q_{\infty }^{*}$ and the zero solution of (3) is uniformly stable it follows that $x\left(t\right)$ is uniformly continuous for $t\in [max(a,0),\infty )$. For every $\phi \left(s\right)\in BV([a-h,a],{\mathbb{R}}^n)$ (see Theorem 4.4 in [16,17]) we have that (24) holds. Then integrating (24) from $max(a,0)$ to t we have

$${\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}\left|x\left(\tau \right)\right|d\tau ={\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}\left|{\displaystyle \underset{a-h}{\overset{a}{\int }}}Q(\tau ,s)d\phi \left(s\right)\right|d\tau$$

(25)

and applying the Fubini theorem to the right side of (25) (see proposition 5.15 in [20] which guaranties the correct application of the theorem) we obtain that

$$\begin{array}{cc}\hfill {\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}\left|x\left(\tau \right)\right|d\tau & ={\displaystyle \underset{a-h}{\overset{a}{\int }}}\left|{\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}Q(\tau ,s)d\tau \right|d\phi \left(s\right)\hfill \\ \hfill & \le {\displaystyle \underset{a-h}{\overset{a}{\int }}}\left({\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}\left|Q(\tau ,s)\right|d\tau \right)dVa{r}_{v\in [a-h,s]}\phi \left(v\right).\hfill \end{array}$$

(26)

Define via (26) the family of linear functionals indexed by $t\in J_a$

$$\left({\mathbf{X}}_t\phi \right):={\displaystyle \underset{a-h}{\overset{a}{\int }}}\left({\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}\left|Q(\tau ,s)\right|d\tau \right)dVa{r}_{v\in [a-h,s]}\phi \left(v\right).$$

Then we have that $|{\mathbf{X}}_t\phi |\le Q_t^{*}{∥\phi ∥}_{Var}$, where ${∥\phi ∥}_{Var}=\phi \left(a\right)+Va{r}_{v\in [a-h,a]}\phi \left(v\right)$ and $Q_t^{*}={\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}\left|Q(\tau ,s)\right|d\tau$.

Then according to Banach–Steinhaus theorem there exists a constant $Q_{\infty }^{*}>0$, such that $Q_t^{*}\le Q_{\infty }^{*}$ for any $t\in J_a$. Thus we have that $\underset{t\to \infty }{lim}{\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}\left|Q(\tau ,s)\right|d\tau \le Q_{\infty }^{*}$. Since $\underset{t\in J_a}{sup}\left|Q(t,s)\right|\le Q_{\infty }^{*}$ for arbitrary $s\in [a-h,a]$ and $Q(t,s)$ is absolutely continuous in $t\in J_a$ (see Corollary 1, above), then we have that

$$\underset{t\in J_a}{sup}|{\displaystyle \underset{max(a,0)}{\overset{t}{\int }}}{\displaystyle \frac{\partial }{\partial \tau }}Q(\tau ,s)d\tau |=\underset{t\in J_a}{sup}|Q(t,s)-Q(max(a,0),s)|\le 2Q_{\infty }^{*}$$

and hence ${\displaystyle \frac{\partial }{\partial t}}Q(·,s)\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$ and then according to a variant of Barbalat’s lemma (see Theorem 5 in ref. [21]) we obtain that $\underset{t\to \infty }{lim}\left|Q(t,s)\right|=0$ uniformly in s for $s\in [a-h,a]$. □

Corollary 5. 

Let the conditions of Theorem 2 hold.

Then the relation $\underset{t\to \infty }{lim}\left|Q(t,s)\right|=0$ uniformly for $s\in [a-h,a]$ is a necessary and sufficient condition for uniform asymptotic stability of the zero solution of (3).

Proof. 

Necessity:

Let of the zero solution of (3) be uniformly asymptotically stable. Then according Theorem 4 we have that $\underset{t\to \infty }{lim}\left|Q(t,s)\right|=0$ uniformly for $s\in [a-h,a]$.

Sufficiency:

Let $\underset{t\to \infty }{lim}\left|Q(t,s)\right|=0$ uniformly for $s\in [a-h,a]$. Then $\underset{t\in J_a}{sup}\left|Q(t,s)\right|<\infty$ and then from Corollary 4 it follows that all solutions of the system (3) are bounded and hence the zero solution of (3) is uniformly stable. Thus the relation $\underset{t\to \infty }{lim}\left|Q(t,s)\right|=0$ implies that the zero solution of (3) is uniformly asymptotically stable. □

4. Comments and Conclusions

The main goal of this work was to study the impact of the Perron condition on the boundedness of the fundamental and extended fundamental matrix of the homogeneous system (3) and the consequences of this boundedness to the stability properties of the system under consideration.

The proofs differ from those in the classical case (for first-order derivatives), which is caused by the nonlocal nature of the fractional derivatives. This consideration requires a change in the basic object (tool) in the constructed mathematical proofs. So the main tool in the integer case is the Lagrange (formal) adjoint system and the appropriate integral representation of the solutions to the system.

In our study the role of the Lagrange (formal) adjoint system is played from the extended fundamental matrix through which in the fractional case we have obtained an appropriate integral representation of all solutions of the system (3). The obtained integral representation plays a key role in the study of the stability properties of the homogeneous system (3).

Note that our results generalize the results obtained for the first-order case.

Now we present some considerations regarding the conditions used in the paper. It is easy to see that the requirements in the conditions (S) are actually of two types: The first of them guaranties the existence and uniqueness of the solution of the IP (4), (2) for any initial function $\phi \in \mathbf{BL}$ and forcing term $f\in B{L}_1^{loc}(J_a,{\mathbb{R}}^n)$. This part of the conditions includes the same conditions as the conditions used in the classical case of the same systems but with first-order derivatives. The second part of the conditions under which the Perron condition is equivalent to the uniform asymptotic stability of the zero solution of the homogeneous system (3), given in conditions (S2), (S3) and condition 1 of Theorem 2, means that all matrices $A_i\left(t\right),i\in ⟨m⟩$ and $A_{*}\left(t\right)$ are uniformly bounded in $t\in J_a$. This is the analogue of the classical condition that the matrix $A\left(t\right)\in BC(J_a,{\mathbb{R}}^{n\times n})$ which is necessary even in the case of a linear system of ordinary differential equations ${\displaystyle \frac{dx}{dt}}=A\left(t\right)x\left(t\right)+f\left(t\right)$ for asymptotic stability of the zero solution of the corresponding homogeneous system, a case considered in the work of Perron [6].

We emphasize that the lost key properties of fractional derivatives such as locality and symmetry, allow us to model processes depending from their history with a non local internal nature. Then, taking into account the geometric and physical interpretation of the fractional approach (see [22,23]) it is possible that fractional models are closer to the view of real world’s phenomena developed in the relativistic theory. Note that, to our knowledge, there is no analysis of the consequences of this view so far.

Finally we can add the following open problem: It is well known that mixed problems for partial differential equations (PDEs) can be rewritten as initial problems for neutral systems with delay in the case of integer-order derivatives (see J. Hale, S. Lunel [24]). By generalizing the stability results obtained in the present article for the case of fractional neutral systems we can apply the obtained stability criteria to check the stability of numerical simulations for fractional PDEs with delays. For example, in [25], studies of tempered fractional heat transfer show that fractional constitutive models require stability checks.