On the Preservation with Respect to Nonlinear Perturbations of the Stability Property for Nonautonomous Linear Neutral Fractional Systems with Distributed Delays

Abstract

In the present paper, sufficient conditions are obtained under which the Cauchy problem for a nonlinearly perturbed nonautonomous neutral fractional system with distributed delays and Caputo type derivatives has a unique solution in the case of initial functions with first-kind discontinuities. For this system, by applying a formula for the integral presentation of the solution of the nonhomogeneous linear neutral fractional system, we found some additional natural conditions to ensure that from the global asymptotically stability of the zero solution of the linear part of the nonlinearly perturbed system, global asymptotic stability of the zero solution of the whole nonlinearly perturbed system follows.

1. Introduction

Delayed fractional systems stand for dynamical systems with both fractional-order derivatives and time delays, where the fractional-order derivatives are used to model nonlocal properties of viscoelastic materials, anomalous diffusion of particles, etc., and time delays usually arise in the modeling of controllers, filters, and actuators. Detailed understandings of the fractional calculus theory and fractional differential equations can be obtained from the remarkable monographs of Kilbas et al. [1] and Podlubny [2]. An important theme of distributed order fractional differential equations is considered in Jiao at al. [3] and an application-oriented exposition is given in Diethelm [4]. The impulsive fractional differential and functional differential equations are represented in Stamova, Stamov [5] where several applications are considered too.

One of the main research topics in the qualitative theory of the fractional differential equations is the problem of stability of the process (in particular, of a stationary state), i.e., the possibility of the process to withstand the impact of a priori, unknown, small in some sense, influences (disturbances). We emphasize that since this property turns out to be of utmost importance, the study of stability is an “evergreen” theme for investigations. From a practical point of view, it is clear that a separate predictable process can be physically realized only if it is stable in some suitable natural sense.

The study of the stability of fractional differential equations and systems with delay is more complicated in comparison with fractional differential equations and systems without delay, i.e., we have the same situation as in the integer case. Different types of fractional differential equations and systems with delays (retarded and neutral) or without delays are studied for several types of stability. An excellent overview of the works related to our theme published up to 2011 is given in [6] and the references therein. From the more recently published works, we refer to [7,8,9,10,11,12,13,14,15]. Since the existence and the integral representation of the solutions of the studied systems play an important role in the study of all types stability results, we refer also to [16,17,18,19,20,21].

In the present work, a nonlinearly perturbed non-autonomous neutral fractional system with distributed delays and Caputo type derivatives is studied. We generalize some results, obtained in [15] for the partial case when the Lebesgue decomposition of the linear part does not include absolutely continuous and singular parts, i.e., the linear part includes only concentrated delays. Moreover, the linear part of the nonlinear perturbed system in the work [15] is autonomous in contrast with the nonautonomous linear part considered in the present work. The aim of this work is first to obtain sufficient conditions under which the Initial Problem (IP) for a nonlinearly perturbed nonautonomous neutral system with distributed delays and Caputo fractional derivatives has a unique solution in the case of initial functions with first-kind discontinuities. Then, for this system, applying the formula for the integral presentation of the solution of the nonhomogeneous linear neutral fractional system with zero initial condition obtained in [21], some additional natural conditions are found to ensure that from global asymptotically stability of the zero solution of the linear part of the nonlinearly perturbed system, the global asymptotic stability of the zero solution of the whole nonlinearly perturbed system follows.

The paper is organized as follows. In Section 2 we recall the definitions of Riemann–Liouville and Caputo fractional derivatives and the needed part of their properties. Some notations are introduced and the problem statement is presented too. Section 3 is devoted to the problem of existence and uniqueness of the solutions of the Cauchy problem for a non-autonomous neutral nonlinear fractional differential system with a discontinuous initial function in the case of incommensurate order Caputo derivatives. In Section 4 we study the stability of a neutral non-autonomous linear fractional differential system nonlinear perturbed in the case of Caputo-type derivatives with incommensurate differential orders. Under some natural conditions it is proved that if the zero solution of the linear part of the nonlinear perturbed system is globally asymptotically stable (GAS), then the zero solution of the perturbed nonlinear system is globally asymptotic stable too.

2. Preliminaries and Problem Statement

As is standard to avoid possible misunderstandings, below we present the definitions of Riemann–Liouville and Caputo fractional derivatives and some needed their properties. More detailed information about this theme can be found in the monographs [1,2].

Let $f\in L_1^{loc}(\mathbb{R},\mathbb{R})$, where $L_1^{loc}(\mathbb{R},\mathbb{R})$ is the real linear space of all locally Lebesgue integrable functions $f:\mathbb{R}\to \mathbb{R}$ and $\alpha \in (0,1)$ is an arbitrary number. For $a\in \mathbb{R}$ and each $t>a$, the left-sided fractional integral operator, the left side Riemann–Liouville and Caputo fractional derivative of order $\alpha$ are defined by:

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

$${}_C{D}_{a+}^{\alpha }f\left(t\right){=}_{RL}{D}_{a+}^{\alpha }\left[f\left(s\right)-f\left(a\right)\right]\left(t\right){=}_{RL}{D}_{a+}^{\alpha }f\left(t\right)-\frac{f\left(a\right)}{\mathsf{\Gamma }(1-\alpha )}{(t-a)}^{-\alpha },$$

respectively. In our exposition the following relations will be used (see [1]):

$$\left(\mathrm{a}\right)\left(D_{a+}^{0}f\right)\left(t\right)=f\left(t\right);\left(\mathrm{b}\right){}_C{D}_{a+}^{\alpha }{D}_{a+}^{-\alpha }f\left(t\right)=f\left(t\right);\left(\mathrm{c}\right)D_{a+}^{-\alpha }{}_C{D}_{a+}^{\alpha }f\left(t\right)=f\left(t\right)-f\left(a\right).$$

As it is standard we denote the one parameter Mittag–Leffler function by:

$$E_{\omega }\left(t\right)=\sum _{k=0}^{\infty }\frac{t^k}{\mathsf{\Gamma }(\omega k+1)},$$

where $\omega \ge 0,t\in \mathbb{R}$ and $\mathsf{\Gamma }$ is the gamma function.

Below the following notations will be used too: ${\mathbb{R}}_{+}=(0,\infty )$, ${\overline{\mathbb{R}}}_{+}=[0,\infty )$, ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, $k\in \langle n\rangle =\{1,\dots ,n\}$, $n\in \mathbb{N}$, $J_a=[a,\infty )$, $a\in \mathbb{R}$, $J_{a+M}=[a,a+M]$, $M\in {\mathbb{R}}_{+}$, $I,\mathsf{\Theta }\in {\mathbb{R}}^{n\times n}$ are the identity and zero matrix respectively and $\mathbf{0}\in {\mathbb{R}}^n$ is the zero vector-column. For $Y:J\to {\mathbb{R}}^{n\times n}$, $Y\left(t\right)={\left\{y_{k,j}\left(t\right)\right\}}_{k,j=1}^n$ we denote $\left|Y\left(t\right)\right|={\sum }_{k,j=1}^n\left|y_{k,j}\left(t\right)\right|$ for $t\in J_a$. As usual for arbitrary fixed $h>0$ with $BV([a-h,a],{\mathbb{R}}^n)$ we denote the linear space of all vector-column functions $\mathsf{\Phi }={({\varphi }_1,\dots ,{\varphi }_n)}^{{}^{\top }}:[a-h,a]\to {\mathbb{R}}^n$ with bounded variation on $[a-h,a]$. The function $\mathsf{\Phi }=({\varphi }_1,\dots ,{\varphi }_n){}^{\top }:[-h,0]\to {\mathbb{R}}^n$ is called piecewise continuous on $[-h,0]$ (noted by $\mathsf{\Phi }\in PC([-h,0],{\mathbb{R}}^n)$), if the interval $[-h,0]$ can be broken into a finite number of subintervals on which $\mathsf{\Phi }\left(t\right)$ is continuous on each open subinterval and has a finite limit at the endpoints of each subinterval. With $S^{\mathsf{\Phi }}$ we denote the set of all jumps points of $\mathsf{\Phi }\left(t\right)$ and with $\mathbf{PC}$ we denote the normed space of all right continuous in the interval $[-h,0]$ vector functions $\mathsf{\Phi }\in PC([-h,0],{\mathbb{R}}^n)$ with norm

$$\parallel \mathsf{\Phi }\parallel =\underset{s\in [-h,0]}{sup}\left|\mathsf{\Phi }\left(s\right)\right|=\underset{s\in [-h,0]}{sup}\sum _{k=1}^n\left|{\varphi }_k\left(s\right)\right|<\infty ,$$

by ${\mathbf{PC}}^{*}=\mathbf{PC}\cap BV([-h,0],{\mathbb{R}}^n)$ is denoted the normed space with the same norm and $\mathbf{C}=\mathbf{C}([-h,0],{\mathbb{R}}^n)$, the space of all continuous functions. Obviously, we have ${\mathbf{PC}}^{*}\subset \mathbf{PC}$, $\mathbf{C}\subset \mathbf{PC}$ and define $\mathbf{E}\left({\mathbf{E}}^{*}\right)=J_a\times \mathbf{PC}\left({\mathbf{PC}}^{*}\right)$.

Consider for $t>a$ the nonlinear perturbed neutral linear delayed system with incommensurate type differential orders and distributed delays in the following general form:

$$D_{a+}^{\alpha }(X\left(t\right)-{\int }_{-h}^0\left[d_{\theta }V(t,\theta )\right]X(t+\theta ))={\int }_{-h}^0\left[d_{\theta }U(t,\theta )\right]X(t+\theta )+W(t,X_t{}^{\top }),$$

(1)

and the corresponding homogeneous (unperturbed) neutral linear delayed system,

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

(2)

where $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$, ${\alpha }_k\in (0,1)$, $V(t,\theta )={\sum }_{l\in 〈r〉}{V}^l(t,\theta )$, $r\in \mathbb{N}$, $U(t,\theta )={\sum }_{i\in {〈m〉}_0}{U}^i(t,\theta )$, $i\in {〈m〉}_0=\{0,1,\dots ,m\}$, $m\in {\mathbb{N}}_0$, $X:J_a\to {\mathbb{R}}^n$, $W:\mathbf{E}\to {\mathbb{R}}^n$, $U^i,V^l:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$, $V^l(t,\theta )={\left\{v_{kj}^l(t,\theta )\right\}}_{k,j=1}^n$, $U^i(t,\theta )={\left\{u_{kj}^i(t,\theta )\right\}}_{k,j=1}^n$, $X\left(t\right)=(x_1\left(t\right),\dots ,x_n\left(t\right)){}^{\top }$, $X_t\left(\theta \right)=$$(x_t^1\left(\theta \right),\dots ,x_t^n\left(\theta \right)){}^{\top }$, $X_t\left(\theta \right)=X(t+\theta )$, $-h\le \theta \le 0$, $h>0$, $W(t,X_t{}^{\top })=(w_1(t,X_t{}^{\top }),\dots ,$$w_n(t,X_t{}^{\top })){}^{\top }$, $D_{a+}^{\alpha }X\left(t\right)=(D_{a+}^{{\alpha }_1}{x}_1\left(t\right),\dots ,D_{a+}^{{\alpha }_n}{x}_n\left(t\right)){}^{\top }$ for every $t>a$, where $D_{a+}^{{\alpha }_k}$ denotes the left side Caputo fractional derivative ${}_C{D}_{a+}^{{\alpha }_k}$. In addition, for $Y(t,\theta ):J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ we denote $\left|Y(t,\theta )\right|={\sum }_{k,j=1}^n\left|y_{k,j}(t,\theta )\right|$ and when $Y(t,·)\in BV([-h,0],{\mathbb{R}}^n)$ then we have

$${\mathrm{Var}}_{[-h,0]}Y(t,·)=\sum _{k,j=1}^n{\mathrm{Var}}_{[-h,0]}{y}_{k,j}(t,·).$$

For every fixed $t\in J_a$ and for $Y\left(t\right)=(y_1\left(t\right),\dots ,y_n\left(t\right)){}^{\top }:J_a\to {\mathbb{R}}^n$, $\beta =({\beta }_1,\dots ,{\beta }_n)$ with ${\beta }_k\in [-1,1]$, $k\in 〈n〉$ we will use the notation $I_{\beta }\left(Y\left(t\right)\right)=\mathrm{diag}(y_1^{{\beta }_1}\left(t\right),\dots ,y_n^{{\beta }_n}\left(t\right))$. Let also denote ${\displaystyle {\alpha }_m=\underset{k\in 〈n〉}{min}\left({\alpha }_k\right)}$ and ${\displaystyle {\alpha }_M=\underset{k\in 〈n〉}{max}\left({\alpha }_k\right)}$.

For clarity, we rewrite the system (2) in a more detailed form:

$$D_{a+}^{{\alpha }_k}\left(x_k\left(t\right)-\sum _{l=1}^r\left(\sum _{j=1}^n{\int }_{-{\tau }_l}^0{x}_j(t+\theta )d_{\theta }{v}_{kj}^l(t,\theta )\right)\right)=\sum _{i=0}^m\left(\sum _{j=1}^n{\int }_{-{\sigma }_i}^0{x}_j(t+\theta )d_{\theta }{u}_{kj}^i(t,\theta )\right),$$

where ${\sigma }_i,{\tau }_l\in {\mathbb{R}}_{+}$, ${\displaystyle \tau =\underset{l\in \langle r\rangle }{max}{\tau }_l}$, ${\displaystyle \sigma =\underset{i\in {\langle m\rangle }_0}{max}{\sigma }_i}$, $h=max(\tau ,\sigma )$.

Introduce for arbitrary $\mathsf{\Phi }\in \mathbf{PC}\left({\mathbf{PC}}^{*}\right)$ the following initial condition:

$$\begin{array}{ccc}\hfill & & X_t\left(\theta \right)=X(t+\theta )=\mathsf{\Phi }(t-a+\theta )\mathrm{for}t+\theta \le a,-h\le \theta \le 0.\hfill \end{array}$$

(3)

We say that for the kernels $U^i,V^l:J_a\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$, $i\in {〈m〉}_0$, $l\in 〈r〉$ the conditions (S) are fulfilled if the following conditions hold (see [22,23]):

(S1) The functions $(t,\theta )\to U^i(t,\theta )$, $(t,\theta )\to V^l(t,\theta )$ are measurable in $(t,\theta )\in J_a\times \mathbb{R}$ and normalized so that $U^i(t,\theta )=0$, $V^l(t,\theta )=0$ for $\theta \ge 0$, $U^i(t,\theta )=U^i(t,-{\sigma }_i)$ for $\theta \le -{\sigma }_i$, $V^l(t,\theta )=V^l(t,-{\tau }_l)$ for $\theta \le -{\tau }_l$, for every $t\in J_a$. The kernels $U^i(t,\theta )$ and $V^l(t,\theta )$ are continuous from left in $\theta$ on $(-{\sigma }_i,0)$ and $(-{\tau }_l,0)$ respectively for $t\in J_a$, $l\in 〈r〉$, $i\in {〈m〉}_0$, $U^i(t,·),V^l(t,·)\in BV([-h,0],{\mathbb{R}}^{n\times n})$.

(S2) For $t\in J_a$, $l\in 〈r〉$, $i\in {〈m〉}_0$, ${\displaystyle {\mathrm{Var}}_{[-h,0]}{U}^i(t,·)\in L_1^{\mathrm{loc}}(J_a,{\mathbb{R}}^{n\times n})}$ is locally bounded for $t\in J_a$, ${\displaystyle {\mathrm{Var}}_{[-h,0]}{V}^l(t,·)\in L_1^{\mathrm{loc}}(J_a,{\mathbb{R}}^{n\times n})}$ is uniformly bounded in t on $J_a$ and $V^l(t,\theta )$ are uniformly nonatomic at zero (see [23]) (i.e., for every $\epsilon >\mathbf{0}$, there exists $\delta \left(\epsilon \right)>\mathbf{0}$ such that for each $t\in J_a$ we have that ${\displaystyle {\mathrm{Var}}_{[-\delta ,0]}{V}^l(t,·)<\epsilon }$).

(S3) For $t\in J_a$ and $\theta \in [-h,0]$ the Lebesgue decompositions of the kernels have the form:

$$U^i(t,\theta )=U_d^i(t,\theta )+U_c^i(t,\theta ),V^l(t,\theta )=V_d^l(t,\theta )+V_c^l(t,\theta ),$$

where $V_c^l(t,\theta )=V_{ac}^l(t,\theta )+V_s^l(t,\theta )$ is the sum of the absolutely continuous and singular part respectively of $V^l(t,\theta )$. We have also

$$U^i(t,\theta )={\left\{a_{kj}^i\left(t\right)H(\theta +{\sigma }_i\left(t\right))\right\}}_{k,j=1}^n,V^l(t,\theta )={\left\{{\overline{a}}_{kj}^i\left(t\right)H(\theta +{\tau }_l\left(t\right))\right\}}_{k,j=1}^n,$$

where $H\left(t\right)$ is the Heaviside function, $A^i\left(t\right)={\left\{a_{kj}^i\left(t\right)\right\}}_{k,j=1}^n\in L_1^{\mathrm{loc}}(J_a,{\mathbb{R}}^{n\times n})$ are locally bounded, ${\overline{A}}^l\left(t\right)={\left\{{\overline{a}}_{kj}^l\left(t\right)\right\}}_{k,j=1}^n\in C(J_a,{\mathbb{R}}^{n\times n})$, ${\tau }_l\left(t\right)\in C(J_a,[0,{\tau }_l]),{\sigma }_i\left(t\right)\in C(J_a,[0,{\sigma }_i])$, ${\sigma }_0\left(t\right)\equiv 0$ for every $t\in J_a$, $l\in 〈r〉$, $i\in 〈m〉$.

(S4) For each $t^{*}\in J_a$ the relations

$$\underset{t\to t^{*}}{lim}{\int }_{-\sigma }^0|U^i(t,\theta )-U^i(t^{*},\theta )|\mathrm{d}\theta =\underset{t\to t^{*}}{lim}{\int }_{-\tau }^0|V^l(t,\theta )-V^l(t^{*},\theta )|\mathrm{d}\theta =0$$

hold and the sets $S_{\mathsf{\Phi }}^l=\{t\in J_a|t-{\tau }_l\left(t\right)\in S^{\mathsf{\Phi }}\},S_{\mathsf{\Phi }}^i=\{t\in J_a|t-{\sigma }_i\left(t\right)\in S^{\mathsf{\Phi }}\}$ do not have limit points (see [23]) for each $l\in 〈r〉$ and $i\in 〈m〉$.

(S5) There exists $\gamma \in {\mathbb{R}}_{+}$ such that, the kernels $V_{ac}^l(t,\theta )$ and $V_s^l(t,\theta )$ are continuous in t, when $t\in [a,a+\gamma ]$, $\theta \in [-h,0]$ and $l\in 〈r〉$.

Remark 1.

The initial condition (3) means that the function $t\to X\left(t\right)$, $t\in J_a$ is considered as one prolongation of the functions $t\to \mathsf{\Phi }(t-a)$, $t\in [a-h,a]$. Everywhere below in our consideration we assume that the conditions (S) hold. It is not difficult to see that Condition (S4) implies that the sets

$$S_{\mathsf{\Phi }}^i\left(M\right)=\{t\in [a,a+M]|t-{\sigma }_i\left(t\right)\in S^{\mathsf{\Phi }}\}\mathrm{and}{S}_{\mathsf{\Phi }}^l\left(M\right)=\{t\in [a,a+M]|t-{\tau }_l\left(t\right)\in S^{\mathsf{\Phi }}\}$$

are finite for all $l\in 〈r〉$, $i\in 〈m〉$ and $M\in {\mathbb{R}}_{+}$.

Definition 1.

The vector function $X\left(t\right)={(x_1\left(t\right),\dots ,x_n\left(t\right))}^{{}^{\top }}$ is a solution of the initial problem (IP) (1), (3) in $J_{a+M}\left(J_a\right)$ if ${X|}_{[a,a+M]}\in C([a,a+M],{\mathbb{R}}^n)$, $(X{|}_{J_a}\in C(J_a,{\mathbb{R}}^n))$ satisfies the system (1) for all $t\in (a,M](t\in (a,\infty \left)\right)$ and the initial condition (3) for $t+\theta \le a$, $-h\le \theta \le 0$.

In our consideration below we need the following auxiliary system:

$$\begin{array}{ccc}\hfill & X\left(t\right)=& C_{\mathsf{\Phi }\left(0\right)}+{\int }_{-h}^0\left[d_{\theta }V(t,\theta )\right]X(t+\theta )\hfill \\ \hfill & & +I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_a^t{I}_{\alpha -1}(t-\tau )\left({\int }_{-h}^0\left[d_{\theta }U(t,\theta )\right]X(\tau +\theta )+W(\tau ,X_t^{{}^{\top }})\right)\mathrm{d}\tau ,\hfill \end{array}$$

(4)

where ${\displaystyle C_{\mathsf{\Phi }\left(0\right)}=\mathsf{\Phi }\left(0\right)-{\int }_{-h}^0\left[d_{\theta }V(a,\theta )\right]\mathsf{\Phi }\left(\theta \right)}$.

Definition 2.

The vector function $X\left(t\right)={(x_1\left(t\right),\dots ,x_n\left(t\right))}^{{}^{\top }}$ is a solution of the initial problem (IP) (4), (3) in $J_{a+M}\left(J_a\right)$ if ${X|}_{[a,a+M]}\in C([a,a+M],{\mathbb{R}}^n)$, $(X{|}_{J_a}\in C(J_a,{\mathbb{R}}^n))$ satisfies the system (4) for all $t\in (a,M](t\in (a,\infty \left)\right)$ and the initial condition (3) for $t+\theta \le a$, $-h\le \theta \le 0$.

We say that the vector valued functional $W:\mathbf{E}\left({\mathbf{E}}^{*}\right)\to {\mathbb{R}}^n$ satisfies the conditions (H) (modified Caratheodory conditions) in $\mathbf{E}\left({\mathbf{E}}^{*}\right)$ if the following conditions hold:

(H1) For almost all fixed $t\in J_a$ the functional $(t,Y)\to W(t,Y)$ is continuous in arbitrary $Y\in \mathbf{PC}\left({\mathbf{PC}}^{*}\right)$ and for each fixed function $Y\in \mathbf{PC}\left({\mathbf{PC}}^{*}\right)$ the function $(t,Y)\to W(t,Y)$ is Lebesgue measurable and locally bounded for $t\in J_a$.

(H2) (Local Lipschitz type condition) For each $(t,Y)\in \mathbf{E}\left({\mathbf{E}}^{*}\right)$ and for some its neighborhood $O(t,Y)\subset \mathbf{E}\left({\mathbf{E}}^{*}\right)$ there exists a locally bounded, Lebesgue measurable function $\ell \in L_1^{\mathrm{loc}}(J_a,{\mathbb{R}}_{+})$ such that the inequalities

$$|W(t,Y_1{}^{\top })-W(t,Y_2{}^{\top })|\le \ell \left(t\right)\left|\right|Y_1-Y_2\left|\right|$$

(5)

hold for every $(t,Y_1),(t,Y_2)\in O(t,Y)$, $\left|W\right(t,\mathbf{0}\left)\right|\equiv 0$ for $t\in J_a$.

Remark 2.

Note that the function $\ell \in L_1^{\mathrm{loc}}(J_a,{\mathbb{R}}_{+})$ in the condition (H2) generally speaking depends on the neighborhood of the chosen point $(t,Y)\in \mathbf{E}\left({\mathbf{E}}^{*}\right)$, i.e., can be different for the different existing neighborhoods according to condition (H2) for the different points.

Lemma 1.

Let the conditions (S) hold and the condition (H1) holds in $\mathbf{E}$.

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

Proof. 

The proof of this lemma is an elementary modification of the proof for the case of continuous initial function and will be omitted. □

Let $M^0\in {\mathbb{R}}_{+}$, $[a-h,a+M^0]$ be an initial interval, with right endpoint $a+M^0$, and the function $\mathsf{\Phi }\in \mathbf{PC}$ be an arbitrary initial function and $X^0\left(t\right)\in C([a,a+M^0],{\mathbb{R}}^n)$ be the unique solution of the IP (4), (3) with the same initial function. Define the following initial function $\overline{\mathsf{\Phi }}\left(t\right)[a,a+M^0]\to {\mathbb{R}}^n$ as follows: $\overline{\mathsf{\Phi }}\left(t\right)=\mathsf{\Phi }(t-a),t\in [a-h,a]$, $\overline{\mathsf{\Phi }}\left(t\right)=X^0\left(t\right),t\in [a,a+M^0]$ and introduce the following initial condition:

$$X\left(t\right)=\overline{\mathsf{\Phi }}\left(t\right)\mathrm{for}t\in [a-h,a+M^0].$$

(6)

Definition 3.

The vector function $X\left(t\right)={(x_1\left(t\right),\dots ,x_n\left(t\right))}^{{}^{\top }}$ is a solution of the IP (4), (6) or of the IP (1), (6) in $(a+M^0,a+M^0+{\overline{M}}^0]$, $\left((a+M^0,\infty )\right)$, $M\in {\mathbb{R}}_{+}$ if

$${X|}_{[a,a+M^0+{\overline{M}}^0]}\in C([a,a+M+{\overline{M}}^0],{\mathbb{R}}^n),(X{|}_{(a,\infty )}\in C((a,\infty ),{\mathbb{R}}^n)),$$

satisfies the system (4) or the system (1) respectively for all $t\in (a+M^0,a+M^0+{\overline{M}}^0]$$\left((a+M^0,\infty )\right)$ and the initial condition (6) for $t\in [a-h,a+M^0]$.

The next lemma treats the same problem as Lemma 1 between IP (4), (6) and IP (1), (6) in the important case when the initial point does not coincide with the lower terminal of the fractional derivative.

Lemma 2.

Let the following conditions be fulfilled:

1.

The conditions (S) hold and the vector valued functional $W:\mathbf{E}\left({\mathbf{E}}^{*}\right)\to {\mathbb{R}}^n$ satisfies the conditions (H) in $\mathbf{E}$.

2.

Let $\mathsf{\Phi }\in \mathbf{PC}$, $M^0\in {\mathbb{R}}_{+}$ be arbitrary and $X^0\left(t\right)\in C([a,a+M^0],{\mathbb{R}}^n)$ be the unique solution of the IP (4), (3).

Then every solution of IP (1), (6) is a solution of IP (4), (6) and vice versa.

Proof. 

The proof is standard and analogical of the proof of Lemma 1 in [13] with the difference that instead of condition 2 there, condition (H1) here implies that $W(t,X_t)\in L_1^{\mathrm{loc}}(J_a,\mathbb{R})$ for each solution $X\left(t\right)$ of (1) or (4). □

For arbitrary fixed $\mathsf{\Phi }\in \mathbf{PC}$ we introduce the following sets:

$${\mathfrak{J}}_a^{\mathsf{\Phi }}=\{G\in C(J_a,{\mathbb{R}}^n)|G\left(a\right)=\mathsf{\Phi }\left(0\right),G(t+\theta )=\mathsf{\Phi }(t-a+\theta ),t+\theta <a,\theta \in [-h,0]\},$$

$${\overline{\mathfrak{J}}}_a^{\mathsf{\Phi }}=\left\{Y\right|Y\left(\theta \right)=G_t\left(\theta \right)=G(t+\theta ):[t-h,t]\to {\mathbb{R}}^n,G\in {\mathfrak{J}}_a^{\mathsf{\Phi }},t\in J_a\}\mathrm{and}\overline{\mathbf{E}}=J_a\times {\overline{\mathfrak{J}}}_a^{\mathsf{\Phi }}.$$

For each $M\in {\mathbb{R}}_{+}$ define the sets:

$${\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}=\left\{Y\right|Y\in {\overline{\mathfrak{J}}}_a^{\mathsf{\Phi }},t\in [a,a+M]\}\mathrm{and}{\overline{\mathbf{E}}}_M=[a,a+M]\times {\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}.$$

Since for each $G=(g^1,\dots ,g^n){}^{\top }\in {\mathfrak{J}}_a^{\mathsf{\Phi }}$ and $t\in J_a$, we have that $G_t\in \mathbf{PC}$ for every $\mathsf{\Phi }\in \mathbf{PC}$, then we have that ${\overline{\mathbf{E}}}_M\subset \overline{\mathbf{E}}\subset \mathbf{E}$.

Let $(t_1,Y_1),(t_2,Y_2)\in \overline{\mathbf{E}}$ be arbitrary and introduce in $\overline{\mathbf{E}}$ the following distance function:

$$d((t_1,Y_1),(t_2,Y_2))=|t_1-t_2|+\parallel {\overline{Y}}_1-{\overline{Y}}_2\parallel ,$$

where ${\overline{Y}}_i=Y_i$ for $t_i\ge a+h$ and for $t_i\in [a,a+h]$, $i=1,2$ we define:

$${\overline{Y}}_i\left(\theta \right)=\left\{\begin{array}{cc}{Y}_i\left(\theta \right),\hfill & a-t_i\le \theta \le 0\hfill \\ Y_i(a-t_i),\hfill & a-h\le \theta \le a-t_i.\hfill \end{array}\right.$$

It is simply to check that in respect to the introduced distance function, the set $\overline{\mathbf{E}}$ is a complete metric space and then the sets ${\overline{\mathbf{E}}}_M$ are complete metric subspaces of $\overline{\mathbf{E}}$ for every $M\in {\mathbb{R}}_{+}$.

In our exposition, we will use the next technical lemma, formulated below.

Lemma 3

([21]). Let the following conditions be fulfilled:

1.

The conditions (S) hold.

2.

${\displaystyle \prod _{l\in 〈r〉}{\tau }_l\left(a\right)>0}$.

Then for every initial function $\mathsf{\Phi }\in \mathbf{PC}$ there exists a constant $M^{*}\in (0,h])$ (eventually depending from Φ) such that $t-{\tau }_l\left(t\right)<a$ and $\mathsf{\Phi }(t-{\tau }_l\left(t\right))$ is continuous for $t\in [a,a+M^{*}]$ and all $l\in 〈r〉$.

3. Existence and Uniqueness of the Solutions of the Initial (Cauchy) Problem (4), (3)

We start this section with an overview of the possible interactions between the concentrated delays in the neutral part of the system and the low terminal a of the fractional derivatives in the case when $a\in S^{\mathsf{\Phi }}$ for some initial function $\mathsf{\Phi }\in \mathbf{PC}$.

Definition 4.

For an initial function $\mathsf{\Phi }\in \mathbf{PC}$ with $a\in S^{\mathsf{\Phi }}$, the low terminal a will be called a regular jump point relative to some delay ${\tau }_l\left(t\right)$, $l\in 〈r〉$ if the equality ${\tau }_l\left(a\right)=0$ implies that there exists a constant $\epsilon \in (0,h]$ (eventually depending on ${\tau }_l$), such that $t-{\tau }_l\left(t\right)\ge a$ for $t\in [a,a+\epsilon ]$.

Definition 5.

For an initial function $\mathsf{\Phi }\in \mathbf{PC}$ with $a\in S^{\mathsf{\Phi }}$, the low terminal a will be called an irregular jump point relative to some delay ${\tau }_l\left(t\right)$, $l\in 〈r〉$, if the equality ${\tau }_l\left(a\right)=0$ implies that there exists a constant $\epsilon \in (0,h]$ (eventually depending from ${\tau }_l$) such that $t-{\tau }_l\left(t\right)<a$ for $t\in (a,a+\epsilon ]$ and $\mathsf{\Phi }(t-{\tau }_l\left(t\right))$ is continuous.

The next lemma is more convenient for the application variant of Lemma 4 in [21].

Lemma 4.

Let the initial function $\mathsf{\Phi }\in \mathbf{PC}$ with $a\in S^{\mathsf{\Phi }}$ be arbitrary and the following conditions hold:

1.

The conditions (S) are fulfilled.

2.

${\displaystyle \prod _{l\in 〈r〉}{\tau }_l\left(a\right)=0}$.

Then for every delay ${\tau }_l\left(t\right)$ with ${\tau }_l\left(a\right)=0$, $l\in 〈r〉$, the point $a\in S^{\mathsf{\Phi }}$ is either a regular or irregular jump point, relative to this delay.

Proof. 

From condition 2 it follows that there exists at least one $\overline{l}\in 〈r〉$ such that ${\tau }_{\overline{l}}\left(a\right)=0$. Condition (S4) implies that there exists a constant $\epsilon \in (0,h]$ (eventually depending from ${\tau }_{\overline{l}}$) such that either $t-{\tau }_{\overline{l}}\left(t\right)\ge a$ for $t\in [a,a+\epsilon ]$, or for $t\in (a,a+\epsilon ]$ we have $t-{\tau }_{\overline{l}}\left(t\right)<a$ and $\mathsf{\Phi }(t-{\tau }_{\overline{l}}\left(t\right))$ is continuous for $t\in (a,a+\epsilon ]$ ($\mathsf{\Phi }(0-)\ne \mathsf{\Phi }\left(0\right)$) which completes the proof. □

Remark 3.

Note that without loss of generality, we can renumber all delays so that those for which $a\in S^{\mathsf{\Phi }}$ is a regular jump point to have the numbers ${\tau }_1\left(t\right),\dots ,{\tau }_q\left(t\right)$, $q\in 〈r〉$. In the next exposition, for convenience we assume that this renumbering is made.

Remark 4.

Considering the case of constant delays, we establish that they do not possess regular or irregular jump points; however, in the case of variable delays the situation is not so convenient. It is simple to see that in the case of variable delays the situation is more complicated, especially in the neutral case, when it is necessary to introduce additional restrictions on the delays to avoid at least the case when the low terminal a is an irregular jump point relative to some delay. Some examples of such type conditions are: ${\tau }_l\left(t\right)>0$ for $t\in J_a$, $l\in 〈r〉$; ${\tau }_l\left(t\right)\in AC(J_a,{\overline{R}}_{+})$, ${\tau }_l^{\prime }\left(t\right)<1$ when $t\in J_a$, $l\in 〈r〉$ but from practical point of view it is much better (less restrictive and simple to check) when the conditions are local (see condition 2 of Lemma 3 for example).

Introduce the vector valued functional

$$Z(t,X_t{}^{\top })=(z_1(t,X_t{}^{\top }),\dots ,z_n(t,X_t{}^{\top })){}^{\top }:\mathbf{E}\to {\mathbb{R}}^n$$

as follows:

$$Z(t,X{}^{\top }(t+\theta ))={\int }_{-h}^0\left[d_{\theta }U(t,\theta )\right]X(t+\theta )+W(t,X{}^{\top }(t+\theta )).$$

(7)

Then the system (4) can be rewritten in the following form:

$$\begin{array}{c}\hfill X\left(t\right)=C_{\mathsf{\Phi }\left(0\right)}+{\int }_{-h}^0\left[d_{\theta }V(t,\theta )\right]X(t+\theta )+I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_a^t{I}_{\alpha -1}(t-\tau )Z(\tau ,X_{\tau }{}^{\top })\mathrm{d}\tau .\end{array}$$

(8)

Remark 5.

It is simple to check that if the conditions (S) hold and the vector valued functional $W:\mathbf{E}\to {\mathbb{R}}^n$ satisfies the conditions (H) in some subset $\tilde{\mathbf{E}}\subseteq \mathbf{E}$, then the vector valued functional $Z:\mathbf{E}\to {\mathbb{R}}^n$ satisfies the conditions (H) in the same subset.

Theorem 1.

Let the following conditions be fulfilled:

1.

The conditions (S) hold and for the vector valued functional $W:\mathbf{E}\to {\mathbb{R}}^n$ the conditions (H) are fulfilled in $\overline{\mathbf{E}}$.

2.

For every initial function $\mathsf{\Phi }\in \mathbf{PC}$ with $a\in S^{\mathsf{\Phi }}$, the low terminal a is at most a regular jump point relative to some delay ${\tau }_l\left(t\right)$, $l\in 〈r〉$.

Then there exists $M^0\in {\mathbb{R}}_{+}$ such that the IP (4), (3) has a unique solution in the interval $[a,a+M^0]$.

Proof. 

Let $M^1=min(h,\delta ,\gamma )$ (see (S3) and (S5)) and $M\in (0,M^1]$ be an arbitrary number. Let $t\in [a,a+M]$ and $(t,G_t)\in {\overline{\mathbf{E}}}_M$ be arbitrary. Then using (8) we define the operator $\left(\mathfrak{R}{G}_t\right)=(\left({\mathfrak{R}}_1{g}_t^1\right),\dots ,\left({\mathfrak{R}}_n{g}_t^n\right)){}^{\top }$ as follows:

$$\begin{array}{ccc}\hfill \left(\mathrm{i}\right)& \left(\mathfrak{R}{G}_t\right)\left(t\right)& =C_{\mathsf{\Phi }\left(0\right)}+{\int }_{-h}^0\left[d_{\theta }V(t,\theta )\right]G(t+\theta )\hfill \\ & & +I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_a^t{I}_{\alpha -1}(t-\tau )Z(\tau ,G_{\tau }{}^{\top })\mathrm{d}\tau ,t\in (a,a+M);\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill & & \left(\mathrm{ii}\right)\left(\mathfrak{R}{G}_t\right)(t+\theta )=\mathsf{\Phi }(t-a+\theta ),t+\theta \in [a-h,a];\hfill \\ & & \left(\mathrm{iii}\right)\left(\mathfrak{R}{G}_{a+M}\right)(a+M)=\underset{t\to a+M-0}{lim}\mathfrak{R}{W}_t\left(t\right).\hfill \end{array}$$

(10)

We will prove that $\mathfrak{R}\left({\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}\right)\subseteq {\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}$.

(a) First we consider the case when ${\displaystyle \prod _{l\in 〈r〉}{\tau }_l\left(a\right)>0}$. Since for the initial function $\mathsf{\Phi }\in \mathbf{PC}$ the low terminal a is neither a regular nor an irregular jump point relative to some delay, then according Lemma 3 there exists a constant $M^2\le M^1$ (eventually depending from $\mathsf{\Phi }$) such that $\mathsf{\Phi }(t-{\tau }_l\left(t\right)-a)$ is continuous in $[a,a+M^2]$ (right continuous at a if $a\in S^{\mathsf{\Phi }}$). Note that the initial function $\mathsf{\Phi }\in \mathbf{PC}$ is bounded in $[-h,0]$, $G\left(t\right)$ is bounded in $[a,a+h]$ as well as $V^l(t,\theta )$ are bounded in $\Delta =[a,a+h]\times [-h,0]$, $l\in 〈r〉$, $t,\overline{t}\in [a,a+M]$ and assume for definiteness that $t\ge \overline{t}$.

Let $M\in (0,M^2]$ be arbitrary and $t\in [a,a+M]$. It is clear that for every M the first and the third addends in (9) are continuous functions in $[a,a+M]$. Moreover, the initial function $\mathsf{\Phi }\in \mathbf{PC}$ is bounded in $[-h,0]$, $G\left(t\right)$ is continuous and hence bounded in $[a,a+h]$, the kernels $V^l(t,\theta )$ are bounded in $\Delta =[a,a+h]\times [-h,0]$ and continuous for $t\in [a,a+M^2]$, $l\in 〈r〉$. Since the function $G_t=G(t+\theta )$ is continuous in $t\in [a,a+M]$ when $t+\theta \ge a$ (right continuous at a and left continuous at $a+M$), then for the second addend in the right side of (9) for every $t,\overline{t}\in [a,a+M]$ and $l\in 〈r〉$ in virtue of (S5), we obtain

$$\begin{array}{ccc}\hfill & & {\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}^l(t,\theta )\right]G(t+\theta )-{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}^l(\overline{t},\theta )\right]G(\overline{t}+\theta )\hfill \\ & & \le |{\overline{A}}^l\left(t\right)(G(t-{\tau }_l\left(t\right))-{\overline{A}}^l\left(\overline{t}\right)G(\overline{t}-{\tau }_l\left(\overline{t}\right))|+\left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }(V_c^l(t,\theta )-V_c^l(\overline{t},\theta ))\right]G(\overline{t}+\theta )\right|\hfill \\ & & +\left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }(V_c^l(t,\theta )-V_c^l(\overline{t},\theta ))\right]G(\overline{t}+\theta )\right|.\hfill \end{array}$$

(11)

The continuity for $t\in [a,a+M^2]$ of the first addend in the right side of (11) follows from Lemma 3, conditions (S3), (S5) and the relation

$$|{\overline{A}}^l\left(t\right)(G(t-{\tau }_l\left(t\right))-{\overline{A}}^l\left(\overline{t}\right)G(\overline{t}-{\tau }_l\left(\overline{t}\right)))|=|{\overline{A}}^l\left(t\right)(\mathsf{\Phi }(t-{\tau }_l\left(t\right))-{\overline{A}}^l\left(\overline{t}\right)\mathsf{\Phi }(\overline{t}-{\tau }_l\left(\overline{t}\right)))|.$$

The first integral in the right side of (11) can be estimated as follows:

$$\begin{array}{ccc}\hfill & & \left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )-V_c^l(\overline{t},\theta )\right]G(\overline{t}+\theta )\right|\hfill \\ & & \le \underset{\theta \in [-h,0]}{sup}|G(\overline{t}+\theta )\left|\right|{\mathrm{Var}}_{\theta \in [-h,0]}{V}_c^l(t,·)-{\mathrm{Var}}_{\theta \in [-h,0]}{V}_c^l(\overline{t},·)|.\hfill \end{array}$$

(12)

Since for arbitrary $\overline{t}\in (a,a+M)$ the kernels $V_c^l(t,\theta )$ are continuous at $\overline{t}$ then ${\mathrm{Var}}_{\theta \in [-h,0]}{V}_c^l(t,·)$ are continuous at $\overline{t}$ too. Then taking into account that ${sup}_{\theta \in [-h,0]}\left|G(\overline{t}+\theta )\right|$$<\infty$, we conclude that the right side of (12) tends to zero when $t\to \overline{t}$. In the case when $\overline{t}=a$ or $\overline{t}=a+M$ the same assertion is true when $t\to a+0$ or $t\to a+M-0$ respectively.

For the second integral in (11) let $M\in (0,M^2]$ be arbitrary, $t,\overline{t}\in [a,a+M]$ and assume for definiteness first that $t\ge \overline{t}$. Then for each $t,\overline{t}\in [a,a+M]$ we have that:

$$\begin{array}{ccc}\hfill & & \left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-G(\overline{t}+\theta ))\right|\le \left|{\int }_{-h}^{a-t-0}\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-G(\overline{t}+\theta ))\right|\hfill \\ \hfill & & +\left|{\int }_{a-t}^{a-\overline{t}}\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-G(\overline{t}+\theta ))\right|+\left|{\int }_{a-\overline{t}}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-G(\overline{t}+\theta ))\right|.\hfill \end{array}$$

(13)

The first and the third integrals in the right side of (13) can be estimated as follows:

$$\begin{array}{ccc}\hfill & & \left|{\int }_{-h}^{a-t-0}\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-G(\overline{t}+\theta ))\right|+\left|{\int }_{a-\overline{t}}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-G(\overline{t}+\theta ))\right|\hfill \\ & & \le V^M\left(\underset{\theta \in [-h,a-t]}{sup}|\mathsf{\Phi }(\overline{t}+\theta )-\mathsf{\Phi }(t+\theta )|+\underset{\theta \in [a-\overline{t},0]}{sup}|G(\overline{t}+\theta )-G(t+\theta )|\right),\hfill \end{array}$$

(14)

where $V^M={max}_{t\in [a,a+M]}\left|{\mathrm{Var}}_{\theta \in [-h,0]}{V}_c^l(t,·)\right|$.

For the second integral in right side of (13) we have that

$$\begin{array}{ccc}\hfill & & \left|{\int }_{a-t}^{a-\overline{t}-0}\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-G(\overline{t}+\theta ))\right|\hfill \\ & & \le {\mathrm{Var}}_{\theta \in [a-t,a-\overline{t}]}{V}_c^l(t,\theta )\underset{\theta \in [-h,0];t,\overline{t}\in [a,a+M]}{sup}\left|\right(G(t+\theta )-G(\overline{t}+\theta )|,\hfill \end{array}$$

(15)

and hence, since ${\mathrm{Var}}_{\theta \in [a-t,a-\overline{t}]}{V}_c^l(t,\theta )$ is continuous at $\overline{t}$, then the right sides of (14) and (15) tend to zero when $t\to \overline{t}$. Thus, we can conclude that:

$$\underset{t\to \overline{t}}{lim}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}^l(t,\theta )\right]G(t+\theta )={\int }_{a-\overline{t}}^0\left[{\mathrm{d}}_{\theta }{V}^l(\overline{t},\theta )\right]G(\overline{t}+\theta ),$$

(16)

when $\overline{t}\in [a,a+M)$ and thus $\left(\mathfrak{R}{G}_t\right)\left(t\right)$ is continuous for $t\in (a,a+M)$ and right continuous at a. Changing the assumption for $t,\overline{t}\in [a,a+M]$ with $t<\overline{t}$ the same way we can prove that $\left(\mathfrak{R}{G}_t\right)\left(t\right)$ is left continuous at $a+M$ and hence we have proved that $\mathfrak{R}\left({\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}\right)\subseteq {\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}$.

(b) Let us consider the case when ${\displaystyle \prod _{l\in 〈r〉\backslash 〈q〉}{\tau }_l\left(a\right)=0}$, $q\in 〈r〉$. Then condition 2 of the theorem implies that the low terminal a is an irregular jump point relative to some delays, ${\tau }_l\left(t\right)$, $l\in 〈r〉\backslash 〈q〉$. Then, according Lemma 4 the equality ${\tau }_l\left(a\right)=0$, $l\in 〈r〉\backslash 〈q〉$ implies that there exists a constant $\epsilon \in (0,h]$ (eventually depending from ${\tau }_l$) such that $t-{\tau }_l\left(t\right)<a$ for $t\in (a,a+\epsilon ]$ and $\mathsf{\Phi }(t-{\tau }_l\left(t\right))$ is continuous for each $l\in 〈q〉$. Then for $t\in [a,a+{\overline{M}}^2]$, ${\overline{M}}^2=min(M^2,\epsilon )$ the equality (11) holds and the continuity of the first addend in the right side of (11) follows from Lemma 4. Both integrals in the right side of (11) can be estimated similarly as in the point $\left(a\right)$ and we obtain that the relation (16) holds for all $l\in 〈q〉$. The cases when $l\in 〈r〉\backslash 〈q〉$ can be treated as the case considered in point $\left(a\right)$ and can be proved that (16) holds for these cases too and hence $\mathfrak{R}\left({\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}\right)\subseteq {\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}$.

Let $M\in (0,M^2]$ and $G_t,{\overline{G}}_t\in {\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}$ for $t\in [a,a+M]$ and $l\in 〈q〉$ be arbitrary. For the second addend in the right side of (9) since according condition (S5) the relation

$$V^l(t,\theta )=V_d^l(t,\theta )+V_c^l(t,\theta )$$

holds, we obtain:

$$\begin{array}{ccc}\hfill & & \sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))=\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_d^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\hfill \\ & & +\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta )).\hfill \end{array}$$

(17)

Then since in the case (a) we have that ${\displaystyle \prod _{l\in 〈r〉}{\tau }_l\left(a\right)>0}$ and hence

$$G(t-\tau \left(t\right))=\overline{G}(t-\tau \left(t\right))=\mathsf{\Phi }(t-\tau \left(t\right))$$

(18)

for each $l\in 〈r〉$ it follows that:

$$\begin{array}{ccc}\hfill & & \sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))=\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_d^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\hfill \\ & & +\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))=\sum _{l\in 〈r〉}{\overline{A}}^l\left(t\right)(G(t+\theta )-\overline{G}(t+\theta ))\hfill \\ & & +\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\hfill \\ & & =\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta )).\hfill \end{array}$$

(19)

In the case (b) when ${\displaystyle \prod _{l\in 〈q〉}{\tau }_l\left(a\right)=0}$, $q\in 〈r〉$ for $t\in [a,a+M]$, $l\in 〈r〉$ taking into account that (18) holds for $l\in 〈q〉$ and thus ${\sum }_{l\in 〈q〉}{\overline{A}}^l\left(t\right)(G(t+\theta )-\overline{G}(t+\theta ))=\mathbf{0}$ from (17) we obtain that:

$$\begin{array}{ccc}\hfill & & \sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))=\sum _{l\in 〈r〉\backslash 〈q〉}{\overline{A}}^l\left(t\right)(G(t+\theta )-\overline{G}(t+\theta ))\hfill \\ & & +\sum _{l\in 〈q〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\hfill \\ & & +\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\hfill \\ & & =\sum _{l\in 〈q〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-{\overline{G}}_t(t+\theta ))\hfill \\ & & +\sum _{l\in 〈r〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G_t(t+\theta )-{\overline{G}}_t(t+\theta )).\hfill \end{array}$$

(20)

Comparing (19) and (20) we can conclude that it is enough to estimate only the right side of (20). Condition (S3) implies that there exists ${\delta }^{*}\in (0,1)$ such that for $t\in [a,a+{\delta }^{*}]$ we have that:

$$\underset{l\in 〈r〉}{sup}\underset{t\in [a,a+\delta ]}{sup}{\mathrm{Var}}_{\theta \in [-\delta ,0]}{V}^l(t,\theta )=\mathbf{V}<\frac{1}{4r}.$$

(21)

Let $M\in (0,M^3]$, $M^3=min({\delta }^{*},M^2)$ be arbitrary. Then for $t\in [a,a+M]$ from (20) and (21) it follows the estimation:

$$\begin{array}{ccc}\hfill & & \left|\sum _{l\in 〈q〉}{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\right|\hfill \\ & & \le \sum _{l\in 〈n〉\backslash 〈q〉}\left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_d^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\right|\hfill \\ & & +\sum _{l\in 〈n〉}\left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }{V}_c^l(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\right|\hfill \\ & & \le \frac{1}{4}{d}_M^{\mathsf{\Phi }}(G_t,{\overline{G}}_t)+\frac{1}{4}{d}_M^{\mathsf{\Phi }}(G_t,{\overline{G}}_t)=\frac{1}{2}{d}_M^{\mathsf{\Phi }}(G_t,{\overline{G}}_t).\hfill \end{array}$$

(22)

For the third addend in (9) for each $t\in [a,a+M]$ with $M=(0,M^3]$ we have:

$$\begin{array}{ccc}\hfill & & \left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_a^t{I}_{\alpha -1}(t-\tau )Z(\tau ,G_{\tau }{}^{\top })\mathrm{d}\tau -I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_a^t{I}_{\alpha -1}(t-\tau )Z(\tau ,{\overline{G}}_{\tau }{}^{\top })\mathrm{d}\tau \right|\hfill \\ & & \le \left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)\right|{\int }_a^t\left|I_{\alpha -1}(t-\tau )Z(\tau ,G_{\tau }{}^{\top })-Z(\tau ,{\overline{G}}_{\tau }{}^{\top })\mathrm{d}\tau \right|\hfill \\ & & \le \left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)\right|{\int }_a^t\left(I_{\alpha -1}(t-\tau ){\int }_{-h}^0\left[{\mathrm{d}}_{\theta }U(\tau ,\theta )\right](G(\tau +\theta )-\overline{G}(\tau +\theta ))\right)\mathrm{d}\tau \hfill \\ & & +\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)\right|{\int }_a^t\left|I_{\alpha -1}(t-\eta )(W(\tau ,G_{\tau }{}^{\top })-W(\tau ,{\overline{G}}_{\tau }{}^{\top }))\mathrm{d}\tau \right|\hfill \\ & & \le C^U{d}_M(G_t,{\overline{G}}_t){\int }_a^t\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)I_{\alpha -1}(t-\tau )\right|\mathrm{d}\tau +L{d}_M(G_t,{\overline{G}}_t){\int }_a^t\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)I_{\alpha -1}(t-\tau )\right|\mathrm{d}\tau \hfill \\ & & \le (C^U+L)d_M(G_t,{\overline{G}}_t){\int }_a^t\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)I_{\alpha -1}(t-\eta )\right|\mathrm{d}\eta ,\hfill \end{array}$$

(23)

where ${\displaystyle C^U=\sum _{i\in {〈m〉}_0}\underset{s\in J_{a+h}}{sup}{\mathrm{Var}}_{\theta \in [-\sigma ,0]}{U}^i(s,\theta )}$ and ${\displaystyle L=\underset{s\in [a,a+h]}{sup}\ell \left(s\right)}$.

The integral in the right side of (23) can be estimated as follows:

$$\begin{array}{ccc}\hfill & & {\int }_a^t\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)I_{\alpha -1}(t-\tau )\right|\mathrm{d}\tau =\sum _{k\in 〈n〉}{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right){\int }_a^t{(t-\tau )}^{{\alpha }_k-1}\mathrm{d}\tau \hfill \\ & & =\sum _{k\in 〈n〉}{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right){\alpha }_k^{-1}{(t-a)}^{{\alpha }_k}=\sum _{k\in 〈n〉}\frac{{(t-a)}^{{\alpha }_k}}{\mathsf{\Gamma }(1+{\alpha }_k)}\hfill \\ & & \le {(t-a)}^{{\alpha }_{\min }}\sum _{k\in 〈n〉}\frac{1}{\mathsf{\Gamma }(1+{\alpha }_k)}={\mathsf{\Gamma }}^{*}{(t-a)}^{{\alpha }_{\min }}\le {\mathsf{\Gamma }}^{*}{M}^{{\alpha }_{\min }},\hfill \end{array}$$

(24)

where ${\displaystyle {\mathsf{\Gamma }}^{*}=\sum _{k\in 〈n〉}\frac{1}{\mathsf{\Gamma }(1+{\alpha }_k)}}$ and ${\displaystyle {\alpha }_{\min }=\underset{k\in 〈n〉}{min}\left({\alpha }_k\right)}$.

Then from (22)–(24) we obtain that for arbitrary $G_t,{\overline{G}}_t\in {\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}$, $t\in [a,a+M]$ and $M\in (0,M^3]$ in both cases ((a) and (b)) the following estimation holds:

$$\begin{array}{ccc}\hfill & & |\left(\mathfrak{R}{G}_t\right)\left(t\right)-\left(\mathfrak{R}{\overline{G}}_t\right)\left(t\right)|\le \frac{1}{2}{d}_M^{\mathsf{\Phi }}(G_t,{\overline{G}}_t)+(C^U+L)d_M(G_t,{\overline{G}}_t){\mathsf{\Gamma }}^{*}{M}^{{\alpha }_{\min }}\hfill \\ & & =\left(\frac{1}{2}+M^{{\alpha }_{min}}{\mathsf{\Gamma }}^{*}(C^U+L)\right)d_M^{\mathsf{\Phi }}(G_t,{\overline{G}}_t).\hfill \end{array}$$

(25)

Let us choose $M^0=min\{M^3,{\left(4{\mathsf{\Gamma }}^{*}(C^U+L)\right)}^{(-{\alpha }_{\min }^{-1})}\}$. Then from (25) it follows that:

$$d_{M^0}(\mathfrak{R}{G}_t,\mathfrak{R}{\overline{G}}_t)\le \frac{3}{4}{d}_{M^0}(G_t,{\overline{G}}_t)$$

and hence the operator $\mathfrak{R}$ is contractive in ${\overline{\mathbf{E}}}_{M^0}$. □

Remark 6.

Actually, in both cases a) and b) in the proof of Theorem 1 we essentially use that there exists $\epsilon >0$ such that for $t\in (a,a+\epsilon ]$ the equation $t-{\tau }_l\left(t\right)=a$ has no roots for each $l\in \langle r\rangle$. Then for $t\in (a,a+\epsilon ]$ and $l\in \langle r\rangle$ we have that either $t-{\tau }_l\left(t\right)>a$ or $t-{\tau }_l\left(t\right)<a$. Under the conditions (S) there are no other possibilities. If we assume that for every $\epsilon >0$ the equation $t-{\tau }_l\left(t\right)=a$ has at least one root for some $l\in \langle r\rangle$, then the set $S_{\mathsf{\Phi }}^l$ (defined in condition (S4)) will have an accumulation point which contradicts with condition (S4).

Remark 7.

It must be noted that Theorem 1 is one of the possible generalizations of Theorems 1 and 2 in [21]. Via one weak restriction on the kernels $V_{ac}^l(t,\theta )$ and $V_s^l(t,\theta )$, $l\in 〈r〉$ given in condition (S5), we can avoid the restrictive condition $\mathsf{\Phi }\in BV([a-\tau ,a],{\mathbb{R}}^n)$ for the initial functions used in Theorems 1 and 2 in [21]. This is important because in most of the applications as standard is widely used the assumption that $V_s^l(t,\theta )\equiv \mathsf{\Theta }$, $l\in 〈r〉$ (i.e., the Lebesgue decomposition of the kernels $V_s^l(t,\theta )$, $l\in 〈r〉$ has not singular term). Actually, if the kernels $V_s^l(t,\theta )\equiv \mathsf{\Theta }$, $l\in 〈r〉$, then the statement of Theorem 1 can be proved via condition (S2), without to assume that the kernels $V_{ac}^l(t,\theta )$ are continuous in t on some vicinity of a. In addition, it is obvious that for the delayed (not neutral) equations condition (S5) is ultimately fulfilled.

The next theorem considers the case when the low terminal a does not coincide with the right endpoint of the initial interval.

Theorem 2.

Let the conditions of Theorem 1 hold.

Then for every initial function $\mathsf{\Phi }\in \mathbf{PC}$ satisfying condition 2 of Theorem 1 there exists a constant ${\overline{M}}^0\in (0,h]$ such that IP (4) and (3) has a unique solution with interval of existence $J_{a+M^0+{\overline{M}}^0}$.

Proof. 

The proof is very similar to the proof of Theorem 1 and therefore we will give only these details which are different from the proof of Theorem 1. Since the initial function $\mathsf{\Phi }\in \mathbf{PC}$ satisfies condition 2 of Theorem 1 there exists a constant $M^0\in (0,h]$ such that the IP (4) and (3) (with the same initial function $\mathsf{\Phi }$) has a unique solution $X^0\left(t\right)$ with interval of existence $J_{a+M^0}$. Define the following initial function $\overline{\mathsf{\Phi }}\left(t\right):[a-h,a+M^0]\to {\mathbb{R}}^n$ as follows: $\overline{\mathsf{\Phi }}\left(t\right)=\mathsf{\Phi }\left(t\right)$, $t\in [a-h,a]$, $\overline{\mathsf{\Phi }}\left(t\right)=X^0\left(t\right)$, $t\in [a,a+M^0]$, where $X^0\left(t\right)\in C([a,a+M^0],{\mathbb{R}}^n)$ and obviously $X^0\left(a\right)=\mathsf{\Phi }\left(a\right)$. Considering the IP (4), (6) with the defined initial function $\overline{\mathsf{\Phi }}\left(t\right):[a-h,a+M^0]\to {\mathbb{R}}^n$ we can conclude that the initial point $a+M^0$ does not coincide with the low terminal a and $\overline{\mathsf{\Phi }}\left(t\right)$ don’t possess jump at the initial point $a+M^0$. Thus, the initial point $a+M^0$ cannot be regular (or irregular) jump point. Note that if $M^0\ge h$, the part of the initial function $\overline{\mathsf{\Phi }}\left(t\right)$ which will be used from the delays is continuous, i.e., $X^0\left(t\right)\in C(M^0-h,M^0]$ and then the IP (4), (6) is a standard IP with a continuous initial function. Therefore, we will consider the most complicated case when $M^0<h$, $M^{*}\in (0,h]$ and $M^0+M^{*}\in (0,h]$. As in the proof of Theorem 1 we define the following sets:

$${\mathfrak{J}}_a^{\mathsf{\Phi }}=\{G:[a-h,\infty )\to {\mathbb{R}}^n{\left|G\right|}_{J_a}\in C(J_a,{\mathbb{R}}^n),G\left(t\right)=\overline{\mathsf{\Phi }}\left(t\right),t\in [a-h,a+M_0]\},$$

$${\overline{\mathfrak{J}}}_{{\overline{M}}_0}^{\overline{\mathsf{\Phi }}}=\{G_t:[t-h,t]\to {\mathbb{R}}^n|G_t=G{|}_{[t-h,t]}G\in {\mathfrak{J}}_a^{\overline{\mathsf{\Phi }}},t\in [a+M^0,a+M^0+M^{*}]\}.$$

Since for each $G=(g^1,\dots ,g^n){}^{\top }\in {\mathfrak{J}}_a^{\mathsf{\Phi }}$ and $t\in J_a$, we have that $G_t=(g_t^1,\dots ,g_t^n){}^{\top }\in$$\mathbf{PC}\left({\mathbf{PC}}^{*}\right)$ for every $\mathsf{\Phi }\in \mathbf{PC}\left({\mathbf{PC}}^{*}\right)$ then for each $M\in (M^0,M^{*}]$ we have: $J_a\times {\mathfrak{J}}_a^{\overline{\mathsf{\Phi }}}\subseteq \overline{\mathbf{E}}$, ${\overline{\mathbf{E}}}_M\subset \overline{\mathbf{E}}$$\subset \mathbf{E}$ and

$$[a+M^0,a+M^0+M]\times {\overline{\mathfrak{J}}}_{M^0+M}^{\overline{\mathsf{\Phi }}}={\tilde{\mathbf{E}}}_{M^0+M}\subseteq {\overline{\mathbf{E}}}_{M^0+M}\subset \overline{\mathbf{E}}\subset \mathbf{E}.$$

Let $M\in (M^0,{\overline{M}}^0]$ be an arbitrary number, $t\in [a+{\overline{M}}^0,a+{\overline{M}}^0+M]$ and $(t,G_t)\in$${\tilde{\mathbf{E}}}_{M^0+M}$ be arbitrary. Then using (8) we define the operator $\left(\mathfrak{R}{G}_t\right)=(\left({\mathfrak{R}}_1{g}_t^1\right),\dots ,\left({\mathfrak{R}}_n{g}_t^n\right)){}^{\top }$ for $t\in (a+M^0,a+M^0+M)$ as follows:

$$\left(\mathrm{i}\right)\left(\mathfrak{R}{G}_t\right)\left(t\right)=C_{\mathsf{\Phi }\left(0\right)}+{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }V(t,\theta )\right]G(t+\theta )+I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_a^t{I}_{\alpha -1}(t-\tau )Z(\tau ,G_{\tau }{}^{\top })\mathrm{d}\tau ,$$

(26)

and

$$\begin{array}{ccc}\hfill & & \left(\mathrm{ii}\right)[a+M^0,a+M^0+{\overline{M}}^0]\times {\overline{\mathfrak{J}}}_{{\overline{M}}^0}^{\overline{\mathsf{\Phi }}}={\tilde{\mathbf{E}}}_{M^0+{\overline{M}}^0}\subseteq {\overline{\mathbf{E}}}_{M^0+{\overline{M}}^0}\subset \overline{\mathbf{E}}\subset \mathbf{E}\hfill \\ & & \left(\mathrm{iii}\right)\left(\mathfrak{R}{G}_t\right)(t+\theta )=\overline{\mathsf{\Phi }}(t+\theta ),t+\theta \in [a+M^0-h,a+M^0].\hfill \end{array}$$

(27)

As above we can prove that: $\mathfrak{R}\left({\overline{\mathfrak{J}}}_{M^0+M}^{\overline{\mathsf{\Phi }}}\right)\subseteq {\overline{\mathfrak{J}}}_{M^0+M}^{\overline{\mathsf{\Phi }}}$ for $M\in (M^0,M^0+{\overline{M}}^1]$ where ${\overline{M}}^1\le {\overline{M}}^0$.

Condition (S3) implies that there exists ${\delta }^{*}\in (0,1)$ such that for $t\in [a+M^0,a+M^0$$+M]$, $M\in (M^0,M^0+{\overline{M}}^2]$, where ${\overline{M}}^2=min({\delta }^{*},{\overline{M}}^1)$ we have that

$$\underset{l\in 〈r〉}{sup}\underset{t\in [a+M^0,a+M^0+{\delta }^{*}]}{sup}{\mathrm{Var}}_{\theta \in [-{\delta }^{*},0]}{V}^l(t,\theta )=\mathbf{V}<\frac{1}{4r}.$$

(28)

Let $M\in (M^0,M^0+{\overline{M}}^2]$, $G_t,{\overline{G}}_t\in {\overline{\mathfrak{J}}}_{M^0+M}^{\overline{\mathsf{\Phi }}}$ be arbitrary. Then for $t\in [a+M^0,a+M^0$$+M]$ and $l\in 〈r〉$ for the second addend in (26) we have

$$\begin{array}{ccc}\hfill & & \left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }V(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\right|\le \left|{\int }_{-h}^{-{\delta }^{*}}\left[{\mathrm{d}}_{\theta }V(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\right|\hfill \\ & & +\left|{\int }_{-{\delta }^{*}}^0\left[{\mathrm{d}}_{\theta }V(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\right|\le \left|{\int }_{-h}^{-{\delta }^{*}}\left[{\mathrm{d}}_{\theta }V(t,\theta )\right](\mathsf{\Phi }(t+\theta )-\mathsf{\Phi }(t+\theta ))\right|\hfill \\ & & +\left|{\int }_{-{\delta }^{*}}^0\left[{\mathrm{d}}_{\theta }V(t,\theta )\right](G(t+\theta )-\overline{G}(t+\theta ))\right|\le \frac{1}{4r}{d}_M^{\mathsf{\Phi }}(G_t,{\overline{G}}_t).\hfill \end{array}$$

(29)

The third addend in (26) can be estimated as in (17) taking into account that for $t+\theta \in [a-h,a+M^0]$ is fulfilled $G(t+\theta )=\overline{G}(t+\theta )=\overline{\mathsf{\Phi }}(t+\theta )$ and then we obtain

$$\begin{array}{ccc}\hfill & & \left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)\right|{\int }_a^t\left|I_{\alpha -1}(t-\tau )Z(\tau ,G_{\tau }{}^{\top })-Z(\tau ,{\overline{G}}_{\tau }{}^{\top })\mathrm{d}\tau \right|\hfill \\ & & \le \left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)\right|{\int }_{a+M^0}^t{I}_{\alpha -1}(t-\tau )\left({\int }_{-h}^0\left[{\mathrm{d}}_{\theta }U(\tau ,\theta )\right](G_{\tau }{}^{\top }(\tau +\theta )-{\overline{G}}_{\tau }{}^{\top }(\tau +\theta ))\right)\mathrm{d}\tau \hfill \\ & & +\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)\right|{\int }_{a+M^0}^t\ell \left(\tau \right)\left|I_{\alpha -1}(t-\tau )\right|\left|(G_{\tau }{}^{\top }(\tau +\theta )-{\overline{G}}_{\tau }{}^{\top }(\tau +\theta ))\right|\mathrm{d}\tau \hfill \\ & & \le (C^U+L)d_M(G_t,{\overline{G}}_{\tau }){\int }_{a+M^0}^t\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)I_{\alpha -1}(t-\tau )\right|\mathrm{d}\tau ,\hfill \end{array}$$

(30)

where $C^U$ and L are the same constants as in Theorem 1. The integral in the right side of (30) can be calculated the same way as in (18):

$${\int }_{a+M^0}^t\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)I_{\alpha -1}(t-\tau )\right|\mathrm{d}\tau \le {\mathsf{\Gamma }}^{*}{(t-(a+M^0))}^{{\alpha }_{\min }}\le {\mathsf{\Gamma }}^{*}{M}^{{\alpha }_{\min }},$$

(31)

where ${\mathsf{\Gamma }}^{*}$ and ${\alpha }_{\min }$ are the same constants as in Theorem 1. Choose

$${\overline{M}}^0=min\{M^0+{\overline{M}}^2,{\left(4{\mathsf{\Gamma }}^{*}(C^U+L)\right)}^{(-{\alpha }_{\min }^{-1})}\}$$

and then from (29)–(31) we obtain that the operator $\mathfrak{R}$ is contractive in ${\overline{\mathbf{E}}}_{{\overline{M}}^0}$. □

Remark 8.

Note that the case when $a\in S^{\mathsf{\Phi }}$ is very important in point of view of application. However, in the case when $a\in S^{\mathsf{\Phi }}$ is an irregular jump point relative some delay ${\tau }_{l^{*}}\left(t\right)$ for $l^{*}\in 〈r〉$ and there exists $\epsilon >0$ such that $t-{\tau }_{l^{*}}\left(t\right)<a$ for $t\in (a,a+\epsilon ]$ and $\mathsf{\Phi }(t-{\tau }_{l^{*}}\left(t\right))$ is continuous. Thus ${\displaystyle \underset{t\to a+0}{lim}\mathsf{\Phi }(t-a-{\tau }_{l^{*}}\left(t\right))=\mathsf{\Phi }(0-)\ne \mathsf{\Phi }\left(0\right)}$ and in this case in general without additional conditions, since the condition (ii) in (10) is not true for $l^{*}\in 〈r〉$ we have that $\mathfrak{R}\left({\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}\right)⊈{\overline{\mathfrak{J}}}_M^{\mathsf{\Phi }}$ for each $M>0$.

Theorem 3.

Let the conditions of Theorem 1 hold.

Then the IP (4) and (3) has a unique solution in $J_a$.

Proof. 

By virtue in Theorem 1 there exists $M^0>0$ such that the IP (4), (3) for some initial function $\mathsf{\Phi }\in \mathbf{PC}$ with $a\in S^{\mathsf{\Phi }}$ and the low terminal a is at most a regular jump point, has a unique solution $X^0\left(t\right)$ with interval of existence $J_{a+M^0}$. Let $X^{max}\left(t\right)$ denote the maximal solution (i.e., a continuation of all other solutions) of the IP (4), (3) with the same initial function and assume that the interval of existence $J_{max}$ is closed from the right, i.e., $J_{max}=[a,a+M^{max}]$. Then define the initial function $\overline{\mathsf{\Phi }}\left(t\right):[a-h,a+M^{max}]\to {\mathbb{R}}^n$ defined as follows: $\overline{\mathsf{\Phi }}\left(t\right)=\mathsf{\Phi }\left(t\right)$, $t\in [a-h,a]$, $\overline{\mathsf{\Phi }}\left(t\right)=X^{max}\left(t\right)$, $t\in [a,a+M^{max}]$. Applying Theorem 2 to the IP (4), (6) with the defined initial function $\overline{\mathsf{\Phi }}\left(t\right)$ we obtain that the unique solution $X^{max}\left(t\right)$ can be prolonged, which is a contradiction with the assumption that $X^{max}\left(t\right)$ is the maximal solution. Thus, we can conclude that interval of existence $J_{max}$ is open from the right, i.e., $J_{max}=[a,a+M^{max})$ and let us assume that $M^{max}<\infty$. Taking into account that $X^{max}\left(t\right)$ is the unique solution for the system (4) for $t\in [a,a+M^{max})$ and since for the right side the limit when $t\to a+M^{max}-0$ exists, then we can conclude that the system (4) holds for $t=a+M^{max}$ too, which contradicts our assumption. Thus $M^{max}=\infty$. □

4. Asymptotic Stability of Nonlinear Perturbed Neutral Linear Fractional System with Distributed Delays

The aim of this section is to obtain sufficient conditions, which guarantee that if the zero solution of the system (2) (i.e., the linear part of the system (1)) is globally asymptotically stable, then the zero solution of the system (1) is globally asymptotically stable too. The same problem in the case of a delayed system with distributed delay is considered in [14] and a partial case for a neutral system in [15].

Let $s\in J_a$ be an arbitrary fixed number and consider the following matrix system

$$D_{a+}^{\alpha }(Y(t,s)-{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }V(t,\theta )\right]Y(t+\theta ,s))={\int }_{-h}^0\left[{\mathrm{d}}_{\theta }U(t,\theta )\right]Y(t+\theta ,s),t\in (max(a,s),\infty ),$$

(32)

with the following initial condition

$$Y(t,s)={\mathsf{\Phi }}^{*}(t,s)\mathrm{for}t\in [s-h,s],\mathrm{when}s\ge a,$$

(33)

where ${\mathsf{\Phi }}^{*}(t,s)={\left\{{\varphi }_{k,j}^{*}(t,s)\right\}}_{k,j\in \langle n\rangle }:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^{n\times n}$ is defined as follows: ${\mathsf{\Phi }}^{*}(t,s)=\mathsf{\Theta }$ for $t<s$ and ${\mathsf{\Phi }}^{*}(t,s)=I$ when $t=s$.

For each fixed $s\ge a$ the matrix valued function $t\to C(t,s)={\left\{{\gamma }_{kj}(t,s)\right\}}_{k,j=1}^n,C(·,s):J_s\to {\mathbb{R}}^{n\times n}$ is called a solution of the matrix IP (32), (33) if $C(·,s):J_s\to {\mathbb{R}}^{n\times n}$ is continuous in t on $J_s$ and satisfies the matrix Equation (32) for $t\in (s,\infty )$, as well as the initial condition (33) too. As in the integer case we call the matrix $C(t,s)$ a fundamental matrix of the system (2).

Lemma 5

([21]). Let the conditions of Theorem 1 hold.

Then for every fixed $s\in J_a$ the IP (32), (33) has a unique solution $C(·,s):J_s\to {\mathbb{R}}^{n\times n}$ in $J_s$.

Everywhere below for simplicity we will assume that $a=0$.

Definition 6.

The function $X\in C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)$ is called Mittag–Leffler (ML)-bounded of order $\omega \in {\mathbb{R}}_{+}$ if there exist constants $C_X,{\overline{C}}_X\in {\mathbb{R}}_{+}$, such that $\left|X\left(t\right)\right|\le C_X{E}_{\omega }\left({\overline{C}}_X\mathsf{\Gamma }\left(\omega \right)t^{\omega }\right)$ for $t\in {\mathbb{R}}_{+}$. By $M{L}_{\omega }({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)$ we denote the subset of all ML-bounded of order ω functions in $C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)$.

Let $\mathsf{\Phi }\in \mathbf{C}$ be an arbitrary initial function and let $X\in C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)$ be the corresponding unique solution of the IP (4), (3) with this initial function and denote $F^X\left(t\right)=(F_1^X\left(t\right),\dots ,$$F_n^X\left(t\right)){}^{\top }=W(t,X_t^{{}^{\top }})$. Then from (4) it follows:

$$\begin{array}{ccc}\hfill & X\left(t\right)=& C_{\mathsf{\Phi }\left(0\right)}+{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }V(t,\theta )X(t+\theta )\right]\hfill \\ & & +I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_a^t{I}_{\alpha -1}(t-\tau )\left({\int }_{-h}^0\left[{\mathrm{d}}_{\theta }U(\tau ,\theta )\right]X(\tau +\theta )+F^X\left(\tau \right)\right)\mathrm{d}\tau .\hfill \end{array}$$

(34)

Theorem 4.

Let the following conditions hold:

1.

The conditions of Theorem 1 hold.

2.

$\underset{t\in {\mathbb{R}}_{+}}{sup}{\mathrm{Var}}_{\theta \in [-h,0]}\left|V(t,\theta )\right|=\mathbf{V}<1$.

Then the solution of IP (4), (3) for arbitrary initial function $\mathsf{\Phi }\in \mathbf{C}$ is Mittag–Leffler (ML) bounded of order ${\alpha }_M$.

Proof. 

Let $\mathsf{\Phi }\in \mathbf{C}$ is an arbitrary initial function and $X\in C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)$ is the corresponding unique solution of the IP (4), (3) with this initial function. Then denote $Y\left(t\right)=max\left({sup}_{s\in [0,t]}\left|X\left(s\right)\right|,\parallel \mathsf{\Phi }\parallel \right)$, from (34) we obtain:

$$\begin{array}{ccc}\hfill & & max\left(\underset{s\in [0,t]}{sup}\left|X\left(s\right)\right|,\parallel \mathsf{\Phi }\parallel \right)=\left|C_{\mathsf{\Phi }\left(0\right)}\right|+\underset{s\in [0,t]}{sup}\left|{\int }_{-h}^0\left[{\mathrm{d}}_{\theta }V(s,\theta )\right]X(s+\theta )\right|\hfill \\ & & +\underset{s\in [0,t]}{sup}\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_0^t{I}_{\alpha -1}(t-\tau ){\int }_{-h}^0\left[{\mathrm{d}}_{\theta }U(\tau ,\theta )\right]X(\tau +\theta )d\tau \right|\hfill \\ & & +\underset{s\in [0,t]}{sup}\left|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right){\int }_0^t{I}_{\alpha -1}(t-\tau )F^X\left(\tau \right)\mathrm{d}\tau \right|\hfill \\ & & \le \left|C_{\mathsf{\Phi }\left(0\right)}\right|+\underset{s\in [0,t]}{sup}{\mathrm{Var}}_{\theta \in [-h,0]}V(s,\theta )max(\underset{s\in [0,t]}{sup}\left|X\left(s\right)\right|,\parallel \mathsf{\Phi }\parallel )\hfill \\ & & +|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)|\underset{s\in [0,t]}{sup}Va{r}_{\theta \in [-h,0]}U(s,\theta )\left|{\int }_0^t|I_{\alpha -1}(t-\tau )|max(\underset{s\in [0,\tau ]}{sup}\left|X\left(s\right)\right|,\parallel \mathsf{\Phi }\parallel )\mathrm{d}\tau \right|\hfill \\ & & +|I_{-1}\left(\mathsf{\Gamma }\left(\alpha \right)\right)|\underset{s\in [0,t]}{sup}\left|{\int }_0^t{I}_{\alpha -1}(t-\tau )F^X\left(\tau \right)\mathrm{d}\tau \right|.\hfill \end{array}$$

(35)

From (35) denoted by

$$Y\left(t\right)=max\left(\underset{s\in [0,t]}{sup}\left|X\left(s\right)\right|,\parallel \mathsf{\Phi }\parallel \right),U\left(t\right)=\underset{s\in [0,t]}{sup}{\mathrm{Var}}_{\theta \in [-h,0]}U(s,\theta )$$

and taking into account condition 2, we have:

$$\begin{array}{ccc}& Y\left(t\right)\le & \left|C_{\mathsf{\Phi }\left(0\right)}\right|+VY\left(t\right)+n{\mathsf{\Gamma }}^{-1}\left({\alpha }_m\right)U\left(t\right)\sum _{k\in 〈n〉}{\int }_0^t{(t-\tau )}^{1-{\alpha }_M}Y\left(\tau \right)\mathrm{d}\tau \hfill \\ & & +n^2{\mathsf{\Gamma }}^{-2}\left({\alpha }_m\right)t^{{\alpha }_M}\underset{s\in [0,t]}{sup}\left|F^X\left(s\right)\right|\hfill \end{array}$$

and hence

$$Y\left(t\right)\le a\left(t\right)+g\left(t\right)\sum _{k\in 〈n〉}{\int }_0^t{(t-\tau )}^{1-{\alpha }_M}Y\left(\tau \right)\mathrm{d}\tau ,$$

(36)

where ${\displaystyle a\left(t\right)=\frac{\left|C_{\mathsf{\Phi }\left(0\right)}\right|n{\mathsf{\Gamma }}^{-1}\left({\alpha }_m\right)}{1-V}+n^2{\mathsf{\Gamma }}^{-2}\left({\alpha }_m\right)t^{{\alpha }_M}\underset{s\in [0,t]}{sup}\left|F^X\left(s\right)\right|}$, ${\displaystyle g\left(t\right)=\frac{n{\mathsf{\Gamma }}^{-1}\left({\alpha }_m\right)}{1-V}U\left(t\right)}$.

Applying Corollary 2 in [24] we obtain:

$$max\left(\underset{\eta \in [0,t]}{sup}\left|X(\eta ,a)\right|,\parallel \mathsf{\Phi }\parallel \right)\le a\left(t\right)E_{{\alpha }_M}\left(g\left(t\right)\mathsf{\Gamma }\left({\alpha }_M\right)t^{{\alpha }_M}\right),$$

which completes the proof. □

Remark 9.

Actually, Theorem 4 establishes that under some natural condition (condition 2 of the theorem) all solutions of IP (1), (3) with initial functions $\mathsf{\Phi }\in \mathbf{C}$ are Mittag–Leffler (ML) bounded of order ${\alpha }_M$.

Definition 7.

The zero solution of the system (1) is said to be:

(a) 

Stable (uniformly) if for any $\epsilon >0$ there is a $\delta \left(\epsilon \right)>0$ such that for every initial function $\mathsf{\Phi }\in \mathbf{C}$ with $\parallel \mathsf{\Phi }\parallel <\delta$ the corresponding solution $X\left(t\right)$ satisfies for each $t\in {\overline{\mathbb{R}}}_{+}$ the inequality $\left|X\right(t\left)\right|\le \epsilon$;

(b) 

Locally asymptotically stable (LAS) if there is a $\Delta \subset \mathbf{C}$ such that for every initial function $\mathsf{\Phi }\in \Delta$, the relation ${\displaystyle \underset{t\to \infty }{lim}\left|X\left(t\right)\right|=0}$ holds for the corresponding solution $X\left(t\right)$;

(c) 

Globally asymptotically stable (GAS) iff for every initial function $\mathsf{\Phi }\in \mathbf{C}$, for the corresponding solution $X\left(t\right)$ we have that ${\displaystyle \underset{t\to \infty }{lim}\left|X\left(t\right)\right|=0}$.

Definition 8.

We say that the vector valued functional $W:\mathbf{E}\to {\mathbb{R}}^n$ is a damper of order ω for the system (1) if the system (2) is asymptotically stable (thus $\underset{t\ge t_0}{sup}(\underset{s\in [t_0,t]}{sup}\left|C(t,·)\right|<\infty$) and for every $X\in M{L}_{\omega }({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)$ the function $F^X\left(t\right)=W(t,X_t^{{}^{\top }})$ is bounded and there exists a point $t_0\in {\mathbb{R}}_{+}$ and a constant $C_{t_0}>0$ (the constant can depend eventually from $t_0$) such that it is fulfilled $|F^X\left(t\right)|\le C_{t_0}{t}^{-(\omega +1)}$.

The proof of the next theorem is based on the following practical variant of the well-known Barbalat’s lemma.

Lemma 6

(see [25], p. 124). Let the differentiable function $\chi :{\overline{\mathbb{R}}}_{+}\to {\overline{\mathbb{R}}}_{+}$ have a finite limit as $t\to \infty$ and its derivative ${\chi }^{\prime }\left(t\right)$ is bounded on ${\overline{\mathbb{R}}}_{+}$.

Then the relation $\underset{t\to \infty }{lim}{\chi }^{\prime }\left(t\right)=0$ holds.

Theorem 5.

Let the following conditions be fulfilled:

1.

The conditions (S) and (H) hold and the zero solution of the system (2) is GAS;

2.

There exists some $\omega >{\alpha }_M$ such that the vector valued functional $W:\mathbf{E}\to {\mathbb{R}}^n$ is a damper of order ω;

3.

$\underset{t\in {\mathbb{R}}_{+}}{sup}{\mathrm{Var}}_{\theta \in [-h,0]}\left|V(t,\theta )\right|<1$.

Then every solution $X\left(t\right)$ of the IP (1), (3), with initial function $\mathsf{\Phi }\in \mathbf{C}$ is GAS.

Proof. 

Let for arbitrary initial function $\mathsf{\Phi }\in \mathbf{C}$, the function $X^{*}\left(t\right)$ be the corresponding unique solution of the IP (1), (3) (IP (4), (3)). According to condition 3 and Theorem 4 we have that $X^{*}\left(t\right)$ is ML bounded of order ${\alpha }_M$. Denote by $X_{\mathsf{\Phi }}\left(t\right)$ the corresponding unique solution of the IP (2), (3) and by ${\overline{X}}^{*}\left(t\right)$ the corresponding unique solution of the IP (34), (3) with zero initial condition, i.e., ${\overline{X}}^{*}(t+\theta )\equiv \mathbf{0}$, $t+\theta <0$, $-h\le \theta \le 0$ existing according to Theorems 1–3, where in (34) we put $F\left(t\right)=F^{*}\left(t\right)=W(t,X_t^{*{}^{\top }}\left(t\right))$, $t\in {\overline{\mathbb{R}}}_{+}$. Then by virtue of the superpositions principle the function ${\overline{X}}^{*}\left(t\right)+X_{\mathsf{\Phi }}\left(t\right)$ is a solution of the IP (1), (3) with the same initial function $\mathsf{\Phi }\in \mathbf{C}$ as $X^{*}\left(t\right)$. Thus, the uniqueness of the solution of the IP (1), (3) implies that ${\overline{X}}^{*}\left(t\right)+X_{\mathsf{\Phi }}\left(t\right)=X^{*}\left(t\right)$. From Theorem 6 in [21] it follows that:

$${\overline{X}}^{*}\left(t\right)={\int }_0^{t}C{(t,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s,$$

(37)

hence integrating both sides of (37) we obtain:

$$\chi \left(t\right)={\int }_0^t\left|{\overline{X}}^{*}\left(\xi \right)\right|\mathrm{d}\xi ={\int }_0^t\left|{\int }_0^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi .$$

(38)

Taking into account that $|{\overline{X}}^{*}\left(t\right)|$ is continuous in ${\overline{\mathbb{R}}}_{+}$ then $\chi \left(t\right)\in C^1({\overline{\mathbb{R}}}_{+},{\overline{\mathbb{R}}}_{+})$ and ${\chi }^{\prime }\left(t\right)=\left|{\overline{X}}^{*}\left(t\right)\right|,t\in {\mathbb{R}}_{+}$.

For $|{\overline{X}}^{*}\left(t\right)|$ from (37) we obtain:

$$\begin{array}{cc}\hfill |{\overline{X}}^{*}\left(t\right)|& =\sum _{k\in \langle n\rangle }\left|x_k^{*}\left(t\right)\right|=\sum _{k,j\in \langle n\rangle }\left|{\int }_0^t{c}_k^j{(t,s)}_{RL}{D}_{0+}^{1-{\alpha }_k}{F}_k^{*}\left(s\right)\mathrm{d}s\right|\hfill \\ \hfill & =\sum _{k,j\in \langle n\rangle }\left|{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right){\int }_0^t{c}_k^j(t,s)\left(\frac{\mathrm{d}}{\mathrm{d}s}{\int }_0^s{(s-\eta )}^{{\alpha }_k-1}{F}_k^{*}\left(\eta \right)\mathrm{d}\eta \right)\mathrm{d}s\right|\hfill \\ \hfill & =\sum _{k,j\in \langle n\rangle }\left|{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right){\int }_0^t{c}_k^j(t,s){\mathrm{d}}_s{\int }_0^s{(s-\eta )}^{{\alpha }_k-1}{F}_k^{*}\left(\eta \right)\mathrm{d}\eta \right|\hfill \\ \hfill & \le \sum _{k,j\in \langle n\rangle }\left|{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right){\int }_0^t\left|c_k^j(t,s)\right|{\mathrm{d}}_s\left|{\int }_0^s{(s-\eta )}^{{\alpha }_k-1}{F}_k^{*}\left(\eta \right)\mathrm{d}\eta \right|\right|\hfill \\ \hfill & \le \underset{s\in [0,t],t\in {\mathbb{R}}_{+}}{sup}\left|C(t,·)\right|\sum _{k\in \langle n\rangle }{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right)\left|{\int }_0^t{\mathrm{d}}_s\left|{\int }_0^s{(s-\eta )}^{{\alpha }_k-1}{F}_k^{*}\left(\eta \right)\mathrm{d}\eta \right|\right|\hfill \\ \hfill & \le \underset{s\in [0,t],t\in {\mathbb{R}}_{+}}{sup}\left|C(t,·)\right|\sum _{k\in \langle n\rangle }{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right)\left|{\int }_0^t{(t-\eta )}^{{\alpha }_k-1}{F}_k^{*}\left(\eta \right)\mathrm{d}\eta \right|.\hfill \end{array}$$

(39)

Since System (2) is asymptotically stable then it is enough to estimate the sum of the integrals on the right side of (39). From condition 2 of the theorem it follows that there exists a point $t_0\in {\mathbb{R}}_{+}$ and a constant $C_{t_0}>0$ such that it is fulfilled $|F^{*}\left(t\right)|\le C_{t_0}{t}^{-(\omega +1)}$, where without loss of generality we can assume that $t_0\ge 1$. Then for this sum of integrals on the right side of (39) we obtain the following estimation:

$$\begin{array}{cc}\hfill & \sum _{k\in \langle n\rangle }{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right)\left|{\int }_0^t{(t-\tau )}^{{\alpha }_k-1}{F}_k^{*}\left(\tau \right)\mathrm{d}\tau \right|\le \sum _{k\in \langle n\rangle }{\alpha }_k^{-1}{\mathsf{\Gamma }}^{-1}\left({\alpha }_k\right){\int }_0^t\left|F_k^{*}(t-\tau )\right|\mathrm{d}{\tau }^{{\alpha }_k}\hfill \\ \hfill & =\sum _{k\in \langle n\rangle }{\mathsf{\Gamma }}^{-1}(1+{\alpha }_k){\int }_0^t\left|F_k^{*}(t-\tau )\right|\mathrm{d}{\tau }^{{\alpha }_k}\hfill \\ \hfill & \le {\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)\sum _{k\in \langle n\rangle }({\int }_0^{t_0}|F_k^{*}(t-\tau )|\mathrm{d}{\tau }^{{\alpha }_k}+{\int }_{t_0}^t|F_k^{*}(t-\tau )\left|\mathrm{d}{\tau }^{{\alpha }_k}\right)\hfill \\ \hfill & \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)t_0^{{\alpha }_M}\underset{\tau \in [0,t_0]}{sup}\left|F_k^{*}(t-\tau )\right|+n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}{\int }_{t_0}^t{(t-\tau )}^{-(\omega +1)}\mathrm{d}{\tau }^{{\alpha }_M}\hfill \\ \hfill & \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)t_0^{{\alpha }_M}\underset{\tau \in [0,t_0]}{sup}\left|F_k^{*}(t-\tau )\right|+n{\alpha }_{max}{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}{t}^{-(\omega +1)}{\int }_{t_0}^t{(1-\frac{\tau }{t})}^{-(\omega +1)}\mathrm{d}{\tau }^{{\alpha }_M}\hfill \\ & \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)t_0^{{\alpha }_M}\underset{\tau \in [0,t_0]}{sup}\left|F_k^{*}(t-\tau )\right|+n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}{t}^{-(\omega +1)}(t^{{\alpha }_M}-t_0^{{\alpha }_M})\hfill \\ \hfill & \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)t_0^{{\alpha }_M}\underset{\tau \in [0,t_0]}{sup}\left|F_k^{*}(t-\tau )\right|+n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}{t}^{-(\omega +1)+{\alpha }_M}.\hfill \end{array}$$

(40)

Then from (39) and (40) it follows that $|{\overline{X}}^{*}\left(t\right)|$ is bounded in ${\overline{\mathbb{R}}}_{+}$.

From (38) since $\left|C\right(t,s\left)\right|=0$ for $t<s$ we obtain:

$$\begin{array}{cc}\hfill \chi \left(t\right)& ={\int }_0^t\left|{\overline{X}}^{*}\left(\xi \right)\right|\mathrm{d}\xi ={\int }_0^t\left|{\int }_0^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi \hfill \\ \hfill & ={\int }_0^{t_0}\left|{\int }_0^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi +{\int }_{t_0}^t\left|{\int }_0^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi \hfill \\ \hfill & \le {\int }_0^{t_0}\left|{\int }_0^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi +{\int }_{t_0}^t\left|{\int }_0^{t_0}C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi \hfill \\ \hfill & +{\int }_{t_0}^t\left|{\int }_{t_0}^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi \hfill \\ \hfill & ={\int }_0^{t_0}\left|{\int }_0^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi +{\int }_{t_0}^t\left|{\int }_{t_0}^{\xi }C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi \hfill \\ \hfill & \le {\int }_0^{t_0}\left|{\int }_0^{t_0}C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi +{\int }_{t_0}^t\left|{\int }_{t_0}^{t}C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi .\hfill \end{array}$$

(41)

For the second integral in the right side of (41) using (39) and (40) we obtain:

$$\begin{array}{cc}\hfill & {\int }_{t_0}^t\left|{\int }_{t_0}^{t}C{(\xi ,s)}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\mathrm{d}s\right|\mathrm{d}\xi \le \underset{s\in [t_0,t],t\ge t_0}{sup}\left|C(t,·)\right|{\int }_{t_0}^t({\int }_{t_0}^t{|}_{RL}{D}_{0+}^{1-\alpha }{F}^{*}\left(s\right)\left|\mathrm{d}s\right)\mathrm{d}\xi \hfill \\ \hfill & \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}\underset{s\in [t_0,t],t\ge t_0}{sup}\left|C(t,·)\right|{\int }_{t_0}^t{t}^{-\omega -1}(t^{{\alpha }_M}-t_0^{{\alpha }_M})\mathrm{d}\xi \hfill \\ \hfill & \le \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}\underset{s\in [t_0,t],t\ge t_0}{sup}\left|C(t,·)\right|t^{-\omega -1+{\alpha }_M}(t-t_0)\hfill \\ \hfill & \le \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}\underset{s\in [t_0,t],t\ge t_0}{sup}\left|C(t,·)\right|t^{-\omega -1+{\alpha }_M+1}(1-\frac{t_0}{t})\hfill \\ \hfill & \le \le n{\mathsf{\Gamma }}^{-1}\left(z_{\min }\right)C_{t_0}\underset{s\in [t_0,t],t\ge t_0}{sup}\left|C(t,·)\right|t^{-\omega +{\alpha }_M}.\hfill \end{array}$$

(42)

Thus, from (41) and (42) it follows that the function $\chi \left(t\right)$ defined by (38) has a finite limit as $t\to \infty$, and hence $\underset{t\to \infty }{lim}{\chi }^{\prime }\left(t\right)=0$, which completes the proof. □

The next example illustrates that condition 2 of Theorem 5 is natural and not very restrictive.

Example 1.

Let the vector–function $W:{\overline{\mathbb{R}}}_{+}\times C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)\to {\mathbb{R}}^n$ be arbitrary which satisfies conditions (H) (or for simplicity be continuous in both arguments and satisfies the Lipschitz condition concerning the second argument), $t^{*}\in {\mathbb{R}}_{+}$ be an arbitrary point and define the function $\overline{W}:{\overline{\mathbb{R}}}_{+}\times C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n)\to {\mathbb{R}}^n$ as follows:

$$\overline{W}(t,X\left(t\right),X(t-h))=\left\{\begin{array}{cc}W(t,X(t),X(t-h\left)\right)\hfill & fort\in [0,t^{*}],X\in C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n),h>0\hfill \\ t^{-(1+\omega )}\frac{X(t-h)-X(t^{*}-h)}{1+\left|X\right(t\left)\right|}+C^{*}\hfill & fort\ge t^{*},X\in C({\overline{\mathbb{R}}}_{+},{\mathbb{R}}^n),h>0,\hfill \end{array}\right.$$

where $C^{*}=W(t^{*},X\left(t^{*}\right),X(t^{*}-h))$.

This is simply to check that this class of functions satisfies condition 2 of Theorem 5.

5. Conclusions and Comments

As a first result, the existence and uniqueness of the solutions of an initial problem (IP) with piecewise continuous initial function for the general case of a nonlinear perturbed linear nonautonomous neutral fractional system with Caputo fractional derivatives of incommensurate order and distributed delays is proved. Then, using the integral representation for the solution of the nonhomogeneous system with zero initial conditions in our former work [21] some natural sufficient conditions are found to ensure that from global asymptotic stability of the zero solution of the linear part of a nonlinearly perturbed system, it follows that global asymptotic stability of the zero solution of the whole nonlinearly perturbed system. These results improve and extend the results obtained in all particular cases for the delayed and neutral fractional systems.