Hyers–Ulam and Hyers–Ulam–Rassias Stability for Linear Fractional Systems with Riemann–Liouville Derivatives and Distributed Delays

Abstract

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems.

1. Introduction

Practically, it is established that many real-world phenomena in various fields of science can be represented more accurately through mathematical models, including fractional differential equations. For more detailed information on fractional calculus theory and fractional differential equations, see the monographs of Kilbas et al. [1] and Podlubny [2]. It is well known that the existence of an integral representation (variation of constants formula) of the solutions of linear fractional differential equations and/or systems (ordinary or delayed) is a main tool in executing their qualitative analysis. In this aspect, the problem of establishing such integral representations (for which the existence of a fundamental matrix is needed) is an important task for stability analysis. It is no surprise that there exist many papers devoted to this problem.

A good historical overview concerning the stability results for fractional differential equations obtained till 2011 can be found in the excellent survey [3] and the references therein. For more recent works, for fractional differential equations and systems without delay, see [4,5]. Integral representation and the stability results in the autonomous case of delayed fractional differential equations mainly with Caputo-type derivatives are given in [6,7] and for the neutral case in [8,9,10]. For the nonautonomous case with variable delay, we refer to [11,12,13] and for the neutral case, to [14,15]. The case with Riemann–Liouville (RL)-type derivatives is studied significantly less often, but the works [16,17,18,19,20], and the references therein, give a good overview of the research in this area. To expand the information concerning the scope of the studied objects, we refer to the new works [21,22,23,24] devoted to the stability analysis of other important kinds of equations such as integro-differential, fuzzy, neural networks, etc.

It must be noted that the difference between the fractional Caputo derivatives and the fractional Riemann–Liouville (RL) derivatives are not only technical but also fundamental, since the Caputo fractional derivative of a constant is equal to zero, while the Riemann–Liouville fractional derivative of a constant is different from zero when the constant is not equal to zero. Thus, the main theorem of integral calculus is not true for the case of fractional Riemann–Liouville derivatives. This fact leads to large complications in many technical and fundamental aspects.

Our work is primarily motivated by the works [17,20]. In the present work, we consider a linear fractional system with distributed delay and derivatives in the RL sense. For these systems, we study two important problems. The first of them is to clear the problem with existence and the uniqueness of the solutions of the initial problem (IP) in the case of discontinuous initial functions. As far as we know (except in the autonomous case), there are no results concerning the initial problem for fractional differential equations with derivatives in the RL sense and distributed delay with discontinuous initial function. This result allows as a consequence to establish a variation of the constants formula for this initial problem. The second one is to introduce a concept for Hyers–Ulam (HU) in time stability and Hyers–Ulam–Rassias (HUR) in time stability (based on the concept of time stability in the Lyapunov sense introduced in the remarkable work [20]) for these systems and to establish some sufficient conditions which guaranty their Hyers–Ulam in time stability.

As far we know this paper is the first to study Hyers–Ulam-type stability and Hyers–Ulam–Rassias-type stability for linear fractional systems with distributed delay and derivatives in the Riemann–Liouville sense.

The paper is organized as follows: In Section 2, we recall some needed definitions and properties concerning the RL and Caputo fractional derivatives and present the problem statement. Section 3 is devoted to the existence and the uniqueness of the solutions of the initial (Cauchy) problem for the linear fractional differential system with distributed delays and RL-type derivatives in the case when the initial function is discontinuous. In Section 4, as a consequence, we prove the existence and uniqueness of a fundamental matrix, which allows us to establish an integral representation of the solution to the initial problem for the corresponding inhomogeneous system. In Section 5, we introduce a concept for HU in time stability and HUR in time stability for the investigated systems. In addition, as an application of the obtained in the previous section’s results, we introduce a new approach via the obtained integral representation (replacing the standard fixed point approach) to establish sufficient conditions for HU in the time stability of these systems. Finally, for the homogeneous systems it is proved that the HU in time stability implies time stability in the Lyapunov sense. As usual, in the last Section 6, we provide some conclusions concerning the obtained results, and some open problems are proposed.

2. Preliminaries and Problem Statement

As is usual to avoid misunderstandings, below we provide the definitions of RL (RL) and Caputo fractional derivatives. For more details and other properties, we refer to [1].

Let $a\in \mathbb{R},\alpha \in (0,1)$ be arbitrary and $g\in L_1^{loc}(\mathbb{R},\mathbb{R})$, where $L_1^{loc}(\mathbb{R},\mathbb{R})$ is the linear space of all locally Lebesgue integrable functions $g:\mathbb{R}\to \mathbb{R}$ and let $B{L}_1^{loc}(\mathbb{R},\mathbb{R})\subset L_1^{loc}(\mathbb{R},\mathbb{R})$ be the subspace of all locally bounded functions.

The left-sided fractional integral operators of order $\alpha \in (0,1)$ for arbitrary $g\in L_1^{loc}(\mathbb{R},\mathbb{R})$ is defined by $\left(I_{a+}^{\alpha }g\right)\left(t\right)=\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{t}{\int }}{(t-s)}^{\alpha -1}g\left(s\right)\mathrm{d}s$, and the corresponding left-side RL fractional derivative by ${(}_{RL}{D}_{a+}^{\alpha }g)\left(t\right)=\frac{d}{dt}\left(I_{a+}^{1-\alpha }g\right)\left(t\right)$, $\left(D_{a+}^{0}g\right)\left(t\right)=g\left(t\right)$ for every $t>a$.

By ${}_C{D}_{a+}^{\alpha }g\left(t\right)={(}_{RL}{D}_{a+}^{\alpha }[g\left(s\right)-g\left(a\right)])\left(t\right)$, we define the Caputo fractional derivative of the same order (see [1]).

Consider the fractional linear system with RL-type derivatives and distributed delays in the following general form:

$${}_{RL}{D}_{a+}^{\alpha }X\left(t\right)=\underset{-h}{\overset{0}{\int }}[{\mathrm{d}}_{\theta }U(t,\theta )]X(t+\theta )+F\left(t\right),$$

(1)

where $J=[a,\infty ),J^0=(a,\infty ),a\in \mathbb{R},k\in ⟨n⟩=\{1,2,\dots ,n\},{⟨m⟩}_0=⟨m⟩\cup \left\{0\right\},h>0,$ $\alpha \in (0,1),$  $X\left(t\right)=col(x_1\left(t\right),\dots ,x_n\left(t\right)):J^0\to {\mathbb{R}}^n,$ $F\left(t\right)=col\left(f_1\left(t\right),\dots ,f_n\left(t\right)\right):J\to {\mathbb{R}}^n$ (the notation $col$ mean column), $U(t,\theta )={\displaystyle \sum _{i\in {⟨m⟩}_0}^{}}{U}^i(t,\theta ),$ $U^i:J\times \mathbb{R}\to {\mathbb{R}}^{n\times n},$ $U^i(t,\theta )={\left\{u_{kj}^i(t,\theta )\right\}}_{k,j=1}^n,$ ${}_{RL}{D}_{a+}^{\alpha }X\left(t\right)=col({}_{RL}{D}_{a+}^{\alpha }{x}_1\left(t\right),\dots \dots {}_{RL}{D}_{a+}^{\alpha }{x}_n\left(t\right)),$ ${}_{RL}{D}_{a+}^{\alpha }$ denotes the left-side RL fractional derivative and $\alpha \in (0,1)$. A more detailed description of the homogenous case of system (1) (i.e., $f_k\left(t\right)\equiv 0,k\in ⟨n⟩$) has the form

$${}_{RL}{D}_{a+}^{\alpha }{x}_k\left(t\right)=\sum _{i\in {⟨m⟩}_0}^m(\sum _{j=1}^n\underset{-\sigma }{\overset{0}{\int }}{x}_j(t+\theta ){\mathrm{d}}_{\theta }{u}_{kj}^i(t,\theta )),k\in ⟨n⟩,n\in \mathbb{N}.$$

(2)

The following standard notations will be used too: ${\mathbb{R}}_{+}^0=(0,\infty ),$ $J_{-h}=[a-h,\infty ),J_b^{-h}=[a-h,a+b],b,h\in {\mathbb{R}}_{+}^0,J_b=[a,a+b],J_b^0=(a,a+b],$ $\mathbf{0}\in {\mathbb{R}}^n$ is the zero vector, and by $\mathrm{I},\Theta \in {\mathbb{R}}^{n\times n}$ are denoted the identity and the zero matrices. For $Y:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n},Y(t,\theta )={\left\{y_j^i(t,\theta )\right\}}_{i,j=1}^n,\left|Y(t,\theta )\right|=\sum _{k,j=1}^n|y_k^j(t,\theta )|,$ $B{V}^{loc}(J\times \mathbb{R},{\mathbb{R}}^{n\times n})$, we denote the linear space of matrix valued functions $Y(t,\theta )$ with bounded variation in $\theta$ on every compact subinterval $K\subset \mathbb{R},$ and $Va{r}_{K}Y(t,·)={\left\{Va{r}_K{y}_k^j(t,·)\right\}}_{k,j=1}^n.$

With $\mathbf{PC}=PC([-h,0],{\mathbb{R}}^n)({\mathbf{PC}}^{*}=\mathbf{PC}\cap BV([-h,0],{\mathbb{R}}^n))$, we denote the Banach spaces of all vector-valued piecewise continuous (piecewise continuous with bounded variation) functions, $\mathrm{\Phi }={\left({\varphi }_1,\dots ,{\varphi }_n\right)}^T:[-h,0]\to {\mathbb{R}}^n$ with norm $\left|\right|\mathrm{\Phi }\left|\right|=\sum _{k\in ⟨n⟩}^{}\underset{s\in [-h,0]}{sup}\left|{\varphi }_k\left(s\right)\right|<\infty$ and for each $\mathrm{\Phi }\in \mathbf{PC}$ by $S^{\mathrm{\Phi }}$, we denote the set of all jump points. In addition, for $\mathrm{\Phi }\in \mathbf{PC}$, we assume that they are right continuous at $t\in S^{\mathrm{\Phi }}$.

For arbitrary $\mathrm{\Phi }\in \mathbf{PC}$, we introduce the following initial condition for the system (1):

$$X\left(t\right)=\mathrm{\Phi }(t-a)(x_k\left(t\right)={\varphi }_k(t-a),k\in ⟨n⟩),t\in [a-h,a]{,}_{RL}{D}_a^{\alpha -1}X(a+0)=\mathrm{\Phi }\left(0\right),h\in {\mathbb{R}}_{+}.$$

(3)

For other types of initial conditions, see [25].

Definition 1

([26] p. 12, [27] p. 167, and [28] p. 100). We say that for the kernels $U^i:{\overline{\mathbb{R}}}_{+}\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$, the conditions (S) are fulfilled if for $i\in {⟨m⟩}_0$, the following conditions hold:

$\left(\mathbf{S}\mathbf{1}\right)$

The function $(t,\theta )\to U^i(t,\theta )$ is measurable in $(t,\theta )\in J\times \mathbb{R}$ and normalized so that $U^i(t,\theta )=0$ for $\theta \ge 0$ and $U^i(t,\theta )=U^i(t,-h)$ for $\theta \le -h,t\in J.$

$\left(\mathbf{S}\mathbf{2}\right)$

For any $t\in J$, the kernel $U^i(t,\theta )$ is continuous from the left in θ on $(-\sigma ,0),U^i(t,·)\in B{V}^{loc}(J\times \mathbb{R},{\mathbb{R}}^{n\times n})$ in θ and $\left|Va{r}_{[-h,0]}{U}^i(t,·)\right|\in B{L}_1^{loc}(J_a^{},{\mathbb{R}}_{+}).$

$\left(\mathbf{S}\mathbf{3}\right)$

The Lebesgue decomposition of the kernel $U^i(t,\theta )$ for $t\in J$ and $\theta \in [-h,0]$ for each $i\in {⟨m⟩}_0$ have the form: $U^i(t,\theta )=U_j^i(t,\theta )+U_{ac}^i(t,\theta )+U_s^i(t,\theta ),$ where the jump part $U_j^i(t,\theta )={\left\{a_{kj}^i\left(t\right)H(\theta +{\sigma }_{kj}^i\left(t\right))\right\}}_{k,j=1}^n,$ $A^i\left(t\right)={\left\{a_{kj}^i\left(t\right)\right\}}_{k,j=1}^n\in B{L}_1^{loc}(J,{\mathbb{R}}^n),H\left(t\right)$ is the Heaviside function and the delays ${\sigma }_{kj}^i\left(t\right)\in C(J_a^{},[0,h]),{\sigma }_{kj}^0\left(t\right)\equiv 0,k,j\in ⟨n⟩,t\in J_a.$ For every fixed $t\in J$, the functions $U_{ac}^i(t,·)\in AC([-h,0],{\mathbb{R}}^{n\times n})$ and $U_s^i(t,·)\in C([-h,0],{\mathbb{R}}^{n\times n})$ in $\theta \in \mathbb{R}.$

$\left(\mathbf{S}\mathbf{4}\right)$

The sets $S_{{}_{\mathrm{\Phi }}}^i=\{t\in J|t-{\sigma }_i\left(t\right)\in S_{{}_{\mathrm{\Phi }}}^{}\}$ do not have limit points and for any $t,t_{*}\in J$, the relation $\underset{-\sigma }{\overset{0}{\int }}|U^i(t,\theta )-U^i(t_{*},\theta )|\mathrm{d}\theta \to 0$ hold.

Definition 2.

The vector function $colX\left(t\right)=(x_1\left(t\right),\dots ,x_n\left(t\right))$ is a solution of the IP (1), (3) in $J_b^0\left(J^0\right)$, if $X{|}_{J_b^0}\in C(J_b^0,{\mathbb{R}}^n)(X{|}_{J^0}\in C(J^0,{\mathbb{R}}^n))$ satisfies the system (1) for all $t\in J_b^0\left(J^0\right)$ and the initial condition (3).

Consider the following auxiliary system for $k\in ⟨n⟩$

$$\begin{array}{cc}\hfill x_k\left(t\right)& =\frac{{\varphi }_k\left(0\right){(t-a)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}[\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}[\sum _{i\in {⟨m⟩}_0}^{}(\sum _{j\in ⟨n⟩}^{}\underset{-h}{\overset{0}{\int }}{x}_j(\eta +\theta ){\mathrm{d}}_{\theta }{u}_{kj}^i(\eta ,\theta ))]d\eta \hfill \\ \hfill & +\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{f}_k\left(\eta \right)\mathrm{d}\eta ].\hfill \end{array}$$

(4)

Definition 3.

The vector function $colX\left(t\right)=(x_1\left(t\right),\dots ,x_n\left(t\right))$ is a solution of the IP (4), (3) in $J_b^0\left(J^0\right)$ if $X{|}_{J_b^0}\in C(J_b^0,{\mathbb{R}}^n)(X{|}_{J^0}\in C(J^0,{\mathbb{R}}^n))$ satisfies the system (4) for all $t\in J_b^0\left(J^0\right)$ and the initial condition (3).

Let $G\left(t\right)=(g_1\left(t\right),\dots ,g_n\left(t\right)):J_b^0\to {\mathbb{R}}^n,$ $b\in {\mathbb{R}}_{+}$ and $\gamma \in [0,1]$ be arbitrary.

Definition 4.

The function $G\left(t\right)=(g_1\left(t\right),\dots ,g_n\left(t\right))\in C(J_b^0,{\mathbb{R}}^n),$ $b\in {\mathbb{R}}_{+},$ $\gamma \in [0,1]$ will be called $\gamma -$ continuous at a if the function $I_{\gamma }(t-a)G\left(t\right)=col({(t-a)}^{\gamma }{g}_1\left(t\right),\dots ,{(t-a)}^{\gamma }{g}_n\left(t\right))\in C(J_b,{\mathbb{R}}^n).$

With ${\mathbf{C}}_b^{\gamma }$, we will denote the real linear space of all $\gamma$-continuous at a functions $G\left(t\right)\in C(J_b^0,{\mathbb{R}}^n)$ and with ${\mathbf{C}}^{\gamma }$ the linear space of all functions $G\left(t\right)\in C(J^0,{\mathbb{R}}^n),$ which are $\gamma$-continuous at $a.$

In our exposition below we will need the following auxiliary results:

Theorem 1

([29] Fixpunktsatz). Let Ω be a complete metric space endowed with metric $d_{\mathrm{\Omega }},$ the operator $T:\mathrm{\Omega }\to \mathrm{\Omega }$ and let the following conditions hold:

 1.

There exists a sequence ${\{{\epsilon }_q\ge 0\}}_{q\in \mathbb{N}},$ with ${\displaystyle \sum _{q=1}^{\infty }}{\epsilon }_q<\infty .$

 2.

For each $q\in \mathbb{N}$ and for arbitrary $x,y\in \mathrm{\Omega }$, the inequality $d_{\mathrm{\Omega }}(T^{q}x,T^{q}y)\le {\epsilon }_q{d}_{\mathrm{\Omega }}(x,y)$ hold.

Then, the operator T has a uniquely fixed point $x^{*}\in \mathrm{\Omega }$, and for every $x^0\in \mathrm{\Omega }$, we have that $\underset{q\to \infty }{lim}{T}^q{x}^0=x^{*}$.

Lemma 1

(Lemma 1 [7]). Let the following conditions be fulfilled.

 1.

The conditions (S) hold.

 2.

The functions $F\in B{L}_1^{loc}(J,{\mathbb{R}}^n).$

Then, every solution $X\left(t\right)$ of IP (1), (3) is a solution of the IP (4), (3) and vice versa.

Lemma 2

(Lemma 3.2 [1]). Let $\alpha \in (0,1)$, and let $y\left(t\right)$ be a Lebesgue measurable function on $J_b$.

 (a)

If there exists a.e. (almost everywhere) the limit $\underset{t\to a+0}{lim}\left[{(t-a)}^{1-\alpha }y\left(t\right)\right]=c\in \mathbb{R}$, then there also exists a.e. the limit $\left(D_a^{\alpha -1}y\right)(a+0)=\left(I_a^{1-\alpha }y\right)(a+0)=\underset{t\to a+0}{lim}\left(I_a^{1-\alpha }y\right)\left(t\right)=c\mathrm{\Gamma }\left(\alpha \right).$

 (b)

If there exist a.e. the limit $\underset{t\to a+0}{lim}\left[{(t-a)}^{1-\alpha }y\left(t\right)\right]$ and $\underset{t\to a+0}{lim}\left(I_a^{1-\alpha }y\right)\left(t\right)=c^{*},$ then we have that $\underset{t\to a+0}{lim}\left[{(t-a)}^{1-\alpha }y\left(t\right)\right]=\frac{c^{*}}{\mathrm{\Gamma }\left(\alpha \right)}.$

Let $\overline{\mathrm{\Phi }}\left(t\right)\in \mathbf{P}{\mathbf{C}}^{*}$ be an arbitrary function. Define the set

$$\overline{M}=\{\mathrm{\Phi }\left(t\right)\in \mathbf{P}{\mathbf{C}}^{*}|\mathrm{\Phi }\left(0\right)=\overline{\mathrm{\Phi }}\left(0\right)\}$$

and introduce for arbitrary ${\mathrm{\Phi }}^1\left(t\right),{\mathrm{\Phi }}^2\left(t\right)\in \overline{M}$ the following metric functions:

$$d_{Var}({\mathrm{\Phi }}^1,{\mathrm{\Phi }}^2)=\left|Va{r}_{t\in [-h,0]}({\mathrm{\Phi }}^1\left(t\right)-{\mathrm{\Phi }}^2\left(t\right))\right|\mathrm{and}{d}_{sup}({\mathrm{\Phi }}^1,{\mathrm{\Phi }}^2)=\underset{t\in [-h,0]}{sup}\left|{\mathrm{\Phi }}^1\left(t\right)-{\mathrm{\Phi }}^2\left(t\right)\right|.$$

Lemma 3

(Lemma 1 [30]). The set $\overline{M}$ is a complete metric space concerning both metrics and they are equivalent, i.e., there exist constant $C\in {\mathbb{R}}_{+}^0$ such that $d_{Var}({\mathrm{\Phi }}^1,{\mathrm{\Phi }}^2)\le C{d}_{sup}({\mathrm{\Phi }}^1,{\mathrm{\Phi }}^2)$ for arbitrary ${\mathrm{\Phi }}^1\left(t\right),{\mathrm{\Phi }}^2\left(t\right)\in \overline{M}$ (the inequality $d_{sup}({\mathrm{\Phi }}^1,{\mathrm{\Phi }}^2)\le d_{Var}({\mathrm{\Phi }}^1,{\mathrm{\Phi }}^2)$ obviously holds).

3. The Initial Problem with Discontinuous Initial Function

Let $\mathrm{\Phi }\in \mathbf{PC}$ be a fixed arbitrary initial function and introduce the set

$$\begin{array}{cc}\hfill {\mathbf{M}}^{1-\alpha }=& \{G:[a-h,\infty )\to {\mathbb{R}}^n|G{|}_J\in {\mathbf{C}}^{1-\alpha },\hfill \\ & G\left(t\right)=\mathrm{\Phi }\left(t\right),t\in [a-h,a],{}_{RL}{D}_a^{\alpha -1}G(a+0)=\mathrm{\Phi }\left(0\right)\}.\hfill \end{array}$$

For every $b\in {\mathbb{R}}_{+}$, define the sets

$${\mathbf{M}}_b^{1-\alpha }=\{G^b=(g_1^b\left(t\right),\dots ,g_n^b\left(t\right))|G^b=G{|}_{[a-h,a+b]},G\in {\mathbf{M}}^{1-\alpha }\}$$

and the metric function $d_b^{\mathrm{\Phi }}:{\mathbf{M}}_b^{1-\alpha }\times {\mathbf{M}}_b^{1-\alpha }\to {\overline{\mathbb{R}}}_{+}$ with

$$d_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b)=\sum _{k=1}^n\underset{t\in J_b}{sup}{(t-a)}^{1-\alpha }|g_k^b\left(t\right)-{\overline{g}}_k^b\left(t\right)|$$

for each $G^b,{\overline{G}}^b\in {\mathbf{M}}_b^{\mathrm{\Phi }}$. It is not so hard to check that the set ${\mathbf{M}}_b^{\mathrm{\Phi }}$ endowed with the metric $d_b^{\mathrm{\Phi }}$ is a complete metric space. Note that for arbitrary $G^b,{\overline{G}}^b\in {\mathbf{M}}_b^{\mathrm{\Phi }}$, according Lemma 2, we have that

$$\underset{t\to a+0}{lim}{(t-a)}^{1-\alpha }{G}^b\left(t\right)=\mathrm{\Phi }\left(0\right)=\underset{t\to a+0}{lim}{(t-a)}^{1-\alpha }{\overline{G}}^b\left(t\right)\mathrm{and}{G}^b\left(a\right)={\overline{G}}^b\left(a\right)=\mathrm{\Phi }\left(0\right).$$

For every $G^b\in {\mathbf{M}}_b^{1-\alpha }$, we define for $t\in J_0$ the operator $\Re =({\Re }_1,...,{\Re }_n)$ as follows:

$$\begin{array}{cc}\hfill {\Re }_k{g}_k^b\left(t\right)& =\frac{{\varphi }_k\left(0\right){(t-a)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}+\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{f}_k\left(\eta \right)\mathrm{d}\eta ]\hfill \\ \hfill & +\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}\left(\sum _{i\in {⟨m⟩}_0}\left(\sum _{j\in ⟨n⟩}\underset{-h}{\overset{0}{\int }}{g}_j^b(\eta +\theta ){\mathrm{d}}_{\theta }{u}_{kj}^i(\eta ,\theta )\left)\right]d\eta \right)\right)\hfill \end{array}$$

(5)

$${\Re }_k{g}_k^b\left(t\right)={\varphi }_k^{}\left(t\right),t\in [a-h,a],k\in ⟨n⟩.$$

(6)

Theorem 2.

Let the following conditions be fulfilled.

 1.

The conditions (S) hold.

 2.

The kernels $U_s^i(t,\theta )\equiv \Theta$ in $J\times \mathbb{R},i\in {⟨m⟩}_0,$ (i.e., in the Lebesgue decomposition of $U^i(t,\theta )$ did not exist a singular part) and ${\displaystyle \sum _{i\in {⟨m⟩}_0}^{}}\underset{\theta \in [-h,0]}{sup}\left|\frac{\partial U^i}{\partial \theta }(·,\theta )\right|\in B{L}_1^{loc}(J,{\mathbb{R}}^{n\times n}).$

 3.

The functions $F\in B{L}_1^{loc}(J,{\mathbb{R}}^n).$

Then, the IP (1), (3) has a unique solution $X\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }$ for arbitrary $b\in {\mathbb{R}}_{.}$

Proof. 

According to Lemma 1, we can instead (1), (3) study the IP (4), (3).

Let $\mathrm{\Phi }\in \mathbf{PC}$ be an arbitrary fixed initial function and $b\in {\mathbb{R}}_{+}$ be an arbitrary fixed number. First, we will prove that $\Re \left({\mathbf{M}}_b^{1-\alpha }\right)\subseteq {\mathbf{M}}_b^{1-\alpha }$. From condition 2 of Theorem 2, it follows that the function $t\to \underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{f}_k\left(\eta \right)\mathrm{d}\eta$ is a continuous function in $J_b$ for each $k\in ⟨n⟩$.

Let $G^b\in {\mathbf{M}}_b^{1-\alpha },$ $k,j\in ⟨n⟩,i\in {⟨m⟩}_0$ be arbitrary and consider the function

${\tilde{g}}^b\left(t\right):=\underset{-h}{\overset{0}{\int }}{g}_j^b(t+\theta ){\mathrm{d}}_{\theta }{u}_{kj}^i(t,\theta )$. Since $G^b\in {\mathbf{M}}_b^{1-\alpha }$, then from the conditions (S), it follows that ${\tilde{g}}^b\left(t\right)\in L_1^{loc}(J_b,\mathbb{R})$. From (5), it follows that ${\Re }_k{g}_k\left(t\right)$ is a continuous function in $t\in J_b^0,k\in ⟨n⟩$. Moreover, the second and third addend in the right side of (5) tends to zero when $t\to a+0$ and then taking into account this fact, from (5), it follows that

$$\begin{array}{cc}\hfill & \underset{t\to a+0}{lim}({(t-a)}^{1-\alpha }\Re g_k)\left(t\right)=\underset{t\to a+0}{lim}\left({(t-a)}^{1-\alpha }\frac{{\varphi }_k\left(0\right){(t-a)}^{\alpha -1}}{\mathrm{\Gamma }\left(\alpha \right)}\right)\hfill \\ \hfill & +\underset{t\to a+0}{lim}\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{(t-a)}^{1-\alpha }[\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}\left(\sum _{i\in {⟨m⟩}_0}(\sum _{j\in ⟨n⟩}\underset{-h}{\overset{0}{\int }}{g}_j^b(\eta +\theta ){\mathrm{d}}_{\theta }{u}_{kj}^i(\eta ,\theta )\left)d\eta \right)\right)\right)\hfill \\ \hfill & +\underset{t\to a+0}{lim}\left(\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{(t-a)}^{1-\alpha }\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{f}_k\left(\eta \right)\mathrm{d}\eta ]\right)=\frac{{\varphi }_k\left(0\right)}{\mathrm{\Gamma }\left(\alpha \right)}\hfill \end{array}$$

(7)

and hence $\Re G^b\left(t\right)\in {\mathbf{C}}_b^{1-\alpha }.$ Since from (7) it follows that $\underset{t\to a+0}{lim}({(t-a)}^{1-\alpha }\Re g_k)\left(t\right)=\frac{{\varphi }_k\left(0\right)}{\mathrm{\Gamma }\left(\alpha \right)},$ then applying Lemma 2, we obtain that $(D_a^{\alpha -1}\Re g_k)(a+0)=\underset{t\to a+0}{lim}(I_a^{1-\alpha }\Re g_k)\left(t\right)={\varphi }_k\left(0\right)$ and thus $\Re G^b\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }$ and satisfies (6). Therefore, the operator ℜ maps ${\mathbf{M}}_b^{1-\alpha }$ into ${\mathbf{M}}_b^{1-\alpha }.$

The rest of the proof is based on some ideas introduced in [15]. In our exposition below, we need the values of the integral

$${\Im }_q\left(t\right)=\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{(\eta -a)}^{q\alpha }\mathrm{d}\eta$$

for each $q\in \mathbb{N}$ and $t\in J.$ Via the substitution $\eta -a=z(t-a)$ and using the relation between the beta and gamma functions we obtain

$$\begin{array}{cc}\hfill {\Im }_q\left(t\right)& =\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{(\eta -a)}^{q\alpha }\mathrm{d}\eta ={(t-a)}^{q\alpha }\underset{0}{\overset{1}{\int }}{(t-\eta )}^{\alpha -1}{\left(\frac{\eta -a}{t-a}\right)}^{q\alpha }\mathrm{d}\eta \hfill \\ \hfill & ={(t-a)}^{q\alpha +1+\alpha -1}\underset{0}{\overset{1}{\int }}{(1-z)}^{\alpha -1}{z}^{2\alpha +1-1}\mathrm{d}z={(t-a)}^{(1+q)\alpha }\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(1+q\alpha )}{\mathrm{\Gamma }(1+(1+q\left)\alpha \right)}\hfill \end{array}$$

(8)

Let us denote $U_b=max\left({\displaystyle \sum _{i\in {⟨m⟩}_0}^{}}\underset{t\in [a,a+b]}{sup}\left|Va{r}_{\theta \in [-h,0]}{U}^i(t,·)\right|,\sum _{i\in {⟨m⟩}_0}\underset{\genfrac{}{}{0pt}{}{t\in [a,a+b],\hfill }{\theta \in [-h,0]\hfill }}{sup}\left|\frac{\partial U^i}{\partial \theta }(t,\theta )\right|\right)$, and then for arbitrary $G^b\left(t\right),{\overline{G}}^b\left(t\right)\in {\mathbf{M}}_b^{1-\alpha },k\in ⟨n⟩$ and $t\in J_b^0$ from (5) and (8) we obtain

$$\begin{array}{cc}\hfill & |{\Re }_k{g}_k^b\left(t\right)-{\Re }_k{\overline{g}}_k^b\left(t\right)|\hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}\sum _{i\in {⟨m⟩}_0}\left(\sum _{j\in ⟨n⟩}\left|\underset{-h}{\overset{0}{\int }}(g_j^b(\eta +\theta )-{\overline{g}}_j^b(\eta +\theta )){\mathrm{d}}_{\theta }{u}_{kj}^i(\eta ,\theta ))\right|\right)\mathrm{d}\eta \hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}\sum _{i\in {⟨m⟩}_0}(\sum _{j\in ⟨n⟩}|\underset{a-\eta }{\overset{0}{\int }}\left|{(\eta +\theta -a)}^{1-\alpha }\right(g_j^b(\eta +\theta )\hfill \\ \hfill & -{\overline{g}}_j^b(\eta +\theta )\left)\right||\frac{\partial u_{kj}^i}{\partial \theta }(t,\theta )\left|{(\eta +\theta -a)}^{\alpha -1}\mathrm{d}\theta )\right|\mathrm{d}\eta \hfill \\ \hfill & \le \frac{U_b}{\alpha \mathrm{\Gamma }\left(\alpha \right)}\sum _{j\in ⟨n⟩}\left(\underset{t\in J_b}{sup}{(t-a)}^{1-\alpha }|g_j^b\left(t\right)-{\overline{g}}_j^b\left(t\right)|\right)\left|\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}(\underset{a-\eta }{\overset{0}{\int }}{\mathrm{d}}_{\theta }{(\eta +\theta -a)}^{\alpha })\mathrm{d}\eta \right|\hfill \\ \hfill & \le \frac{U_b}{\mathrm{\Gamma }(1+\alpha )}{d}_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b)\left|\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{(\eta -a)}^{\alpha }\mathrm{d}\eta \right|\hfill \\ \hfill & \le \frac{U_b}{\mathrm{\Gamma }(1+\alpha )}{d}_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b){(t-a)}^{2\alpha }\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(1+\alpha )}{\mathrm{\Gamma }(1+2\alpha )}={(t-a)}^{2\alpha }\frac{\mathrm{\Gamma }\left(\alpha \right)U_b}{\mathrm{\Gamma }(1+2\alpha )}{d}_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b)\hfill \end{array}$$

(9)

Note that for $\eta +\theta \le a$ we have that $g_j^b(\eta +\theta )-{\overline{g}}_j^b(\eta +\theta )=0$ for $j\in ⟨n⟩.$

We will prove that for each $t\in J_b^0$ and $k\in ⟨n⟩$ the inequalities

$$\left|{\Re }_k^q{g}_k^b\left(t\right)-{\Re }_k^q{\overline{g}}_k^b\left(t\right)\left(t\right)\right|\le \frac{{(t-a)}^{(1+q)\alpha }\mathrm{\Gamma }\left(\alpha \right)U_b^q}{\mathrm{\Gamma }(1+(1+q\left)\alpha \right)}{d}_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b)$$

(10)

hold for any $q\in \mathbb{N}.$ From (9), it follows that the hypothesis (10) holds for $q=1,t\in J_b^0$, and suppose that for each $t\in J_b^0$ and $k\in ⟨n⟩$, the inequality (10) holds for some $q\ge 1.$ Then for arbitrary $G^b\left(t\right),{\overline{G}}^b\left(t\right)\in {\mathbf{M}}_b^{1-\alpha },k\in ⟨n⟩$ and $t\in J_b^0$ from (5), (8) and (10) for $q+1$, we obtain that

$$\begin{array}{cc}\hfill & |{\Re }_k^{q+1}{g}_k^b\left(t\right)-{\Re }_k^{q+1}{\overline{g}}_k^b\left(t\right)|=|\Re \left({\Re }_k^q{g}_k^b\right)\left(t\right)-\Re \left({\Re }_k^q{\overline{g}}_k^b\right)\left(t\right)|\hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}\sum _{i\in {⟨m⟩}_0}\left(\sum _{j\in ⟨n⟩}\left|\underset{-h}{\overset{0}{\int }}({\Re }_j\left({\Re }_j^q{g}_j^b\right)(\eta +\theta )-{\Re }_j\left({\Re }_j^q{\overline{g}}_j^b\right)(\eta +\theta )){\mathrm{d}}_{\theta }{u}_{kj}^i(\eta ,\theta )\right|\right)\mathrm{d}\eta \hfill \\ \hfill & \le \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}\sum _{i\in {⟨m⟩}_0}\left(\sum _{j\in ⟨n⟩}\left|\underset{a-\eta }{\overset{0}{\int }}\left|{\Re }_j^q{g}_j^b(\eta +\theta )-{\Re }_j^q{\overline{g}}_j^b(\eta +\theta ))\right|\left|\frac{\partial u_{kj}^i(\eta ,\theta )}{\partial \theta }\right|d\theta \right|\right)\mathrm{d}\eta \hfill \\ \hfill & \le \frac{U_b{U}_b^q\mathrm{\Gamma }\left(\alpha \right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(1+q\alpha )}{d}_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b)\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}\left(\underset{a-\eta }{\overset{0}{\int }}{(\eta +\theta -a)}^{q\alpha }\mathrm{d}\theta \right)\mathrm{d}\eta \hfill \\ \hfill & \le \frac{U_b^{1+q}}{(1+(1+q\left)\alpha \right)\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(1+(1+q\left)\alpha \right)}{d}_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b)\underset{a}{\overset{t}{\int }}{(t-\eta )}^{\alpha -1}{(\eta -a)}^{(1+q)\alpha }\mathrm{d}\eta \hfill \\ \hfill & \le \frac{U_b^{1+q}{(t-a)}^{(2+q)\alpha }}{\mathrm{\Gamma }(2+(1+q\left)\alpha \right)}{d}_b^{\mathrm{\Phi }}(G^b,{\overline{G}}^b)\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(2+(1+q\left)\alpha \right)}{\mathrm{\Gamma }(1+(2+q\left)\alpha \right)}=\frac{{U_b}^{1+q}\mathrm{\Gamma }\left(\alpha \right){(t-a)}^{(2+q)\alpha }}{\mathrm{\Gamma }(1+(2+q\left)\alpha \right)}.\hfill \end{array}$$

(11)

Thus, (11) implies that hypothesis (10) holds for each $q\in \mathbb{N},t\in J_b$, and hence, from (10), it follows that for any $q\in \mathbb{N}$ the estimation

$$d_b^{\mathrm{\Phi }}({\Re }^q{G}^b,{\Re }^q{\overline{G}}^b)\le \frac{n\mathrm{\Gamma }\left(\alpha \right){(b-a)}^{(1+q)\alpha }{U}_b^q}{\mathrm{\Gamma }(1+\alpha (q+1\left)\right)}{d}_b^{\mathrm{\Phi }}(G,\overline{G}),$$

(12)

holds.

Then, consider the Mittag–Leffler function $E_{\alpha ,1}\left(z\right)={\displaystyle \sum _{q=1}^{\infty }}\frac{z^q}{\mathrm{\Gamma }(1+\alpha q)}$; using (12), we define the sequence ${\left\{{\epsilon }_q\right\}}_{q\in \mathbb{N}}$ appearing in Theorem 1 for each $q\in \mathbb{N}$ as follows:

$${\epsilon }_{q+1}=\left(\frac{n\mathrm{\Gamma }\left(\alpha \right)}{U_b}\right)\frac{{\left({(b-a)}^{\alpha }{U}_b\right)}^{q+1}}{\mathrm{\Gamma }(1+\alpha (q+1\left)\right)}$$

(13)

It is simple to see that the series ${\displaystyle \sum _{q=1}^{\infty }}\frac{{\left({(b-a)}^{\alpha }{U}_b\right)}^{q+1}}{\mathrm{\Gamma }(1+\alpha (q+1\left)\right)}$ is the value of the considered Mittag–Leffler function calculated at the point $z={(b-a)}^{\alpha }{U}_b$, and hence, it is convergent. Thus, for the series ${\displaystyle \sum _{q=1}^{\infty }}{\epsilon }_q$ defined with (13), we have that ${\displaystyle \sum _{q=1}^{\infty }}{\epsilon }_q=\left(\frac{n\mathrm{\Gamma }\left(\alpha \right)}{U_b}\right)\left(\sum _{q=1}^{\infty }\frac{{\left({(b-a)}^{\alpha }{U}_b\right)}^{q+1}}{\mathrm{\Gamma }(1+\alpha (q+1\left)\right)}\right)<\infty .$

Therefore, from Theorem 1, it follows that the IP (4), (3), and according to Lemma 1, the IP (1), (3), has a unique solution $X\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }$ for arbitrary $b\in {\mathbb{R}}_{+}$. □

Corollary 1.

Let the conditions of Theorem 2 hold.

Then, the IP (1), (3) has a unique solution $X\left(t\right)\in {\mathbf{M}}^{1-\alpha }.$

Proof. 

Denote for each $q\in \mathbb{N}$ the unique solution of the IP (1), (3) by $X_q\left(t\right)\in {\mathbf{M}}_q^{1-\alpha }$ with the interval of existence $J_q^0$ existing according to Theorem 2. From the uniqueness, it follows that the solution $X_{(q+1}\left(t\right)$ is a continuous prolongation of the solution $X_q\left(t\right).$ Then, we define for arbitrary $t\in J^0$ global solution $X\left(t\right)$ as $X\left(t\right){|}_{t\in J_{q+1}^0}=X_{q+1}\left(t\right),$ where $q=\left[t\right]$ and hence $X\left(t\right)\in {\mathbf{M}}^{1-\alpha }$ is the unique solution of IP (1), (3), with the interval of existence $J^0.$ □

4. Fundamental Matrix and Integral Representation

Consider for every arbitrary fixed number $s\in J$ the following matrix system

$${}_{RL}{D}_{a+}^{\alpha }W(t,s)=\underset{-h}{\overset{0}{\int }}[{\mathrm{d}}_{\theta }U(t,\theta )]W(t+\theta ,s),t>s$$

(14)

and the initial condition:

$$W(t,s)=\Theta ,t\in [s-h,s);W(t,s)=I,t=s$$

(15)

For every arbitrary fixed number $\overline{s}\in (-\infty ,a]$, define

$$\mathrm{\Phi }(t,\overline{s})=\left\{\begin{array}{c}\mathbf{I},a-h\le \overline{s}\le t\le a\hfill \\ \Theta ,t<\overline{s}\hfill \\ \Theta ,\overline{s}<a-h\hfill \end{array}\right.$$

and with ${\mathrm{\Phi }}^j(t,\overline{s})$, denote the $j$-th column of the $\mathrm{\Phi }(t,\overline{s}).$

Introduce the following initial condition:

$$W(t,\overline{s})=\mathrm{\Phi }(t,\overline{s}),\mathrm{where}\overline{s}\in (-\infty ,a]\mathrm{is}\mathrm{an}\mathrm{arbitrary}\mathrm{fixed}\mathrm{number}.$$

(16)

Definition 5.

For some fixed $s\in J$, the matrix valued function $t\to C(t,s)={\left\{c_{kj}(t,s)\right\}}_{k,j=1}^n$ is called a solution of the IP (14), (15) if $C(·,s):J_s^0=(s,\infty )\to {\mathbb{R}}^{n\times n}$ is continuous for $t\in J_s^0$ and satisfies the matrix equation (14) in $J_s^0,$ as well as the initial condition (15). The matrix $C(t,s)$ will be called the fundamental (or Cauchy) matrix for the system (2).

Remark 1.

Since $C(a,s)=\Theta$, according the condition (15) for all $s\in J^0$, then we have that ${}_{RL}{D}_{a+}^{\alpha }C(t,s)={}_C{D}_{a+}^{\alpha }C(t,s)$ (i.e., both derivatives coincide when $s\in J^0$). Then, Theorem 6 in [31] implies that for any $j\in ⟨n⟩,$ $C^j(t,s)=col(c_{1j}(t,s),...,c_{nj}(t,s))$ is the unique solution of IP (2), (3) with initial function $\mathrm{\Phi }\left(0\right)=I^j,\mathrm{\Phi }(t-a)=\mathbf{0},t\in [a-h,a),$ where $I^j$ denotes the $j$-th column of the identity matrix $\mathbf{I}\in {\mathbb{R}}^{n\times n}$ and hence the IP (14), (15) has a unique solution $C(t,s)=(C^1(t,s),\dots ,C^n(t,s)).$ In the case when $\overline{s}=a$ for arbitrary $j\in ⟨n⟩$, according to Corollary 1, the IP (2), (3) has a unique solution $C^j(t,s)=col(c_{1j}(t,s),\dots ,c_{nj}(t,s))\in {\mathbf{M}}^{1-\alpha }$ with initial function ${\mathrm{\Phi }}_{}^j(t,a)\in \mathbf{P}{\mathbf{C}}^{*}$; then $C(t,s)=(C^1(t,s),\dots ,C^n(t,s))$, is obviously the unique solution of IP (14), (15) in this case.

Let $\overline{s}\in [a-h,a]$ be an arbitrary fixed number and consider the matrix IP (14), (16).

Definition 6.

The matrix-valued function $t\to Q(t,\overline{s})={\left\{q_{kj}(t,\overline{s})\right\}}_{k,j=1}^n:\mathbb{R}\times (-\infty ,a]\to {\mathbb{R}}^{n\times n}$ is called a solution of the IP (14), (16) for any fixed $\overline{s}\in [a-h,a],$ if $Q(·,\overline{s})\in {\mathbf{M}}^{1-\alpha }$ and satisfies the matrix equation (14) for $t\in J^0,$ as well as the initial condition (16).

Since ${\mathrm{\Phi }}^j(t,s)\in \mathbf{P}{\mathbf{C}}^{*}$ for any fixed $s\in (-\infty ,a]$ and $j\in ⟨n⟩$ then in virtue of the IP (4), (3), it has a unique solution $Q^j(t,s)=col(q_{1j}(t,s),\dots ,q_{nj}(t,s))\in {\mathbf{M}}^{1-\alpha }$ with ${\mathrm{\Phi }}_a^j(t,s)$ as the initial function. Since $j\in ⟨n⟩$ is arbitrary, then the matrix $Q(t,s)=(Q^1(t,s),\dots ,Q^n(t,s))$ is the unique solution of the IP (14), (16) with $({\mathrm{\Phi }}^1(t,s),...,{\mathrm{\Phi }}^n(t,s))$ as the initial matrix function.

Note that $C(t,a)=Q(t,a)$ since the Equations (14) and (16) are the same and the initial functions of both IP coincide with $s=a.$

Define the vector function

$$X_F^0\left(t\right)\left(t\right)=\underset{a}{\overset{t}{\int }}C(t,s){}_{RL}{D}^{1-\alpha }F\left(s\right)\mathrm{d}s$$

(17)

and for shortness denote $Z(t,s)=C(t,s)R\left(s\right),R\left(s\right){=}_{RL}{D}^{1-\alpha }F\left(s\right).$

As in the Caputo case (see [13]), we will prove that $X_F^0\left(t\right)$ is the unique solution of the IP (1), (3) with initial function $\mathrm{\Phi }(t-a)\equiv 0,t\in [-h,0].$

Theorem 3.

Let the conditions of Theorem 2 be fulfilled and $F\left(a\right)=\mathbf{0}.$

Then, the function $X_F^0\left(t\right)$ defined with the equality (17) is the unique solution of the IP (1), (3) with initial condition $\mathrm{\Phi }(t-a)\equiv 0,t-a\in [-h,0].$

Proof. 

Let us denote with $Z^{*}(t,s)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }\mathrm{Z}(\eta ,s)\mathrm{d}\eta .$ Then, since $C(t,s)=0$ for $t<s,$ via the Fubini–Tonelli theorem and (Formula (2.211) [2]), we obtain that

$$\begin{array}{cc}\hfill & {}_{RL}{D}_{a+}^{\alpha }{X}_F^0\left(t\right)=\left({}_{RL}{D}_{a+}^{\alpha }\underset{a}{\overset{t}{\int }}\mathrm{Z}(t,s)\mathrm{d}s\right)\left(t\right)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\frac{d}{dt}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }(\underset{a}{\overset{\eta }{\int }}\mathrm{Z}(\eta ,s))\mathrm{d}s)\mathrm{d}\eta \hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(1-\alpha )}\frac{d}{dt}\underset{a}{\overset{t}{\int }}(\underset{s}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }\mathrm{Z}(\eta ,s)\mathrm{d}\eta )\mathrm{d}s=\frac{\mathrm{d}}{\mathrm{d}t}\underset{a}{\overset{t}{\int }}(\frac{1}{\mathrm{\Gamma }(1-\alpha )}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }\mathrm{Z}(\eta ,s)\mathrm{d}\eta )\mathrm{d}s\hfill \\ \hfill & =\frac{\mathrm{d}}{\mathrm{d}t}\underset{a}{\overset{t}{\int }}{\mathrm{Z}}^{*}(t,s)\mathrm{d}s=\underset{a}{\overset{t}{\int }}\frac{\partial }{\partial t}{Z}^{*}(t,s)\mathrm{d}s+\underset{s\to t-0}{lim}{Z}^{*}(t,s)=\underset{a}{\overset{t}{\int }}{}_{RL}{D}_{a+}^{\alpha }Z(t,s)\mathrm{d}s\hfill \\ \hfill & +\underset{s\to t-0}{lim}{D}_{a+}^{\alpha -1}Z(t,s)=\underset{a}{\overset{t}{\int }}R{\left(s\right)}_{RL}{D}_{a+}^{\alpha }C(t,s)\mathrm{d}s+\underset{s\to t-0}{lim}{D}_{a+}^{\alpha -1}Z(t,s).\hfill \end{array}$$

(18)

Taking into account that $C(t,s)$ is the unique solution of IP (14), (15) and $C(a,s)=0$ when $a<s$ for the first addend on the right side of (18), we obtain

$$\begin{array}{cc}\hfill & \underset{a}{\overset{t}{\int }}R{\left(s\right)}_{RL}{D}_{a+}^{\alpha }C(t,s)\mathrm{d}s=\underset{a}{\overset{t}{\int }}R\left(s\right)\underset{-h}{\overset{0}{\int }}[{\mathrm{d}}_{\theta }U(t,\theta )]C(t+\theta ,s)\mathrm{d}s\hfill \\ \hfill & =\underset{-h}{\overset{0}{\int }}\left[{\mathrm{d}}_{\theta }U(t,\theta )\right](\underset{a}{\overset{t}{\int }}C(t+\theta ,s)R\left(s\right)\mathrm{d}s=\underset{-h}{\overset{0}{\int }}\left[{\mathrm{d}}_{\theta }U(t,\theta )\right]X_F^0(t+\theta ).\hfill \end{array}$$

(19)

For the second addend in the right side of (19), taking into account that $F\left(a\right)=\mathbf{0}$ and using (Lemma 3.2 [1]), we obtain that

$$\begin{array}{cc}\hfill & \underset{s\to t-0}{lim}{D}_{a+}^{\alpha -1}Z(t,s)=\underset{s\to t-0}{lim}{I}_{a+}^{1-\alpha }Z(t,s)=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\underset{a+}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }\underset{s\to \eta -0}{lim}Z(\eta ,s)\mathrm{d}\eta \hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(1-\alpha )}\underset{a+}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }C(\eta ,\eta )R\left(\eta \right)\mathrm{d}\eta =\frac{1}{\mathrm{\Gamma }(1-\alpha )}\underset{a+}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }{D}_{a+}^{1-\alpha }F\left(\eta \right)\mathrm{d}\eta \hfill \\ \hfill & =D_{a+}^{\alpha -1}{}_{RL}{D}_{a+}^{1-\alpha }F\left(t\right)=F\left(t\right).\hfill \end{array}$$

(20)

Then, from (18)–(20), it follows that $X_F^0\left(t\right)$ defined with the equality (17) is the unique solution of the IP (1), (3) with initial condition $\mathrm{\Phi }(t-a)\equiv 0,t-a\in [-h,0].$ □

Let $s\in [a-h,a]$ be an arbitrary number, $Q(t,s)$ be the corresponding unique solution of IP (14), (16) similar to the case of Caputo derivatives (see [15]), we introduce the vector function

$$X_0^{\mathrm{\Phi }}\left(t\right)=\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)$$

(21)

for all $\mathrm{\Phi }\in \mathbf{P}{\mathbf{C}}^{*},$ where $\overline{\mathrm{\Phi }}(s-a)\equiv \mathrm{\Phi }(s-a)$ for $s\in (a-h,a]$ and $\overline{\mathrm{\Phi }}(-h)=\mathbf{0}.$

Theorem 4.

Let the following conditions be fulfilled.

 1.

The conditions of Theorem 2 hold.

 2.

The function $F\left(t\right)\equiv 0$ for $t\in J.$

Then, for each initial function $\mathrm{\Phi }\in \mathbf{P}{\mathbf{C}}^{*}$ and $t\in J^0$, the vector function $X_0^{\mathrm{\Phi }}\left(t\right)$ defined by equality (21) is a unique solution of the IP (2), (3).

Proof. 

Since $Q(t,s)$ is a continuous function for $t\in J^0,$ $\mathrm{\Phi }\in \mathbf{P}{\mathbf{C}}^{*}$ and hence according to (Lemma 1 [26]) $X_0^{\mathrm{\Phi }}\left(t\right)$ defined via (21) is continuous in the same interval too. Then, similar as in (18), via the Fubini–Tonelli theorem, we obtain that

$$\begin{array}{cc}\hfill {}_{RL}{D}_{a+}^{\alpha }{X}_0^{\mathrm{\Phi }}\left(t\right)& =\left({}_{RL}{D}_{a+}^{\alpha }\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)\right)\left(t\right)\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(1-\alpha )}\frac{\mathrm{d}}{\mathrm{d}t}\underset{a}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }(\underset{a-h}{\overset{a}{\int }}Q(\eta ,s)d_s\overline{\mathrm{\Phi }}(s-a))\mathrm{d}\eta \hfill \\ \hfill & =\underset{a-h}{\overset{a}{\int }}\left(\frac{1}{\mathrm{\Gamma }(1-\alpha )}\frac{\mathrm{d}}{\mathrm{d}t}\underset{s}{\overset{t}{\int }}{(t-\eta )}^{-\alpha }Q(\eta ,s)\mathrm{d}\eta )\right){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)\hfill \\ \hfill & =\underset{a-h}{\overset{a}{\int }}{}_{RL}{D}_{a+}^{\alpha }Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)\hfill \end{array}$$

(22)

For arbitrary fixed $t\in J^0$, denote by $m_{\theta }$ and $m_s$ the Lebesgue–Stieltjes measures corresponding to $U(t,\theta )$ and $\overline{\mathrm{\Phi }}\left(s\right).$ Then, for the rectangle $\rho =[-h,0]\times [a-h,a]$ and the product measure $m_{\theta }\times m_s$, the equality $m_{\theta }\times m_s\left(\rho \right)=m_{\theta }\left(\rho \right)m_s\left(\rho \right)$ holds. Thus,

$$\left|\underset{\rho }{\int \int }Q(t+\theta ,s)m_{\theta }\times m_s\left(\rho \right)\right|<\infty$$

and for each fixed $t\in J^0$, $(\theta ,s)\in \rho$ the matrix function $Q(t+\theta ,s)\in L_1^{loc}(\rho ,\mathbb{R})$ is locally bounded. Then, in virtue of (Proposition 5.4 [32]), we can correctly apply the Fubini–Tonelli theorem and for the right side of (2) we obtain

$$\begin{array}{cc}\hfill & \underset{-h}{\overset{0}{\int }}\left[{\mathrm{d}}_{\theta }{U}^i(t,\theta )\right]X_{\mathrm{\Phi }}(t+\theta )=\underset{-h}{\overset{0}{\int }}\left[{\mathrm{d}}_{\theta }{U}^i(t,\theta )\right]\left(\underset{a-h}{\overset{a}{\int }}Q(t+\theta ,s){\mathrm{d}}_s\mathrm{\Phi }(s-a)\right)\hfill \\ \hfill & =\underset{a-h}{\overset{a}{\int }}\left(\underset{-h}{\overset{0}{\int }}\left[d_{\theta }{U}^i(t,\theta )\right]Q(t+\theta ,s))\right){\mathrm{d}}_s\mathrm{\Phi }(s-a),\hfill \end{array}$$

(23)

and hence from (22), (23) it follows that $X_0^{\mathrm{\Phi }}\left(t\right)$ satisfies (2) for $t\in J^0.$

Let $s^{*}\in [a-h,a]$ be an arbitrary fixed number. Then, for $t=s^{*}$ from (22), we have that

$$\begin{array}{cc}& X_0^{\mathrm{\Phi }}\left(s^{*}\right)=\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)=\underset{s^{*}}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)+\underset{a-h}{\overset{s^{*}}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)\hfill \\ & =-\underset{s^{*}}{\overset{a-h}{\int }}I{\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a)=-\overline{\mathrm{\Phi }}(-h)+\overline{\mathrm{\Phi }}(s^{*}-a)=\mathrm{\Phi }(s^{*}-a),\hfill \end{array}$$

i.e., $X_0^{\mathrm{\Phi }}\left(t\right)$ satisfies the initial condition (3), which completes the proof. □

Corollary 2.

Let the following conditions hold.

 1.

The conditions of Theorem 4 hold.

 2.

The Lebesgue decomposition of the function $\mathrm{\Phi }\in \mathbf{P}{\mathbf{C}}^{*}$ does not possess a singular term.

Then, the vector function $X_0^{\mathrm{\Phi }}\left(t\right)$ defined by equality (21) has the representation in the form

$$\begin{array}{cc}\hfill X_0^{\mathrm{\Phi }}\left(t\right)& =(Q(t,a)(\mathrm{\Phi }(0+)-\mathrm{\Phi }\left(0\right))+\sum _i^{}Q(t,s_i)(\mathrm{\Phi }(s_i-a+0)-\mathrm{\Phi }(s_i-a-0))\hfill \\ \hfill & +\underset{a-h}{\overset{a}{\int }}Q(t,s){{\mathrm{\Phi }}^{\prime }}_{ac}(s-a)\mathrm{d}s),\hfill \end{array}$$

(24)

where the summation is over all jump points $s_i\in [a-h,a)$ and the sum is finite.

Proof. 

Since $\mathrm{\Phi }\in \mathbf{P}{\mathbf{C}}^{*}$ has finite many jump points then (24) immediately follows from (21). □

Corollary 3.

Let the conditions of Theorem 4 hold.

Then, for each initial function $\mathrm{\Phi }\in \mathbf{P}{\mathbf{C}}^{*}$, the unique solution $X_{\mathrm{\Phi }}^F\left(t\right)$ of the IVP (1), (3) for every $t\in J^0$ has the following representation

$$X_F^{\mathrm{\Phi }}\left(t\right)=\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }F\left(s\right)\mathrm{d}s+\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Phi }}(s-a),$$

where $\overline{\mathrm{\Phi }}(s-a)\equiv \mathrm{\Phi }(s-a)$ for $s\in (a-h,a]$ and $\overline{\mathrm{\Phi }}(-h)=\mathbf{0}.$

Proof. 

The statement of Corollary 3 immediately follows from the superposition principle and Theorems 3 and 4. □

5. Hyers–Ulam and Hyers–Ulam–Rassias in Time Stability

It is well known that the standard definitions of stability used in the systems with integer order or fractional Caputo-type derivatives are not directly applicable to the systems with fractional Riemann–Liouville-type derivatives, since the modulus of the solutions of the systems with Riemann–Liouville-type derivatives tends to infinity, when the independent variable tends to the initial point from the right, i.e., $\underset{t\to a+0}{lim}\left|X\left(t\right)\right|=\infty$. That is why new types of definitions for the different kinds of stabilities applicable to systems with Riemann–Liouville-type derivatives are needed.

The aim of this section is to introduce definitions of time stability, Hyers–Ulam (HU) in time stability, and Hyers–Ulam–Rassias (HUR) in time stability for fractional systems (equations) with RL-type derivatives and to establish some sufficient conditions which guarantee the HU in time stability of the studied systems.

As was mentioned, our concept uses the idea of the concept “stability in time” in the Lyapunov sense introduced in the remarkable work [20] for fractional equations with Riemann–Liouville-type derivatives.

Definition 7

([20]). The zero solution of the IP (2), (3) (i.e., with $\mathrm{\Phi }(t-a)\equiv \mathbf{0},t-a\in [-h,0]$ as initial function) is said to be:

 (i)

Stable in time in (Lyapunov in time stable) if for arbitrary $\epsilon >0$, there exist a point $t_{\epsilon }\in J^0$ and number $\delta (\epsilon ,t_{\epsilon })>0$ such that for any initial functions $\mathrm{\Phi }\left(t\right)\in \mathbf{PC}$ with $∥\mathrm{\Phi }∥<\delta ,$ the corresponding solution $X_{\mathrm{\Phi }}^0\left(t\right)$ of the IP (2), (3) satisfies $\left|X_{\mathrm{\Phi }}^0\left(t\right)\right|\le \epsilon$ for $t\ge t_{\epsilon }.$

 (ii)

Asymptotically stable in time if it is stable in time and additionally $\underset{t\to \infty }{lim}\left|X_{\mathrm{\Phi }}^0\left(t\right)\right|=0.$

With the next definitions, we introduce a concept for HU and HUR in time stability for fractional systems (equations) with RL-type derivatives.

Definition 8.

The system (1) is said to be Hyers–Ulam (HU) in time stable on $J_b^0\left(J^0\right),$ $b\in {\mathbb{R}}_{+}^0$ if there exists a constant $C>0$ such that for any $\epsilon >0$ and function $Y\left(t\right):J_b^{-h}\to {\mathbb{R}}^n(J^{-h}\to {\mathbb{R}}^n),$ with $Y\left(t\right){|}_{[a-h,a]}=\mathrm{\Psi }(t-a)\in {\mathbf{PC}}^{*},$ $t\in [a-h,a],$ $Y\left(t\right){|}_{J^0}=Z_Y\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }\left({\mathbf{M}}^{1-\alpha }\right)$ for which there exists a function ${\mathrm{\Phi }}^{\epsilon }\left(t\right)\in \mathbf{P}{\mathbf{C}}^{*}$ with $|\mathrm{\Psi }(t-a)-{\mathrm{\Phi }}^{\epsilon }(t-a)|\le \epsilon$ for $t\in [a-h,a]$ and $t_{\epsilon }\in (a,a+b),(t_{\epsilon }\in J^0)$ such that for $t\in [t_{\epsilon },a+b](t\in [t_{\epsilon },\infty ))$, the following inequalities hold

$$\left|{}_{RL}{D}_{a+}^{\alpha }Y\left(t\right)-\underset{-h}{\overset{0}{\int }}[{\mathrm{d}}_{\theta }U(t,\theta )]Y(t+\theta )-F\left(t\right)\right|\le \epsilon$$

(25)

then, there exists a unique solution $X_{{\mathrm{\Phi }}^{\epsilon }}^F\left(t\right)$ of the IP (1), (3) (with initial function ${\mathrm{\Phi }}^{\epsilon }\left(t\right)$) for which the inequality

$$\left|Y\left(t\right)-X_{{\mathrm{\Phi }}^{\epsilon }}^F\left(t\right)\right|\le C\epsilon ,$$

(26)

holds for any $t\in [t_{\epsilon },a+b](t\in [t_{\epsilon },\infty )).$

Let $b\in {\mathbb{R}}_{+}^0$ and $\phi \left(t\right)\in C(J_b^{-h},{\mathbb{R}}_{+}^0)\left(C(J^{-h},{\mathbb{R}}_{+}^0)\right)$ be arbitrary.

Definition 9.

The system (1) is said to be Hyers–Ulam–Rassias (HUR) in time stable on $J_b^0\left(J^0\right),$ $b\in {\mathbb{R}}_{+}^0$ with respect to $\phi \left(t\right)$ if there exists a constant $c_{\phi }>0$ such that for arbitrary function $Y\left(t\right):J_b^{-h}\to {\mathbb{R}}^n(J^{-h}\to {\mathbb{R}}^n),$ with $Y\left(t\right){|}_{J^0}\in {\mathbf{M}}_b^{1-\alpha }\left({\mathbf{M}}^{1-\alpha }\right),$ $Y\left(t\right){|}_{[a-h,a]}=\mathrm{\Psi }(t-a)\in \mathbf{P}{\mathbf{C}}^{*}$ for which there exist a function ${\mathrm{\Phi }}^{\phi }\left(t\right)\in \mathbf{P}{\mathbf{C}}^{*}$ with $|\mathrm{\Psi }(t-a)-{\mathrm{\Phi }}^{\phi }(t-a)|\le c_{\varphi }\varphi \left(t\right)$, $t\in [a-h,a]$ and $t_{\phi }\in J_b^0(t_{\phi }\in J^0)$ such that for $t\in [t_{\phi },a+b](t\in [t_{\phi },\infty ))$ the following inequality holds

$$\left|{}_{RL}{D}_{a+}^{\alpha }Y\left(t\right)-\underset{-h}{\overset{0}{\int }}[{\mathrm{d}}_{\theta }U(t,\theta )]Y(t+\theta )-F\left(t\right)\right|\le \phi \left(t\right),$$

(27)

then, there exists a unique solution $X_{{\mathrm{\Phi }}^{\phi }}^F\left(t\right)$ of the IP (1), (3) (with initial function ${\mathrm{\Phi }}^{\phi }\left(t\right)$) such that the inequality

$$\left|Y\left(t\right){|}_{J^0}-X_{{\mathrm{\Phi }}^{\phi }}^F\left(t\right)\right|\le c_{\varphi }\phi \left(t\right),$$

holds for any $t\in [t_{\phi },a+b](t\in [t_{\phi },\infty )).$

Remark 2.

We note that in (25) and (27), we assume that as initial function is used $Y\left(t\right){|}_{[a-h,a]}=\mathrm{\Psi }(t-a),$ which is mentioned explicitly. It seems that our Definitions 8 and 9 are stated in the sense of the classical definitions for delayed equations with integer-order derivatives (see [33,34]).

Theorem 5.

Let the following conditions be fulfilled.

 1.

The conditions of Theorem 4 hold.

 2.

$b\in {\mathbb{R}}_{+}^0$ is an arbitrary number.

Then, the system (1) is HU and time stable on $J_b^0.$

Proof. 

Let $t,s\in J_b$, and consider the fundamental matrix $C(t,s).$ Accordingly (Theorem 6 [14]), $C(t,s)$ is a continuous function in s and t for $s>a$ and $s\ne t.$ When $s>a$ and $s=t$, then $C(t,s)$ has a first-kind jump. If $s=a$, and $s\ne t,$ then $C(t,s)$ has a first kind jump at $s=a$, and if $t=s=a$ and $C(t,s)$ has a second kind jump at $t=a$ but is Lebesgue integrable (more precisely, for $t\to a+0$) we have that $C(t,s)=O\left({(t-a)}^{\alpha -1}\right).$ Since $Q(t,a)=C(t,a)$, then $Q(t,a)$ has the same properties as $C(t,a).$ When $s\in [a-h,a),$ then $Q(t,s)$ has an integrable second kind jump at $t=a,$ i.e., for $t\to a+0$ we have that $Q(t,s)=O\left({(t-a)}^{\alpha -1}\right).$ Taking into account (16) for $t\in J^0\cup [a-h,a)$ and $s\ne t,Q(t,s)$ is a continuous function in s and $t.$ When $s=t$, then $Q(t,s)$ has a first kind jump. Thus, we can conclude that for every $\overline{t}\in J^0,C(t,s)$ is bounded for $t\in [\overline{t},b],s\in J_b$ and Lebesgue integrable in s on $J_b.$ For every $\overline{t}\in J^0$ $Q(t,s)$ is bounded for $t\in [\overline{t},b],$ $s\in [a-h,a]$ and Lebesgue integrable in s on $s\in [a-h,a].$ Note that $C(t,s)$ and $Q(t,s)$ are constructed via the system (2) and do not depend on the choice of the vector function $F\left(t\right)$ in system (1).

Let $b\in {\mathbb{R}}_{+}^0,\epsilon >0$ and the arbitrary function $Y\left(t\right):J_b^{-h}\to {\mathbb{R}}^n,$ with $Y\left(t\right){|}_{[a-h,a]}=\mathrm{\Psi }(t-a)\in \mathbf{P}{\mathbf{C}}^{*},$ and $Y\left(t\right){|}_{J_b^0}=Z_Y\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }$ and satisfy the inequality (25) for $t\in [t_{\epsilon },a+b],$ $t_{\epsilon }\in (a,a+b).$ Since $\mathrm{\Psi }(t-a)\in \mathbf{P}{\mathbf{C}}^{*}$, then defining ${\mathrm{\Phi }}^{\epsilon }(t-a)=\mathrm{\Psi }(t-a)+col{\underbrace{(\frac{\epsilon }{2n},\dots ,\frac{\epsilon }{2n})}}_n$, we obtain that ${\mathrm{\Phi }}^{\epsilon }(t-a)\in \mathbf{P}{\mathbf{C}}^{*}$ and for $t\in [a-h,a]$, the functions satisfy the inequality $|\mathrm{\Psi }(t-a)-{\mathrm{\Phi }}^{\epsilon }(t-a)|\le \epsilon .$

Denote for $t\in [t_{\epsilon },a+b]$

$$\mathrm{H}\left(t\right)={}_{RL}{D}_{a+}^{\alpha }Y\left(t\right)-\underset{-h}{\overset{0}{\int }}[{\mathrm{d}}_{\theta }U(t,\theta )]Y(t+\theta )-F\left(t\right)$$

(28)

and assume that $H\left(t\right)$ is prolonged on $[a,t_{\epsilon }]$ as a continuous function with $\mathrm{H}\left(a\right)=\mathbf{0}$ and $\left|\mathrm{H}\left(t\right)\right|\le \epsilon$ for $t\in [a,t_{\epsilon }].$

Consider the IP (1), (3) with right side (1) $\tilde{F}\left(t\right)=\mathrm{H}\left(t\right)+F\left(t\right)$ for $t\in J_b^0,$ and initial function $\mathrm{\Psi }(t-a).$ Note that from (25) and the prolongation, it follows that $\left|\mathrm{H}\left(t\right)\right|\le \epsilon$ for $t\in [a,a+b].$ Since $\tilde{F}\left(t\right)\in B{L}_1^{loc}(J_b,{\mathbb{R}}^n)$ in virtue of Theorem 2, we obtain that the considered IP (1), (3) has a unique solution $\tilde{X}\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }.$ Thus, $\tilde{X}\left(t\right)$ coincides with $Z_Y\left(t\right)$ for $t\in J_b^0$ and hence in virtue of Corollary 3, it has the following integral representation

$$\begin{array}{cc}\hfill & \tilde{X}\left(t\right)=Z_Y\left(t\right)=\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }\tilde{F}\left(s\right)\mathrm{d}s+\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Psi }}(s-a)\hfill \\ \hfill & =\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }F\left(s\right)\mathrm{d}s+\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }\mathrm{H}\left(s\right)\mathrm{d}s+\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\overline{\mathrm{\Psi }}(s-a)\hfill \end{array}$$

(29)

Analogically in virtue of Theorem 2, we obtain that the IP (1), (3) with right side (1) $F\left(t\right)$ for $t\in J_b^0,$ and initial function ${\mathrm{\Phi }}^{\epsilon }(t-a)$, has a unique solution $X_{{\mathrm{\Phi }}^{\epsilon }}^F\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }$ for $t\in J_b^0$ which it has the representation

$$X_F^{{\mathrm{\Phi }}^{\epsilon }}\left(t\right)=\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }F\left(s\right)\mathrm{d}s+\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s{\overline{\mathrm{\Phi }}}^{\epsilon }(s-a),$$

(30)

where ${\overline{\mathrm{\Phi }}}^{\epsilon }(s-a)\equiv {\mathrm{\Phi }}^{\epsilon }(s-a),$ $\overline{\mathrm{\Psi }}(t-a)\equiv \mathrm{\Psi }(t-a)$ for $s\in (a-h,a]$ and $\overline{\mathrm{\Phi }}(-h)=\overline{\mathrm{\Psi }}(-h)=\mathbf{0}.$

Denote

$${\overline{C}}_b=\underset{t\in [t_{\epsilon },a+b],s\in J_b}{sup}\left|C(t,s)\right|,\overline{Q_b}=\underset{t\in [t_{\epsilon },a+b],s\in [a-h,a]}{sup}\left|Q(t,s)\right|$$

and from (29) and (30), we obtain for $t\in [t_{\epsilon },a+b]$ that

$$\left|Y\left(t\right){|}_{J^0}-X_{{\mathrm{\Phi }}^{\phi }}^F\left(t\right)\right|\le \left|\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }\mathrm{H}\left(s\right)\mathrm{d}s\right|+\left|\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s(\overline{\mathrm{\Psi }}(s-a)-{\overline{\mathrm{\Phi }}}^{\epsilon }(s-a))\right|.$$

(31)

For the second addend in the right side of (31) in virtue of Lemma 3, we have

$$\begin{array}{cc}\hfill \left|\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\left(\overline{\mathrm{\Phi }}(s-a)-\overline{\mathrm{\Psi }}(s-a)\right)\right|& \le {\overline{Q}}_b\left|Va{r}_{s\in [a-h,a]}\left(\overline{\mathrm{\Phi }}(s-a)-\overline{\mathrm{\Psi }}(s-a)\right)\right|\hfill \\ \hfill & \le C{Q}_{t_{\epsilon }}\underset{s\in [a-h,a]}{sup}\left|\overline{\mathrm{\Phi }}(s-a)-\overline{\mathrm{\Psi }}(s-a)\right|\le C{Q}_b\epsilon .\hfill \end{array}$$

(32)

Estimating the first addend on the right side of (31), we obtain that

$$\begin{array}{cc}\hfill & \left|\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }H\left(s\right)\mathrm{d}s\right|=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\left|\underset{a}{\overset{t}{\int }}C(t,s)\left(\frac{\mathrm{d}}{\mathrm{d}s}\underset{a}{\overset{s}{\int }}{(s-z)}^{-\alpha }H\left(z\right)dz\right)\mathrm{d}s\right|\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(2-\alpha )}\left|\underset{a}{\overset{t}{\int }}C(t,s)\left(\frac{\mathrm{d}}{\mathrm{d}s}\underset{a}{\overset{s}{\int }}H\left(z\right)d_z{(s-z)}^{1-\alpha }\right)\mathrm{d}s\right|\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(2-\alpha )}\left|\underset{a}{\overset{t}{\int }}C(t,s){\mathrm{d}}_s\left(\underset{a}{\overset{s}{\int }}H\left(z\right){\mathrm{d}}_z{(s-z)}^{1-\alpha }\right)\right|\hfill \\ \hfill & \le \frac{\epsilon {\overline{C}}_b}{\mathrm{\Gamma }(2-\alpha )}\underset{a}{\overset{t}{\int }}{d}_{s}Va{r}_{\eta \in [a,s]}\left(\underset{a}{\overset{\eta }{\int }}{\mathrm{d}}_z{(\eta -z)}^{1-\alpha }\right)\le \frac{\epsilon {\overline{C}}_b}{\mathrm{\Gamma }(3-\alpha )}\underset{a}{\overset{t}{\int }}{\mathrm{d}}_{s}Va{r}_{\eta \in [a,s]}{(\eta -a)}^{2-\alpha }\hfill \\ \hfill & \le \frac{\epsilon {\overline{C}}_b}{\mathrm{\Gamma }(3-\alpha )}Va{r}_{\eta \in [a,t]}\underset{a}{\overset{t}{\int }}{\mathrm{d}}_s{(s-a)}^{2-\alpha }\hfill \\ \hfill & \le \frac{\epsilon {\overline{C}}_b}{\mathrm{\Gamma }(3-\alpha )}{(t-a)}^{2-\alpha }\le \frac{\epsilon {\overline{C}}_b{b}^{2-\alpha }}{\mathrm{\Gamma }(3-\alpha )}\hfill \end{array}$$

(33)

Then, from (31)–(33), we obtain that

$$\begin{array}{cc}\hfill & \left|Y\left(t\right){|}_{J^0}-X_{{\mathrm{\Phi }}^{\epsilon }}^F\left(t\right)\right|\le \left(C{\overline{Q}}_b+\frac{{\overline{C}}_b{b}^{2-\alpha }}{\mathrm{\Gamma }(3-\alpha )}\right)\epsilon \hfill \end{array}$$

and then (26) holds for $t\in [t_{\epsilon },a+b],$ with $\tilde{C}=C{\overline{Q}}_b+\frac{{\overline{C}}_b{b}^{2-\alpha }}{\mathrm{\Gamma }(3-\alpha )}.$ □

Theorem 6.

Let the following conditions be fulfilled.

 1.

The conditions of Theorem 4 hold.

 2.

For some $r\in {\mathbb{R}}_{+}^0$, we have that $Q_r=\underset{t\in [a+r,\infty )}{sup}\left(\underset{s\in [a-h,a]}{sup}\left|Q(t,s)\right|\right)<\infty .$

 3.

For some $r\in {\mathbb{R}}_{+}^0$, the relation $C_{\infty }=\underset{t\in [a+r,\infty )}{sup}{(t-a)}^{2-\alpha }C\left(t\right)<\infty$ hold where

$C\left(t\right)=\underset{s\in [a+r,t]}{sup}\left|C(t,s)\right|.$

Then, the system (1) is HU in time stable on $J^0.$

Proof. 

This proof uses the same approach as the proof of Theorem 5, and hence, the matching details will only be sketched. First, we see that condition 3 implies that $C_r=\underset{t\in [a+r,\infty )}{sup}C\left(t\right)<\infty .$

Let $\epsilon >0$ and the function $Y\left(t\right):J_b^{-h}\to {\mathbb{R}}^n,$ with $Y\left(t\right){|}_{J^0}=Z_Y\left(t\right)\in {\mathbf{M}}_b^{1-\alpha }$ and $Y\left(t\right){|}_{[a-h,a]}=\mathrm{\Psi }(t-a)\in \mathbf{P}{\mathbf{C}}^{*},$ be arbitrary, which satisfies the inequality (25) for $t\in [t_{\epsilon },\infty )$ and define the function ${\mathrm{\Phi }}^{\epsilon }(t-a)$ in the same way as in Theorem 5. Since $Z_Y\left(t\right)$ satisfies (25), for $t\in [t_{\epsilon },\infty )$, we can define the function $H\left(t\right)$ via (28) and as in the above, we assume that $H\left(t\right)$ is prolonged on $[a,t_{\epsilon }]$ as a continuous function with $H\left(a\right)=\mathbf{0}$ and $\left|\mathrm{H}\left(t\right)\right|\le \epsilon$ for $t\in [a,t_{\epsilon }].$ Then, from (25) and the prolongation, it follows that $\left|\mathrm{H}\left(t\right)\right|\le \epsilon$ for $t\in J.$

As above, consider the IP (1), (3) with right side (1), the function $\tilde{F}\left(t\right)=\mathrm{H}\left(t\right)+F\left(t\right)$ for $t\in J,$ and initial function $\mathrm{\Psi }(t-a).$

Since $\tilde{F}\left(t\right)\in B{L}_1^{loc}(J,{\mathbb{R}}^n)$ in virtue of Corollary 1, we obtain that the considered IP (1), (3) has a unique solution $\tilde{X}\left(t\right)\in {\mathbf{M}}^{1-\alpha }.$ From the uniqueness, it follows that $\tilde{X}\left(t\right)$ coincides with $Z_Y\left(t\right)$ for $t\in J^0$, and hence, in virtue of Corollary 3, it has the integral representation (29). Analogically, in virtue of Corollary 3, we obtain that the IP (1), (3) with right side (1) $F\left(t\right)$ for $t\in J^0,$ and initial function ${\mathrm{\Phi }}^{\epsilon }(t-a)$, has a unique solution $X_{{\mathrm{\Phi }}^{\epsilon }}^F\left(t\right)\in {\mathbf{M}}^{1-\alpha }$ for $t\in J^0$, which has the representation (30).

Note that for arbitrary $\overline{r}\in [a,r]$, according the consideration at the beginning of the proof of Theorem 5, we conclude that $C_{\overline{r}}=\underset{t\in [a+\overline{r},\infty )}{sup}\left(\underset{s\in [a+\overline{r},t]}{sup}\left|C(t,s)\right|\right)<\infty ,$

$Q_{\overline{r}}=\underset{t\in [a+\overline{r},\infty )}{sup}\left(\underset{s\in [a-h,a]}{sup}\left|Q(t,s)\right|\right)<\infty$ and hence from conditions 2 and 3 of Theorem 5, it follows that $C_{t_{\epsilon }}=\underset{t\in [t_{\epsilon },\infty )}{sup}\left(\underset{s\in [t_{\epsilon },t]}{sup}\left|C(t,s)\right|\right)<\infty$ and $Q_{t_{\epsilon }}=\underset{t\in [t_{\epsilon },\infty )}{sup}\left(\underset{s\in [a-h,a]}{sup}\left|Q(t,s)\right|\right)<\infty .$

Then, as above, we obtain the estimation (31), and hence, for the second addend in the right side of (31) in virtue of Lemma 3, we have that

$$\left|\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s\left(\overline{\mathrm{\Phi }}(s-a)-\overline{\mathrm{\Psi }}(s-a)\right)\right|\le Q_{t_{\epsilon }}\left|Va{r}_{s\in [a-h,a]}\left(\overline{\mathrm{\Phi }}(s-a)-\overline{\mathrm{\Psi }}(s-a)\right)\right|\le C{Q}_{t_{\epsilon }}\epsilon$$

(34)

For the second addend in the right side of (31) taking into account condition 3 of Theorem 5, we obtain

$$\begin{array}{cc}\hfill & \left|\underset{a}{\overset{t}{\int }}C{(t,s)}_{RL}{D}^{1-\alpha }H\left(s\right)\mathrm{d}s\right|=\frac{1}{\mathrm{\Gamma }(1-\alpha )}\left|\underset{a}{\overset{t}{\int }}C(t,s)\left(\frac{\mathrm{d}}{\mathrm{d}s}\underset{a}{\overset{s}{\int }}{(s-z)}^{-\alpha }H\left(z\right)\mathrm{d}z\right)\mathrm{d}s\right|\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(2-\alpha )}\left|\underset{a}{\overset{t}{\int }}C(t,s)\left(\frac{\mathrm{d}}{\mathrm{d}s}\underset{a}{\overset{s}{\int }}H\left(z\right)\mathrm{d}{(s-z)}^{1-\alpha }\right)\mathrm{d}s\right|\hfill \\ \hfill & =\frac{1}{\mathrm{\Gamma }(2-\alpha )}\left|\underset{a}{\overset{t}{\int }}C(t,s){\mathrm{d}}_s\left(\underset{a}{\overset{s}{\int }}H\left(z\right)\mathrm{d}{(s-z)}^{1-\alpha }\right)\right|\hfill \\ \hfill & \le \frac{\epsilon C\left(t\right)}{\mathrm{\Gamma }(2-\alpha )}Va{r}_{s\in [a,t]}\left(\underset{a}{\overset{s}{\int }}{(s-z)}^{1-\alpha }\mathrm{d}z\right)\le \frac{\epsilon C\left(t\right)}{\mathrm{\Gamma }(3-\alpha )}Va{r}_{s\in [a,t]}{(s-a)}^{2-\alpha }\hfill \\ \hfill & \le \frac{\epsilon C\left(t\right)}{\mathrm{\Gamma }(3-\alpha )}{(t-a)}^{2-\alpha }\le \frac{\epsilon C_{\infty }}{\mathrm{\Gamma }(3-\alpha )}\hfill \end{array}$$

(35)

Then, from (31), (34), and (35), it follows that for $t\in [t_{\epsilon },\infty )$, we obtain the estimation

$$\left|Y\left(t\right){|}_{J^0}-X_{{\mathrm{\Phi }}^{\epsilon }}^F\left(t\right)\right|\le \left(C{Q}_{t_{\epsilon }}+\frac{C_{\infty }}{\mathrm{\Gamma }(3-\alpha )}\right)\epsilon$$

and then (26) holds for $t\in [t_{\epsilon },\infty )$ with $\tilde{C}=\left(C{Q}_{t_{\epsilon }}+\frac{C_{\infty }}{\mathrm{\Gamma }(3-\alpha )}\right).$ □

Theorem 7.

Let the system (2) be HU in time stable on $J^0.$

Then, the system (2) is time stable in the Lyapunov sense (in the sense of Definition 7).

Proof. 

Let us consider the function $Z\left(t\right):J^{-h}\to {\mathbb{R}}^n,Z\left(t\right)\equiv \mathbf{0},t\in J^{-h}$ and let $\epsilon >0,$ $\delta \in (0,\epsilon ]$ be arbitrary numbers.

Introduce the initial function ${\mathrm{\Phi }}^{\delta }(t-a)\in {\mathbf{PC}}^{*}$ with $∥{\mathrm{\Phi }}^{\delta }∥<\delta$ and then in virtue of Corollary 3, the IP (2), (3) has a unique solution $X_{{\mathrm{\Phi }}^{\delta }}^0\left(t\right)\in {\mathbf{M}}^{1-\alpha }$, which has the representation

$$X_{{\mathrm{\Phi }}^{\delta }}^0\left(t\right)=\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s{\overline{\mathrm{\Phi }}}^{\delta }(s-a)$$

(36)

Since the function $Z\left(t\right)$ satisfies the inequality (25) for $t\in [t_{\epsilon },\infty )$ where $t_{\epsilon }\in J^0,$ $|{\mathrm{\Phi }}^{\delta }(t-a)|<\delta$ for $t\in [a-h,a]$ and the system (2) is HU in time stable, then we obtain that $X_{{\mathrm{\Phi }}^{\delta }}^0\left(t\right)$ satisfies (26) for $t\in [t_{\epsilon },\infty )$. Thus, from (26) and (36), it follows that for the $t\in J_{t_{\epsilon }}$, if follows the estimation

$$\left|X_{{\mathrm{\Phi }}^{\delta }}^0\left(t\right)\right|=\left|\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s{\overline{\mathrm{\Phi }}}^{\delta }(s-a)\right|\le \epsilon$$

(37)

and hence, $Q_{\epsilon }=\underset{t\in [t_{\epsilon },\infty )}{sup}\left(\underset{s\in [a-h,a]}{sup}\left|Q(t,s)\right|\right)<\infty .$ Then, choosing $\delta =min(\epsilon {\left(Q_{\epsilon }C\right)}^{-1},\epsilon )$ and estimating the integral in (37) for $t\in [t_{\epsilon },\infty )$, we obtain that

$$\left|\underset{a-h}{\overset{a}{\int }}Q(t,s){\mathrm{d}}_s{\overline{\mathrm{\Phi }}}^{\delta }(s-a)\right|\le Q_{\epsilon }Va{r}_{s\in [a-h,a]}{\overline{\mathrm{\Phi }}}^{\delta }(s-a)\le Q_{\epsilon }C∥{\mathrm{\Phi }}^{\delta }∥\le Q_{\epsilon }C\delta <\epsilon ,$$

holds for any function ${\mathrm{\Phi }}^{\delta }(t-a)\in {\mathbf{PC}}^{*}$ with $∥{\mathrm{\Phi }}^{\delta }∥<\delta =min(\epsilon {\left(Q_{\epsilon }C\right)}^{-1},\epsilon )$ which implies that the zero solution of (2) is stable in time. □

6. Conclusions and Comments

In the present paper for linear fractional systems with Riemann–Liouville (RL)-type derivatives and distributed delays, we obtained three main results.

The first is that under natural assumptions we proved the existence and uniqueness of the solutions of the initial problem (IP) for these systems with discontinuous initial functions. Note that the used assumptions are similar to these used for the same result in the case when the derivatives in the system are first (integer) order. As a consequence of this result, we also prove the existence of a unique fundamental matrix for the homogeneous system.

The second main result is the existence of a unique fundamental matrix to obtain integral representations of the solutions of the IP for the inhomogeneous systems as well as the solutions of the IP for the corresponding inhomogeneous system.

To obtain our third main result, first we introduce concepts for HU in time stability and HUR in time stability for the studied systems with Riemann–Liouville fractional derivatives, in which concepts the are based on the concept for Lyapunov in time stability proposed in [20]. Furthermore, to obtain our stability results, instead of the standard approach based on some concrete fixed-point theorem chosen by the researcher, we introduce a new approach based on the integral representation of the solutions for the studied systems in the corresponding linear case, which is a consequence of our results obtained in Section 3 and Section 4 above. Our approach can be used in all cases (without the case of fuzzy equations, where additional work must be done) in which the standard approach based on some fixed-point theorem is applicable and without the difference of fractional derivative types included in the studied class equations (systems). The only restriction is that the equation must possess at least one continuous solution of the Cauchy problem for a class initial function, which can also be discontinuous with finitely many jumps of the first kind. Moreover, the applicability of our approach is regardless of the chosen technique for the proof of the solution’s existence (fixed point theorems, topological methods, successive approximations, etc.). Generally speaking, the nonlinear case can be considered with the proposed approach in a similar way, after transforming it in the form of the nonlinear perturbed linear system under some natural assumptions on the nonlinearity term as in the integer case. As a third main result, using the proposed approach, we establish sufficient conditions which guarantee HU in time stability of the investigated systems. Finally, we prove that the HU in time stability leads to Lyapunov in time stability for the studied homogeneous systems.

As a comment, we note that the fact of existence and uniqueness of the fundamental matrix established in the present work, as well as the introduced new approach based on the integral presentations of the solutions of IP for the studied systems with initial function $\mathrm{\Phi }\in {\mathbf{PC}}^{*}$, lead to some interesting open problems:

1.

To establish sufficient conditions, which guarantee system (1) to be HUR in time stable on $J_b^0$ for arbitrary $b\in {\mathbb{R}}_{+}^0,$ with respect to some $\phi \left(t\right)\in C(J_b^{-h},{\mathbb{R}}_{+}^0)\left(C(J^{-h},{\mathbb{R}}_{+}^0)\right)$.

2.

To establish sufficient conditions which guarantee system (1) to be HUR in time stable on $J^0$ with respect to some $\phi \left(t\right)\in C(J_b^{-h},{\mathbb{R}}_{+}^0)\left(C(J^{-h},{\mathbb{R}}_{+}^0)\right).$

3.

To prove or disprove the conjecture that if the system (2) is HUR in time stable on $J^0$ with respect to some appropriate $\phi \left(t\right)\in C(J_b^{-h},{\mathbb{R}}_{+}^0)\left(C(J^{-h},{\mathbb{R}}_{+}^0)\right)$, then the zero solution of (2) is asymptotically stable in time in sense of Definition 7.