Lipschitz Stability in Terms of Two Measures for Kurzweil Equations and Applications

Abstract

For generalized ordinary differential equations, sufficient criteria are given for the Lipschitz stability in terms of two measures of the trivial solutions. As an application, we apply our main results by studying the Lipschitz stability for measure differential equations and impulsive differential equations. Compared to the classical ones, the conditions here regarding the functions are more general.

1. Introduction

In the qualitative theory of differential equations, the stability of solutions is one of the main problems. There are various concepts of stability. One type of stability, called Lipschitz stability, is very useful in the theory of physics, chemistry, biology, optimal control, technology and many other branches of science, etc. This type of stability is introduced by F. Dannan and S. Elaydi [1] and is only significant for nonlinear systems, as it is consistent with uniform Lyapunov stability in linear systems. Significant contributions to Lipschitz stability have been made by many authors of various types of differential equations [2,3,4,5,6,7,8]. On the other hand, to unify various concepts of stability and to provide a general framework for study, stability in terms of two measures was defined and studied in [9] and intensively in [10,11,12,13,14].

The theory of Kurzweil equations or generalized ordinary differential equations (generalized ODEs for short) was introduced in 1957 by J. Kurzweil [15]. Since then, this theory has undergone several developments and has been generalized into different directions [16,17,18,19,20,21,22]. As we know, generalized ODEs include some classical differential equations, such as ordinary differential Equations (ODEs for short) [23,24], functional differential equations [25,26], impulsive differential Equations (IDEs for short) [27], measure differential Equations (MDEs for short) [28], measure functional differential equations [29], dynamic equations on time scales [30], Fredholm–Stieltjes and Volterra–Stieltjes equations [31,32], and so on.

For classical differential equations, the theory of Lipschitz stability is well developed, but for generalized ODEs, the literature on Lipschitz stability is scarce, which is essential in practice and is also an extremely difficult problem. As a result of the above, this paper concentrates on the generalized ODEs of the form:

$$\frac{dx}{d\tau }=DF(x,t),$$

(1)

where the function F is integrable in the sense of Kurzweil. This is done by providing sufficient criteria on the Lipschitz stability of the trivial solution for Equation (1); then, by using the relationship between the solutions of generalized ODEs, MDEs, and IDEs, we also present the Lipschitz stability of trivial solutions for the following MDEs and IDEs:

$$\begin{array}{cc}\hfill & Dx=f(x,t)Dg,\hfill \end{array}$$

(2)

and

$$\left\{\begin{array}{c}\dot{x}=f(x,t),t\ne t_k,\hfill \\ \Delta x\left(t\right)=I_k\left(x\left(t\right)\right),k=1,2,\dots ,\hfill \end{array}\right.$$

(3)

where $Dx$ and $Dg$ are the distributional derivatives of x and g in the sense of L. Schwartz, respectively, $t_k$ and $k=1,2,\dots$ are pre-assigned moments of impulse, and $\Delta x\left(t_k\right):=x(t_k+)-x(t_k-)=x(t_k+)-x\left(t_k\right)$.

We note that the conditions of theorem for Equations (1)–(3) are weaker, and they do not need the Lipschitz condition on the Lyapunov function, which is usually used in the classical literature [18,33]. Moreover, all the conditions we have assumed are in the integral of the function f (the integral is the Kurzweil–Henstock–Stieltjes sense), rather than conditions directly on the function f; hence, f may be highly oscillatory, have many points of discontinuities, or be of unbounded variation. On the other hand, the concepts and criteria of Lipschitz stability are investigated by using two measures, including the classical Lipschitz stability, the Lipschitz stability of the prescribed motion, and the Lipschitz stability of the invariant, so that the study is carried out in a unified way that makes the proof simpler and the conclusions more concise.

This paper is structured as follows. In Section 2, we give some definitions, lemmas, and conclusions on Kurzweil integration and generalized ODEs. Section 3 is devoted to recalling the notions of Lipschitz stability for the generalized ODEs of Equation (1), and to prove new results on the Lipschitz stability of trivial solutions of the generalized ODEs of Equation (1). Finally, in Section 4, by using the relationship between the generalized ODEs of Equation (1), the MDEs of Equation (2), and the IDEs of Equation (3), we present the Lipschitz stability for the MDEs of Equation (2) and the IDEs of Equation (3).

2. Preliminaries

Let $(X,\parallel ·\parallel )$ be Banach space and $\mathbb{R}$ and ${\mathbb{R}}^{+}$ stand for the set of real numbers and nonnegative real numbers, respectively; we present the following notations:

• $C([t_0,+\infty )\times X,{\mathbb{R}}^{+})$: the set of continuous functions from $[t_0,+\infty )\times X$ to ${\mathbb{R}}^{+}$ with the supremum norm.
• $\Gamma =\{h:h\in C([t_0,+\infty )\times X,{\mathbb{R}}^{+})andinfh(s,x)=0\}.$
• $G\left(\right[a,b],X)$: the set of functions $f:[a,b]\to X$ such that the one-side limits exist:

  $$\underset{s\to t-}{lim}f\left(s\right)=f(t-)\in X,t\in (a,b]$$

  and

  $$\underset{s\to t+}{lim}f\left(s\right)=f(t+)\in X,t\in [a,b).$$
  It is a Banach space with the norm $\parallel f\parallel =\underset{a\le t\le b}{sup}\parallel f\left(t\right)\parallel$, and f is called the regulated function.
• $G^{-}([a,b],X)$: the subspace of $G\left(\right[a,b],X)$, which is left-continuous functions.
• $G([t_0,+\infty ),{\mathbb{R}}^n)$: the set of functions $f:[t_0,+\infty )\to {\mathbb{R}}^n$ such that ${f|}_{[\alpha ,\beta ]}\in G([\alpha ,\beta ],{\mathbb{R}}^n)$ for all $[\alpha ,\beta ]\subset [t_0,+\infty )$.
• $G_0([t_0,+\infty ),{\mathbb{R}}^n)$: the set of functions $f\in G([t_0,+\infty ),{\mathbb{R}}^n)$ such that

  $$\underset{s\in [t_0,+\infty )}{sup}{e}^{-(s-t_0)}\left|f\left(s\right)\right|<+\infty .$$
  It is a Banach space with the norm ${\parallel f\parallel }_{[t_0,+\infty )}=\underset{s\in [t_0,+\infty )}{sup}{e}^{-(s-t_0)}\left|f\left(s\right)\right|$.

In this part, we will give some properties and basic concepts of the Kurzweil integral, which is essential for studying generalized ODEs; one can see [27] for more details.

A tagged division of a compact interval $[a,b]\subset \mathbb{R}$ is a finite collection of point-interval pairs $({\tau }_i,[s_{i-1},s_i])$, where $a=s_0\le s_1\le \cdots \le s_n=b$ is a division of $[a,b]$, ${\tau }_i\in [s_{i-1},s_i]$, and $i=1,2,\dots ,n$.

A gauge on a set $E\subset [a,b]$ is any function $\delta :E\to (0,+\infty )$. Given a gauge $\delta$ on $[a,b]$, a tagged division $d=({\tau }_i,[s_{i-1},s_i])$ is $\delta$-fine if, for every i, one has

$$[s_{i-1},s_i]\subset \{t\in [a,b]:|t-{\tau }_i|<\delta \left({\tau }_i\right)\}.$$

Definition 1

([27]). A function $U:[a,b]\times [a,b]\to X$ is said to be Kurzweil integrable on $[a,b]$, if there exists a unique element $I\in X$ such that given $\epsilon >0$, there exits a gauge δ on $[a,b]$ such that

$$∥\sum _i\left[U({\tau }_i,s_i)-U({\tau }_i,s_{i-1})\right]-I∥<\epsilon$$

for every δ-fine tagged division $({\tau }_i,[s_{i-1},s_i])$ of $[a,b]$. In this case, I is called a Kurzweil integral of U on $[a,b]$, and it is denoted by ${\int }_a^{b}DU(\tau ,t)$.

Similar to the Riemann integral, the Kurzweil integral has the properties of linearity, integrability on subintervals, additivity with respect to adjacent intervals, etc.

Remark 1.

(i)

Similarly, we can give the Definition 1 to the function U defined in unbounded intervals; see, for instance, [34].

(ii)

If we choose $U(\tau ,t)=f\left(\tau \right)g\left(t\right)$, then it is the Perron–Stieltjes (or Kurzweil–Henstock–Stieltjes) integral of $f:[a,b]\to X$ with respect to $g:[a,b]\to \mathbb{R}$, which will be denoted by ${\int }_a^{b}f\left(s\right)dg\left(s\right)$. Moreover, if for any $s_1,s_2\in [t_0,+\infty )$, f is a Kurzweil–Henstock–Stieltjes integrable in $[s_1,s_2]$, then f is said locally to be a Kurzweil–Henstock–Stieltjes integrable function.

(iii)

The well-known Perron (or Kurzweil–Henstock) integral of $f:[a,b]\to X$, which is obtained by letting $U(\tau ,t)=f\left(\tau \right)t$, will be denoted by ${\int }_a^{b}f\left(s\right)ds$.

Following this, we recall the concepts of the solution of the generalized ODEs and the generalized ODEs with an initial condition. Suppose $F:\Omega \to X$ is defined for each $(x,t)\in \Omega$, where $\Omega =O\times [t_0,+\infty )$, $O\subset X$ is an open subset.

Definition 2

([27]). $x:[\alpha ,\beta ]\to X$ is said to be a solution of the generalized ODEs (or Kurzweil equations)

$$\frac{dx}{d\tau }=DF(x,t)$$

(4)

on $[\alpha ,\beta ]\subset [t_0,+\infty )$ if $\left(x\right(t),t)\in \Omega$ for every $t\in [\alpha ,\beta ]$ and

$$x\left(s_2\right)-x\left(s_1\right)={\int }_{s_1}^{s_2}DF(x\left(\tau \right),t)$$

(5)

for every $s_1,s_2\in [\alpha ,\beta ]$. The integral on the right-hand side of (5) is the Kurzweil integral, according to Definition 1, where the integral is obtained by letting $U(\tau ,t)=F\left(x\right(\tau ),t)$.

Definition 3

([27]). $x:[\alpha ,\beta ]\to X$ is said to be a solution of Equation (4) with $x\left(s_0\right)=x_0$ on $[\alpha ,\beta ]\subset [t_0,+\infty )$ if $s_0\in [\alpha ,\beta ]$, $\left(x\right(t),t)\in \Omega$ for every $t\in [\alpha ,\beta ]$ and

$$\begin{array}{c}\hfill x\left(s\right)-x_0={\int }_{s_0}^{s}DF(x\left(\tau \right),t)\end{array}$$

for every $s\in [\alpha ,\beta ]$.

Definition 2 is defined in the bounded interval $[\alpha ,\beta ]$. Indeed, one can extend it if the solution of Equation (4) is defined in a nondegenerated interval I that is not necessary for $[\alpha ,\beta ]$.

Definition 4

([27]). $x:I\to X$, where I is a nondegenerated subinterval of $[t_0,+\infty )$, is said to be a solution of Equation (4) on I if $\left(x\right(t),t)\in \Omega$ for every $t\in I$ and satisfies the following equality

$$x\left(s_2\right)-x\left(s_1\right)={\int }_{s_1}^{s_2}DF(x\left(\tau \right),t)$$

for every $s_1,s_2\in I$.

Let $F(t,x)=tH\left(x\right)$; H is a continuous function, and the equation

$$\frac{dx}{d\tau }=DF(x,t)=D\left[tH\left(x\right)\right]$$

(6)

is the generalized ODEs associate with autonomous ODEs

$$\frac{dx}{dt}=H\left(x\right).$$

(7)

By the definition of generalized ODEs and Kurzweil integral, x is a solution of Equation (6) if and only if x is a solution of Equation (7).

Assume that $\tilde{h}:[t_0,+\infty )\to \mathbb{R}$ is a nondecreasing function.

Definition 5

([27]). A function $F:\Omega \to X$ is of the class $\mathcal{F}(\Omega ,\tilde{h})$ if

$$\begin{array}{c}\hfill \parallel F(x,t)-F(x,s)\parallel \le |\tilde{h}\left(t\right)-\tilde{h}\left(s\right)|\end{array}$$

(8)

for all $(x,t),(x,s)\in \Omega$ and if

$$\begin{array}{cc}\hfill & \parallel F(x,t)-F(x,s)-F(y,t)+F(y,s)\parallel \hfill \\ \hfill & \le \parallel x-y\parallel ·|\tilde{h}\left(t\right)-\tilde{h}\left(s\right)|\hfill \end{array}$$

for all $(x,t),(x,s),(y,t),(y,s)\in \Omega$.

The below consequences provide us with useful properties regarding the solutions of Equation (4), which can be seen in [27].

Lemma 1

([27]). Assume that $F:\Omega \to X$ satisfies (8), $\left(x\right(t),t)\in \Omega$ for every $t\in [\alpha ,\beta ]$, $[\alpha ,\beta ]\subset [t_0,+\infty )$, and, if Kurzweil integral ${\int }_{\alpha }^{\beta }DF(x\left(\tau \right),t)$ exists, then we have for every $s_1,s_2\in [\alpha ,\beta ]$,

$$∥{\int }_{s_1}^{s_2}DF(x\left(\tau \right),t)∥\le |h\left(s_2\right)-h\left(s_1\right)|.$$

Lemma 2

([27]). Assume that $F:\Omega \to X$ satisfies (8), $x:[\alpha ,\beta ]\to X$ is a solution of Equation (4), $[\alpha ,\beta ]\subset [t_0,+\infty )$, then the inequality

$$\parallel x\left(t\right)-x\left(s\right)\parallel \le |\tilde{h}\left(t\right)-\tilde{h}\left(s\right)|$$

is satisfied for every $t,s\in [\alpha ,\beta ]$. Moreover, every point in $[\alpha ,\beta ]$ at which the function $\tilde{h}$ is continuous is a continuity point of the solution x.

Corollary 1

([27]). Assume that $F:\Omega \to X$ satisfies (8), $x:[\alpha ,\beta ]\to X$ is a solution of Equation (4), $[\alpha ,\beta ]\subset [t_0,+\infty )$; then, x is a function of bounded variation over $[\alpha ,\beta ]$ and

$${var}_{\alpha }^{\beta }x\le \tilde{h}\left(\beta \right)-\tilde{h}\left(\alpha \right)<+\infty .$$

The following lemma shows the existence of the Kurzweil integral. In the case of $X={\mathbb{R}}^n$, the proof can be seen in [27], and the conclusion is also correct in the more general Banach space X.

Lemma 3

([27]). Assume that $F\in \mathcal{F}(\Omega ,\tilde{h})$ and x is regulated in $[\alpha ,\beta ]\subset [t_0,+\infty )$, $\left(x\right(t),t)\in \Omega$ for $t\in [\alpha ,\beta ]$, then the Kurzweil integral ${\int }_{\alpha }^{\beta }DF(x\left(\tau \right),t)$ exists and the function $s\to {\int }_{\alpha }^{s}DF(x\left(\tau \right),t)$ is of bounded variation (hence regulated).

The next result shows the existence and uniqueness of a maximal solution of Equation (4) and can be found in [28]. Through this paper, $x\left(t\right)=x(t,s_0,x_0)$ and $t\in [s_0,\omega (s_0,x_0))$ denote the unique maximal solution of Equation (4).

Lemma 4

([28]). If $\Omega =X\times [t_0,+\infty )$, $F\in \mathcal{F}(\Omega ,\tilde{h})$, where the function $\tilde{h}$ is nondecreasing and left-continuous, then for every $(x_0,s_0)\in \Omega$, there exists a unique maximal solution of Equation (4) with $x\left(s_0\right)=x_0$ defined in $[s_0,+\infty )$.

3. Stability of Generalized ODEs

Some criteria are given on stability for generalized ODEs in this section. First of all, we define the conceptions of stability in terms of two measures for $x=0$ of generalized ODEs.

Consider the following generalized ODEs:

$$\frac{dx}{d\tau }=DF(x,t)$$

(9)

with

$$x\left(t_0\right)=x_0$$

where $(x_0,t_0)\in \Omega =B_c\times [t_0,+\infty ),$ $B_c=\{x\in X:\parallel x\parallel \le c\}$, and $c>0$. Let $x\left(t\right)=x(t,t_0,x_0)$ be the solution of Equation (9) which is defined on $[t_0,+\infty )$. Assume that

$$F(0,t)-F(0,s)=0\mathrm{for} \mathrm{every}t,s\in [t_0,+\infty );$$

then, for every $[\alpha ,\beta ]\subset [t_0,+\infty )$, we obtain

$${\int }_{\alpha }^{\beta }DF(0,t)=F(0,\beta )-F(0,\alpha )=0;$$

that is, $x=0$ is a solution of Equation (9) on $[t_0,+\infty )$.

Then, we present a few new concepts regarding the Lipschitz stability of $x=0$ of Equation (9).

Definition 6.

$x=0$ of Equation (9) is called uniformly regularly Lipschitz stable if there exist $M>0$, $\delta >0$ such that $\overline{x}:[\alpha ,\beta ]\to B_c$, $t_0\le \alpha <\beta <+\infty$ is a regulated function on $[\alpha ,\beta ]$ and left-continuous on $(\alpha ,\beta ]$ which satisfies

$$\parallel \overline{x}\left(\alpha \right)\parallel <\delta$$

and

$$\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥<\delta ,$$

then

$$\parallel \overline{x}\left(t\right)\parallel \le M\parallel \overline{x}\left(\alpha \right)\parallel ,\mathit{for} \mathit{all}t\in [\alpha ,\beta ].$$

Remark 2.

Note that if $x=0$ of Equation (9) is uniformly regularly Lipschitze stable, then it is variationally Lipschitz stable in the sense defined in [33].

Definition 7.

Let $h_0,h\in \Gamma$; then, $x=0$ of (9) is called:

• $(h_0,h)$- uniformly regularly Lipschitz stable if there exist $M>0$ and $\delta >0$ such that $\overline{x}:[\alpha ,\beta ]\to B_c$, $t_0\le \alpha <\beta <+\infty$ is a regulated function on $[\alpha ,\beta ]$ and left-continuous on $(\alpha ,\beta ]$, which satisfies

  $$h_0(\alpha ,\overline{x}\left(\alpha \right))<\delta$$

  and

  $$\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥<\delta ,$$

  then

  $$h(t,\overline{x}\left(t\right))\le M·h_0(\alpha ,\overline{x}\left(\alpha \right)),for allt\in [\alpha ,\beta ].$$
• $(h_0,h)$-uniformly Lipschitz stable  if there exist $M>0$, $\delta >0$ and $x:[\alpha ,\beta ]\to B_c$, $t_0\le \alpha <\beta <+\infty$ is a solution of Equation (9) on $[\alpha ,\beta ]$; then,

  $$h_0(\alpha ,x\left(\alpha \right))<\delta$$

  implies

  $$h(t,x\left(t\right))<M·h_0(\alpha ,x\left(\alpha \right)),for allt\in [\alpha ,\beta ].$$
• globally $(h_0,h)$-uniformly Lipschitz stable  if there exists $M>0$ and $x:[\alpha ,\beta ]\to B_c$, $t_0\le \alpha <\beta <+\infty$ is a solution of Equation (9) on $[\alpha ,\beta ]$; then,

  $$h(t,x\left(t\right))<M·h_0(\alpha ,x\left(\alpha \right)),for allt\in [\alpha ,\beta ].$$

Remark 3.

(i)

If we choose $h_0(t,x)=h(t,x)=\parallel x\parallel$ in Definition 7, then we can omit $(h_0,h)$ and simply refer to $x=0$ of Equation (9) as being uniformly regularly Lipschitz stable, uniformly Lipschitz stable [33], and globally uniformly Lipschitz stable [33].

(ii)

It is easy to see that if $x=0$ of Equation (9) is $(h_0,h)$-uniformly regularly Lipschitz stable, then it is $(h_0,h)$-uniformly Lipschitz stable.

The following Lemma comes from [35], which is useful for the proof of our results.

Lemma 5

([35]). Let $F\in \mathcal{F}(\Omega ,\tilde{h})$, where $\tilde{h}:[t_0,+\infty )\to \mathbb{R}$ is nondecreasing and left-continuous; assume that there exists a function $V:[t_0,+\infty )\times X\to \mathbb{R}$ satisfies the following conditions:

$\left(V_1\right)$

For each left-continuous function $z:[\alpha ,\beta ]\to X$ on $(\alpha ,\beta ]$, $[\alpha ,\beta ]\subset [t_0,+\infty )$, the function $t\to V(t,z(t\left)\right)$ is left-continuous on $(\alpha ,\beta ]$.

$\left(V_2\right)$

For all regulated function $x,y:[\alpha ,\beta ]\to X$, $[\alpha ,\beta ]\subset [t_0,+\infty )$ and every $\alpha \le s<t\le \beta$, one has

$$\begin{array}{cc}\hfill & |V(t,x\left(t\right))-V(t,y\left(t\right))-V(x,x\left(s\right))+V(s,y\left(s\right))|\le \underset{\xi \in [s,t]}{sup}\parallel x\left(\xi \right)-y\left(\xi \right)\parallel .\hfill \end{array}$$

$\left(V_3\right)$

For solution $x:[t_0,\omega (t_0,x_0))\to X$ of Equation (9) with $x\left(t_0\right)=x_0$, the function $t\to V(t,x(t\left)\right)$ is nonincreasing along every solution x.

If $\overline{x}:[\alpha ,\beta ]\to X$ is regulated and left-continuous on $(\alpha ,\beta ]$, then

$$\begin{array}{cc}\hfill V(v,\overline{x}\left(v\right))& \le V(\gamma ,\overline{x}\left(\gamma \right))+2\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥\hfill \end{array}$$

for all $v,\gamma \in [\alpha ,\beta ]$ with $\gamma \le v$.

The next results give us the conditions to ensure that $x=0$ of Equation (9) is $(h_0,h)$-uniformly Lipschitz stable and globally $(h_0,h)$-uniformly Lipschitz stable, where $\tilde{h}:[t_0,+\infty )\to \mathbb{R}$ is nondecreasing and left-continuous.

Theorem 1.

Assume that $F\in \mathcal{F}(\Omega ,\tilde{h})$ and $\left(V_1\right)$–$\left(V_3\right)$ hold; moreover, the function $V:[t_0,+\infty )\times B_{\rho }\to {\mathbb{R}}^{+}$, $0<\rho <c$ satisfies the following condition:

$\left(V_4\right)$

There exist two monotone increasing functions $b,p:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $b\left(0\right)=p\left(0\right)=0$,

$$b\left(h(t,z)\right)\le V(t,z)\le p\left(h_0(t,z)\right),$$

for every $(t,z)\in [t_0,+\infty )\times B_{\rho }$, $h_0,h\in \Gamma$;

then, the trivial solution $x=0$ of Equation (9) is $(h_0,h)$-uniformly Lipschitz stable.

Proof. 

Let $\overline{x}:[\alpha ,\beta ]\to B_{\rho }$ be left-continuous on $(\alpha ,\beta ]$ and regulated on $[\alpha ,\beta ]\subset [t_0,+\infty )$. By Lemma 5, for all $t\in [\alpha ,\beta ]$, one has

$$\begin{array}{cc}\hfill V(t,\overline{x}\left(t\right))& \le V(\alpha ,\overline{x}\left(\alpha \right))+2\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥.\hfill \end{array}$$

Let $\epsilon >0$ and $b\left(\epsilon \right)>0$; there exists $\delta >0$ such that $p\left(\delta \right)+2\delta <b\left(\epsilon \right)$; if

$$h_0(\alpha ,\overline{x}\left(\alpha \right))<\delta$$

and

$$\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥<\delta ,$$

then

$$\begin{array}{cc}\hfill V(t,\overline{x}\left(t\right))& \le V(\alpha ,\overline{x}\left(\alpha \right))+2\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥\hfill \\ \hfill & \le p\left(h_0(\alpha ,\overline{x}\left(\alpha \right))\right)+2\delta \hfill \\ \hfill & \le p\left(\delta \right)+2\delta \hfill \\ \hfill & <b\left(\epsilon \right)\hfill \\ \hfill & <b(h_0(\alpha ,\overline{x}\left(\alpha \right))+\epsilon ).\hfill \end{array}$$

Moreover, by $\left(V_4\right)$, we have

$$V(t,\overline{x}\left(t\right))\ge b\left(h(t,\overline{x}\left(t\right))\right),t\in [\alpha ,\beta ];$$

then,

$$h(t,\overline{x}\left(t\right))<h_0(\alpha ,\overline{x}\left(\alpha \right))+\epsilon ,t\in [\alpha ,\beta ];$$

hence, $x=0$ of Equation (9) is $(h_0,h)$-uniformly regularly Lipschitz stable, where $M=1$ in Definition 7, so it is $(h_0,h)$-uniformly Lipschitz stable. □

Example 1.

Let $X=\mathbb{R}$ with norm $|·|$ (absolute value), $\Omega =(-1,1)\times [0,+\infty )$, and define $F:\Omega \to X$ by $F(x,t)=-tx$. It is not difficult to see that $\mathcal{F}(\Omega ,\tilde{h})$, where $\tilde{h}\left(s\right)=s$, $s\in [0,+\infty )$ is a nondecreasing and left-continuous function on $(0,+\infty )$. Consider the following generalized ODEs and autonomous ODEs:

$$\begin{array}{cc}\hfill & \frac{dx}{d\tau }=DF(x,t)=D[-tx]\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \frac{dx}{dt}=-x\hfill \end{array}$$

(11)

By the definition of generalized ODEs and Kurzweil integral, x is a solution of Equation (10) if and only if x is a solution of Equation (11). Hence, $x\left(t\right)=x_0{e}^{-t}$ is the solution of Equation (10) with the initial condition $x\left(0\right)=x_0\in [0,1]$.

For any given $s_0>0$, let

$$k\left(t\right)=\left\{\begin{array}{cc}1\hfill & for0\le t\le s_0,\hfill \\ 2\hfill & fort>s_0.\hfill \end{array}\right.$$

Define $V:[0,+\infty )\times [-1,1]\to {\mathbb{R}}^{+}$ by

$$V(t,x)=\frac{1}{4\pi }k\left(t\right){arctan}^{2}x.$$

Now we show that all the conditions of Theorem 1 are satisfied, where $h_0(s,x)=h(s,x)=\left|x\right|$. Since $k\left(t\right)$ is a left-continuous function, then $V(·,x):[0,+\infty )\to {\mathbb{R}}^{+}$ is left-continuous on $(0,+\infty )$ for all $x\in [-1,1]$, i.e., $\left(V_1\right)$ holds. For $x,y:[\alpha ,\beta ]\to [-1,1]$, $[\alpha ,\beta ]\subset [0,+\infty )$, and $0\le s<t<+\infty$, we have

$$\begin{array}{cc}\hfill & \left|V\right(t,x\left(t\right))-V(t,y\left(t\right))-V(x,x\left(s\right))+V(s,y\left(s\right)\left)\right|\hfill \\ \hfill & =\frac{1}{4\pi }|k\left(t\right){arctan}^{2}x\left(t\right)-k\left(t\right){arctan}^{2}y\left(t\right)-k\left(s\right){arctan}^{2}x\left(s\right)+k\left(s\right){arctan}^{2}y\left(s\right)|\hfill \\ \hfill & \le \frac{1}{2\pi }|{arctan}^{2}x\left(t\right)-{arctan}^{2}y\left(t\right)|+\frac{1}{2\pi }|{arctan}^{2}x\left(s\right)-{arctan}^{2}y\left(s\right)|\hfill \\ \hfill & \le \frac{1}{2}|arctanx\left(t\right)-arctany\left(t\right)|+\frac{1}{2}|arctanx\left(s\right)-arctany\left(s\right)|\hfill \\ \hfill & \le \frac{1}{2}|x\left(t\right)-y\left(t\right)|+\frac{1}{2}|x\left(s\right)-y\left(s\right)|\hfill \\ \hfill & \le \underset{\xi \in [s,t]}{sup}|x\left(\xi \right)-y\left(\xi \right)|;\hfill \end{array}$$

that is, $\left(V_2\right)$ holds. If x is a solution of Equation (10) with $x\left(0\right)=x_0\in [0,1]$, then

$$D^{+}V(t,x\left(t\right))=\underset{\eta \to 0^{+}}{lim\; sup}\frac{V(t+\eta ,x(t+\eta \left)\right)-V(t,x(t\left)\right)}{\eta }$$

• $t>s_0,t+\eta >s_0$

  $$\begin{array}{cc}\hfill D^{+}V(t,x\left(t\right))& =\underset{\eta \to 0^{+}}{lim\; sup}\frac{\frac{1}{2\pi }{(arctan{x}_0{e}^{-(t+\eta )})}^2-\frac{1}{2\pi }{(arctan{x}_0{e}^{-t})}^2}{\eta }\hfill \\ \hfill & =-\frac{x_0{e}^{-t}arctan\left(x_0{e}^{-t}\right)}{\pi (1+x_0^2{e}^{-2t})}\le 0,\hfill \end{array}$$
• $t\le s_0,t+\eta \le s_0$

  $$\begin{array}{cc}\hfill D^{+}V(t,x\left(t\right))& =\underset{\eta \to 0^{+}}{lim\; sup}\frac{\frac{1}{4\pi }{(arctan{x}_0{e}^{-(t+\eta )})}^2-\frac{1}{4\pi }{(arctan{x}_0{e}^{-t})}^2}{\eta }\hfill \\ \hfill & =-\frac{x_0{e}^{-t}arctan\left(x_0{e}^{-t}\right)}{2\pi (1+x_0^2{e}^{-2t})}\le 0,\hfill \end{array}$$
• $t\le s_0,t+\eta >s_0$

  $$\begin{array}{cc}\hfill D^{+}V(t,x\left(t\right))& =\underset{\eta \to 0^{+}}{lim\; sup}\frac{\frac{1}{2\pi }{(arctan{x}_0{e}^{-(t+\eta )})}^2-\frac{1}{4\pi }{(arctan{x}_0{e}^{-t})}^2}{\eta }\hfill \\ \hfill & =-\frac{x_0{e}^{-t}arctan\left(x_0{e}^{-t}\right)}{\pi (1+x_0^2{e}^{-2t})}\le 0;\hfill \end{array}$$

that is, the right derivative of V is non-positive along solution of Equation (10); hence, $\left(V_3\right)$ holds. Consider the monotone increasing functions $b,p:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ given by

$$b\left(s\right)=\frac{1}{2}{arctan}^{2}s,p\left(s\right)=\frac{5}{2}{arctan}^{2}s,s\in {\mathbb{R}}^{+},$$

so $b\left(0\right)=p\left(0\right)=0$ and $b\left(\right|z\left|\right)\le V(t,z)\le p\left(\right|z\left|\right),$ for every $(t,z)\in [0,+\infty )\times [-1.1]$. Whence it follows that the conditions of Theorem 1 are satisfied. So $x=0$ of Equation (10) is $(h_0,h)$-uniformly regularly Lipschitz stable. Then, it is also $(h_0,h)$-uniformly Lipschitz stable.

Theorem 2.

Assume that $F\in \mathcal{F}(\Omega ,\tilde{h})$, there exists a function $V:[t_0,+\infty )\times B_{\rho }\to {\mathbb{R}}^{+}$ that satisfies $\left(V_1\right)$–$\left(V_4\right)$ and the following condition:

$\left(V_5\right)$

There exists a constant $M>0$ such that $p\left(s\right)\le b\left(Ms\right)$ for all $s\in {\mathbb{R}}^{+}$.

Then the trivial solution $x=0$ of Equation (9) is globally $(h_0,h)$-uniformly Lipschitz stable.

Proof. 

Let $x:[\alpha ,\beta ]\to B_{\rho }$ be a solution of Equation (9) on $[\alpha ,\beta ]$; then,

$$x\left(s\right)=x\left(\alpha \right)+{\int }_{\alpha }^{s}DF(x\left(\tau \right),t),\mathrm{for}s\in [\alpha ,\beta ].$$

By Lemma 5, one has

$$\begin{array}{cc}\hfill V(t,x(t\left)\right)& \le V(\alpha ,x\left(\alpha \right))+2\underset{s\in [\alpha ,\beta ]}{sup}∥x\left(s\right)-x\left(\alpha \right)-{\int }_{\alpha }^{s}DF(x\left(\tau \right),t)∥\hfill \\ \hfill & \le p\left(h_0(\alpha ,x\left(\alpha \right))\right)\hfill \\ \hfill & \le b(M·h_0(\alpha ,x\left(\alpha \right))).\hfill \end{array}$$

Moreover, by $\left(V_4\right)$, we get

$$V(t,x(t\left)\right)\ge b\left(h\right(t,x\left(t\right)\left)\right),t\in [\alpha ,\beta ];$$

hence,

$$h(t,x\left(t\right))\le M·h_0(\alpha ,x\left(\alpha \right));$$

that is, $x=0$ of Equation (9) is globally $(h_0,h)$-uniformly Lipschitz stable. □

Remark 4.

Let $X={\mathbb{R}}^n$. Consider the following ODEs (Carathéodory equations):

$$\frac{dx}{dt}=f(x,t)$$

(12)

where $f:B_c\times [a,b]\to {\mathbb{R}}^n$, $f(0,t)=0$ and satisfies the following conditions:

$\left(i\right)$

$f(x,·)$ is a Lebesgue measure on $[a,b]$;

$\left(ii\right)$

$f(·,t)$ is continuous for every $t\in [a,b]$;

$\left(iii\right)$

There exists a Lebesuge measurable function $m:[a,b]\to \mathbb{R}$ such that ${\int }_a^{b}m\left(t\right)dt<+\infty$ and $\parallel f(x,t)\parallel \le m\left(t\right)$ for every $(x,t)\in B_c\times [a,b]$.

A solution $x:[\alpha ,\beta ]\to {\mathbb{R}}^n$ is called a solution of Equation (12) (in the sense of Carathéodory) if x is absolutely continuous on $[\alpha ,\beta ]\subset [a,b]$, $x\left(t\right)\in B_c$ for almost all $t\in [\alpha ,\beta ]$ and for almost all $t\in [\alpha ,\beta ]$, the equality $x^{\prime }=f(x\left(t\right),t)$ is satisfied.

Define

$$\begin{array}{c}\hfill F(x,t)={\int }_{t_0}^{t}f(x,t)dt,x\in B_c,t,t_0\in [a,b]\end{array}$$

(13)

By [27], a function x is a solution of Equation (12) (in the sense of Carathéodory) if and only if x is a solution of Equation (9), where F is given by Equation (13). Hence by Theorems 1 and 2, if $\left(V_1\right)$–$\left(V_4\right)$ hold, then $x=0$ of Equation (12) is $(h_0,h)$-uniformly Lipschitz stable; if $\left(V_1\right)$–$\left(V_5\right)$ hold, then $x=0$ of Equation (12) is globally $(h_0,h)$-uniformly Lipschitz stable.

Now, we consider the perturbed generalized ODEs:

$$d\tau =D\left[F\right(x,t)+P(t\left)\right],$$

(14)

where $F:B_c\times [t_0,+\infty )\to X$, $P:[t_0,+\infty )\to X$, $F\in \mathcal{F}(\Omega ,\tilde{h})$, and $P\in G^{-}([\alpha ,\beta ],X)$, $[\alpha ,\beta ]\subset [t_0,+\infty )$. Now, we introduce the following definition about Lipschitz stability with respect to perturbations and give the relationship between these two definitions of stability.

Definition 8.

Let $h_0,h\in \Gamma$; $x=0$ of Equation (9) is called $(h_0,h)$-uniformly regularly Lipschitz stable with respect to perturbations if there exist $M>0$ and $\delta >0$ such that

$$h_0(\alpha ,\tilde{x}\left(\alpha \right))<\delta and\underset{t\in [\alpha ,\beta ]}{sup}∥P\left(t\right)-P\left(\alpha \right)∥<\delta ;$$

then,

$$h(t,\tilde{x}\left(t\right))\le M·h_0(\alpha ,\tilde{x}\left(\alpha \right)),for allt\in [\alpha ,\beta ],$$

where $\tilde{x}:[\alpha ,\beta ]\to B_c$ is a solution of Equation (14).

Proposition 1.

$x=0$ of Equation (9) is $(h_0,h)$-uniformly regularly Lipschitz stable if and only if it is $(h_0,h)$-uniformly regularly Lipschitz stable with respect to perturbations.

Proof. 

$\left(i\right)$ Assume that $x=0$ of Equation (9) is $(h_0,h)$-uniformly regularly Lipschitz stable, and $\tilde{x}$ is a solution of Equation (14) over $[\alpha ,\beta ]$; then, $\tilde{x}$ is a regulated function on $[\alpha ,\beta ]$ and is left-continuous over $(\alpha ,\beta ]$. By Definitions 1 and 2, we obtain

$$\begin{array}{c}\hfill \tilde{x}\left(s\right)-\tilde{x}\left(\alpha \right)={\int }_{\alpha }^{s}DF(\tilde{x}\left(\tau \right),t)+P\left(s\right)-P\left(\alpha \right).\end{array}$$

(15)

Suppose $h_0(\alpha ,\tilde{x}\left(\alpha \right))<\delta$ and $\underset{s\in [\alpha ,\beta ]}{sup}∥P\left(s\right)-P\left(\alpha \right)∥<\delta ;$ so, Equation (15) implies that

$$\begin{array}{cc}\hfill & \underset{s\in [\alpha ,\beta ]}{sup}∥\tilde{x}\left(s\right)-\tilde{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\tilde{x}\left(\tau \right),t)∥\hfill \\ \hfill & =\underset{s\in [\alpha ,\beta ]}{sup}∥P\left(s\right)-P\left(\alpha \right)∥<\delta ;\hfill \end{array}$$

then, by Definition 7, we have

$$h(t,\tilde{x}\left(t\right))\le M·h_0(\alpha ,\tilde{x}\left(\alpha \right)),\mathrm{for} \mathrm{all}t\in [\alpha ,\beta ],$$

which implies that $x=0$ of Equation (9) is $(h_0,h)$-uniformly regularly Lipschitz stable with respect to perturbations.

$\left(ii\right)$ Conversely, if $x=0$ is $(h_0,h)$-uniformly regularly Lipschitz stable with respect to perturbations, let $\overline{x}:[\alpha ,\beta ]\to B_c$, $t_0\le \alpha <\beta <+\infty$ be a regulated function on $[\alpha ,\beta ]$ and left-continuous on $(\alpha ,\beta ]$ such that

$$h_0(\alpha ,\overline{x}\left(\alpha \right))<\delta$$

and

$$\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥<\delta ,$$

where $\delta =\delta \left(\epsilon \right)>0$ from Definition 7.

Define

$$P\left(s\right)=P\left(\alpha \right)+\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t),s\in [\alpha ,\beta ].$$

Then, for all $s_1,s_2\in [\alpha ,\beta ]$

$$\overline{x}\left(s_2\right)-\overline{x}\left(s_1\right)={\int }_{s_1}^{s_2}DF(\overline{x}\left(\tau \right),t)+P\left(s_2\right)-P\left(s_1\right);$$

hence, $\overline{x}$ is a solution of Equation (14) on $[\alpha ,\beta ]$. Moreover,

$$\begin{array}{cc}\hfill & \underset{s\in [\alpha ,\beta ]}{sup}∥P\left(s\right)-P\left(\alpha \right)∥\hfill \\ \hfill & =\underset{s\in [\alpha ,\beta ]}{sup}∥\overline{x}\left(s\right)-\overline{x}\left(\alpha \right)-{\int }_{\alpha }^{s}DF(\overline{x}\left(\tau \right),t)∥<\delta ;\hfill \end{array}$$

by Definition 8, we have

$$h(t,\overline{x}\left(t\right))\le M·h_0(\alpha ,\overline{x}\left(\alpha \right)),\mathrm{for} \mathrm{all}t\in [\alpha ,\beta ],$$

which implies that $x=0$ of Equation (9) is $(h_0,h)$-uniformly regularly Lipschitz stable. □

4. Applications

4.1. Measure Differential Equations (MDEs)

Consider the following MDEs and perturbed MDEs:

$$\begin{array}{cc}\hfill & Dx=f(x,t)Dg,t\ge t_0\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & Dx=f(x,t)Dg+p\left(t\right)Du,t\ge t_0,\hfill \end{array}$$

(17)

where $Dx$, $Dg$, and $Du$ are the distributional derivatives of x, g, and u in the sense of L. Schwartz, respectively; $f:O\times [t_0,+\infty )\to {\mathbb{R}}^n$, $O\subset {\mathbb{R}}^n$ be an open set, and $p:[t_0,+\infty )\to {\mathbb{R}}^n$, $g,u:[t_0,+\infty )\to \mathbb{R}$ are nondecreasing and left-continuous functions. Assume that $f(0,t)=0$ for every $t\in [t_0,+\infty )$ so that $x=0$ is a solution of Equation (16) on $[t_0,+\infty )$.

By [36], Equations (16) and (17) are equivalent to the integral form of MDEs of the type

$$\begin{array}{cc}\hfill & x\left(t\right)=x\left(t_0\right)+{\int }_{t_0}^{t}f(x\left(s\right),s)dg\left(s\right),t\ge t_0,\hfill \\ \hfill & x\left(t\right)=x\left(t_0\right)+{\int }_{t_0}^{t}f(x\left(s\right),s)dg\left(s\right)+{\int }_{t_0}^{t}p\left(s\right)du\left(s\right),t\ge t_0,\hfill \end{array}$$

where the integrals in the right-hand side are in the sense of Lebesgue–Stieltjes. Here, we consider the integrals in the general sense, that is, the integrals in the right-hand side of the above integral equations are in the sense of Kurzweil–Henstock–Stieltjes.

Now, we make the assumptions:

$\left(H_1\right)$

The following Kurzweil–Henstock–Stieltjes integral

$${\int }_{s_1}^{s_2}f(x\left(t\right),t)dg\left(t\right)$$

exists for all $x\in G([t_0,+\infty ),O)$ and all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(H_2\right)$

There exists a locally Kurzweil–Henstock–Stieltjes integrable function $M:[t_0,+\infty )\to {\mathbb{R}}^{+}$ such that

$$\left|{\int }_{s_1}^{s_2}f(x\left(s\right),s)dg\left(s\right)\right|\le {\int }_{s_1}^{s_2}M\left(s\right)dg\left(s\right)$$

for all $x\in G([t_0,+\infty ),O)$ and all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(H_3\right)$

There exists a locally Kurzweil–Henstock–Stieltjes integrable function $L:[t_0,+\infty )\to {\mathbb{R}}^{+}$ such that

$$\begin{array}{cc}\hfill & \left|{\int }_{s_1}^{s_2}[f(y\left(s\right),s)-f(z\left(s\right),s)]dg\left(s\right)\right|\hfill \\ \hfill & \le {\parallel y-z\parallel }_{[t_0,+\infty )}·{\int }_{s_1}^{s_2}L\left(s\right)dg\left(s\right),\hfill \end{array}$$

for all $y,z\in G_0([t_0,+\infty ),O)$ and all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(H_4\right)$

The following Kurzweil–Henstock–Stieltjes integral

$${\int }_{s_1}^{s_2}p\left(s\right)du\left(s\right)$$

exists for all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(H_5\right)$

There exists a locally Kurzweil–Henstock–Stieltjes integrable function $K:[t_0,+\infty )\to {\mathbb{R}}^{+}$ such that

$$\left|{\int }_{s_1}^{s_2}p\left(s\right)du\left(s\right)\right|\le {\int }_{s_1}^{s_2}K\left(s\right)du\left(s\right)$$

for all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

By [28], if $\left(H_1\right)$–$\left(H_5\right)$ are satisifed, the solution x of Equation (17) is regulated and left-continuous, that is, $x\in G^{-}([t_0,+\infty ),{\mathbb{R}}^n)$.

Define

$$\begin{array}{cc}\hfill & F(x,t)={\int }_{t_0}^{t}f(x\left(s\right),s)dg\left(s\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & P\left(t\right)={\int }_{t_0}^{t}p\left(s\right)du\left(s\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & G(x,t)=F(x,t)+P\left(t\right).\hfill \end{array}$$

(20)

Consider the following generalized ODEs and perturbed generalized ODEs:

$$\begin{array}{cc}\hfill & \frac{dx}{d\tau }=DF(x,t),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \frac{dx}{d\tau }=DG(x,t)=D[F(x,t)+P\left(t\right)],\hfill \end{array}$$

(22)

where the functions $F(x,t)$ and $P\left(t\right)$ are defined in (18) and (19), respectively.

Let ${\tilde{h}}_1,{\tilde{h}}_2:[t_0,+\infty )\to \mathbb{R}$ be defined by

$$\begin{array}{cc}\hfill & {\tilde{h}}_1\left(t\right)={\int }_{t_0}^t[M\left(s\right)+L\left(s\right)]dg\left(s\right),\hfill \\ \hfill & {\tilde{h}}_2\left(t\right)={\int }_{t_0}^t[M\left(s\right)+L\left(s\right)]dg\left(s\right)+{\int }_{t_0}^{t}K\left(s\right)du\left(s\right);\hfill \end{array}$$

hence, ${\tilde{h}}_1$, ${\tilde{h}}_2$ are nondecreasing and left-continuous. Moreover, by [28], we obtain $F\in \mathcal{F}(\Omega ,{\tilde{h}}_1)$, $G\in \mathcal{F}(\Omega ,{\tilde{h}}_2)$, and $\Omega =O\times [t_0,+\infty )$. The following results show a relationship between the solutions of Equation (17) and the solutions of Equation (22), where G is given by (20).

Theorem 3

([28]). Assume that $\left(H_1\right)$–$\left(H_5\right)$ hold. The function $x:I\to {\mathbb{R}}^n$ is a solution of the Equation (17) on $I\subset [t_0,+\infty )$ if and only if x is a solution of the generalized ODEs:

$$\begin{array}{c}\hfill \frac{dx}{d\tau }=DG(x,t)\end{array}$$

on I with the function G defined in (20).

Here, we define the concepts and give the sufficient criteria for Lipschitz stability of the trivial solution for Equation (16) using theories of generalized ODEs, where $E_c=\{x\in {\mathbb{R}}^n:\left|x\right|\le c\}$, $c>0$.

Definition 9.

Let $h,h_0\in \Gamma$ and $x=0$ of Equation (16) be called:

• $(h_0,h)$-uniformly integrally Lipschitz stable  if there exist $M>0$ and $\delta >0$ such that, for every solution $\tilde{x}(t,\alpha ,{\tilde{x}}_0)$ of the perturbed MDEs Equation (17) with $\tilde{x}\left(\alpha \right)={\tilde{x}}_0$, the inequality

  $$h(t,\tilde{x}\left(t\right))\le M·h_0(\alpha ,{\tilde{x}}_0),for allt\in [\alpha ,\beta ]\subset [t_0,+\infty )$$

  holds if

  $$h_0(\alpha ,{\tilde{x}}_0)<\delta and\underset{t\in [\alpha ,\beta ]}{sup}\left|{\int }_{\alpha }^{t}p\left(s\right)du\left(s\right)\right|<\delta .$$
• $(h_0,h)$-uniformly Lipschitz stable  if there exist $M>0$, $\delta >0$ and $x:[\alpha ,\beta ]\to E_c$, $t_0\le \alpha <\beta <+\infty$ is a solution of MDEs Equation (16) on $[\alpha ,\beta ]$, then

  $$h_0(\alpha ,x\left(\alpha \right))<\delta$$

  implies

  $$h(t,x\left(t\right))<M·h_0(\alpha ,x\left(\alpha \right)),for allt\in [\alpha ,\beta ].$$
• globally $(h_0,h)$-uniformly Lipschitz stable  if there exists $M>0$ and $x:[\alpha ,\beta ]\to E_c$, $t_0\le \alpha <\beta <+\infty$ is a solution of MDEs Equation (16) on $[\alpha ,\beta ]$, then

  $$h(t,x\left(t\right))<M·h_0(\alpha ,x\left(\alpha \right)),for allt\in [\alpha ,\beta ].$$

Remark 5.

Note that if $x=0$ of Equation (16) is $(h_0,h)$-uniformly integrally Lipschitz stable, then it is $(h_0,h)$-uniformly Lipschitz stable.

Proposition 2.

Assume that $\left(H_1\right)$–$\left(H_5\right)$ hold, then $x=0$ of Equation (16) is $(h_0,h)$-uniformly integrally Lipschitz stable if and only if $x=0$ of Equation (21) is $(h_0,h)$-uniformly regularly Lipschitz stable.

Proof. 

$\left(i\right)$ Suppose $x=0$ of Equation (16) is $(h_0,h)$-uniformly integrally Lipschitz stable; then, there exist $M>0$, $\delta >0$ such that if $h_0(\alpha ,{\tilde{x}}_0)<\delta$ and

$$\underset{t\in [\alpha ,\beta ]}{sup}\left|{\int }_{\alpha }^{t}p\left(s\right)du\left(s\right)\right|<\delta ,$$

then

$$h(t,\tilde{x}\left(t\right))\le M·h_0(\alpha ,{\tilde{x}}_0),t\in [\alpha ,\beta ]\subset [t_0,+\infty ),$$

(23)

where $\tilde{x}\left(t\right)=\tilde{x}(t,\alpha ,{\tilde{x}}_0)$ is a solution of Equation (17) with $\tilde{x}\left(\alpha \right)={\tilde{x}}_0$.

Then, we will prove that $x=0$ of Equation (21) is $(h_0,h)$-uniformly regularly Lipschitz stable with respect to perturbations, then the conclusion will be followed by Proposition 1. By Theorem 3, $\tilde{x}\left(t\right)$ is a solution of Equation (22) with $\tilde{x}\left(\alpha \right)={\tilde{x}}_0$; moreover, for the above $\delta >0$, assume that

$$h_0(\alpha ,{\tilde{x}}_0)<\delta and\underset{t\in [\alpha ,\beta ]}{sup}∥P\left(t\right)-P\left(\alpha \right)∥<\delta ,$$

where $P\in G^{-}([\alpha ,\beta ],{\mathbb{R}}^n)$ is defined in (19). Since

$$\begin{array}{cc}\hfill & ∥P\left(t\right)-P\left(\alpha \right)∥\hfill \\ \hfill & =\underset{t\in [\alpha ,\beta ]}{sup}\left|{\int }_{t_0}^{t}p\left(s\right)du\left(s\right)-{\int }_{t_0}^{\alpha }p\left(s\right)du\left(s\right)\right|\hfill \\ \hfill & =\underset{t\in [\alpha ,\beta ]}{sup}\left|{\int }_{\alpha }^{t}p\left(s\right)du\left(s\right)\right|,\hfill \end{array}$$

then

$$\underset{t\in [\alpha ,\beta ]}{sup}\left|{\int }_{\alpha }^{t}p\left(s\right)du\left(s\right)\right|<\delta ;$$

by (23), one has

$$h(t,\tilde{x}\left(t\right))\le M·h_0(\alpha ,{\tilde{x}}_0),\mathrm{for} \mathrm{all}t\in [\alpha ,\beta ]\subset [t_0,+\infty ),$$

which implies that $x=0$ of Equation (21) is $(h_0,h)$-uniformly regularly Lipschitz stable with respect to perturbations. Hence, $x=0$ of Equation (21) is $(h_0,h)$-uniformly regularly Lipschitz stable by Proposition 1.

$\left(ii\right)$ Conversely, the proof is similar to $\left(i\right)$; here we omit it. □

Theorem 4.

Suppose $\left(H_1\right)$–$\left(H_5\right)$ hold and $U:[t_0,+\infty )\times E_{\rho }\to {\mathbb{R}}^{+}$ satisfies the following conditions:

$\left(U_1\right)$

For each left-continuous function $z:[\alpha ,\beta ]\to E_{\rho }$ on $(\alpha ,\beta ]$, $[\alpha ,\beta ]\subset [t_0,+\infty )$, the function $t\to U(t,z(t\left)\right)$ is left-continuous on $(\alpha ,\beta ]$.

$\left(U_2\right)$

For all regulated functions $x,y:[\alpha ,\beta ]\to E_{\rho }$, $[\alpha ,\beta ]\subset [t_0,+\infty )$ and every $\alpha \le s<t\le \beta$, we have

$$\begin{array}{cc}\hfill & |U(t,x\left(t\right))-U(t,y\left(t\right))-U(s,x\left(s\right))+U(s,y\left(s\right))|\le \underset{\xi \in [s,t]}{sup}|x\left(\xi \right)-y\left(\xi \right)|.\hfill \end{array}$$

$\left(U_3\right)$

For the solution of $x:[t_0,\omega (t_0,x_0))\to E_{\rho }$ of Equation (16) with $x\left(t_0\right)=x_0$, the function $t\to U(t,x(t\left)\right)$ is nonincreasing along every solution x.

$\left(U_4\right)$

There exist two monotone increasing functions $b,p:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $b\left(0\right)=p\left(0\right)=0$,

$$b\left(h(t,z)\right)\le U(t,z)\le p\left(h_0(t,z)\right),$$

for every $(t,z)\in [t_0,+\infty )\times E_{\rho }$, $h_0,h\in \Gamma .$

Then, the trivial solution $x=0$ of MDEs Equation (16) is $(h_0,h)$-uniformly Lipschitz stable.

Proof. 

Define the function F as in (18); then, $F\in \mathcal{F}(\Omega ,{\tilde{h}}_1)$. It is easy to see that the function U satisfies the conditions $\left(V_1\right)$, $\left(V_2\right)$, $\left(V_3\right)$, and $\left(V_4\right)$; that is, U satisfies all the conditions of Theorem 1. By Theorem 3, $x=0$ of Equation (21) is $(h_0,h)$ uniformly regularly Lipschitz stable. Thus, by Proposition 2, $x=0$ of Equation (16) is $(h_0,h)$-uniformly integrally Lipschitz stable; therefore, is $(h_0,h)$-uniformly Lipschitz stable. □

By Theorems 2 and 3, we have

Theorem 5.

Assume that $\left(H_1\right)$–$\left(H_5\right)$ hold and there exists a function $U:[t_0,+\infty )\times E_{\rho }\to {\mathbb{R}}^{+}$ that satisfies the conditions $\left(U_1\right)$–$\left(U_4\right)$ and $\left(V_5\right)$, then $x=0$ of MDEs Equation (16) is globally $(h_0,h)$-uniformly Lipschitz stable.

4.2. Impulsive Differential Equations (IDEs)

Consider the following IDEs and perturbed IDEs:

$$\left\{\begin{array}{c}\dot{x}=f(x,t),t\ne t_k,\hfill \\ \Delta x\left(t\right)=I_k\left(x\left(t\right)\right),k=1,2,\dots ,\hfill \end{array}\right.$$

(24)

and

$$\left\{\begin{array}{c}\dot{x}=f(x,t)+p\left(t\right),t\ne t_k,\hfill \\ \Delta x\left(t\right)=I_k\left(x\left(t\right)\right),k=1,2,\dots ,\hfill \end{array}\right.$$

(25)

satisfy the initial condition $x\left(t_0\right)=x_0$, where $t_k$ and $k=1,2,\dots$ are pre-assigned moments of impulse, $t_0<t_k$, $t_k<t_{k+1}$ for each $k=1,2,\dots$ and $t_k\to +\infty$ as $k\to +\infty$. $\Delta x\left(t_k\right):=x(t_k+)-x(t_k-)=x(t_k+)-x\left(t_k\right)$, and the impulsive operator $I_k$ are bounded continuous functions from ${\mathbb{R}}^n$ to ${\mathbb{R}}^n$, $k=1,2,\dots$. Without the loss of generality, here we assume that the initial time $t_0$ is not a time of impulse of system Equations (24) and (25) and $f(0,t)=0$, $I_k\left(0\right)=0$ for every $t\in [t_0,+\infty )$, $k=1,2,\dots$, so that $x=0$ is a solution of Equation (24) on $[t_0,+\infty )$.

First, we make the assumptions:

$\left(A_1\right)$

The following Lebesgue integral

$${\int }_{s_1}^{s_2}f(x\left(t\right),t)dt$$

exists for all $x\in G^{-}([t_0,+\infty ),{\mathbb{R}}^n)$ and all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(A_2\right)$

There exists a locally Lebesgue integrable function ${\lambda }_1:[t_0,+\infty )\to {\mathbb{R}}^{+}$ such that

$$\left|{\int }_{s_1}^{s_2}f(x\left(s\right),s)ds\right|\le {\int }_{s_1}^{s_2}{\lambda }_1\left(s\right)ds$$

for all $x\in G([t_0,+\infty ),{\mathbb{R}}^n)$ and all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(A_3\right)$

There exists a locally Lebesgue integrable function ${\lambda }_2:[t_0,+\infty )\to \mathbb{R}$ such that

$$\begin{array}{cc}\hfill & \left|{\int }_{s_1}^{s_2}[f(x\left(s\right),s)-f(y\left(s\right),s)]ds\right|\hfill \\ \hfill & \le {\parallel x-y\parallel }_{[t_0,+\infty )}·{\int }_{s_1}^{s_2}{\lambda }_2\left(s\right)ds\hfill \end{array}$$

for all $x,y\in G_0([t_0,+\infty ),{\mathbb{R}}^n)$ and all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(A_4\right)$

The following Lebesgue integral

$${\int }_{s_1}^{s_2}p\left(s\right)ds$$

exists for all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(A_5\right)$

There exists a locally Lebesgue integrable function ${\lambda }_3:[t_0,+\infty )\to {\mathbb{R}}^{+}$ such that

$$\left|{\int }_{s_1}^{s_2}p\left(s\right)ds\right|\le {\int }_{s_1}^{s_2}{\lambda }_3\left(s\right)ds$$

for all $s_1,s_2\in [t_0,+\infty )$, $s_1\le s_2$.

$\left(B_1\right)$

There exists a constant $K_1>0$ such that

$$|I_k\left(x\right)|\le K_1$$

for all $k=1,\dots ,x\in G([t_0,+\infty ),{\mathbb{R}}^n).$

$\left(B_2\right)$

There exists a constant $K_2>0$ such that

$$|I_k\left(x\right)-I_k\left(y\right)|\le K_2·{\parallel x-y\parallel }_{[t_0,+\infty )}$$

for all $k=1,\dots ,x,y\in G_0([t_0,+\infty ),{\mathbb{R}}^n).$

Remark 6.

$\left(i\right)$

Note that impulsive phenomena appear widely in the real world, such as many evolution processes that are subject to sudden perturbations, changes at certain moments, human harvesting, or stocking often happens instantaneously, which should be described by impulsive functions instead of continuous functions. IDEs arise from real world problems to describe the dynamics of processes in which discontinuous jumps suddenly occur. They are extensively used as models in biology, chemistry, physics, engineering, and other sciences; one can see [2,6,19,37,38] for more details.

$\left(ii\right)$

The mapping $t\to f(x,t)$ does not require piecewise continuous functions, and it only assumes locally Lebesgue integrability. Note that the Lebesgue integral can be replaced by the Kurzweil–Henstock integral [27], so the mapping may not only be discontinuous functions, but it may also be of unbounded variation.

It is easy to see that Equations (24) and (25) are equivalent to the following “integral” equations:

$$\begin{array}{cc}\hfill x\left(t\right)& =x\left(t_0\right)+{\int }_{t_0}^{t}f(x\left(s\right),s)ds+\sum _{t_0<t_k\le t}{I}_k\left(x\left(t_k\right)\right),t\ge t_0,\hfill \\ \hfill x\left(t\right)& =x\left(t_0\right)+{\int }_{t_0}^{t}f(x\left(s\right),s)ds+{\int }_{t_0}^{t}p\left(s\right)ds\hfill \\ \hfill & +\sum _{t_0<t_k\le t}{I}_k\left(x\left(t_k\right)\right),t\ge t_0,\hfill \end{array}$$

where the integrals in the right hand side exist in the Lebesgue sense.

Remark 7.

The sum

$$\sum _{t_0<t_k\le t}{I}_k\left(x\left(t_k\right)\right)$$

can be rewritten as

$$\sum _{k=1}^{+\infty }{H}_{t_k}\left(t\right)I_k\left(x\left(t_k\right)\right),$$

where $H_{t_k}$ denotes the left-continuous Heavyside function concentrated at $t_k$, that is

$$H_{t_k}\left(t\right)=\left\{\begin{array}{cc}0\hfill & for{t}_0\le t\le t_k,\hfill \\ 1\hfill & fort>t_k.\hfill \end{array}\right.$$

By Remark 7, Equations (24) and (25) are equivalent to

$$\begin{array}{cc}\hfill x\left(t\right)& =x\left(t_0\right)+{\int }_{t_0}^{t}f(x\left(s\right),s)ds+\sum _{k=1}^{+\infty }{H}_{t_k}\left(t\right)I_k\left(x\left(t_k\right)\right).\hfill \\ \hfill x\left(t\right)& =x\left(t_0\right)+{\int }_{t_0}^{t}f(x\left(s\right),s)ds+{\int }_{t_0}^{t}p\left(s\right)ds\hfill \\ \hfill & +\sum _{k=1}^{+\infty }{H}_{t_k}\left(t\right)I_k\left(x\left(t_k\right)\right),\hfill \end{array}$$

where the integrals are in the Lebesgue sense.

Define

$$\begin{array}{cc}\hfill & F(x,t)={\int }_{t_0}^{t}f(x\left(s\right),s)ds+\sum _{k=1}^{+\infty }{H}_{t_k}\left(t\right)I_k\left(x\left(t_k\right)\right)\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & P\left(t\right)={\int }_{t_0}^{t}p\left(s\right)ds,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & G(x,t)=F(x,t)+P\left(t\right).\hfill \end{array}$$

(28)

Consider the following generalized ODEs and perturbed generalized ODEs:

$$\begin{array}{cc}\hfill & \frac{dx}{d\tau }=DF(x,t),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \frac{dx}{d\tau }=DG(x,t)=D[F(x,t)+P\left(t\right)],\hfill \end{array}$$

(30)

where the functions $F(x,t)$ and $P\left(t\right)$ are defined in (26) and (27), respectively.

Let ${\tilde{h}}_3,{\tilde{h}}_4:[t_0,+\infty )\to \mathbb{R}$ be defined by

$$\begin{array}{cc}\hfill {\tilde{h}}_3\left(t\right)& ={\int }_{t_0}^t\left[{\lambda }_1\left(s\right)+{\lambda }_2\left(s\right)\right]ds+max\{K_1,K_2\}\sum _{k=1}^{+\infty }{H}_{t_k}\left(t\right),\hfill \\ \hfill {\tilde{h}}_4\left(t\right)& ={\int }_{t_0}^t\left[{\lambda }_1\left(s\right)+{\lambda }_2\left(s\right)+{\lambda }_3\left(s\right)\right]ds\hfill \\ \hfill & +max\{K_1,K_2\}\sum _{k=1}^{+\infty }{H}_{t_k}\left(t\right),\hfill \end{array}$$

It is easy to verify that ${\tilde{h}}_3$, ${\tilde{h}}_4$ are nondecreasing and left-continuous. We also know from [27] that $F\in \mathcal{F}(\Omega ,{\tilde{h}}_3)$ and $G\in \mathcal{F}(\Omega ,{\tilde{h}}_4)$.

By [27], we have the following consequences describing a relationship between the solutions of Equation (25) and the solutions of Equation (30), where G is given by (28).

Theorem 6

([27]). Assume that $\left(A_1\right)$–$\left(A_5\right)$ and $\left(B_1\right)$–$\left(B_2\right)$ hold. The function $x:I\to {\mathbb{R}}^n$ is a solution of Equation (25) on $I\subset [t_0,+\infty )$ if and only if x is a solution of the generalized ODEs:

$$\begin{array}{c}\hfill \frac{dx}{d\tau }=DG(x,t)\end{array}$$

on I with the function G defined in (28).

Remark 8.

The above theorem shows the correspondence between the solutions of IDEs and the solutions of generalized ODEs. On the other hand, IDEs can always transformed to MDEs. IDEs are a particular case of MDEs; one can see [39] for more details.

Here, we present the notions and give the criteria on the Lipschitz stability for Equation (24) using theories of generalized ODEs.

Definition 10.

Let $h,h_0\in \Gamma$ and $x=0$ of Equation (24) be called:

• $(h_0,h)$-uniformly integrally Lipschitz stable if there exist $M>0$ and $\delta >0$ such that, for every solution $\tilde{x}(t,t_0,{\tilde{x}}_0)$ of the perturbed IDEs Equation (25) with $\tilde{x}\left(t_0\right)={\tilde{x}}_0$, the inequality

  $$h(t,\tilde{x}\left(t\right))\le M·h_0(t_0,{\tilde{x}}_0),for allt\in [t_0,\beta ]$$

  holds if

  $$h_0(t_0,{\tilde{x}}_0)<\delta and\underset{t\in [t_0,\beta ]}{sup}\left|{\int }_{t_0}^{t}p\left(s\right)ds\right|<\delta .$$
• $(h_0,h)$-uniformly Lipschitz stable if there exist $M>0$, $\delta >0$ and $x:[t_0,\beta ]\to E_c$, $t_0<\beta <+\infty$ is a solution of IDEs Equation (24) on $[t_0,\beta ]$, then

  $$h_0(t_0,x\left(t_0\right))<\delta$$

  implies

  $$h(t,x\left(t\right))<M·h_0(t_0,x\left(t_0\right)),for allt\in [t_0,\beta ].$$
• globally$(h_0,h)$-uniformly Lipschitz stable if there exists $M>0$ and $x:[t_0,\beta ]\to E_c$, $t_0<\beta <+\infty$ is a solution of IDEs Equation (24) on $[t_0,\beta ]$, then

  $$h(t,x\left(t\right))<M·h_0(t_0,x\left(t_0\right)),for allt\in [t_0,\beta ].$$

Remark 9.

Note that if $x=0$ of Equation (24) is $(h_0,h)$-uniformly integrally Lipschitz stable, then it is $(h_0,h)$-uniformly Lipschitz stable.

Since

$$\begin{array}{cc}\hfill ∥P\left(t\right)-P\left(t_0\right)∥& =\underset{t\in [t_0,\beta ]}{sup}\left|{\int }_{t_0}^{t}p\left(s\right)ds\right|,\hfill \end{array}$$

is similar to Proposition 2, by Proposition 1, we have

Proposition 3.

Assume that $\left(A_1\right)$–$\left(A_5\right)$ and $\left(B_1\right)$–$\left(B_2\right)$ hold. Then, $x=0$ of Equation (24) is $(h_0,h)$-uniformly integrally Lipschitz stable if and only if $x=0$ of Equation (29) is $(h_0,h)$-uniformly regularly Lipschitz stable.

Theorem 7.

Suppose $\left(A_1\right)$–$\left(A_5\right)$ and $\left(B_1\right)$–$\left(B_2\right)$ hold and $W:[t_0,+\infty )\times E_{\rho }\to {\mathbb{R}}^{+}$ satisfies the following conditions:

$\left(W_1\right)$

 For each left-continuous function $z:[\alpha ,\beta ]\to E_{\rho }$ on $(\alpha ,\beta ]$, $[\alpha ,\beta ]\subset [t_0,+\infty )$, the function $t\to W(t,z(t\left)\right)$ is left-continuous on $(\alpha ,\beta ]$.

$\left(W_2\right)$

 For every regulated $x,y:[\alpha ,\beta ]\to E_{\rho }$, $[\alpha ,\beta ]\subset [t_0,+\infty )$, and every $\alpha \le s<t\le \beta$, we have

$$\begin{array}{cc}\hfill & |W(t,x\left(t\right))-W(t,y\left(t\right))-W(s,x\left(s\right))+W(s,y\left(s\right))|\le \underset{\xi \in [s,t]}{sup}|x\left(\xi \right)-y\left(\xi \right)|.\hfill \end{array}$$

$\left(W_3\right)$

 For every solution $x:[t_0,\omega (t_0,x_0))\to E_{\rho }$ of Equation (24) with $x\left(t_0\right)=x_0$, the function $t\to W(t,x(t\left)\right)$ is nonincreasing along every solution x.

$\left(W_4\right)$

 There exist two monotone increasing functions $b,p:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $b\left(0\right)=p\left(0\right)=0$ and

$$b\left(h(t,z)\right)\le W(t,z)\le p\left(h_0(t,z)\right)$$

for every $(t,z)\in [t_0,+\infty )\times E_{\rho },h_0,h\in \Gamma .$

Then, the trivial solution $x=0$ of IDEs Equation (24) is $(h_0,h)$-uniformly Lipschitz stable.

Proof. 

Define the function F as in (26), then $F\in \mathcal{F}(\Omega ,{\tilde{h}}_3)$, and W satisfies all the conditions of Theorem 1; hence, $x=0$ of Equation (29) is $(h_0,h)$-uniformly regularly Lipschitz stable. Thus, by Proposition 3, $x=0$ of Equation (24) is $(h_0,h)$-uniformly integrally Lipschitz stable; therefore, is $(h_0,h)$-uniformly Lipschitz stable. □

By Theorems 2 and 6, we have

Theorem 8.

Assume that $\left(A_1\right)$–$\left(A_5\right)$ and $\left(B_1\right)$–$\left(B_2\right)$ hold and there exists a function $W:[t_0,+\infty )\times E_{\rho }\to {\mathbb{R}}^{+}$ that satisfies the conditions $\left(W_1\right)$–$\left(W_4\right)$ and $\left(V_5\right)$; then, $x=0$ of IDEs Equation (24) is globally $(h_0,h)$-uniformly Lipschitz stable.

5. Discussion and Conclusions

In this paper, our results extend and improve related contributions to the stability of measure differential equations and impulsive differential equations. To unify a variety of stability notions and offer a general framework for investigation, the stability in terms of two measures has been investigated. On the other hand, the function $f(t,x)$ in the differential equations does not need to be continuous; it only assumes local Lebesgue integrability, which weakens the conditions for classical results. Note that the Lebesgue integral can be replaced by the Kurzweil–Henstock integral; hence, it may also be of unbounded variation.

We give the applications of Kurzweil equations in autonomous ODEs and Carathéodory equations, which give the sufficient criteria for the Lipschitz stability in terms of two measures of trivial solutions for these equations, but some conditions are relatively stronger. The question of how to weaken relevant conditions and how to apply conclusions to mathematical models in biology is a challenging problem. These are the directions of our future research.