 Research
 Open Access
Lyapunov functions and strict stability of Caputo fractional differential equations
 Ravi Agarwal^{1, 2},
 Snezhana Hristova^{3}Email author and
 Donal O’Regan^{4}
https://doi.org/10.1186/s1366201506745
© Agarwal et al. 2015
Received: 29 September 2015
Accepted: 15 October 2015
Published: 6 November 2015
Abstract
One of the main properties studied in the qualitative theory of differential equations is the stability of solutions. The stability of fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. However, the Lyapunov approach to fractional differential equations causes many difficulties. In this paper a new definition (based on the Caputo fractional Dini derivative) for the derivative of Lyapunov functions to study a nonlinear Caputo fractional differential equation is introduced. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are given. Examples are presented to illustrate the theory.
Keywords
 strict stability
 Lyapunov functions
 Caputo derivatives
 fractional differential equations
MSC
 34A34
 34A08
 34D20
1 Introduction
One of the main problems in the qualitative theory of differential equations is stability of solutions. However, the usual stability concepts do not give any information concerning the rate of decay of solutions, and hence are not strict concepts. As a result, strict stability was defined, and criteria for such notions were discussed (see, for example, [1–5]).
Fractional differential equations play an important role not only in mathematics but also in physics, control systems, dynamical systems, engineering and in particular in mathematical modeling of many natural physical phenomena. For example, fractional derivatives are used in modeling mechanical and electrical properties of real materials, in the description of properties of gases, liquids and rocks, and in many other fields (see, for example, [6, 7]).

continuously differentiable Lyapunov functions (see, for example, the papers [9–15]). Different types of stability are discussed using the Caputo derivative of Lyapunov functions which depends significantly on the unknown solution of the fractional equation;

continuous Lyapunov functions (see, for example, the papers [16–18]) in which the authors use the derivative of a Lyapunov function which is similar to the Dini derivative of Lyapunov functions.
In this paper the strict stability of nonlinear nonautonomous Caputo fractional differential equations is defined and studied using continuous Lyapunov functions. The Caputo fractional Dini derivative of a Lyapunov function is defined in an appropriate way. Note that this type of derivative is introduced in [19] and used to study the stability and asymptotic stability of Caputo fractional differential equations. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are obtained.
The manuscript is organized as follows. In Section 2 some preliminaries of fractional calculus are mentioned. Section 3 presents basic definitions concerning strict stability and the new definition of the Caputo fractional Dini derivative of Lyapunov functions among the nonlinear fractional differential equations. In Section 4 some comparison results are given. Section 5 presents some sufficient conditions for strict stability and uniform strict stability.
2 Notes on fractional calculus
Fractional calculus generalizes the derivative and the integral of a function to a noninteger order [20, 21], and there are several definitions of fractional derivatives and fractional integrals. In engineering, the fractional order q is often less than 1, so we restrict our attention to \(q\in(0,1)\).
Example 1
The relation between the two types of fractional derivatives is given by the equality \({}_{t_{0}}^{c}D^{q}m(t)={}_{t_{0}}^{RL}D^{q}[m(t)m(t_{0})]\).
Proposition 1
(Theorem 2.25 [23])
Let \(m\in C^{1}[t_{0},b]\). Then, for \(t \in(t_{0},b]\), \({}_{t_{0}}^{GL}D^{q}m(t)={}_{t_{0}}^{RL}D^{q}m(t)\).
Also, according to Lemma 3.4 [23], the equality \({}_{t_{0}}^{c}D_{t}^{q}m(t)={}_{t_{0}}^{RL}D_{t}^{q}m(t)m(t_{0})\frac {(tt_{0})^{q}}{\Gamma(1q)}\) holds.
Definition 1
([16])
We say \(m\in C^{q}([t_{0},T],\mathbb{R}^{n})\) if \(m(t)\) is differentiable (i.e., \(m'(t)\) exists), the Caputo derivative \({}_{t_{0}}^{c}D^{q}m(t)\) exists and satisfies (1) for \(t\in[t_{0},T]\).
Remark 1
If \(m\in C^{q}([t_{0},T],\mathbb{R}^{n})\), then \({}_{t_{0}}^{c}D_{+}^{q}m(t) ={}_{t_{0}}^{c}D^{q}m(t)\).
3 Statement of the problem
We will assume in the paper that the function \(f\in C[\mathbb{R}_{+}\times\mathbb{R}^{n},\mathbb{R}^{n}]\) is such that for any initial data \((t_{0},x_{0})\in\mathbb{ R}_{+}\times \mathbb{R}^{n}\), FrDE (4) has a solution \(x(t;t_{0},x_{0})\in C^{q}([t_{0},\infty),\mathbb{R}^{n})\). Note that some sufficient conditions for global existence of solutions of (4) are given in [17, 23, 24].
The goal of the paper is to study strict stability of FrDE (4). Strict stability for fractional equations is studied in [25], but the definitions and conditions are not clear. Now we will define strict stability for fractional equations following the idea for ordinary differential equations (see, for example, [3]).
Definition 2

strictly stable if for given \(\epsilon_{1}>0\) and \(t_{0}\in \mathbb{R}_{+}\) there exists \(\delta_{1}=\delta_{1}(t_{0},\epsilon_{1})>0\) such that for any initial point \(x_{0}\in \mathbb{R}^{n}\) the inequality \(\x_{0}\<\delta_{1}\) implies \(\x(t;t_{0},x_{0})\<\epsilon_{1}\), \(t\geq t_{0}\), and for any \(\delta_{2}=\delta_{2}(t_{0},\epsilon_{1})\), \(\delta_{2}\in(0,\delta_{1}]\) there exists \(\epsilon_{2}=\epsilon_{2}(t_{0},\delta_{2})\), \(\epsilon_{2} \in (0,\delta_{2}]\) such that the inequality \(\delta_{2}<\x_{0}\\) implies \(\epsilon_{2}<\x(t;t_{0},x_{0})\\) for \(t\geq t_{0}\) where \(x(t;t_{0},x_{0})\) is a solution of the IVP for FrDE (4);

uniformly strictly stable if for any given \(\epsilon _{1}>0\) there exists \(\delta_{1}=\delta_{1}(\epsilon_{1})>0\) such that for any initial time \(t_{0}\in \mathbb{R}_{+}\) and any initial point \(x_{0}\in \mathbb{R}^{n}\) the inequality \(\x_{0}\<\delta_{1}\) implies \(\x(t;t_{0},x_{0})\<\epsilon_{1}\), \(t\geq t_{0}\), and for any \(\delta_{2}\in(0,\delta_{1}]\) there exists \(\epsilon_{2} \in (0,\delta_{2}]\), \(\varepsilon _{2}=\varepsilon _{2}(\delta_{2})\), such that the inequality \(\delta_{2}<\x_{0}\\) implies \(\epsilon_{2}<\x(t;t_{0},x_{0})\\) for \(t\geq t_{0}\) where \(x(t;t_{0},x_{0})\) is a solution of the IVP for FrDE (4).
Remark 2
The strict stability immediately implies that the zero solution is not asymptotically stable.
Example 2
(Strict stability)
Consider the ODE \(x'=\frac{1}{(t+1)^{2}}x\), \(x(t_{0})=x_{0}\) with a solution \(x(t)=x_{0}e^{\frac{1}{t_{0}+1}}e^{\frac{1}{t+1}}\). Since \(1< e^{\frac{1}{t+1}}\leq e\) for \(t\in \mathbb{R}_{+}\), it follows that for any \(\varepsilon _{1}\) if \(x_{0}<\frac{\varepsilon _{1}}{e}=\delta_{1}\) then \(x(t)<\varepsilon _{1}\), and for any \(\delta_{2}\in(0,\delta_{1})\) the inequality \(x_{0}>\delta_{2}\) implies \(x(t)>\varepsilon _{2}=\delta_{2}e^{\frac{1}{t_{0}+1}}\). Therefore, the zero solution of the considered ODE is strictly stable.
 (H1)
For any compact interval \([t_{0},T]\subset \mathbb{R}_{+}\), there exists a small enough number \(L_{t_{0}}>0\) such that for any \(\eta< L_{t_{0}}\) the IVP for FrDE \({}_{t_{0}}^{c}D^{q}u=g_{i} ( t,u )+\eta\), \(u(t_{0})=u_{0}\) has a solution \(u(t;t_{0},u_{0},\eta)\in C^{q}([t_{0},T],\mathbb{R})\) where \(u_{0}\in \mathbb{R}\).
We now introduce the strict stability of the couple of Caputo fractional differential equations as follows.
Definition 3

strictly stable in couple if for given \(\epsilon_{1}>0\) and \(t_{0}\in \mathbb{R}_{+}\) there exists \(\delta_{1}=\delta_{1}(t_{0},\epsilon_{1})>0\) and for any \(\delta_{2}=\delta_{2}(t_{0},\epsilon_{1})\), \(\delta_{2}\in(0,\delta_{1}]\) there exists \(\epsilon_{2}=\epsilon_{2}(t_{0},\delta_{2})\), \(\epsilon_{2} \in(0,\delta _{2}]\) such that the inequalities \(u_{0}<\delta_{1}\) and \(\delta_{2}<v_{0}\) imply \(u(t;t_{0},u_{0})<\epsilon_{1}\) and \(\epsilon_{2}<v(t;t_{0},v_{0})\) for \(t\geq t_{0}\), where the couple of functions \((u(t;t_{0},u_{0}), v(t;t_{0},u_{0}) )\) is a solution of the IVP for FrDE (6);

uniformly strictly stable in couple if for any given \(\epsilon_{1}>0\) there exists \(\delta_{1}=\delta_{1}(\epsilon_{1})>0\) and for any \(\delta_{2}\in (0,\delta_{1}]\) there exists \(\epsilon_{2} \in(0,\delta_{2}]\), \(\varepsilon _{2}=\varepsilon _{2}(\delta_{2})\), such that for any initial time \(t_{0}\in \mathbb{R}_{+}\) the inequalities \(u_{0}<\delta_{1}\) and \(\delta_{2}<v_{0}\) imply \(u(t;t_{0},u_{0})<\epsilon_{1}\) and \(\epsilon_{2}<v(t;t_{0},v_{0})\) for \(t\geq t_{0}\), where the couple of functions \((u(t;t_{0},u_{0}), v(t;t_{0},u_{0}) )\) is a solution of the IVP for FrDE (6).
Example 3
(Uniform strict stability in couple)
The solution of (7) is \(( u_{0}E_{q}(A(tt_{0})^{q}), v_{0}E_{q}(B(tt_{0})^{q}) )\), where the MittagLeffler function (with one parameter) is defined by \(E_{q}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(qk+1)}\).
Case 1. Let \(A<0\), \(B> 0\). From the inequalities \(0< E_{q}(z)\leq1\) for \(z\leq0\) and \(E_{q}(z)\geq1\) for \(z\geq0\) it follows that \(u_{0}E_{q}(A(tt_{0})^{q})\lequ_{0}\) and \(v_{0}E_{q}(B(tt_{0})^{q})\geqv_{0}\), i.e., the zero solution of the couple of FrDE (7) is uniformly strictly stable in couple.
Case 2. Let \(A, B= 0\). Then the solution of (7) is \(( u_{0}, v_{0} )\) which shows that the zero solution of the couple of FrDE (7) is uniformly strictly stable in couple.
Example 4
(Strict stability in couple)
We now introduce the class Λ of Lyapunovlike functions which will be used to investigate the strict stability of the system of FrDE (4).
Definition 4
Let \(t_{0}, T\in \mathbb{R}_{+}: T>t_{0}\), and \(\Delta\subset \mathbb{R}^{n}\), \(0\in\Delta\). We will say that the function \(V(t,x):[t_{0},T)\times \Delta\rightarrow\mathbb{R}_{+}\) belongs to the class \(\Lambda ([t_{0},T),\Delta)\) if \(V(t,x)\in C[[t_{0},T)\times \Delta, \mathbb{R}_{+}]\), and it is locally Lipschitzian with respect to its second argument.
Remark 3
In the case when the Lyapunov function does not depend on the time t, i.e., \(V(t,x)=V(x)\), \(V\in C[\Delta, \mathbb{R}_{+}]\) and it is locally Lipschitzian, we denote the class introduced in Definition 4 by \(\Lambda(\Delta)\).
Proposition 2
Proof
Example 5
Let \(V\in\Lambda(\mathbb{R}_{+},\mathbb{R})\) be such that \(V(t,x)=m^{2}(t)x^{2}\), where \(m\in C^{1}(\mathbb{R}_{+},\mathbb{R})\) and \(x\in \mathbb{R}\).
Note that the Caputo fractional Dini derivative \({}_{(\mbox{4})}^{c}D_{+}^{q}V(t,x;t_{0}, x_{0}) \) depends significantly not only on the order q of the fractional differential equation but also on the initial data.
Let \(m(t)\equiv1\) (i.e., we consider the quadratic Lyapunov function \(V(x)=x^{2}\)). Note that for \(q\to1\) the limit \({}_{(\mbox{4})}^{c}D_{+}^{q}V(t,x;t_{0}, x_{0})=2x f(t,x)\) in (17) coincides with the corresponding derivative \(D_{+}V(x)=2x f(t,x)\) in the ordinary case (18).
Let \(m(t)\not\equiv1\). For \(q\to1\), the limit \({}_{(\mbox{4})}^{c}D_{+}^{q}V(t,x;t_{0}, x_{0})=2x m^{2}(t)f(t,x)+x^{2}{}_{t_{0}}^{C}D^{q} (m^{2}(t) )\) in (17) is similar to \(D_{+}V(t,x)\) in the ordinary case (18) where the ordinary derivative of \(m^{2}(t)\) is replaced by the fractional one.
The Caputo fractional Dini derivative given by formula (11) seems to be the natural generalization of the Dini derivative (9) for ordinary differential equations.
4 Fractional differential inequalities and comparison results for the scalar FrDE
Again in this section we assume \(0< q<1\). Now we will give some comparison results. Note that similar results were obtained by the authors in paper [19].
Lemma 1
([19])
Let \(m\in C[[ t_{0},T],\mathbb{R}] \) and suppose that there exists \(t^{*}\in(t_{0},T]\) such that \(m(t^{*})=0\) and \(m(t)<0\) for \(t_{0}\leq t< t^{*}\). Then, if the Caputo fractional Dini derivative (3) of m exists at \(t^{*}\), then the inequality \({}_{t_{0}}^{c}D_{+}^{q}m(t^{*})>0\) holds.
We will use the following comparison result, which generalizes the result in [19].
Lemma 2
(Comparison result by Caputo fractional Dini derivative)
 (1)
The function \(x^{*}(t)=x(t;t_{0},x_{0})\), \(x^{*}\in C^{q}([t_{0},T],\Delta)\), is a solution of FrDE (4), where \(\Delta\subset \mathbb{R}^{n}\), \(0\in\Delta \), \(t_{0}, T\in \mathbb{R}_{+}: t_{0}< T\) are given constants, \(x_{0}\in\Delta\).
 (2)
The function \(g \in C[[t_{0},T]\times \mathbb{R},\mathbb{R}]\) and satisfies condition (H1).
 (3)The function \(V\in\Lambda([t_{0},T],\Delta)\) and, for any \(t \in[t_{0},T]\), the inequalityholds.$${}_{(4)}^{c}D_{+}^{q}V \bigl(t,x^{*}(t);t_{0},x_{0}\bigr)\leq(\geq) g\bigl(t,V \bigl(t,x^{*}(t)\bigr)\bigr) $$
 (4)
The function \(u^{*}(t)=u(t;t_{0},u_{0})\), \(u^{*}\in C^{q}([t_{0},T],\mathbb{R})\), is the maximal solution (minimal solution) of the initial value problem (5).
Then the inequality \(V(t_{0},x_{0})\leq(\geq) u_{0}\) implies \(V(t,x^{*}(t))\leq(\geq) u^{*}(t)\) for \(t\in[t_{0},T]\).
Proof
Case 1. Suppose that all inequalities are ≤. This case is proved in [19].
Case 2. Suppose that all inequalities are ≥. Let the function \(u^{*}(t)=u(t;t_{0},u_{0})\), \(u^{*}\in C^{q}([t_{0},T],\mathbb{R})\), be the minimal solution of the initial value problem (5) such that \(V(t_{0},x_{0})\geq u_{0}\).
If \(g(t,x)\equiv0\) in Lemma 2, we obtain the following result.
Corollary 1
 (1)
The function \(x^{*}(t)=x(t;t_{0},x_{0})\), \(x^{*}\in C^{q}([t_{0},T],\Delta)\), is a solution of FrDE (4) where \(\Delta\subset \mathbb{R}^{n}\), \(0\in\Delta\).
 (2)The function \(V\in\Lambda([t_{0},T],\Delta)\) and for \(t \in[t_{0},T]\) the inequalityholds.$${}_{(4)}^{c}D_{+}^{q}V \bigl(t,x^{*}(t);t_{0},x_{0}\bigr)\leq(\geq)0 $$
Then, for \(t\in[t_{0},T]\), the inequality \(V(t,x^{*}(t))\leq(\geq )V(t_{0},x_{0}) \) holds.
Note that a similar result to Corollary 1 for Dini fractional derivatives is proved in [18] (Corollary 2.2 considers the case when all inequalities are ≤). We note that (10) could lead to some problems.
Example 6
Let \(V:\mathbb{R}_{+}\times \mathbb{R}\to \mathbb{R}_{+}\) be given by \(V(t,x)=\sin^{2}t x^{2}\) and \(t_{0}=0\). It is locally Lipschitz with respect to its second argument x.
Let \(f(t,x)\equiv0\). The solution of (4) for \(n=1\) and \(t_{0}=0\) is \(x(t)\equiv x_{0}\), \(t\geq0\) and \(V(t,x(t))=x_{0}^{2} \sin^{2}t\). Note that \({}^{c}{ D}_{+}^{q}V(t,x)=0\) and all the conditions of Corollary 2.2 [18] are satisfied, so the inequality \(V(t,x(t))\leq V(t_{0},x_{0})\), \(t\geq t_{0}\), has to hold. However, the inequality \(x_{0}^{2} \sin^{2}t\leq x_{0}^{2} \sin^{2}0=0\) is not satisfied for all \(t\geq t_{0}\).
Now let \(V:\mathbb{R}_{+}\times \mathbb{R}\to \mathbb{R}_{+}\) be given by \(V(t,x)= x^{2}\). According to (16) for \(m(t)\equiv1\), we get \({}_{(\mbox{4})}^{c}D_{+}^{q}V(t,x;0,x_{0}) = \frac{x^{2}x_{0}^{2}}{t^{q}\Gamma(1q)} + 2xf(t,x)\). Let \(f(t,x)=\frac{x}{t^{q} \Gamma(1q)}\). Then \({}_{(\mbox{4})}^{c}D_{+}^{q}V(t,x;0,x_{0}) \leq0\), and according to Corollary 1 the inequality \(x(t;0,x_{0})\leqx_{0}\), \(t\geq0\), holds for any solution \(x(t;0,x_{0})\) of (4).
The result of Lemma 2 is also true on the half line (see [19] for the case when all inequalities are ≤).
Corollary 2
 (1)
The function \(x^{*}(t)=x(t;t_{0},x_{0})\), \(x^{*}\in C^{q}([t_{0},\infty ),\Delta)\), is a solution of FrDE (4) where \(\Delta\subset \mathbb{R}^{n}\), \(0\in\Delta\).
 (2)
The function \(g \in C[[t_{0},\infty)\times \mathbb{R},\mathbb{R}]\) and satisfies condition (H1).
 (3)The function \(V\in\Lambda([t_{0},\infty),\Delta)\), and for any points \(t \geq t_{0}\) and \(x\in\Delta\) the inequalityholds.$${}_{(4)}^{c}D_{+}^{q}V \bigl(t,x^{*}(t);t_{0},x_{0}\bigr)\leq(\geq)g\bigl(t,V \bigl(t,x^{*}(t)\bigr)\bigr) $$
 (4)
The function \(u^{*}(t)=u(t;t_{0},u_{0})\), \(u^{*}\in C^{q}([t_{0},\infty),\mathbb{R})\) is the maximal solution (minimal solution) of the initial value problem (5).
Then the inequality \(V(t_{0},x_{0})\leq(\geq) u_{0}\) implies \(V(t,x^{*}(t))\leq(\geq) u^{*}(t)\) for \(t\geq t_{0}\).
5 Main results
We obtain sufficient conditions for strict stability of the system FrDE (4). Again we assume \(0< q<1\).
Theorem 1
 (1)
The functions \(g_{i} \in C[\mathbb{R}_{+}\times \mathbb{R}, \mathbb{R}]\), \(g_{i}(t,0)\equiv0\), \(i=1,2\), and satisfy condition (H1).
 (2)There exists a function \(V_{1}\in\Lambda(\mathbb{R}_{+}, \mathbb{R}^{n}) \) such that \(V_{1}(t,0)\equiv0\) for \(t\in \mathbb{R}_{+}\) and
 (i)the inequalityholds for any \(t_{0},t\in \mathbb{R}_{+}\), \(t\geq t_{0}\) and \(x,x_{0}\in \mathbb{R}^{n}\);$${}_{(4)}^{c}D_{+}^{q}V_{1}(t,x;t_{0},x_{0}) \leq g_{1}\bigl(t,V_{1}(t,x)\bigr) $$
 (ii)
\(a(\x\)\leq V_{1}(t,x)\) for \(t\in \mathbb{R}_{+}\), \(x\in \mathbb{R}^{n}\), where \(a \in\mathcal{K}\).
 (i)
 (3)There exists a function \(V_{2}\in\Lambda(\mathbb{R}_{+}, \mathbb{R}^{n}) \) such that
 (iii)the inequalityholds for any \(t_{0},t\in \mathbb{R}_{+}\), \(t\geq t_{0}\) and \(x,x_{0}\in \mathbb{R}^{n}\);$${}_{(4)}^{c}D_{+}^{q}V_{2}(t,x;t_{0}, x_{0}) \geq g_{2}\bigl(t,V_{2}(t,x)\bigr) $$
 (iv)
\(c(\x\)\leq V_{2}(t,x)\leq b(\x\)\) for \(t\in \mathbb{R}_{+}\), \(x\in \mathbb{R}^{n}\), where \(b,c\in\mathcal{K}\).
 (iii)
 (4)
The zero solution of FrDE (6) is strictly stable in couple.
Proof
Since \(V_{1}(t_{0},0)=0\), there exists \(\delta_{3}=\delta_{3}(t_{0},\varepsilon )\), \(\delta_{3}\in(0,\delta_{1})\) such that \(V_{1}(t_{0},x)<\delta_{1}\) for \(\x\<\delta_{3}\). Let \(\delta_{4}\in(0,\delta_{3}]\) be an arbitrary number. Then there exists \(\delta_{5} \in(0,\delta_{1}]\) such that \(c(\delta_{4})>\delta_{5}\). According to the above, for \(\delta_{5}\) there exists \(\varepsilon _{2}\in(0,\delta_{5}]\) such that \(v_{0}>\delta_{5}\) implies (33). Choose \(\varepsilon _{3}\in(0,\delta_{4}]\) such that \(b(\varepsilon _{3})<\varepsilon _{2}\). Choose \(x_{0} \in \mathbb{R}^{n}\) with \(\delta_{4}<\x_{0}\<\delta_{3}\). Let \(x^{*}(t)=x(t;t_{0},x_{0})\) be a solution of the IVP for FrDE (4) for the initial data \((t_{0},x_{0})\).
Let \(u_{0}=V_{1}(t_{0},x_{0})\) and \(v_{0}=V_{2}(t_{0},x_{0})\). Let the couple \((u(t;t_{0},u_{0}),v(t;t_{0},v_{0}) )\) be the solution of FrDE (6) for the initial values \((u_{0},v_{0})\) such that the components are maximal and minimal solutions of the first and second equation, respectively. From the choice of \(x_{0}\) it follows that \(u_{0}<\delta_{1}\). Therefore, the component \(u(t;t_{0},u_{0})\) satisfies inequality (32). From the choice of \(x_{0}\) and condition (3)(iv) it follows that \(v_{0}=V_{2}(t_{0},x_{0})\geq c(\x_{0}\)>c(\delta_{4})>\delta_{5}\). Therefore, the component \(v(t;t_{0},v_{0})\) satisfies inequality (33).
According to Corollary 2 applied to the solution \(x^{*}(t)\) and condition (2)(ii), we obtain \(a(\x^{*}(t)\)\leq V(t, x^{*}(t))\leq u(t;t_{0},u_{0})< a(\varepsilon _{1})\), \(t\geq t_{0}\). Therefore \(\x^{*}(t)\<\varepsilon _{1}\) for \(t\geq t_{0}\).
According to Corollary 2 applied to the solution \(x^{*}(t)\) and condition (3)(iv), we obtain \(b(\x^{*}(t)\)\geq V(t, x^{*}(t))\geq v(t;t_{0},v_{0})>\varepsilon _{2}>b(\varepsilon _{3})\), \(t\geq t_{0}\). Therefore \(\x^{*}(t)\>\varepsilon _{3}\) for \(t\geq t_{0}\). □
Theorem 2
 (1)
The functions \(g_{i} \in C[\mathbb{R}_{+}\times \mathbb{R}, \mathbb{R}]\), \(g_{i}(t,0)\equiv0\), \(i=1,2\), and satisfy condition (H1).
 (2)There exists a function \(V_{1}\in\Lambda(\mathbb{R}_{+}, B(A)) \) such that
 (i)the inequalityholds for any \(t_{0},t\in \mathbb{R}_{+}\), \(t\geq t_{0}\) and \(x,x_{0}\in B(A)\), where \(A>0\) is a given number;$$ {}_{(4)}^{c}D_{+}^{q}V_{1}(t,x;t_{0},x_{0}) \leq g_{1}\bigl(t,V_{1}(t,x)\bigr) $$
 (ii)
\(a(\x\)\leq V_{1}(t,x)\leq b(\x\)\) for \(t\in \mathbb{R}_{+}\), \(x\in B(A)\), where \(a,b \in\mathcal{K}\).
 (i)
 (3)There exists a function \(V_{2}\in\Lambda(\mathbb{R}_{+}, B(A)) \) such that
 (iii)the inequalityholds for any \(t_{0},t\in \mathbb{R}_{+}\), \(t\geq t_{0}\) and \(x,x_{0}\in B(A)\);$$ {}_{(4)}^{c}D_{+}^{q}V_{2}(t,x;t_{0}, x_{0}) \geq g_{2}\bigl(t,V_{2}(t,x)\bigr) $$
 (iv)
\(c(\x\)\leq V_{2}(t,x)\leq d(\x\)\) for \(t\in \mathbb{R}_{+}\), \(x\in B(A)\), where \(c,d\in\mathcal{K}\).
 (iii)
 (4)
The zero solution of the couple of FrDE (6) is uniformly strictly stable in couple.
Proof
Choose \(\delta_{3}\in(0, A)\) such that \(b(\delta_{3})<\delta_{1}\). Choose \(x_{0} \in \mathbb{R}^{n}\) with \(\x_{0}\<\delta_{3}\), and let \(x^{*}(t)=x(t;t_{0},x_{0})\) be the solution of the IVP for FrDE (4) for the initial data \((t_{0},x_{0})\).
Let \(u_{0}=V_{1}(t_{0},x_{0})\). Let \(u(t;t_{0},u_{0})\) be the maximal solution of the first equation of the couple of FrDE (6). According to condition (2)(ii) and the choice of \(x_{0}\), we obtain \(u_{0}=V_{1}(t_{0}, x_{0})\leq b(\x_{0}\)< b(\delta_{3})<\delta_{1}\). Therefore the first component \(u(t;t_{0},u_{0})\) of the solution of (6) satisfied (34).
Let \(\delta_{4}\in(0,\delta_{3}]\) be an arbitrary number. Then there exists \(\delta_{5} \in(0,\delta_{4}]\) such that \(c(\delta_{4})>\delta_{5}\). According to the above, for \(\delta_{5}\) there exists \(\varepsilon _{2}^{*}\in (0,\delta_{5}]\) such that \(v_{0}>\delta_{5}\) implies inequality (35), \(\varepsilon _{2}=\varepsilon _{2}^{*}\), for the second component \(v(t;t_{0},v_{0})\) of the solution of (6). Let the initial value \(x_{0} \in \mathbb{R}^{n}\) additionally satisfy \(\x_{0}\>\delta_{4}\). From the above \(x^{*}(t)\in B(A)\) for \(t\geq t_{0}\). Choose \(\varepsilon _{3}\in(0,\delta_{4}]\) such that \(d(\varepsilon _{3})\leq \varepsilon _{2}^{*}\).
Let \(v_{0}=V_{2}(t_{0},x_{0})\) and let \(v(t;t_{0},v_{0})\) be the minimal solution of the second equation of the couple of FrDE (6). According to condition (3)(iv), it follows that \(v_{0}=V_{2}(t_{0},x_{0})\geq c(\x_{0}\)>c(\delta_{4})>\delta_{5}\). Therefore, the component \(v(t;t_{0},v_{0})\) of the solution of FrDE (6) satisfies inequality (35) for \(\varepsilon _{2}=\varepsilon _{2}^{*}\).
Corollary 3
Let the conditions (2) and (3) of Theorem 2 be satisfied with \(g_{i}(t,x)\equiv0\), \(i=1,2\).
Then the zero solution of the system of FrDE (4) is uniformly strictly stable.
Sufficient conditions for strict stability could be obtained in the case of one Lyapunov function.
Theorem 3
 (1)
The function \(g_{i} \in C[\mathbb{R}_{+}\times \mathbb{R}, \mathbb{R}]\), \(g_{i}(t,0)\equiv0\), \(i=1,2\), \(g_{1}(t,u)\geq g_{2}(t,u)\) for \(t\in \mathbb{R}_{+}\), \(u\in \mathbb{R}\) and satisfies condition (H1).
 (2)There exists a function \(V\in\Lambda(\mathbb{R}_{+}, B(A)) \) such that
 (i)the inequalitieshold for any \(t_{0},t\in \mathbb{R}_{+}\), \(t\geq t_{0}\) and \(x,x_{0}\in \mathbb{R}^{n}\);$$g_{2}\bigl(t,V(t,x)\bigr)\leq {}_{(4)}^{c}D_{+}^{q}V(t,x;t_{0},x_{0}) \leq g_{1}\bigl(t,V(t,x)\bigr) $$
 (ii)
\(a(\x\)\leq V(t,x)\leq b(\x\)\) for \(t\in \mathbb{R}_{+}\), \(x\in B(A)\), where \(a,b \in\mathcal{K}\), \(A>0\) is a given number.
 (i)
 (3)
The zero solution of the couple of FrDE (6) is strictly stable (uniformly strictly stable) in couple.
The result of Theorem 3 is a partial case of Theorem 1 and Theorem 2.
Declarations
Acknowledgements
Research was partially supported by the Fund NPD, Plovdiv University, No. MU15FMIIT008.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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