1 Introduction

In this paper, we study stability for a very general class of fractional delay integro-differential equations. We use a recently introduced concept [11] of the so-called generalized Caputo proportional fractional derivative. This derivative is a generalization of the Caputo fractional derivative. For the study of various types of fractional differential equations, we refer the reader to the classical book [19]. In [4, 6, 7, 9, 13, 14, 20], Lyapunov functions and functionals are used to study the stability of various fractional-order nonlinear dynamic systems with classical Caputo and Riemann–Liouville fractional derivatives and for ordinary differential equations with state delays in [21]. Following the introduction [11] of the generalized Caputo proportional fractional derivative, some studies concerning this derivative were contained in [1–3, 5, 10, 15–18].

In this paper, we apply Lyapunov functions to study some stability properties of integro-differential delay equations with generalized Caputo proportional fractional derivatives. We consider the case of bounded delays that are variable in time. We develop some necessary tools for generalized Caputo proportional fractional derivatives, starting with an important inequality concerning an estimate of that derivative of quadratic functions. A modified Razumikhin condition is developed and used to show the main result, conditions that guarantee the stability of the zero solution with zero initial function. From there, we give results concerning the boundedness of all solutions, as well as results about exponential stability and asymptotic stability.

The organization of the paper is as follows. In Sect. 2, the problem is stated and essential assumptions and definitions are given. In Sect. 3, we prove several auxiliary results about generalized Caputo proportional fractional derivatives. Section 4 contains the main results as well as a comparison with the recent paper [8]. The paper concludes with Sect. 5, in which five detailed examples are offered.

2 Problem statement

Consider the fractional delay integro-differential equation with generalized Caputo proportional fractional derivative

$$ \begin{aligned} &\bigl({}_{t_{0}}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }x \bigr) (t) =-F\bigl(t,x(t)\bigr)+ \sum _{k=1}^{n}\psi _{k}(t,x(t),x \bigl(t-g_{k}(t)\bigr) \\ &\hphantom{\bigl({}_{t_{0}}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }x \bigr) (t) ={}}{}+\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t}C_{k}(t,s)f_{k} \bigl(s,x(s)\bigr) \,\mathrm{d}s,\quad t>t_{0}, \\ &x(t_{0}+\theta )=\phi (\theta ), \quad \theta \in [-\tau ,0], \end{aligned} $$
(2.1)

where we assume throughout this paper

\((\mathrm{C}_{0})\):

\(t_{0}\geq 0\), \(\alpha \in (0,1)\), \(\rho \in (0,1]\),

\({}_{t_{0}}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }x\) is the generalized Caputo proportional fractional derivative of \(x\in \mathrm{C}^{1}([t_{0},\infty ),\mathbb{R})\),

  • \(g_{k}:[t_{0},\infty )\to [0,\tau _{k}]\), \(\tau _{k}\geq 0\), \(k=1,2,\ldots ,n\),

  • \(\phi \in \mathrm{C}([-\tau ,0],\mathbb{R})\), \(\tau =\max_{k=1,2,\ldots ,n}\tau _{k}\),

  • \(F:[t_{0},\infty )\times \mathbb{R}\to \mathbb{R}\), and

  • \(f_{k}:[t_{0}-\tau ,\infty )\times \mathbb{R}\to \mathbb{R}\), \(C_{k}:[t_{0},\infty )\times [t_{0}-\tau ,\infty )\to \mathbb{R}\), \(\psi _{k}:[t_{0},\infty )\times \mathbb{R}^{2}\to \mathbb{R}\) for \(k=1,2,\ldots ,n\).

Remark 2.1

Everywhere in this paper, rather than giving conditions that imply the existence and uniqueness of solutions of (2.1), we will assume that for any initial function \(\phi \in \mathrm{C}([-\tau ,0],\mathbb{R})\), the initial value problem (2.1) has a unique solution defined for \(t\geq t_{0}\).

We recall that the generalized proportional fractional integral and the generalized Caputo proportional fractional derivative of a function \(u:[a,\infty )\to \mathbb{R}\) are defined, respectively, by (as long as all integrals are well defined, see [11])

$$ ({}_{a}\!{\mathcalligra{I}}^{\beta ,\rho }u )(t)= \frac{1}{\rho ^{\beta }\Gamma (\beta )} \int _{a}^{t}e^{ \frac{\rho -1}{\rho }(t-s)}(t-s)^{\beta -1}u(s)\,\mathrm{d}s, \quad t>a $$

and

$$\begin{aligned} \bigl({}_{a}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }u \bigr) (t) &= \bigl({}_{a}\!{\mathcalligra{I}}^{1-\alpha ,\rho } \bigl({ \mathcalligra{D}}^{\rho }u \bigr) \bigr) (t) \\ &=\frac{1}{\rho ^{1-\alpha }\Gamma (1-\alpha )} \int _{a}^{t}e^{ \frac{\rho -1}{\rho }(t-s)} \frac{ ({\mathcalligra{D}}^{\rho }u )(s)}{(t-s)^{\alpha }} \,\mathrm{d}s, \quad t>a, \end{aligned}$$

where

$$ \mathcalligra{D}^{\rho }u=(1-\rho )u+\rho u'. $$

Remark 2.2

The generalized proportional fractional integral \({}_{a}\!{\mathcalligra{I}}^{\beta ,\rho }u\) is defined also for \(\beta \in (-1,0)\) since \(\Gamma (\beta )<0\) exists.

Remark 2.3

If \(\rho =1\), then the generalized Caputo proportional fractional derivative is reduced to the classical Caputo fractional derivative.

Remark 2.4

If \(u(t)=e^{\frac{\rho -1}{\rho }(t-c)}\) for \(t\geq a\), where \(c\in \mathbb{R}\), then the relation

$$ \bigl({}_{a}^{\mathrm{C}}\!{ \mathcalligra{D}}^{\alpha ,\rho }u \bigr)(t)=0 \quad \text{for } t\geq a $$
(2.2)

is known from [11, Remark 3.2].

3 Auxiliary results

The next result about the generalized Caputo proportional fractional derivative is new, and it is used twice in the remainder of the paper. It also will be of valuable use any time the generalized Caputo proportional fractional derivative is studied.

Lemma 3.1

If there exists \(t>a\) such that \(u\in \mathrm{C}^{1}([a,t],\mathbb{R})\) satisfies \(u(t)=0\), then

$$ \bigl({}_{a}^{\mathrm{C}}\!{ \mathcalligra{D}}^{\alpha ,\rho }u \bigr) (t) = \bigl({}_{a} \!{ \mathcalligra{I}}^{-\alpha ,\rho }u \bigr) (t) - \frac{\rho ^{\alpha }}{\Gamma (1-\alpha )}e^{\frac{\rho -1}{\rho }(t-a)} \frac{u(a)}{(t-a)^{\alpha }}. $$
(3.1)

Proof

Let \(u\in \mathrm{C}^{1}([a,t],\mathbb{R})\) and \(u(t)=0\). We denote

$$ f(s):=e^{\frac{\rho -1}{\rho }(t-s)} \quad \text{and}\quad g(s):= \frac{u(s)}{(t-s)^{\alpha }}\quad \text{for } s\in [a,t). $$

Now, we integrate by parts to obtain

$$\begin{aligned} \bigl({}_{a}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha , \rho }u \bigr) (t) &=\frac{1}{\rho ^{1-\alpha }\Gamma (1-\alpha )} \int _{a}^{t}f(s)\frac{(1-\rho )u(s)+\rho u'(s)}{(t-s)^{\alpha }} \, \mathrm{d}s \\ &=\frac{\rho ^{\alpha }}{\Gamma (1-\alpha )} \int _{a}^{t} \biggl\{ \frac{1-\rho }{\rho }f(s)g(s)+f(s) \frac{u'(s)}{(t-s)^{\alpha }} \biggr\} \,\mathrm{d}s \\ &=\frac{\rho ^{\alpha }}{\Gamma (1-\alpha )} \int _{a}^{t}\biggl\{ f'(s)g(s) +f(s) \biggl[g'(s)-\frac{\alpha u(s)}{(t-s)^{\alpha +1}} \biggr] \biggr\} \, \mathrm{d}s \\ &=\frac{\rho ^{\alpha }}{\Gamma (1-\alpha )} \int _{a}^{t}(fg)'(s) \, \mathrm{d}s-\frac{\alpha \rho ^{\alpha }}{\Gamma (1-\alpha )} \int _{a}^{t}f(s) \frac{u(s)}{(t-s)^{\alpha +1}}\, \mathrm{d}s \\ &=\frac{\rho ^{\alpha }}{\Gamma (1-\alpha )} \Bigl\{ \lim_{s\to t^{-}}f(s)g(s)-f(a)g(a) \Bigr\} \\ &\quad {}+\frac{\rho ^{\alpha }}{\Gamma (-\alpha )} \int _{a}^{t}f(s) \frac{u(s)}{(t-s)^{\alpha +1}}\, \mathrm{d}s \\ &= \bigl({}_{a}\!{\mathcalligra{I}}^{-\alpha ,\rho }u \bigr) (t) - \frac{\rho ^{\alpha }}{\Gamma (1-\alpha )}f(a)g(a), \end{aligned}$$

where we used L’Hôpital’s rule to find

$$ \lim_{s\to t^{-}}\frac{f(s)u(s)}{(t-s)^{\alpha }} =\lim_{s\to t^{-}} \frac{f'(s)u(s)+u'(s)f(s)}{\alpha (t-s)^{\alpha -1}}=0, $$

completing the proof. □

Now, we prove three results that will be used in the derivation of our main results in the next section.

Lemma 3.2

If \(u\in \mathrm{C}^{1}([a,\infty ),\mathbb{R})\), then

$$ \bigl({}_{a}^{\mathrm{C}}\!{ \mathcalligra{D}}^{\alpha ,\rho }u^{2} \bigr) (t) \leq 2u(t) \bigl({}_{a}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }u \bigr) (t) \quad \textit{for all } t>a. $$
(3.2)

Proof

We fix \(t>a\) and define the function \(v:[a,t]\to [0,\infty )\) by

$$ v(s)= \bigl(u(t)-u(s) \bigr)^{2}. $$

Note that \(v\in \mathrm{C}^{1}([a,t],\mathbb{R})\) and \(v(t)=0\), so v satisfies the assumptions of Lemma 3.1, and therefore, from (3.1) and the inequalities \(v(s)\geq 0\) and \(\Gamma (-\alpha )<0\) for \(\alpha \in (0,1)\), it follows \(({}_{a}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }v )(t)\leq 0\). Thus, since

$$\begin{aligned} &\bigl({\mathcalligra{D}}^{\rho }u^{2} \bigr) (s)-2u(t) \bigl({ \mathcalligra{D}}^{\rho }u \bigr) (s) \\ &\quad =(1-\rho )u^{2}(s)+\rho \bigl(u^{2} \bigr)'(s)-2u(t) \bigl\{ (1-\rho )u(s)+\rho u'(s) \bigr\} \\ &\quad =(1-\rho )u^{2}(s)+2\rho u(s)u'(s)-2(1-\rho )u(t)u(s)-2\rho u'(s)u(t) \\ &\quad =(1-\rho ) \bigl(u(s)-u(t) \bigr)^{2}+2\rho u'(s) \bigl(u(s)-u(t) \bigr)-(1-\rho )u^{2}(t) \\ &\quad =(1-\rho )v(s)+\rho v'(s)-(1-\rho )u^{2}(t) \\ &\quad = \bigl({\mathcalligra{D}}^{\rho }v \bigr) (s)-(1-\rho )u^{2}(t) \\ &\quad \leq \bigl({\mathcalligra{D}}^{\rho }v \bigr) (s), \end{aligned}$$

we obtain

$$ \bigl({}_{a}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }u^{2} \bigr) (t)-2u(t) \bigl({}_{a}^{\mathrm{C}}\!{ \mathcalligra{D}}^{\alpha ,\rho }u \bigr) (t) \leq \bigl({}_{a}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }v \bigr) (t)\leq 0, $$

where the last inequality follows from Lemma 3.1 because \(v(s)\geq 0\) and since \(\Gamma (-\alpha )<0\) for \(\alpha \in (0,1)\). □

Lemma 3.3

Let \(\phi \in \mathrm{C}([-\tau ,0],\mathbb{R})\) and assume that

$$ x=x(\cdot ,t_{0},\phi )\in \mathrm{C}^{1}\bigl([-\tau , \infty ),\mathbb{R}\bigr) $$

is a solution of (2.1). If for any \(t\geq t_{0}\) such that

$$ x^{2}(t+\theta )e^{\frac{1-\rho }{\rho }\theta }\leq x^{2}(t) \quad \textit{for all } \theta \in [-\tau ,0), $$
(3.3)

the inequality

$$ \bigl({}_{t_{0}}^{\mathrm{C}}\!{ \mathcalligra{D}}^{\alpha ,\rho }x^{2} \bigr) (t) \leq 0 $$
(3.4)

holds, then

$$ \bigl\vert x(s) \bigr\vert \leq \sqrt{\max _{\theta \in [-\tau ,0]}\phi ^{2}(\theta )e^{ \frac{\rho -1}{\rho }(s-t_{0})}}\quad \textit{for all } s\geq t_{0}. $$
(3.5)

Proof

Let \(x=x(\cdot ,t_{0},\phi )\) be a solution of (2.1). We define

$$ \eta (s):=x^{2}(s)-Be^{\frac{\rho -1}{\rho }(s-t_{0})}-\varepsilon e^{ \frac{\rho -1}{\rho }s} \quad \text{for } s\geq t_{0}, $$

where

$$ B:=\max_{\theta \in [-\tau ,0]}\phi ^{2}(\theta )\geq 0. $$

Observe that

$$ \eta (t_{0})=x^{2}(t_{0})-B-\varepsilon e^{\frac{\rho -1}{\rho }t_{0}} < x^{2}(t_{0})-B\leq B-B=0. $$

We claim

$$ \eta (s)< 0 \quad \text{for all } s\geq t_{0}, $$
(3.6)

from which (3.5) would follow. Assuming that (3.6) is not true, there exists \(t>t_{0}\) such that

$$ \eta (s)< 0\quad \text{for all } s\in [t_{0},t) \quad \text{and} \quad \eta (t)=0. $$
(3.7)

Note that \(\eta \in \mathrm{C}^{1}([t_{0},t],\mathbb{R})\) and \(\eta (t)=0\), hence η satisfies the assumptions of Lemma 3.1. According to Lemma 3.1 and the choice of the point t, we have \(({}_{t_{0}}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }\eta )(t)>0\). For any \(\theta \in [-\tau ,0]\), we have

$$\begin{aligned} x^{2}(t+\theta )e^{\frac{1-\rho }{\rho }\theta } & = \bigl\{ \eta (t+\theta )+Be^{\frac{\rho -1}{\rho }(t+\theta -t_{0})}+ \varepsilon e^{\frac{\rho -1}{\rho }(t+\theta )} \bigr\} e^{ \frac{1-\rho }{\rho }\theta } \\ & =\eta (t+\theta )e^{\frac{1-\rho }{\rho }\theta }+Be^{ \frac{\rho -1}{\rho }(t-t_{0})}+\varepsilon e^{\frac{\rho -1}{\rho }t} \\ & =\eta (t+\theta )e^{\frac{1-\rho }{\rho }\theta }+x^{2}(t)-\eta (t) \\ & =x^{2}(t)+\eta (t+\theta )e^{\frac{1-\rho }{\rho }\theta } \\ & < x^{2}(t), \end{aligned}$$

so for this t, (3.3) holds, and thus

$$ 0\stackrel{\text{(3.4)}}{\geq } \bigl({}_{t_{0}}^{\mathrm{C}} \!{ \mathcalligra{D}}^{\alpha ,\rho }x^{2} \bigr) (t) \stackrel{\text{(2.2)}}{=} \bigl({}_{t_{0}}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }\eta \bigr) (t)>0, $$

where the last inequality follows from Lemma 3.1, contradicting (3.7), proving (3.6), and hence implying (3.5). □

Remark 3.4

The condition “if for a point t such that (3.3) holds, then (3.4) is satisfied” is a (modified) Razumikhin condition. In the case of the Caputo fractional derivative, i.e., \(\rho =1\), this condition reduces to the classical Razumikhin condition (see, for example [12]).

Remark 3.5

The main characteristic of Lyapunov functions applied to delay differential equations is the condition for its derivative, the so-called Razhumikhin condition (for example, the negativity of the derivative). It allows us to assume a restriction on their derivatives only at some point, not for all values of t. This is different from the applications to ordinary differential equations and to Caputo fractional differential equations without delays.

Corollary 3.6

If the conditions of Lemma 3.3are satisfied, then

$$ \bigl\vert x(s) \bigr\vert \leq \sqrt{\max _{\theta \in [-\tau ,0]}\phi ^{2}(\theta )} \quad \textit{for all } s\geq t_{0}. $$
(3.8)

Proof

This follows from (3.5), \((\mathrm{C}_{0})\), and \(e^{\frac{\rho -1}{\rho }(s-t_{0})}\leq 1\) for \(s\geq t_{0}\). □

4 Stability and boundedness properties

We introduce the following assumptions.

\((\mathrm{C}_{1})\):

\(F\in \mathrm{C}([t_{0},\infty )\times \mathbb{R},\mathbb{R})\), and for \(k=1,2,\ldots ,n\),

  • \(\psi _{k}\in \mathrm{C}([t_{0},\infty )\times \mathbb{R}^{2}, \mathbb{R})\),

  • \(f_{k}\in \mathrm{C}([t_{0}-\tau _{k},\infty )\times \mathbb{R},[0, \infty ))\),

  • \(C_{k}\in \mathrm{C}([t_{0},\infty )\times [t_{0}-\tau ,\infty ), \mathbb{R})\).

\((\mathrm{C}_{2})\):

\(F(t,0)=0\), and for \(k=1,2,\ldots ,n\),

  • \(\psi _{k}(t,0,0)=0\) for \(t\geq t_{0}\),

  • \(f_{k}(t,0)=0\) for \(t\geq t_{0}-\tau _{k}\).

\((\mathrm{C}_{3})\):

There exist \(b\in \mathrm{C}((t_{0},\infty ),[0,\infty ))\) and, for \(k=1,2,\ldots ,n\), \(A_{k},B_{k}\in \mathrm{C}((t_{0},\infty ),[0,\infty ))\), \(\ell _{k}\in \mathrm{C}((t_{0}-\tau _{k},\infty ),[0,\infty ))\) such that

  • \(xF(t,x)\geq b(t)x^{2}\) for \(t\geq t_{0}\), \(x\in \mathbb{R}\),

  • \(|\psi _{k}(t,x,y)|\leq A_{k}(t)|x|+B_{k}(t)|y|\) for \(t\geq t_{0}\), \(x,y\in \mathbb{R}\),

  • \(|f_{k}(t,x)|\leq \ell _{k}(t)|x|\) for \(t\geq t_{0}-\tau _{k}\), \(x\in \mathbb{R}\).

\((\mathrm{C}_{4})\):

The inequality

$$\begin{aligned} b(t)\geq {}&\sum_{k=1}^{n} \biggl[A_{k}(t)+ \frac{1+e^{\frac{1-\rho }{\rho }g_{k}(t)}}{2}B_{k}(t) \\ &{}+ \int _{t-g_{k}(t)}^{t} \frac{1+e^{\frac{1-\rho }{\rho }(t-s)}}{2} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s) \,\mathrm{d}s \biggr] \end{aligned}$$

holds for all \(t\geq t_{0}\).

Our first main result is concerned with stability. Note that, in contrast to ordinary derivatives, fractional derivatives depend significantly on the initial time point \(t_{0}\), which is equal to the lower limit of the derivative. Hence, any change of the initial time leads to a change of the fractional derivative. For this reason, we will not consider the case of uniform stability and study only the cases of stability, exponential stability, and asymptotic stability.

Theorem 4.1

(Stability)

If \((\mathrm{C}_{1})\)–\((\mathrm{C}_{4})\) hold, then the zero solution of (2.1) with zero initial function is stable.

Proof

Let \(x=x(\cdot ,t_{0},\phi )\) be any solution of (2.1). We will show that the assumptions of Lemma 3.3 are satisfied. To this end, suppose \(t\geq t_{0}\) is such that (3.3) is satisfied. If \(s\in [t-g_{k}(t),t]\subset [t-\tau ,t]\), then \(\theta :=s-t\in [-\tau ,0]\), and from (3.3), we obtain

$$ x^{2}(s)e^{\frac{1-\rho }{\rho }(s-t)}\leq x^{2}(t) \quad \text{for all } s\in \bigl[t-g_{k}(t),t\bigr]. $$
(4.1)

In the calculation below, we also use the “trivial” inequality

$$ \vert 2ab \vert \leq a^{2}+b^{2}\quad \text{for all } a,b\in \mathbb{R}. $$
(4.2)

Now,

$$\begin{aligned} \bigl({}_{t_{0}}^{\mathrm{C}}\!{\mathcalligra{D}}^{ \alpha ,\rho }x^{2} \bigr) (t) &\stackrel{\text{(3.2)}}{\leq }2x(t) \bigl({}_{t_{0}}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }x \bigr) (t) \\ &=-2x(t)F\bigl(t,x(t)\bigr)+2\sum_{k=1}^{n}x(t) \psi _{k}(t,x(t),x\bigl(t-g_{k}(t)\bigr) \\ &\quad {}+2\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t}C_{k}(t,s)x(t)f_{k} \bigl(s,x(s)\bigr) \,\mathrm{d}s \\ &\leq -2x(t)F\bigl(t,x(t)\bigr)+2\sum_{k=1}^{n} \bigl\vert x(t) \bigr\vert |\psi _{k}(t,x(t),x \bigl(t-g_{k}(t)\bigr) \vert \\ &\quad {}+2\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \bigl\vert x(t) \bigr\vert \bigl\vert f_{k} \bigl(s,x(s)\bigr) \bigr\vert \,\mathrm{d}s \\ &\stackrel{\text{$(\mathrm{C}_{3})$}}{\leq }-2b(t)x^{2}(t) +2\sum _{k=1}^{n} \bigl\vert x(t) \bigr\vert \bigl\{ A_{k}(t) \bigl\vert x(t) \bigr\vert +B_{k}(t) \bigl\vert x\bigl(t-g_{k}(t)\bigr) \bigr\vert \bigr\} \\ &\quad {}+2\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s)|x(t)x(s) \vert \,\mathrm{d}s \\ &=-2b(t)x^{2}(t) +\sum_{k=1}^{n}2A_{k}(t)x^{2}(t)+ \sum_{k=1}^{n}B_{k}(t) \bigl\vert 2x(t)x\bigl(t-g_{k}(t)\bigr) \bigr\vert \\ &\quad {}+\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s) \bigl\vert 2x(t)x(s) \bigr\vert \, \mathrm{d}s \\ &\stackrel{\text{(4.2)}}{\leq }-2b(t)x^{2}(t) +\sum _{k=1}^{n}2A_{k}(t)x^{2}(t)+ \sum_{k=1}^{n}B_{k}(t) \bigl\{ x^{2}(t)+x^{2}\bigl(t-g_{k}(t)\bigr) \bigr\} \\ &\quad {}+\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s) \bigl\{ x^{2}(t)+x^{2}(s) \bigr\} \,\mathrm{d}s \\ &=-2b(t)x^{2}(t) +\sum_{k=1}^{n} \bigl\{ 2A_{k}(t)+B_{k}(t) \bigr\} x^{2}(t)+ \sum_{k=1}^{n}B_{k}(t)x^{2} \bigl(t-g_{k}(t)\bigr) \\ &\quad {}+\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s) \bigl\{ x^{2}(t)+x^{2}(s) \bigr\} \,\mathrm{d}s \\ &\stackrel{\text{(4.1)}}{\leq }-2b(t)x^{2}(t) +\sum _{k=1}^{n} \bigl\{ 2A_{k}(t)+B_{k}(t) \bigr\} x^{2}(t) \\ &\quad {}+\sum_{k=1}^{n}B_{k}(t)x^{2}(t)e^{\frac{\rho -1}{\rho }(t-g_{k}(t)-t)} \\ &\quad {}+\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s) \bigl\{ x^{2}(t)+x^{2}(t)e^{\frac{\rho -1}{\rho }(s-t)} \bigr\} \,\mathrm{d}s \\ &=2x^{2}(t) \Biggl[-b(t) +\sum_{k=1}^{n} \biggl\{ A_{k}(t)+ \frac{1+e^{\frac{1-\rho }{\rho }g_{k}(t)}}{2}B_{k}(t) \biggr\} \\ &\quad {}+\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s)\frac{1+e^{\frac{1-\rho }{\rho }(t-s)}}{2} \,\mathrm{d}s \Biggr] \\ &\stackrel{\text{$(\mathrm{C}_{4})$}}{\leq }0, \end{aligned}$$

i.e., (3.4) holds. Hence, the assumptions of Lemma 3.3 are satisfied. According to Corollary 3.6, (3.8) holds. This completes the proof. □

Corollary 4.2

(Boundedness)

If \((\mathrm{C}_{1})\), \((\mathrm{C}_{3})\), and \((\mathrm{C}_{4})\) hold, then the solutions of (2.1) are bounded.

Proof

As in the proof of Theorem 4.1, we obtain (3.8) for any solution \(x=x(\cdot ,t_{0},\phi )\) of (2.1), i.e., boundedness is established. □

Remark 4.3

The above-obtained sufficient conditions for stability and boundedness are true for the case of delay Caputo fractional integro-differential equations (\(\rho =1\)). Note that a partial case of (2.1), namely F in a special form, \(\psi _{1}\) not depending on the third variable, \(\psi _{k}=0\) for all \(2\leq k\leq n\), constant delays, and the Caputo fractional derivative is studied in [8], and the stability of the zero solution and boundedness of solutions are proved. A modified Razhumikhin condition and results known in the literature (see [8, Lemma 1, Lemma 4]) are used. However, this condition is not applied adequately in the proof of the result for stability (see [8, Theorem 1]). In the proof of the boundedness [8, Theorem 2], the negativeness of the Caputo fractional derivative of the Lyapunov function is applied for all points t instead of only for those for which the Razhumikhin condition is true (compare with Remark 3.5). Hence, our results do not only generalize theorems from [8], they also remove the inconsistencies in the case of regular Caputo fractional differential equations.

In connection with the above-mentioned remarks and for future reference, we state our results for delay integro-differential equations with Caputo fractional derivative (with \(\rho =1\))

$$ \begin{aligned} &\bigl({}_{t_{0}}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,1}x \bigr) (t) =-F\bigl(t,x(t)\bigr)+ \sum _{k=1}^{n}\psi _{k}(t,x(t),x \bigl(t-g_{k}(t)\bigr) \\ &\hphantom{\bigl({}_{t_{0}}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,1}x \bigr) (t) ={}}{}+\sum_{k=1}^{n} \int _{t-g_{k}(t)}^{t}C_{k}(t,s)f_{k} \bigl(s,x(s)\bigr) \,\mathrm{d}s,\quad t>t_{0}, \\ &x(t_{0}+\theta )=\phi (\theta ),\quad \theta \in [-\tau ,0]. \end{aligned} $$
(4.3)

As a partial case of Theorem 4.1 and Corollary 4.2, we obtain the following result.

Corollary 4.4

Assume \((\mathrm{C}_{1})\)–\((\mathrm{C}_{3})\) hold and

$$ b(t)\geq \sum_{k=1}^{n} \biggl[A_{k}(t)+B_{k}(t) + \int _{t-g_{k}(t)}^{t} \bigl\vert C_{k}(t,s) \bigr\vert \ell _{k}(s)\,\mathrm{d}s \biggr]. $$
(4.4)

Then,

  • the solutions of (4.3) are bounded and the bound depends on the initial function;

  • the zero solution of (4.3) with zero initial function is stable.

From Theorem 4.1 and Corollary 4.2, we obtain the following results.

Corollary 4.5

(Exponential Stability)

Let \(\rho \in (0,1)\). If \((\mathrm{C}_{1})\)–\((\mathrm{C}_{4})\) hold, then the zero solution of (2.1) is exponentially stable.

Proof

Let \(x=x(\cdot ,t_{0},\phi )\) be any solution of (2.1), and suppose the point \(t\geq t_{0}\) is such that (3.3) holds. If \(s\in [t-g_{k}(t),t]\subset [t-\tau ,t]\), then

$$ \theta :=s-t\in [-\tau ,0], $$

and from (3.3), we obtain (4.1), i.e.,

$$ x^{2}(s)e^{\frac{1-\rho }{\rho }(s-t)}\leq x^{2}(t) \quad \text{for all } s\in \bigl[t-g_{k}(t),t\bigr] $$

holds. Following the ideas of the proof of Theorem 4.1, we obtain

$$ \bigl({}_{t_{0}}^{\mathrm{C}}\!{\mathcalligra{D}}^{\alpha ,\rho }x^{2} \bigr) (t) \leq 0, $$

i.e., (3.4) holds. Therefore, the assumptions of Lemma 3.3 are satisfied, and thus

$$ \bigl\vert x(s) \bigr\vert \leq \sqrt{\max _{\theta \in [-\tau ,0]}\phi ^{2}(\theta )}e^{ \frac{\rho -1}{2\rho }(s-t_{0})} \quad \text{for all } s\geq t_{0} $$
(4.5)

holds, i.e., since \(\rho \in (0,1)\), exponential stability is established. □

Corollary 4.6

(Asymptotic Stability)

Let \(\rho \in (0,1)\). If \((\mathrm{C}_{1})\)–\((\mathrm{C}_{4})\) hold, then the zero solution of (2.1) is asymptotically stable.

Proof

The proof follows from \(\rho \in (0,1)\) and (4.5). □

Remark 4.7

The above-proved results about exponential stability and asymptotic stability are true only for \(\rho \in (0,1)\), i.e., we do not provide sufficient conditions for the case of the Caputo fractional derivative (with \(\rho =1\)). For Caputo fractional derivatives, results concerning the asymptotic stability [8, Theorem 3] and Mittag-Leffler stability [8, Theorem 4] are studied, but the negativeness of the Caputo fractional derivative of the Lyapunov function is applied for all points instead of only for those for which the Razumikhin condition holds (compare with Remark 3.5). Hence, the question about asymptotic stability and Mittag-Leffler stability is still open for delay Caputo fractional integro-differential equations.

5 Examples

We first look at [8, Example 1] with appropriate modifications.

Example 5.1

Consider

$$ \begin{aligned} &\bigl({}_{0.25}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,1}x \bigr) (t) =- \biggl(24+\frac{1}{1+t} \biggr) \bigl(4x(t)+x(t)e^{-x^{2}(t)-t^{2}}\bigr) \\ &\hphantom{\bigl({}_{0.25}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,1}x \bigr) (t) ={}}{}-3x(t)-\sin \bigl(x(t)\bigr)+\frac{\sin (x(t))}{1+e^{t}} \\ &\hphantom{\bigl({}_{0.25}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,1}x \bigr) (t) ={}}{}+ \int _{t-0.25}^{t}e^{s-t} \frac{\sin (x(s))}{1+e^{s^{2}}}\,\mathrm{d}s,\quad t>0.25, \\ &x(0.25+\theta )=\phi (\theta ), \quad \theta \in [-0.25,0]. \end{aligned} $$
(5.1)

Problem (5.1) is a special case of (2.1) with

$$\begin{aligned}& n=1, \qquad \rho =1,\qquad t_{0}=0.25,\qquad g_{1}(t)\equiv \tau _{1}=0.25, \\& C_{1}(t,s)=e^{s-t},\qquad f_{1}(t,x)= \frac{\sin (x)}{1+e^{t^{2}}},\qquad \psi _{1}(t,x,y)=\frac{\sin (x)}{1+e^{t}}, \\& F(t,x)= \biggl(24+\frac{1}{1+t} \biggr) \bigl(4x+xe^{-x^{2}-t^{2}} \bigr)+3x+\sin (x). \end{aligned}$$

It is easy to check that

$$\begin{aligned}& F(t,0)=0, \qquad xF(t,x)\geq \biggl(98+\frac{4}{1+t} \biggr)x^{2},\qquad b(t)=98+\frac{4}{1+t}, \\& \psi _{1}(t,0,0)=0, \qquad \bigl\vert \psi _{1}(t,x,y) \bigr\vert \leq \frac{ \vert x \vert }{1+e^{t}}, \qquad A_{1}(t)=\frac{1}{1+e^{t}},\qquad B_{1}(t)\equiv 0, \\& f_{1}(t,0)=0, \qquad \bigl\vert f_{1}(t,x) \bigr\vert \leq \frac{ \vert x \vert }{1+e^{t^{2}}}\leq \frac{ \vert x \vert }{2},\qquad \ell _{1}(t)= \frac{1}{2}. \end{aligned}$$

Therefore, \((\mathrm{C}_{1})\)–\((\mathrm{C}_{3})\) are satisfied. Moreover, we have (recall that \(B_{1}(t)\equiv 0\))

$$\begin{aligned} A_{1}(t)+ \int _{t-g_{1}(t)}^{t} \bigl\vert C_{1}(t,s) \bigr\vert \ell _{1}(s) \,\mathrm{d}s =&\frac{1}{1+e^{t}}+ \frac{1}{2} \int _{t-0.25}^{t}e^{s-t} \,\mathrm{d}s \\ =&\frac{1}{1+e^{t}}+\frac{1}{2 (1-e^{-0.25} )}< 3.3< 98< 98+ \frac{4}{1+t}=b(t), \end{aligned}$$

so that (4.4) is fulfilled. According to Corollary 4.4, the zero solution is stable.

Example 5.2

Let \(\alpha \in (0,1)\) and let \(\rho \in (0.0247292,1)\). Consider (5.1) with the generalized Caputo proportional fractional derivative instead of the Caputo fractional derivative, i.e., consider

$$ \begin{aligned} &\bigl({}_{0.25}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }x \bigr) (t) =- \biggl(24+\frac{1}{1+t} \biggr) \bigl(4x(t)+x(t)e^{-x^{2}(t)-t^{2}}\bigr) \\ &\hphantom{\bigl({}_{0.25}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }x \bigr) (t) ={}}{}-3x(t)-\sin \bigl(x(t)\bigr)+\frac{\sin (x(t))}{1+e^{t}} \\ &\hphantom{\bigl({}_{0.25}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }x \bigr) (t) ={}}{}+ \int _{t-0.25}^{t}e^{s-t} \frac{\sin (x(s))}{1+e^{s^{2}}}\,\mathrm{d}s,\quad t>0.25, \\ &x(0.25+\theta )=\phi (\theta ), \quad \theta \in [-0.25,0]. \end{aligned} $$
(5.2)

As in Example 5.1, \((\mathrm{C}_{1})\)–\((\mathrm{C}_{3})\) are satisfied. Let us define

$$ \kappa :=\frac{1}{4\rho }-\frac{1}{2}\leq 9.6095062. $$

We have (recall that \(B_{1}(t)\equiv 0\))

$$\begin{aligned} &A_{1}(t)+ \int _{t-g_{1}(t)}^{t} \frac{1+e^{\frac{1-\rho }{\rho }(t-s)}}{2} \bigl\vert C_{1}(t,s) \bigr\vert \ell _{1}(s)\,\mathrm{d}s \\ &\quad =\frac{1}{1+e^{t}}+ \int _{t-0.25}^{t} \frac{1+e^{\frac{1-\rho }{\rho }(t-s)}}{2}e^{s-t} \frac{\mathrm{d}s}{2} \\ &\quad < 1+\frac{1}{4} \int _{t-0.25}^{t} \bigl(e^{s-t}+e^{-4\kappa (s-t)} \bigr)\,\mathrm{d}s \\ &\quad =1+\frac{1}{4} \bigl(1-e^{-0.25} \bigr)+ \frac{1}{4} \int _{t-0.25}^{t}e^{-4 \kappa (s-t)}\,\mathrm{d}s \\ &\quad =1+\frac{1}{4} \bigl(1-e^{-0.25} \bigr) + \frac{1}{16} \textstyle\begin{cases} 1& \text{if } \kappa =0, \\ \frac{e^{\kappa }-1}{\kappa }& \text{otherwise} \end{cases}\displaystyle \\ &\quad < 97.995812< 98< 98+\frac{4}{1+t}=b(t), \end{aligned}$$

and the first inequality in the last line is due to the fact that the function f defined by \(f(x)=(e^{x}-1)/x\) for \(x\in \mathbb{R}\setminus \{0\}\) and \(f(0)=1\) is strictly increasing. Thus, the condition \((\mathrm{C}_{4})\) is fulfilled. According to Corollary 4.5, the zero solution of (5.2) is exponentially stable.

Example 5.3

Consider

$$ \begin{aligned} &\bigl({}_{0}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,1}x \bigr) (t) =- \frac{2+t}{1+t}x(t)+ \frac{x(t)}{1+t^{2}} + \int _{t-1}^{t}(t-s)x(s)\,\mathrm{d}s,\quad t>0, \\ &x(\theta )=\phi (\theta ),\quad \theta \in [-1,0]. \end{aligned} $$
(5.3)

Problem (5.3) is a special case of (2.1) with

$$\begin{aligned}& n=1, \qquad \rho =1, \qquad t_{0}=0,\qquad g_{1}(t)\equiv \tau _{1}=1, \\& C_{1}(t,s)=t-s,\qquad f_{1}(t,x)=x,\qquad \psi _{1}(t,x,y)= \frac{x}{1+t^{2}}, \\& F(t,x)=\frac{2+t}{1+t}x. \end{aligned}$$

It is easy to check that

$$\begin{aligned}& F(t,0)=0,\qquad xF(t,x)=\frac{2+t}{1+t}x^{2}, \qquad b(t)= \frac{2+t}{1+t}, \\& \psi _{1}(t,0,0)=0, \qquad \bigl\vert \psi _{1}(t,x,y) \bigr\vert \leq \frac{ \vert x \vert }{1+t^{2}},\qquad A_{1}(t)=\frac{1}{1+t^{2}},\qquad B_{1}(t)\equiv 0, \\& f_{1}(t,0)=0, \qquad \bigl\vert f_{1}(t,x) \bigr\vert = \vert x \vert , \qquad \ell _{1}(t)=1. \end{aligned}$$

Therefore, \((\mathrm{C}_{1})\)–\((\mathrm{C}_{3})\) are satisfied. Moreover, we have (recall that \(B_{1}(t)\equiv 0\))

$$\begin{aligned} A_{1}(t)+ \int _{t-g_{1}(t)}^{t} \bigl\vert C_{1}(t,s) \bigr\vert \ell _{1}(s) \,\mathrm{d}s&=\frac{1}{1+t^{2}}+ \int _{t-1}^{t} \vert t-s \vert \,\mathrm{d}s \\ &=\frac{1}{1+t^{2}}+\frac{1}{2}< \frac{2+t}{1+t}=b(t), \end{aligned}$$

so that (4.4) is fulfilled. According to Corollary 4.4, the zero solution is stable.

Example 5.4

Let \(\alpha \in (0,1)\) and let \(\rho \in (0.3420471,1)\). Consider (5.3) with the generalized Caputo proportional fractional derivative instead of the Caputo fractional derivative, i.e., consider

$$ \begin{aligned} &\bigl({}_{0}^{\mathrm{C}} \!{\mathcalligra{D}}^{\alpha ,\rho }x \bigr) (t) =- \frac{2+t}{1+t}x(t)+ \frac{x(t)}{1+t^{2}} + \int _{t-1}^{t}(t-s)x(s)\,\mathrm{d}s, \quad t>0, \\ &x(\theta )=\phi (\theta ), \quad \theta \in [-1,0]. \end{aligned} $$
(5.4)

As in Example 5.3, \((\mathrm{C}_{1})\)–\((\mathrm{C}_{3})\) are satisfied. Let us define

$$\begin{aligned}& \kappa :=\frac{1}{\rho }-1\leq 1.9235739,\qquad g(t):= \frac{1}{1+t^{2}}- \frac{2+t}{1+t}, \\& \mu :=\max_{t>0}g(t)=-1-2 \biggl(\frac{1}{2}+ \frac{\sqrt{5}}{2}-\sqrt{ \frac{1+\sqrt{5}}{2}} \biggr)^{2}. \end{aligned}$$

We have for \(t>0\) (recall that \(B_{1}(t)\equiv 0\))

$$\begin{aligned} &A_{1}(t)+ \int _{t-g_{1}(t)}^{t} \frac{1+e^{\frac{1-\rho }{\rho }(t-s)}}{2} \bigl\vert C_{1}(t,s) \bigr\vert \ell _{1}(s)\,\mathrm{d}s \\ &\quad =\frac{1}{1+t^{2}}+ \int _{t-1}^{t}\frac{1+e^{\kappa (t-s)}}{2}(t-s) \, \mathrm{d}s \\ &\quad =\frac{1}{1+t^{2}}+\frac{1}{2} \int _{0}^{1} \bigl(v+ve^{\kappa v} \bigr) \,\mathrm{d}v \\ &\quad =\frac{1}{1+t^{2}}+\frac{1}{2} \biggl\{ \frac{1}{2}+ \frac{(\kappa -1)e^{\kappa }+1}{\kappa ^{2}} \biggr\} \\ &\quad =g(t)+b(t)+\frac{1}{2} \biggl\{ \frac{1}{2}+ \frac{(\kappa -1)e^{\kappa }+1}{\kappa ^{2}} \biggr\} \\ &\quad \leq \mu +\frac{1}{2} \biggl\{ \frac{1}{2}+ \frac{(\kappa -1)e^{\kappa }+1}{\kappa ^{2}} \biggr\} +b(t) \\ &\quad < b(t), \end{aligned}$$

and the last inequality is due to the fact that the function f defined by \(f(x)= ((x-1)e^{x}+1)/x^{2}\) for \(x>0\) is strictly increasing. Thus, the condition \((\mathrm{C}_{4})\) is fulfilled. According to Corollary 4.5, the zero solution of (5.4) is exponentially stable.

Example 5.5

Consider

$$ \begin{aligned} &\bigl({}_{0}^{\mathrm{C}} \!{\mathcalligra{D}}^{0.3,0.8}x \bigr) (t) =- \frac{2+t}{1+t}x(t)+0.03x(t-1) \cos ^{2}t \\ &\hphantom{\bigl({}_{0}^{\mathrm{C}} \!{\mathcalligra{D}}^{0.3,0.8}x \bigr) (t) ={}}{}+\frac{x(t-1-\sin t)}{10+t^{2}} \\ &\hphantom{\bigl({}_{0}^{\mathrm{C}} \!{\mathcalligra{D}}^{0.3,0.8}x \bigr) (t) ={}}{}+ \int _{t-1}^{t}(t-s)x(s)\,\mathrm{d}s, \quad t>0, \\ &x(\theta )=\phi (\theta ),\quad \theta \in [-2,0]. \end{aligned} $$
(5.5)

Problem (5.5) is a special case of (2.1) with

$$\begin{aligned}& n=2, \qquad \rho =0.8, \qquad \alpha =0.3,\qquad t_{0}=0, \\& g_{1}(t)\equiv 1, \qquad g_{2}(t)=1+\sin t,\qquad \tau _{1}=1,\qquad \tau _{2}=2,\qquad \tau =2, \\& C_{1}(t,s)=t-s, \qquad C_{2}(t,s)\equiv 0, \qquad f_{1}(t,x)=f_{2}(t,x)=x, \\& \psi _{1}(t,x,y)=0.03y\cos ^{2}t,\qquad \psi _{2}(t,x,y)= \frac{y}{10+t^{2}},\qquad F(t,x)=\frac{2+t}{1+t}x. \end{aligned}$$

If we put

$$\begin{aligned}& b(t)=\frac{2+t}{1+t},\qquad \ell _{1}(t)=\ell _{2}(t) \equiv 1, \\& A_{1}(t)=A_{2}(t)\equiv 0,\qquad B_{1}(t)=0.03,\qquad B_{2}(t)=0.1, \end{aligned}$$

then \((\mathrm{C}_{1})\)–\((\mathrm{C}_{3})\) are satisfied. We put \(\kappa :=(1-\rho )/\rho =0.25\) and estimate for \(t>0\) (recall that \(A_{1}(t)=A_{2}(t)=C_{2}(t)\equiv 0\))

$$\begin{aligned} &\frac{1+e^{\kappa g_{1}(t)}}{2}B_{1}(t) + \frac{1+e^{\kappa g_{2}(t)}}{2}B_{2}(t) + \int _{t-g_{1}(t)}^{t} \frac{1+e^{\kappa (t-s)}}{2} \bigl\vert C_{1}(t,s) \bigr\vert \ell _{1}(s) \,\mathrm{d}s \\ &\quad =\frac{1+e^{\kappa }}{2}0.03+\frac{1+e^{\kappa (1+\sin t)}}{2}0.1 + \int _{t-1}^{t}\frac{1+e^{\kappa (t-s)}}{2}(t-s)\, \mathrm{d}s \\ &\quad \leq \frac{1+e^{\kappa }}{2}0.03+\frac{1+e^{2\kappa }}{2}0.1 + \frac{1}{2} \int _{0}^{1} \bigl(v+ve^{\kappa v} \bigr) \,\mathrm{d}v \\ &\quad =\frac{1+e^{\kappa }}{2}0.03+\frac{1+e^{2\kappa }}{2}0.1 +\frac{1}{2} \biggl\{ \frac{1}{2}+\frac{(\kappa -1)e^{\kappa }+1}{\kappa ^{2}} \biggr\} \\ &\quad < 0.712544< 1< 1+\frac{1}{1+t}=b(t), \end{aligned}$$

and thus \((\mathrm{C}_{4})\) is fulfilled. From Corollary 4.5, follows the exponential stability and therefore the asymptotic stability of the zero solution of (5.5), and according to (4.5), any solution of (5.5) satisfies

$$ \bigl\vert x(s) \bigr\vert \leq \sqrt{\max_{\theta \in [-2,0]}\phi ^{2}(\theta )}e^{-0.125s}\quad \text{for all } s\geq 0. $$

6 Conclusions

A scalar nonlinear fractional integro-differential equation with time-variable bounded delays and the generalized Caputo proportional fractional derivative is studied. Some qualitative properties such as stability, exponential stability, asymptotic stability, and boundedness are investigated. The main apparatus is the application of Lyapunov functions and a Razumikhin method, modified for our special situation. Five examples are provided to illustrate the usefulness of the obtained sufficient conditions for generalized Caputo proportional fractional delay integro-differential equations, as well as for classical Caputo fractional delay integro-differential equations. Also, the main lemma about the inequality for quadratic functions offers a tool for successful study of stability of various types of models described by generalized Caputo proportional fractional differential equations with and without delays.