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Exponential stability for a class of set dynamic equations on time scales
Journal of Inequalities and Applications volume 2022, Article number: 135 (2022)
Abstract
We first present a new definition for some form of exponential stability of solutions, including Hexponential stability, Hexponentially asymptotic stability, Huniformly exponential stability, and Huniformly exponentially asymptotic stability for a class of set dynamic equations on time scales. Employing Lyapunovlike functions on time scales, we provide the sufficient conditions for the exponential stability of the trivial solution for such set dynamic equations.
1 Introduction
In this paper, we denote the ndimensional real number set, natural number set, and (positive) integer number set by \(\mathbb{R}^{n}\), \(\mathbb{N}\) and (\(\mathbb{Z}_{+}\)) \(\mathbb{Z}\), respectively, and stipulate \(\mathbb{R}=\mathbb{R}^{1}\), \(\mathbb{R}_{+}=\{r\in \mathbb{R}: r\ge 0 \}\). A time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of \(\mathbb{R}\). The set \(K_{c}(\mathbb{R}^{n}) \) consists of all nonempty compact convex subsets of \(\mathbb{R}^{n} \) and \(K_{c}^{m}(\mathbb{R}^{n})= K_{c}(\mathbb{R}^{n}) \times K_{c}( \mathbb{R}^{n})\times \cdots \times K_{c}(\mathbb{R}^{n})\) (mtimes). The purpose of this paper is to discuss the exponential stability of the set dynamic equations (SDE) on time scales
where \(X=(X_{1},X_{2},\ldots,X_{m})^{T}\) with \(m\in \mathbb{Z}_{+}\) and \(X_{i}: \mathbb{T}\to K_{c}(\mathbb{R}^{n})\) for \(1\le i\le m\), \(t_{0}\in \mathbb{T}\) is given, \(X^{\Delta }(t)\) is the Δderivative of X at the moment \(t \in \mathbb{T}\) and F is a setvalued function from \(\mathbb{T}\times K_{c}^{m}(\mathbb{R}^{n})\) into \(K_{c}^{m}(\mathbb{R}^{n})\).
The study of set differential and difference equations has been initiated as an independent subject and some results of interest can be found in [1–13]. More attention has been paid to the stability criteria for such equation’s solutions in recent years. For instance, the comparison results and the stability considerations for hybrid dynamic systems were discussed in [14]. Since then, much progress has been made in studying various fundamental aspects of the stability of set differential or difference equations (see [3–6, 8, 15–20]). For instance, certain Lyapunovlike functions were used to study their stability criteria by Lakshmikantham in [15], Bhaskar and Devi [5] studied the Lyapunov stability for the solutions of set differential equations, using Lyapunovlike functions that are continuous. Moreover, in [5] the authors employed an important comparison result in the light of Lyapunov functions to characterize various stability behaviors of the solutions of the initialvalue problem for a class of set differential equations, such as, the equistability, equiasymptotic stability, uniform, and uniformly asymptotic stability. In [18] the authors obtained the necessary and sufficient conditions of the globally asymptotic stability for a class of nonlinear neutral setvalued functional differential equations via the fixedpoint method.
On the other hand, according to the references [21, 22], the theory of time scales was introduced by Stefan Hilger in his PhD thesis as a mean of unifying structure for the study of differential equations in the continuous case and the study of finitedifference equations in the discrete case. The stability theory of dynamic systems on time scales recently received much attention and is undergoing rapid development (see [23–28]). We mention that in [28] Martynyuk et al. researched the stability of a family of dynamic equations on time scales and proposed efficient sufficient conditions for several stability types of the sets of trajectories on time scales by means of scalar and vector Lyapunovlike functions constructed on the basis of matrixvalued functions. Very recently, since the derivative of setvalued functions on the time scale has been established (see Hong [29]), the qualitative problems of set differential equations have received extensive attention (see [11, 28, 30–34]). However, we observe that there are very few results for the stability to set differential equations on time scales. For example, Ahmad and Sivasundaram [35] discussed some basic problems of set differential equations on time scales and obtained some stability criteria. Wang and Sun [34] obtained a comparison principle by introducing a notion of upper quasimonotone nondecreasing provided the practical stability criteria for set differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the comparison principle. In [36], notions of stability for the solutions of set dynamic equations on time scales are considered by using Lyapunovlike functions. Moreover, criteria for the equistability, equiasymptotic stability, uniform, and uniformly asymptotic stability are developed. In [33], the authors considered the exponential stability, exponentially asymptotic stability, uniformly exponential stability, and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using Lyapunovlike functions.
In this paper, inspired by the abovementioned literature, we also consider the exponential stability for the solutions of set dynamic equations on time scales. More precisely, applying the method of matrixvalued functions in the theory of stability of classical dynamic equations on time scales described in [28], we similarly define appropriate matrixvalued Lyapunovlike functions and then formulate certain inequalities on these functions. Moreover, we employ these results to provide a generalized stability called the Hexponential stability in the paper, as well as Hexponentially asymptotic stability, Huniformly exponential stability, and Huniformly exponentially asymptotic stability of trivial solutions to a class of set dynamic equations on time scales. In addition, we present some sufficient conditions for the exponential stability for the trivial solution to SDE (1).
2 Preliminaries
In this section, some hypotheses and background materials are given that are necessary in this paper. We first recall the notion of the time scale built by Hilger and Bohner. For more details, we refer the reader to [21, 22].
Definition 2.1
For any \(t \in \mathbb{T}\), the forward jump operator is defined by
while the backward jump operator is given by
The distance from an arbitrary element \(t \in \mathbb{T}\) to the nearest element on the right is called the graininess of the time scale \(\mathbb{T}\) and denoted by \(\mu (t)\), i.e.,
In Definition 2.1, it is assumed that \(\inf{\emptyset}=\sup \mathbb{T} \) (i.e., \(\sigma (t)=t\) if \(\mathbb{T}\) contains the largest element t) and \(\sup{\emptyset}=\inf \mathbb{T} \) (i.e., \(\rho (t)=t\) if \(\mathbb{T}\) contains the smallest element t).
Definition 2.2
Using the operators \(\sigma :\mathbb{T}\rightarrow \mathbb{T}\) and \(\rho :\mathbb{T}\rightarrow \mathbb{T}\), the points t on time scale \(\mathbb{T}\) are classified as follows: if \(\sigma (t)= t \), then t is said to be rightdense, while the point t is said to be rightscattered if \(\sigma (t)> t \). Similarly, the point t is called leftdense if \(\rho (t)= t\), while t is called leftscattered if \(\rho (t)< t \).
Unless otherwise stated, we stipulate that \(\mathbb{T}\) stands for \(\mathbb{T} \backslash \{\hat{t}\}\) if \(\mathbb{T}\) contains the leftscattered point maximum t̂.
Definition 2.3
([22])
A function f is right(left)densecontinuous (rd(ld)continuous, for short) if f is continuous at each right(left)dense point in \(\mathbb{T}\) and its left(right)sided limits exist at each left(right)dense point in \(\mathbb{T}\). By \(C_{\mathrm{rd}}(\mathbb{T}, \mathbb{R})\) and \(C_{\mathrm{ld}}(\mathbb{T}, \mathbb{R})\) we denote the set of all right and leftdense continuous functions from \(\mathbb{T}\) to \(\mathbb{R}\), respectively.
We say that a function \(p :\mathbb{T} \rightarrow \mathbb{R} \) (a \(n\times n\)matrixvalued function \(A:\mathbb{T}\to \mathbb{R}^{n\times n}\)) is regressive provided
for all \(t \in \mathbb{T}\). The set of all regressive rdcontinuous functions \(p :\mathbb{T} \rightarrow \mathbb{R}(\mathbb{R}_{+}) \) (a regressive rdcontinuous matrixvalued function) is denoted by
Let \(p ,q (A, B) \in \Re \). For all \(t\in \mathbb{T}\), we define
and
Definition 2.4
([17], Definition 1.10)
Assume that \(g:\mathbb{T}\to \mathbb{R}\) is a function and let \(t\in \mathbb{T}\). Then, we define \(g^{\Delta}(t)\) to be the number (provided it exists) with the property that given any \(\delta >0\), there is a neighborhood U of t (i.e., \(U=(t\delta , t+\delta )\cap \mathbb{T}\) for some \(\delta >0\)) such that
for all \(s\in U\). We call \(g^{\Delta}(t)\) the Δderivative of g at t. Moreover, we say that g is Δdifferentiable on \(\mathbb{T}\) provided \(g^{\Delta}(t)\) exists for all \(t\in \mathbb{T}\).
If a single valued function g is Δdifferentiable and its Δderivative \(g^{\Delta}(t)\) at \(t\in \mathbb{T}\) equals \(f(t)\), then we define the Cauchy integral by
In this case, we say f is Δintegrable on interval \([a, t]\cap{\mathbb{T}}\).
Definition 2.5
([22], Definition 2.30)
If \(p \in \Re \), then we define the exponential function by
where \(\xi _{h}(z)\) with \(h > 0 \) is the cylinder transformation from the set \(\{z\in \mathbb{C} :z \neq \frac{1}{h} \}\) (\(\mathbb{C}\) stands for the complex number set) into the strip \(\{z\in \mathbb{C} :\frac{\pi}{h}<\operatorname{Im}(z)\leq \frac{\pi}{h} \}\) defined by
Lemma 2.1
([22],Theorem 2.36)
Let \(p ,q \in \Re \). Then,

(i)
\(e_{p}(t,t)\equiv 1\), \(e_{0}(t,s)\equiv 1\), \(e_{p}(t,s)=1/e_{p}(s,t)\), and \(e_{p}(t,s)=e_{p}(t,u)e_{p}(u,s)\);

(ii)
\(e_{p}(\sigma (t),s)=(1+\mu (t)p(t))e_{p}(t,s)\), \(e_{p}(s,\sigma (t))=\frac{e_{p}(s,t)}{1+\mu (t)p(t)}\);

(iii)
\(e_{p}^{\triangle}(\cdot ,s)=pe_{p}(\cdot ,s)\), \(e_{p}^{\triangle}(s,\cdot )=(\ominus p)e_{p}(s,\cdot )\);

(iv)
\(e_{p \oplus q}=e_{p}e_{q}\) and \(e_{p \ominus q}=e_{p}/e_{q}\);

(v)
if \(p \in \Re _{+} \), then \(e_{p}(t,s) > 0 \) for all \(t, s \in \mathbb{T} \);

(vi)
\(e_{\alpha}(t,s)\geq 1+\alpha (ts)\) for \(\alpha \in \Re _{+} \), \(\alpha \in \mathbb{R}_{+}\) and any \(t, s \in \mathbb{T}\) with \(t \geq s\).
We continue with a description of the basic known results for Hausdorff metrics, continuity, and differentiability for setvalued mappings on time scales and their corresponding properties within the framework of time scales. We refer readers to [7, 29] for details. The following operations can be naturally defined on \(K_{c}(\mathbb{R}^{n})\): for \(X,Y\in K_{c}(\mathbb{R}^{n})\),
In addition, the set \(Z \in K_{c}(\mathbb{R}^{n})\) satisfying \(X=Y+Z\) is known as the geometric difference of the sets X and Y and is denoted by the symbol \(XY\). It is worth noting that the geometric difference of two sets does not always exist, but if it does then it is unique.
We define the Hausdorff metric as
where \(d(x,Y)=\inf \{d(x,y) : y\in Y \}\) and X, Y are bounded subsets of \(\mathbb{R}^{n}\). Denote \(\X\=D[X, \Theta ]\).
A setvalued function \(F: \mathbb{T} \rightarrow K_{c}(\mathbb{R}^{n})\) is said to be continuous at \(t_{0}\in \mathbb{T}\) if \(D[F(t), F(t_{0})]\to 0\) whenever \(t\to t_{0}\).
Definition 2.6
([31], Definition 3.1)
Assume that \(F : \mathbb{T} \rightarrow K_{c}(\mathbb{R}^{n}) \) is a setvalued function and \(t \in \mathbb{T} \). Let \(F^{\Delta}(t) \) be an element of \(K_{c}(\mathbb{R}^{n}) \) with the property that for a given \(\varepsilon > 0 \), there exists a neighborhood \(U_{\mathbb{T}}\) of t (i.e., \(U_{\mathbb{T}}=(t\delta ,t+\delta )\cap \mathbb{T} \) for some \(\delta > 0 \)) such that
for all \(th,t+h\in U_{\mathbb{T}} \) with \(0 \leq h < \delta \). Then, F is called Δderivable or Δdifferentiable at t and \(F^{\Delta}(t)\) is called the Δderivative of F at t.
A function F is called Δdifferentiable on \(\mathbb{T}\) if its Δderivative exists at each \(t \in \mathbb{T} \).
Lemma 2.2
([31], Theorem 3.5)
Let \(F : \mathbb{T} \rightarrow K_{c}(\mathbb{R}^{n})\) and the following results hold:

(1)
If the Δderivative of F exists, then it is unique. Hence, the Δderivative is well defined.

(2)
If F is Δdifferentiable at \(t\in \mathbb{T}\), then F is continuous at t.

(3)
Let F be continuous at \(t\in \mathbb{T}\). Then, we have the following.

(i)
If t is rightscattered, then F is Δdifferentiable at t with
$$ F^{\Delta}(t)= \frac{ F(\sigma (t))F(t)}{\mu (t)}. $$ 
(ii)
If t is rightdense, then F is Δdifferentiable at t if and only if the limits
$$ \lim_{h \rightarrow 0^{+}}\frac{ F(t+h)F(t)}{ h } ,\qquad \lim_{h \rightarrow 0^{+}} \frac{ F(t)F(th)}{ h } $$exist and satisfy the equations
$$ \lim_{h \rightarrow 0^{+}}\frac{ F(t+h)F(t)}{ h } = \lim_{h \rightarrow 0^{+}} \frac{ F(t)F(th)}{ h } = F^{\Delta}(t). $$

(i)
Definition 2.7
Let \(\mathbb{I}\subset \mathbb{T}\) be a subset. The setvalued function \(F : \mathbb{I} \rightarrow K_{c}(\mathbb{R}^{n}) \) is called rdcontinuous on \(\mathbb{I}\), if it is continuous at all rightdense points of \(\mathbb{I}\) and lefthand limits exist and are finite numbers at all leftdense points of \(\mathbb{I}\). By \(C_{\mathrm{rd}}(\mathbb{I}, K_{c}(\mathbb{R}^{n}))\) we denote the set consisting of all rdcontinuous setvalued functions on \(\mathbb{I}\).
At the end of this section, for fixed \(m\in \mathbb{Z}_{+}\), we define
Moreover, \(K_{c}^{m}(\mathbb{R}^{n})\) is endowed with the distance as follows
for \(X,Y \in K_{c}^{m}(\mathbb{R}^{n})\). It is not difficult to check that \((K_{c}^{m}(\mathbb{R}^{n}), D_{0})\) is a metric space.
3 Main results
Let \(F \in C_{\mathrm{rd}}(\mathbb{T}\times K_{c}^{m}(\mathbb{R}^{n}),K_{c}^{m}( \mathbb{R}^{n}))\) and \(\mathscr{X}=\{X: X\in C_{\mathrm{rd}}(\mathbb{T}, K_{c}^{m}(\mathbb{R}^{n}))\text{ with } X(t)=X(t, t_{0}, X_{0})\text{ is a solution of SDE (1)}\}\) be nonempty. The generalized direct Lyapunov’s method for families of equations is discussed in a number of articles on the basis of scalar and vector auxiliary functions \(v(t,X)\in C(\mathbb{R}_{+}\times K_{c}(\mathbb{R}^{n}),\mathbb{R}_{+})\) (respectively, \(V(t,X)\in C(\mathbb{R}_{+}\times K_{c}(\mathbb{R}^{n}),\mathbb{R}_{+}^{m})\)), constructed on the basis of twoindexed systems of functions, as a suitable manner for constructing a Lyapunovlike matrix function. Next, together with SDE (1), we will consider the matrixvalued function \(U:\mathbb{T} \times K_{c}^{m}(\mathbb{R}^{n})\to \mathbb{R}^{m \times m}\) defined by
with
In the following, we suppose that \(\operatorname{det}U(t, \Theta _{0})=0\), where \(\operatorname{det}U(t, X)\) stands for the determinant of \(U(t, X)\) and \(\Theta _{0}\) stands for the zero element in \(K_{c}^{m}(\mathbb{R}^{n})\). Let \(\mathcal{A}(t,X)\) denote the set consisting of all solutions of the homogeneous linear system
It is clear that system \(U(t, \Theta _{0})a=0\) has nontrivial solutions. More precisely, \(\mathcal{A}(t, \Theta _{0})\) contains at least a nonvanishing vector for \(t\in \mathbb{T}\).
Based on the matrix function (2) we define a scalar function
Clearly, \(v\in C_{\mathrm{rd}}(\mathbb{T} \times K_{c}^{m}(\mathbb{R}^{n}) \times \mathbb{R}_{+}^{m},\mathbb{R}_{+})\) and \(v(t,\Theta _{0}, a)=0 \) for each \(t \in \mathbb{T} \) and each \(a\in \mathcal{A}(t, \Theta _{0})\).
For the function (3) we will consider Δderivatives with respect to \(t\in \mathbb{T}\), that is,
with
for any given \(A \in C_{\mathrm{rd}}(\mathbb{T}, K_{c}^{m}(\mathbb{R}^{n}))\).
The following definition is due to [36].
Definition 3.1
Let \(v \in C_{\mathrm{rd}}(\mathbb{T} \times K_{c}^{m}(\mathbb{R}^{n}) \times \mathbb{R}_{+}^{m},\mathbb{R}_{+})\). We call \(\Delta ^{r}v(t,A,a)\) and \(\Delta _{r}v(t,A,a)\) the right upper(ru) and the right lower(rl) derivatives of v with respect to t at \((t,A(t),a)\) for \(A \in C_{rd}(\mathbb{T},K_{c}^{m}(\mathbb{R}^{n}))\), \(a\in \mathbb{R}_{+}^{m}\), \(t \in \mathbb{T} \), respectively, if
In this case, v is said to be a matrixvalued Lyapunovlike function on \(\mathbb{T} \times K_{c}^{m}(\mathbb{R}^{n})\times \mathbb{R}_{+}^{m}\).
Theorem 3.1
Let \(v(t,X,a)=a^{T}U(t,X)a\) be a matrixvalued Lyapunovlike function. Then, its ru and rl derivatives exist. Moreover, for any fixed \(A \in C_{rd}(\mathbb{T}, K_{c}^{m}(\mathbb{R}^{n}))\), \(a\in \mathbb{R}_{+}^{m}\), the Δderivative of v with respect to \(t \in \mathbb{T} \) exists and
The proof of this theorem is similar to that of Theorem 3.1 of [33], and therefore we omit it.
In [28] the authors introduced two sets of functions Q and \(Q_{0}\) that characterize the current and initial states of the set of solutions of SDE (1), respectively, as follows:
They established some stability conditions under two different measures based on a class of matrixvalued Lyapunov functions for SDE (1), such as (\(H_{0}\), H)stable, (\(H_{0}\), H)uniformly stable, and (\(H_{0}\), H)asymptotically stable, etc.
In this section, we will develop the Hexponential stability, Hexponentially asymptotic stability, Huniformly exponential stability, and Huniformly exponentially asymptotic stability for the trivial solution of SDE (1). To this end, we assume that the initial value of SDE (1) \(t_{0}\in \mathbb{T} \) is positive and \(\mathbb{T}\) is not bounded above.
In what follows, we consider two sets \(\mathcal{M}\) and \(\mathcal{M}_{0}\) consisting of functions that characterize the current and initial states of the set \(\mathscr{X}\) of solutions to SDE (1), respectively, that is,
In addition, we need the following notations:
Definition 3.2
Let \(H, H_{0}\in \mathcal{M}\), \(X\in \mathscr{X}\) and a constant \(p\in (0,+\infty )\). The trivial solution of SDE (1) is said to be

(I)
Hexponentially stable on \(\mathbb{T}\) if there exist \(\alpha \in \Gamma \) and a function \(\varrho :\mathbb{R}_{+} \times \mathbb{T} \rightarrow \mathbb{R}_{+}\) such that
$$\begin{aligned} \alpha \bigl( \bigl(H\bigl(t,X(t)\bigr) \bigr)^{p} \bigr) \leq \varrho (h_{0},t_{0}) \bigl(e_{ \ominus M}(t,t_{0}) \bigr)^{d}, \quad t \in [t_{0},\infty )_{\mathbb{T}}, \end{aligned}$$(4)where \(M\in \Re _{+}\), \(d\in (0, +\infty )\) and \(h_{0}=H_{0}(t_{0},X_{0})\);

(II)
Huniformly exponentially stable if (I) holds with the function ϱ independent of \(t_{0}\);

(III)
Hexponentially asymptotically stable if (I) holds, as well as, for any \(\varepsilon > 0\), there exists a positive real number T such that
$$ \alpha \bigl(\bigl(H\bigl(t,X(t)\bigr)\bigr)^{p} \bigr) < \varepsilon\quad \text{for all } t\in [t_{0}+T, \infty )_{\mathbb{T}}; $$ 
(IV)
Huniformly exponentially asymptotically stable if there exists \(\alpha \in \Gamma \) such that (II) and (III) hold simultaneously.
Theorem 3.2
Assume that v is a matrixvalued Lyapunovlike function on \(\mathbb{T} \times K_{c}^{m}(\mathbb{R}^{n})\times \mathbb{R}_{+}^{m}\) and satisfies the following conditions: for \(H \in \mathcal{M}\), \(X\in \mathscr{X}\), a constant \(p>0\) and a vector \(a\in \mathbb{R}_{+}^{m}\),

(i)
there exist functions \(\lambda _{1}, \lambda _{2}\in \Lambda \) such that
$$ \lambda _{1} \bigl(\bigl(H\bigl(t,X(t)\bigr)\bigr)^{p} \bigr) \leq v\bigl(t,X(t),a\bigr) \leq \lambda _{2} \bigl(\bigl(H \bigl(t,X(t)\bigr)\bigr)^{p} \bigr)\quad {\textit{for all }} t\in \mathbb{T}; $$ 
(ii)
there exist a nondecreasing continuous function \(\lambda _{3} : \mathbb{R}_{+}\rightarrow \mathbb{R}\), functions \(\gamma \in \Gamma \), \(\delta \in \Re \), \(M\in \Re _{+}\) with \(\lambda _{2}(t)\le \gamma (t)\) for \(t\in \mathbb{T}_{+}\) and constants L, r with \(r>0\) such that
$$\begin{aligned}& v^{\Delta}\bigl(t,X(t),a\bigr)\leq \frac{\lambda _{3} ((H(t,X(t)))^{r} )L(M \ominus \delta )(t)e_{\ominus \delta}(t,0)}{1+M\mu (t)}, \end{aligned}$$(5)$$\begin{aligned}& Mv\bigl(t,X(t),a\bigr) \\& \quad \le \lambda _{3} \bigl( \bigl[\gamma ^{1}\bigl(v\bigl(t,X(t),a\bigr)\bigr) \bigr]^{r/p} \bigr)+L(M \ominus \delta ) (t)e_{\ominus \delta}(t,0)\quad \textit{for all } t\in \mathbb{T}. \end{aligned}$$(6)
Then, the trivial solution of SDE (1) is Hexponentially stable on \([t_{0}, \infty )_{\mathbb{T}}\).
Proof
Let \(\bar{\gamma}\in \Gamma \) satisfy \(\bar{\gamma}((H(t,X(t)))^{p})\le \lambda _{1}((H(t,X(t)))^{p})\) and \(\lambda _{2}((H(t,X(t)))^{p})\le \gamma ((H(t, X(t)))^{p})\). Combined with the condition (i), we have
for all \(t\in \mathbb{T}\). To fulfill our wish, it is sufficient to verify (4). The remainder of the proof is divided into three steps.
Step 1. We first verify that \(v(t,X(t),a)e_{M}(t,t_{0})\) is nonincreasing in \(t\in [t_{0}, +\infty )_{\mathbb{T}}\). Indeed, by means of (5), together with Lemma 2.1(ii) and Theorem 3.1, we have
From (7) it follows that \([\gamma ^{1}(v(t,X(t),a)) ]^{r/p} \leq (H(t,X(t)) )^{r}\). By virtue of the monotonicity of \(\lambda _{3}\), we have \(\lambda _{3} ( [\gamma ^{1}(v(t,X(t),a)) ]^{r/p} ) \geq \lambda _{3}( (H(t,X(t)) )^{r})\). Thus, combining (6) and (7), implies that
Consequently, \(v(t,X(t),a)e_{M}(t,t_{0})\) is nonincreasing in t.
Step 2. For the sake of convenience, let \(N>1\) be a given constant and \(u(t_{0}, X_{0})=Nv(t_{0},X_{0},a)\). We claim that
Suppose that the inequality (8) does not hold. Then, there exists \(t\in [t_{0},\infty )_{\mathbb{T}}\) such that
Set \(\bar{t}=\inf \{t\in [t_{0},\infty )_{\mathbb{T}}  v(t,X(t),a)> u(t_{0},X_{0})e_{\ominus M}(t,t_{0}) \}\). From Step 1 it follows that \(\bar{t}>t_{0}\) (otherwise, our claim is achieved). Without loss of generality, assume that
Next, let us choose \(\varphi \in \Gamma \) to satisfy \(s < \varphi (s)\le Ns\) for any \(s\ge 0\). Then, we have
and
Note that the set \(\{t\in [t_{0},\bar{t}]_{\mathbb{T}}  \varphi (v(t,X(t),a) )\le u(t_{0},X_{0})e_{\ominus M}(t,t_{0}) \}\) is nonempty since it includes at least the element \(t_{0}\), we can define
Thus, we deduce that
Step 1 guarantees that \(v(t,X(t),a)e_{M}(t,t_{0})\) is nonincreasing in \(t\in [\tilde{t}, \bar{t}]_{\mathbb{T}}\), which implies that
On the other hand, from (9) and (11) it follows that
This is a contradiction and hence (8) is true.
Step 3. Finally, according to the condition (i), we derive
Let \(\alpha =\bar{\gamma}\in \Gamma \), \(\varrho (h_{0},t_{0})=u(t_{0},X_{0})=Nv(t_{0},X_{0},a)\), as well as, \(h_{0}=H_{0}(t_{0},X_{0})=v(t_{0}, X_{0}, a)\) with \(H_{0}\in \mathcal{M}\). Consequently, inequality (13) guarantees that (4) is satisfied. Hence, the trivial solution of SDE (1) is Hexponentially stable on \([t_{0},\infty )_{\mathbb{T}}\). This proof is complete. □
Corollary 3.1
Let v be a matrixvalued Lyapunovlike function on \(\mathbb{T} \times K_{c}^{m}(\mathbb{R}^{n})\times \mathbb{R}_{+}^{m}\). For \(H \in \mathcal{M}\), \(X\in \mathscr{X}\), suppose that the following conditions are satisfied:

(i)
there exist positive functions \(\mu _{1}, \mu _{2}\in \Lambda \) and positive constants p, q such that
$$ \mu _{1}(t) \bigl(H\bigl(t,X(t)\bigr)\bigr)^{p} \leq v \bigl(t,X(t),a\bigr) \leq \mu _{2}(t) \bigl(H\bigl(t,X(t)\bigr) \bigr)^{q} \quad {\textit{for all }} t\in \mathbb{T}; $$ 
(ii)
there exist a positive function \(\lambda _{3}: \mathbb{T}_{+}\to \mathbb{R}\mathbbm{_{+}}\) with \(M=:\inf_{s \geq t_{0}}{\lambda _{3}(s)}/[{\mu _{2}(s)]^{r/q}}\in \Re _{+}\), function \(\delta \in \Re \), and constants L, r with \(r>0\) such that
$$\begin{aligned}& v^{\Delta}\bigl(t,X(t),a\bigr)\leq \frac{\lambda _{3}(t)(H(t,X(t)))^{r}L(M \ominus \delta )(t)e_{\ominus \delta}(t,0)}{1+M\mu (t)}, \\& M \bigl(v\bigl(t,X(t),a\bigr)v^{r/q}\bigl(t,X(t),a\bigr) \bigr)\le L(M \ominus \delta ) (t)e_{\ominus \delta}(t,0), \end{aligned}$$where q is given as in (i).
Then, the trivial solution of SDE (1) is Hexponentially stable on \([t_{0},\infty )_{\mathbb{T}}\).
Proof
In fact, let us take
Then, the condition (i) of Theorem 3.2 holds. It is easy to verify that the remaining conditions of Theorem 3.2 are satisfied. By virtue of Theorem 3.2, the trivial solution of SDE (1) is Hexponentially stable on \([t_{0},\infty )_{\mathbb{T}}\). The proof is completed. □
Corollary 3.2
Let \(Y\in C_{\mathrm{rd}}^{1}(\mathbb{T} \times K_{c}(\mathbb{R}), \mathbb{R}_{+})\) be a Lyapunovlike function satisfying the following conditions on \(\mathbb{T} \times K_{c}(\mathbb{R})\) for \(X\in \mathscr{X}\),

(i)
there exist positive functions \(\mu _{1},\mu _{2}\in \Lambda \) and positive constants p, q such that
$$ \mu _{1}(t) \bigl\Vert X(t) \bigr\Vert ^{p} \leq Y \bigl(t,X(t)\bigr) \leq \mu _{2}(t) \bigl\Vert X(t) \bigr\Vert ^{q}\quad { \textit{for all }} t\in \mathbb{T}; $$ 
(ii)
there exist a positive function \(\lambda _{3}\) on \(\mathbb{T}\) with \(M=:\inf_{s \geq t_{0}}{\lambda _{3}(s)}/[{\mu _{2}(s)]^{r/q}}\in \Re _{+}\), function \(\delta \in \Re \), and constants L, r with \(r>0\) such that
$$\begin{aligned}& Y^{\Delta}\bigl(t,X(t)\bigr)\leq \frac{\lambda _{3}(t) \Vert X(t) \Vert ^{r}L(M \ominus \delta )(t)e_{\ominus \delta}(t,0)}{1+M\mu (t)}, \\& M \bigl(Y\bigl(t,X(t)\bigr)Y^{r/q}\bigl(t,X(t)\bigr) \bigr)\le L(M \ominus \delta ) (t)e_{ \ominus \delta}(t,0), \end{aligned}$$where q is given as in (i).
Then, the trivial solution of the following SDE
is Hexponentially stable on \([t_{0},\infty )_{\mathbb{T}}\).
Proof
Set \(v(t,X(t),a)=Y(t,X(t))\), \(H(t,X(t))=\X(t)\\), \(H_{0}(t_{0},X_{0})=\X_{0}\\) and \(m=1\), \(n=1\), we see from Corollary 3.1 that the trivial solution of (14) is Hexponentially stable on \([t_{0},\infty )_{\mathbb{T}}\). □
Remark 3.1
Corollary 3.2 is essentially an extension and improvement of Theorem 4.2 in [33].
Theorem 3.3
Let v be a matrixvalued Lyapunovlike function on \(\mathbb{T} \times K_{c}^{m}(\mathbb{R}^{n})\times \mathbb{R}_{+}^{m}\) and the function \(\bar{v}:K_{c}^{m}(\mathbb{R}^{n})\times \mathbb{R}_{+}^{m}\to \mathbb{R}_{+}\) satisfy \(Nv(t, X_{0}, a)\le \bar{v}(X_{0}, a)\) for the constant \(N>1\). Moreover, suppose that the following conditions hold: for \(H \in \mathcal{M}\), \(X\in \mathscr{X}\),

(i)
there exist constants \(k_{1}, p>0\) such that
$$ k_{1}\bigl(H\bigl(t,X(t)\bigr)\bigr)^{p} \leq v \bigl(t,X(t),a\bigr)\quad {\textit{for all }} t\in \mathbb{T}; $$ 
(ii)
there exist constants \(k_{2}\), L, and functions \(\varepsilon \in \Re _{+}\), \(\delta \in \Re \) such that
$$\begin{aligned}& v^{\Delta}\bigl(t,X(t),a\bigr)\leq \frac{k_{2}v(t,X(t),a)L(\varepsilon \ominus \delta )(t)e_{\ominus \delta}(t,0)}{1+\varepsilon \mu (t)}, \end{aligned}$$(15)$$\begin{aligned}& (\varepsilon k_{2})v\bigl(t,X(t),a\bigr)\le L(\varepsilon \ominus \delta ) (t)e_{ \ominus \delta}(t,0) \quad {\textit{for }} t\in \mathbb{T}. \end{aligned}$$(16)
Then, the trivial solution of SDE (1) is Huniformly exponentially stable on \([t_{0},\infty )_{\mathbb{T}}\).
Proof
Let \(\gamma \in \Gamma \) satisfy \(\gamma (s)\le k_{1}s\) for \(s\ge 0\). By the assumption (i), we have
In the light of Definition 3.2 and (17), it suffices to check that
for \(t\in [t_{0},\infty )_{\mathbb{T}}\) and fixed \(a\in \mathbb{R}_{+}^{m}\). If the above inequality is false, then there exists \(t\in [t_{0},\infty )_{\mathbb{T}}\) such that \(v(t,X(t),a)> N\bar{v}(X_{0}, a)e_{\ominus \varepsilon}(t,t_{0})> \bar{v}(X_{0}, a)e_{\ominus \varepsilon}(t,t_{0}) \) since \(N>1\). This implies that \(\bar{t}=\inf \{t\in [t_{0},\infty )_{\mathbb{T}}  v(t,X(t),a)> \bar{v}(X_{0}, a)e_{\ominus \varepsilon}(t,t_{0}) \}\) exists. Therefore, we obtain
Again, we consider the function \(\varphi \in \Gamma \) with \(s < \varphi (s)\le Ns\) for any \(s\ge 0\). Then, by (19) we have
However, we observe that \(\varphi (v(t_{0},X_{0},a) )\le \varphi (N^{1} \bar{v}(X_{0}, a) )\le \bar{v}(X_{0}, a)\). This implies that the set
is nonempty and hence its supremum, denoted by t̃, exists. Thus, we have
On the other hand, similar to the proof of Step 1 in Theorem 3.2, we obtain that \(v(t,X(t),a)e_{\varepsilon}(t,t_{0})\) is nonincreasing in \(t\in [\tilde{t}, \bar{t}]\), which infers that
a contradiction. Consequently, (18) holds and hence the trivial solution of SDE (1) is Huniformly exponentially stable on \([t_{0},\infty )_{\mathbb{T}}\). The proof is completed. □
Theorem 3.4
Let the assumptions of Theorem 3.2hold, except that function M is changed into a constant and the estimate of (5) is strengthened to
where \(\lambda _{3}: \mathbb{R}\mathbbm{_{+}}\to \mathbb{R}_{+}\) is a positive nondecreasing function with \(\lim_{t\rightarrow{+}\infty}\lambda _{3}(t)={+}\infty \) and constant \(r>0\). Then, the trivial solution of SDE (1) is Hexponentially asymptotically stable.
Proof
Theorem 3.2 has proved the Hexponential stability of the trivial solution of SDE (1), that is, (4) holds. By Lemma 2.1(vi) and the Bernoulli inequality, we obtain
This means that
Integrating both sides of (23) from \(t_{0}\) to t, one has
We observe that (23) guarantees \(v(t,X(t),a)\) to be nonincreasing in \(t\in \mathbb{T}\). In addition, \(v(t,X(t),a)\ge 0\). From the monotone convergence theorem, it follows that there exists β with \(\lim_{t \rightarrow \infty}v(t,X(t),a)=\beta \).
Now, we will prove that \(\beta =0\). Suppose that this is false. Then, we have \(\beta >0\). By virtue of the monotonicity of \(v(t,X(t),a)\), we have \(v(t,X(t),a) \geq \beta >0\) for all \(t \in [t_{0},\infty )_{\mathbb{T}}\). On the other hand, since the function \(\lambda _{3}\) is positive nondecreasing and \(\lim_{t\rightarrow{+}\infty}\lambda _{3}(t)={+}\infty \), we see that \(\int ^{t}_{t_{0}}\lambda _{3}((H(s,X(s)))^{r}) \Delta s\) is larger than \(v(t_{0},X_{0},a)\) when \(t\in \mathbb{T}\) is sufficiently large. Combining with (24), we obtain that \(v(t,X(t),a)<0\), which contradicts \(v(t,X(t),a)\ge 0\). Hence, \(\beta =0\), that is \(\lim_{t \rightarrow \infty}v(t,X(t),a)=0\).
Next, we will prove that \(\lim_{t \rightarrow \infty}H(t,X(t))=0\). If this is not true, then there exists \(\varepsilon _{0}>0\) such that \(H(t_{m},X(t_{m}))>\varepsilon _{0}>0\) for any \(m\in \mathbb{Z}_{+}\) and some \(t_{m} \in \mathbb{T}\) with \(t_{m}\geq m\). From this, combined with (7), we have
This contradicts \(\lim_{t \rightarrow \infty}v(t,X(t),a)=0\). Hence, \(\lim_{t \rightarrow \infty}H(t,X(t))=0\). By virtue of the definition of \(\bar{\gamma}\in \Gamma \), we obtain \(\lim_{t \rightarrow \infty}\bar{\gamma}((H(t,X(t)))^{p} )=0\). This guarantees that for any \(\varepsilon >0\), there exists a positive real number T such that
Thus, the trivial solution of SDE (1) is Hexponentially asymptotically stable and the proof is completed. □
Theorem 3.5
Let the assumptions of Theorem 3.3hold except that function ε changes into a constant and the estimate of (15) is strengthened to
In addition, assume that \(v(t,\Theta _{0}, a)=0 \) for each \(t \in \mathbb{T} \) and each \(a\in \mathcal{A}(t, \Theta _{0})\). Then, the trivial solution of SDE (1) is Huniformly exponentially asymptotically stable.
Proof
Theorem 3.3 has proved the Huniformly exponential stability, so that (4) holds. As an analogy of the proof of Theorem 3.4, we have
and
Thus, the trivial solution of SDE (1) is Huniformly exponentially asymptotically stable. The proof is completed. □
At the end of this section, we will start with some simple examples of SDEs to illustrate that our approach and results are applicable. For the sake of convenience, we only consider \(\mathbb{T}_{+}\) instead of \(\mathbb{T}\) as the working platform.
Example 3.1
The trivial solution of the following SDE
is Hexponentially stable, where the function \(\zeta \in C(\mathbb{T}_{+}, \mathbb{R}_{+})\cap \Re _{+}\) is nondecreasing and satisfies that \(w\le \zeta (t)\) on \(\mathbb{T}_{+}\) with the constant \(w>0\).
Proof
From Lemma 12 in [31] it follows that SDE (26) has a unique solution \(X:\mathbb{T}_{+}\to K_{c}^{m}(\mathbb{R}^{n})\) as follows
Choose \(v(t,X(t),a)=H(t,X(t))=\X(t)\\) for \(t \in {\mathbb{T}_{+}}\). Next, we prove that the conditions in the Theorem 3.2 are satisfied. It is easy to see that the condition (i) is satisfied with \(\lambda _{1}(s)=s\), \(\lambda _{2}(s)=s\) for \(s\ge 0\), \(p=1\). In order to verify condition (ii), we first calculate the Δderivative of \(v(t,X(t),a)\). By Lemma 2.1(iii), we obtain
Next, we take
for all \(t\in \mathbb{T}_{+}\). Then, the condition (ii) is satisfied. Consequently, the trivial solution of SDE (26) is Hexponentially stable. □
Example 3.2
The trivial solution of the following SDE
is Huniformly exponentially stable, where the functions \(\eta \in C(\mathbb{T}_{+}, \mathbb{R}_{+})\cap \Re _{+}\) is nondecreasing and satisfies that \(w\le \eta (t)\) on \(\mathbb{T}_{+}\) with the constant \(w>0\) and \(F\in C_{\mathrm{rd}}(\mathbb{T}_{+}, K_{c}^{m}(\mathbb{R}^{n}))\) satisfies
Proof
From Lemma 13 in [31] it follows that SDE (27) has a unique solution \(X:\mathbb{T}_{+}\to K_{c}^{m}(\mathbb{R}^{n})\) as follows
Thus,
Let
for \(t \in {\mathbb{T}_{+}}\) and \(a\in \mathbb{R}_{+}^{m}\). Our hypothesis guarantees that \(v(t, X(t),a)\ge 0\). Next, we prove that the conditions in the Theorem 3.2 are satisfied. It is easy to see that the condition (i) holds with \(\lambda _{1}(s)=s\), \(\lambda _{2}(s)=s\) for \(s\ge 0\), \(p=1\). In order to verify condition (ii), we first calculate the Δderivative of \(v(t,X(t),a)\). By Lemma 2.1(iii) we obtain
Next, we take
Then, the condition (ii) is satisfied. Consequently, the trivial solution of SDE (27) is Hexponentially stable.
Finally, it is not difficult to see that all conditions in Theorem 3.3 are satisfied if we choose \(k_{1}=\frac{1}{2}\) and \(\varepsilon (t)=k_{2}=w\). Therefore, Theorem 3.3 guarantees that the trivial solution of SDE (27) is Huniformly exponentially stable. □
Example 3.3
SDE (27) is Hexponentially asymptotically stable and Huniformly exponentially asymptotically stable under the hypotheses of Example 3.2.
Proof
Let \(v(t,X(t),a)\) be given as in (29). It has been shown that the assumptions of Theorem 3.2 are fulfilled with \(M=\omega \) (a positive constant) in the proof of Example 3.2. Let \(\lambda _{3}(s)=\frac{w}{1+w\mu (t)}s\). Then, \(\lambda _{3}\) satisfies the hypothesis of Theorem 3.4 and the inequality (23) holds by (30). Now, Theorem 3.4 guarantees that the trivial solution of SDE (27) is Hexponentially asymptotically stable.
Let \(\varsigma =\max_{t\in \mathbb{T}_{+}}\mu (t)\). To check the conditions of Theorem 3.5, let us take constants \(k_{1}=\frac{1}{2}\), \(k_{2}\), ε, and the function δ as follows
for all \(t\in \mathbb{T}_{+}\). Thus, if \(\varsigma <+\infty \), similar to the derivation in the proof of Example 3.2, we obtain
for all \(t\in \mathbb{T}_{+}\). If \(\varsigma =+\infty \), we have
for all \(t\in \mathbb{T}_{+}\). Consequently, (25) holds.
From (28) and (29) it follows that \(v(t,\Theta _{0}, a)=0\) for each \(t \in \mathbb{T}_{+} \) and each \(a\in \mathcal{A}(t, \Theta _{0})\). In addition, all the conditions in Theorem 3.3 are clearly satisfied under our assumptions. Now, Theorem 3.5 guarantees that the trivial solution of SDE (27) is Huniformly exponentially asymptotically stable. □
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We gratefully thank the referees for carefully reading the paper and for the suggestions that greatly improved the presentation of the paper. The authors really appreciate the collaboration of the editorial board.
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This paper is supported by the National Natural Science Foundation of China (71771068, 71471051).
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Jia, K., Hong, S., Cao, X. et al. Exponential stability for a class of set dynamic equations on time scales. J Inequal Appl 2022, 135 (2022). https://doi.org/10.1186/s13660022028750
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DOI: https://doi.org/10.1186/s13660022028750