Linearized Stability Analysis of Nonlinear Delay Differential Equations with Impulses

Abstract

This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses. Furthermore, the classical results of linearized stability for nonlinear semigroups are generalized to periodic DDEs with impulses. A significant challenge arises from the need for a discontinuous initial function to obtain periodic solutions. To address this, first-kind discontinuous spaces $\mathcal{R}([a,b],{\mathbb{R}}^n)$ are introduced for defining DDEs with impulses, providing key existence and uniqueness results. This study also establishes linear stability results by linearizing the Poincaré operator for DDEs with impulses. Additionally, the stability properties of equilibrium solutions for these equations are analyzed, highlighting their importance due to the wide range of applications in various scientific fields.

1. Introduction

Impulsive systems have found extensive applications in physics, biology, medicine, and other scientific fields [1,2,3]. The theory concerning impulsive differential systems has significantly developed. However, the general theory regarding the stability of impulsive delay differential equations (DDEs) remains inadequately explored [4]. This study aimed to investigate the linearized stability of nonlinear delay differential equations with impulses. Specifically, the focus was on this type of DDE with impulses, formulated as follows:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dx(t)}{dt}}& =& f(x_t),\mathrm{a}.\mathrm{e}.t\in {\mathbb{R}}^{+},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill x(jT)& =& x(j{T}^{-}),x(j{T}^{+})=h(x(jT)),\forall j\ge 1,j\in \mathbb{N}\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill x_0& =& \varphi ,x(0^{+})=\varphi (0^{+})\in {\mathbb{R}}^n,\hfill \end{array}$$

(3)

We denote by $x(j{T}^{-})$ and $x(j{T}^{+})$ the left limit and right limit of $x(t)$ as t tends to $jT$, respectively. Here, T is a positive constant, and

$$x_t(\theta )=x(t+\theta ),\theta \in [-r,0].$$

Additionally, function f satisfies conditions $(H_1)$ and $(H_2)$, and e function h satisfies condition $(H_3)$; see [4,5] for more details about such conditions.

This type of problem has already been investigated for differential equations without delay for various types of impulses [6,7,8,9,10,11]. This problem began to receive attention, mostly in eastern Europe, between 1960 and 1970, and received significant attention during the 1970s. Later on, several investigations and important monographs appeared with more details, emphasizing the importance of studying such systems.

In recent years, many investigations have focused on applications to life sciences, such as the periodic treatment of some biomedical applications, where the impulses correspond to the administration of a drug treatment at certain given times [2].

The aim of this study was to extend the classical results on the existence of periodic solutions for ordinary differential equations to delay differential equations. Moreover, I aimed to extend the classical result of linearized stability for nonlinear semigroups by Desch and Schappacher [12] to such periodic delay differential equations with impulses. The main difficulty arises from the fact that if $x(t)$ is discontinuous at different points $jT$, where $j\ge 1$, then $x_t(\theta )$ is discontinuous at $T-\theta ,2T-\theta ,\dots ,NT-\theta ,\dots$, where $t\in {\mathbb{R}}^{+}$ and $\theta \in [-r,0]$. Since we are looking for periodic solutions in the case of delay differential equations with impulses, we need to start with regulated initial functions. For more details about regulated functions, see [4,13,14]. To the best of my knowledge, no other study has taken this approach. My method requires working within a regulated space and involves intensive functional analysis, which I have detailed thoroughly in my paper.

To define delay differential equations with impulses, it is convenient to introduce the regulated space $\mathcal{R}([a,b],{\mathbb{R}}^n)$, where we to not need to specify the points of discontinuity. For each $a,b\in \mathbb{R}$, with $a<b$, we denote the regulated space as demonstrated in [4] as

$$\mathcal{R}([a,b],{\mathbb{R}}^n)=\{\phi :[a,b]\to {\mathbb{R}}^n\mid \phi ,\mathrm{which}\mathrm{has}\mathrm{left}\mathrm{and}\mathrm{right}\mathrm{limits}\mathrm{at}\mathrm{every}\mathrm{point}[a,b]\},$$

and the shift space

$$\mathcal{R}=\mathcal{R}([-r,0],{\mathbb{R}}^n)=\{\phi :[-r,0]\to {\mathbb{R}}^n\mid \phi \mathrm{has}\mathrm{left}\mathrm{and}\mathrm{right}\mathrm{limits}\mathrm{at}\mathrm{every}\mathrm{point}\mathrm{of}[-r,0]\}.$$

I introduce the stability properties of the equilibrium solution for the nonlinear problem defined by Equations (1)–(3). I consider the linearization of the nonlinear problem at the equilibrium function $\psi$ with impulses. The stability analysis of the equilibrium is conducted using the Fréchet derivative of the Poincaré operator J at $\psi$. Conditions are established under which the Fréchet derivative of J at $\psi$, denoted as $J_T^{\prime }(\psi )$, is equal to the Poincaré operator $U_T$ of the linearized problem defined by Equations (13) and (14). This investigation provides valuable insights into the behavior of the equilibrium under small perturbations.

Next, I explore the Fréchet derivative of the Poincaré operator J at the equilibrium function $\psi$. By analyzing the linearized problem defined by Equations (13) and (14), the conditions under which $J_T^{\prime }(\psi )$ is equal to the Poincaré operator $U_T$ can be established. This analysis provides a rigorous framework for studying the stability of equilibria in systems governed by delay differential equations with impulses.

Let us then investigate the decomposition of the Poincaré operator $U_T$ into closed linear manifolds of ${\mathcal{R}}^{+}=\mathcal{R}\times {\mathbb{R}}^n$. By applying the decomposition theorem, I analyze the spectral properties of the decomposed operators. This decomposition provides valuable insights into the spectral behavior of the Poincaré operator, which in turn sheds light on the stability and long-term behavior of equilibria in delay differential equations with impulses.

Finally, I conduct a detailed spectral analysis of the decomposed Poincaré operator to understand its behavior and stability properties. By separating the spectrum of the operator into distinct parts, we gain a deeper understanding of its spectral behavior. This analysis enables the identification ofthehte conditions under which the equilibrium solution may lose stability under small perturbations, providing crucial insights for the analysis of delay differential equations with impulses. Additionally, I present an example to illustrate the stability analysis under different circumstances.

2. Existence and Uniqueness

The study of delay differential equations (DDEs) with impulses is essential for understanding the various complex dynamic systems found in science and engineering. These equations include both delays and sudden changes (impulses), making them more realistic for representing real-world phenomena like biological processes and control systems, where delays and abrupt events are common. In this section, I explore the fundamental aspects of these systems by providing definitions, lemmas, and theorems that establish the basis for analyzing their stability and uniqueness properties.

Let $T>0$, $\varphi \in \mathcal{R}$, and let $x(t,\varphi )$ denote the solution of (1)–(3) for which the following definition is provided:

Definition 1. 

$x(t,\varphi ):[-r,\infty )\times \mathcal{R}\to {\mathbb{R}}^n$ is called a solution of (1)–(3) if:

(1) 

$x(\varphi )$ is absolutely continuous with respect to the Lebesgue measure, is differentiable on the complement of a countable subset of ${\mathbb{R}}^{+}\setminus {\cup }_{j\ge 1}\left\{jT\right\}$, and satisfies Equation (1) almost everywhere over ${\mathbb{R}}^{+}$.

(2) 

$x(t,\varphi )$ satisfies the impulse condition (2) at each point $jT,j\in \mathbb{N}$, and the initial value function $\varphi \in \mathcal{R}$ satisfies (3).

Next, I introduce two lemmas for which I provide proofs. First, I show that for any $g\in \mathcal{R}([a,b],{\mathbb{R}}^n)$, g is bounded, and then I prove that $\mathcal{R}([a,b],{\mathbb{R}}^n)$ is a Banach space.

Lemma 1. 

For every element $g\in \mathcal{R}([a,b],{\mathbb{R}}^n)$, g is bounded.

Proof. 

Let $g\in \mathcal{R}([a,b],{\mathbb{R}}^n)$, $t\in (a,b)$, and $ϵ(t)$ be a positive real function such that

$$|g(\tau )-\underset{s\to t^{-}}{lim}g(s)|<1,\mathrm{if}\tau \in (t-ϵ(t),t)$$

and

$$|g(\tau )-\underset{s\to t^{+}}{lim}g(s)|<1,\mathrm{if}\tau \in (t,t+ϵ(t))$$

Since $|g(s)|$ is finite, I can show that g is bounded on $(t-ϵ(t),t+ϵ(t))=V_t$. Let $ϵ(a)$ and $ϵ(b)$ be small enough such that g is bounded on $(a-ϵ(a),a+ϵ(a))=V_a$ and $(b-ϵ(t),b+ϵ(b))=V_b$. Then,

$$\{V_t\mid t\in [a,b]\}$$

is an open covering of the compact set $[a,b]$ and contains a finite open covering $U_{t_1},\cdots ,U_{t_p}$. Since g is bounded on each $V_i$, $i\in \{1,\cdots ,p\}$; g is bounded on $[a,b].$ □

I define the following norm:

$${\parallel g\parallel }_{\infty }=\underset{t\in [a,b]}{sup}\left|g(t)\right|.$$

It is clear that this norm is finite for each g in $\mathcal{R}([a,b],{\mathbb{R}}^n)$ by using Lemma 1.

Lemma 2. 

The space $\mathcal{R}([a,b],{\mathbb{R}}^n)$ is complete with respect to the ${\parallel ·\parallel }_{\infty }$ norm.

Proof. 

We suppose that $g_m\in \mathcal{R}([a,b],{\mathbb{R}}^n)$ form a Cauchy sequence, i.e.,

$$\forall ϵ>0,\exists n_0>0,\forall m,p>n_0,{\parallel g_m-g_p\parallel }_{\infty }<ϵ.$$

In particular, for each fixed $t\in [a,b]$, $g_m(t)$ is a Cauchy sequence in ${\mathbb{R}}^n$ and therefore converges to ${\displaystyle g(t)=\underset{m\to \infty }{lim}{g}_m(t)}$. It is easily shown that

$${\displaystyle \underset{m\to \infty }{lim}{\parallel g_m-g\parallel }_{\infty }=0.}$$

g is an element of $\mathcal{R}([a,b],{\mathbb{R}}^n).$ Letting $t\in [a,b)$, ${lim}_{s\to t^{+}}g(s)$ exists. Set

$${\gamma }_m=\underset{s\to t^{+}}{lim}{g}_m(s).$$

Since

$$\underset{m,p\to \infty }{lim}{\parallel g_m-g_p\parallel }_{\infty }=0,$$

we have

$$|{\gamma }_m-{\gamma }_p|\to 0,\mathrm{as}m,p\to \infty ,$$

then ${\gamma }_n$ converges to a limit $\gamma .$ We show that ${\displaystyle \gamma =\underset{s\to t^{+}}{lim}g(s).}$ Letting $ϵ>0,$ choose m to be sufficiently large such that

$$\parallel g_m{-g\parallel }_{\infty }<{\displaystyle \frac{ϵ}{3}}\mathrm{and}|{\gamma }_m-\gamma |<{\displaystyle \frac{ϵ}{3}}.$$

Then, pick a value of $\eta >0$ that is sufficiently small such that

$$|g_m(s)-{\gamma }_m| <{\displaystyle \frac{ϵ}{3}}\mathrm{if}s\in (t,t+\eta ).$$

Then, for $s\in (t,t+\eta )$, we have

$$\begin{array}{ccc}\hfill |g_m(s)-\gamma |& \le & |g_m(s)-g(s)|+|g(s)-{\gamma }_m|+|{\gamma }_m-\gamma |\hfill \\ & \le & {\displaystyle \frac{ϵ}{3}}+{\displaystyle \frac{ϵ}{3}}+{\displaystyle \frac{ϵ}{3}}=ϵ\hfill \end{array}$$

□

Lemma 3. 

For any $g\in \mathcal{R}([a,b],{\mathbb{R}}^n)$, g is measurable.

Proof. 

For any function g in $\mathcal{R}([a,b],{\mathbb{R}}^n)$, g is measurable. We need to prove that the set

$$M=\{x\in [a,b]\mid \parallel g(x){\parallel }_{\infty }>r\},$$

is measurable for any real number r. First, define the set

$$M_0{=\{x\in [a,b]\mid \exists ϵ>0\mathrm{such}\mathrm{that}\forall y\in (x-ϵ,x+ϵ),\parallel g(y)\parallel }_{\infty }>r\}.$$

I show that M can be written as the union of $M_0$ and a set that is, at most, countable. Now, consider the sets

$$V_{\eta }=\{x\in [a,b]\setminus M_0\mid {\parallel g(x)\parallel }_{\infty }>r+\eta \},$$

for any positive $\eta$.

Note that

$$M\setminus M_0=\bigcup _{k=1}^{\infty }{V}_{1/k}.$$

I show that each $V_{\eta }$ is finite. If $V_{\eta }$ is not finite, it would contain a convergent sequence. Let us extract a monotonic subsequence from $V_{\eta }$:

$$x_k\in V_{\eta }\mathrm{such}\mathrm{that}\underset{k\to \infty }{lim}{x}_k=x.$$

Assume $x_k$ is increasing. Since

$$\parallel g(x_k){\parallel }_{\infty }>r+\eta \mathrm{and}\underset{y\to x^{-}}{lim}{\parallel g(y)\parallel }_{\infty }\mathrm{exists},$$

we have

$$\underset{y\to x^{-}}{lim}{\parallel g(y)\parallel }_{\infty }\ge r+\eta .$$

Therefore, there is an interval $(x-\xi ,x)$, with $\xi >0$, such that ${\parallel g(y)\parallel }_{\infty }>r$ for all $y\in (x-\xi ,x)$. For large k, $x_k\in (x-\xi ,x)$, which means $x_k\in M_0$. This contradicts the fact that $x_k\in V_{\eta }$.

So, $V_{\eta }$ must be finite, making $M\setminus M_0$, at most, countable. Since $M_0$ and countable sets are measurable, M is measurable. Therefore, g is measurable. □

In what follows, I make the following assumptions:

(H₁)

The map $f:\mathcal{R}\to {\mathbb{R}}^n$ satisfies

$$f(0)={0\mathrm{and}\parallel f(\phi )-f(\varphi )\parallel }_{\infty }\le K{\parallel \phi -\varphi \parallel }_{\infty },\mathrm{for}\mathrm{all}\phi ,\varphi \in \mathcal{R}.$$

(H₂)

For each regulated map $x:[a,b]\to {\mathbb{R}}^n$, with $b-a>r$, the map $t\to f(x_t)$ is measurable over $[a+r,b]$.

(H₃)

$h_j:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a continuously differentiable map for each $j\ge 1$.

Before I demonstrate the uniqueness of the solution to the delay differential equation, I introduce the following lemma. This lemma establishes that the problem without impulses, restricted to the interval $[0,T]$ with a given initial condition, possesses a unique solution.

Lemma 4 

([5]). Let $f:\mathcal{R}\to {\mathbb{R}}^n$ be a map that satisfies $(H_1)$ and $(H_2)$. Let T be a positive real constant. Then, for each $\phi \in \mathcal{R}$ and $\xi \in {\mathbb{R}}^n$, the problem

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dx(t)}{dt}}& =& f(x_t),a.e.t\in [0,T]\hfill \\ \hfill (x_0,x(0^{+}))& =& (\varphi ,\varphi (0^{+}))\in {\mathcal{R}}^{+},\hfill \end{array}$$

where ${\mathcal{R}}^{+}=\mathcal{R}\times {\mathbb{R}}^n$ has a unique solution.

The theorem presented below establishes that the delay differential equation defined by (1)–(3) has a unique solution.

Theorem 1 

([5]). Let $f:\mathcal{R}\to {\mathbb{R}}^n$ be a map satisfying $(H_1)$–$(H_3)$. Then, the problem (1)–(3) has a unique solution.

In the subsequent section, my aim is to establish certain linear results regarding stability. I achieve this by linearizing the Poincaré operator in the case of delay differential equations with impulses.

3. Linearized Stability

In this section, I aim to provide a solid foundation for analyzing the stability of delay differential equations with impulses. This includes exploring the linearization of the Poincaré operator using Fréchet derivatives and investigating the conditions under which equilibrium solutions are stable or unstable. To this end, consider the following nonlinear delay differential equation on ${\mathcal{R}}^{+}$:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dx(t)}{dt}}& =& f(x_t),\mathrm{a}.\mathrm{e}.t\in [0,T]\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill x(T)& =& x(T^{-}),x(T^{+})=h(x(T)),\forall j\ge 1,j\in \mathbb{N}\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill (x_0,x(0^{+}))& =& (\varphi ,\varphi (0^{+}))\in {\mathcal{R}}^{+}\hfill \end{array}$$

(6)

In this section, I investigate the stability properties of equilibrium solutions for (4)–(6) by linearizing the Poincaré operator given by

$$\begin{array}{ccc}\hfill J_t:{\mathcal{R}}^{+}& \to & {\mathcal{R}}^{+}\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \Phi & \to & x_t(\Phi )\hfill \end{array}$$

(8)

where $\Phi =(\varphi ,\varphi (0^{+}))\in {\mathcal{R}}^{+}=\mathcal{R}\times {\mathbb{R}}^n$, and

$$x_t(\Phi )=(x_t(\varphi ),x(\varphi )(t^{+}))=(x_{t+T}(\varphi ),x(\varphi )(t^{+}+T))=x_{t+T}(\Phi )\in {\mathcal{R}}^{+}$$

is the solution of (4)–(6) starting from $\varphi \in \mathcal{R}$, where

$$\begin{array}{c}{x}_t(\varphi )(\theta )=\left\{\begin{array}{cc}\varphi (t+\theta ),\hfill & \mathrm{if}-r\le t+\theta \le 0,\hfill \\ x(\varphi )(t+\theta ),\hfill & \mathrm{if}t+\theta \ge 0.\hfill \end{array}\right.\end{array}$$

(9)

and we have $\forall k\in \mathbb{N},J_{kT}=J_T^k.$ We have the following basic lemma for the delay differential equation with impulses:

Lemma 5. 

Under assumptions $(H_1)$–$(H_3)$, the Poincaré operator $J_t:{\mathcal{R}}^{+}\to {\mathcal{R}}^{+}$ is continuous.

Applying the theorem of continuous dependence, the result is as follows:

Remark 1. 

If $J_t:{\mathcal{R}}^{+}\to C\times {\mathbb{R}}^n$ is called an equilibrium solution if $J_t(\psi )=\psi$. To begin, recall the well-known concept of stability.

Designate ${\Phi }_e\in {\mathcal{R}}^{+}$ as an equilibrium solution or fixed-point solution if it satisfies condition $J_t({\Phi }_e)={\Phi }_e$. Before proceeding, it is advantageous to revisit the concept of stability. For a comprehensive understanding, refer to [12], where the fundamental principles of stability analysis in the Lyapunov sense are extensively discussed.

Definition 2 

([12]). An equilibrium ${\Phi }_e$ is called stable (in the Lyapunov sense) if, for any neighborhood V of ψ, there exists a neighborhood V of ψ such that $J_T^k(V)\subset U$ for all $k\in \mathbb{N}$. If, in addition, $J_T^k\psi$ converges to ${\Phi }_e$ as $k\to \infty$ for any ${\Phi }_e\in V$, we call ${\Phi }_e$ asymptotically stable. In case this convergence is exponential, we refer to ${\Phi }_e$ as an exponentially asymptotically stable equilibrium. If ${\Phi }_e$ is not stable, we call it unstable.

The concept of a derivative is a fundamental tool for analyzing various types of functions. When dealing with vector-valued functions, there are two primary versions of derivatives: Gâteaux (or weak) derivatives and Fréchet (or strong) derivatives, as mentioned by Lindenstrauss et al. [15]. For an operator J mapping from a Banach space X into a Banach space Y, the Gâteaux derivative at $\psi \in X$ is defined as a bounded linear operator $L:X\to Y$ such that for every $\phi \in X$,

$$\underset{t\to 0}{lim}{\displaystyle \frac{J(\psi +t\phi )-J(\psi )}{t}}=J^{\prime }(\psi )(\phi ).$$

The operator $J^{\prime }$ is referred to as the Fréchet derivative of J at $\psi$ if it is a Gâteaux derivative of J at $\psi$, and the limit in the above equation holds uniformly for $\phi$ in the unit ball (or unit sphere) in X. Alternatively, it can be defined as

$$J(\psi +\phi )=J(\psi )+J^{\prime }(\psi )(\phi )+o(\parallel \phi \parallel )\mathrm{as}\parallel \phi \parallel \to 0.$$

Thus, L provides the natural linear approximation of J in a neighborhood of $\psi$. Sometimes, L is called the first variation of J at $\psi$. Now, the following definition of Fréchet differentiability can be given:

Definition 3 

([15]). Let J be a transformation defined on an open domain D in a normed space X and having range in a normed space Y. If for a fixed $\psi \in D$ and each $h\in X$, there exists a linear and continuous function $U(\psi ):X\to Y$ such that

$$\underset{\parallel \phi \parallel \to 0}{lim}{\displaystyle \frac{\parallel J(\psi +\phi )-J(\psi )-J^{\prime }(\psi )(\phi )\parallel }{\parallel \phi \parallel }}=0,$$

then J is said to be Fréchet-differentiable at ψ, and $U(\psi )(\phi )$ is said to be the Fréchet derivative of J at ψ with increment φ.

Theorem 2 

([12]). Let $J(·)$ be a nonlinear operator semigroup in ${\mathcal{R}}^{+}$, and let ψ be a fixed point of J. Suppose that $J(·)$ is Fréchet-differentiable at ψ with $U=J^{\prime }(\psi )$ and that the zero solution is exponentially asymptotically stable with respect to this linearized semigroup $U(·)$. Then, ψ is exponentially asymptotically stable with respect to $J(·)$.

The main concern now is what happens if the linearized semigroup is not exponentially stable. In addressing this concern, we can arrive at the following result, with more details available in [12].

Proposition 1 

([12]). Let $J_T$ be a nonlinear operator in ${\mathcal{R}}^{+}$, let ψ be a fixed point of $J_T$, and suppose that $J_T$ is continuously Fréchet-differentiable at ψ. Let $U_T=J_T^{\prime }\left[\psi \right]$ and assume that $X^{+}$ can be decomposed as ${\mathcal{R}}^{+}={\mathcal{R}}_1^{+}\oplus {\mathcal{R}}_2^{+}$, where ${\mathcal{R}}_i^{+}$ are $U_T$-invariant subspaces of ${\mathcal{R}}^{+}$, and there exist positive integer m and real $1<\theta <\eta$ such that for ${\psi }_i\in {\mathcal{R}}_i^{+}$, $i=1,2$, we have

$$\parallel U_T^m{\psi }_1{\parallel }_{{\mathcal{R}}^{+}}\ge \eta \parallel {\psi }_1{\parallel }_{{\mathcal{R}}^{+}}and\parallel U_T^m{\psi }_2{\parallel }_{X^{+}}\le \theta {\parallel {\psi }_2\parallel }_{X^{+}}.$$

Then, there exist constant $\epsilon >0$ and sequences $(k_p)$ of positive integers, $({\psi }_{k_p})\to \psi$ such that

$$\parallel J_T^{m_{k_p}}{\psi }_{k_p}{-\psi \parallel }_{{\mathcal{R}}^{+}}\ge \epsilon .$$

Theorem 3 

([12]). Let $J_T$ be the Poincare operator in ${\mathcal{R}}^{+}$ defined by (7)–(8). Let $U(·)$ denote the Fréchet derivative of $J_T(·)$ at ψ, and suppose that for some fixed t, we have a splitting of $\mathcal{R}$ as $\mathcal{R}={\mathcal{R}}_1\oplus {\mathcal{R}}_2$, where e ${\mathcal{R}}_i$ are invariant with respect to $U_t$, ${\mathcal{R}}_1^{+}$ is finite-dimensional, and with

$$\omega :=\underset{k\to 0}{lim}{\displaystyle \frac{1}{kT}}ln{\parallel {U_{kT}}_{{\mid }_{X_2^{+}}}\parallel }_{X_2^{+}},$$

we have

$$\mu :=inf\left\{\right|\lambda |\mid \lambda \in \sigma (U_T{|}_{X_1^{+}})\}>e^{\omega T}.$$

Then, there exist a constant $>0$ and sequences $({\psi }_k)$ converging to ${\psi }_0$ and $(t_k)$ of positive reals such that

$$\parallel J_{t_k}{\psi }_k-{\psi }_0\parallel \ge ϵ$$

In this section, U investigate the stability properties of the equilibrium solution for (1)–(2). * consider the linearization of the nonlinear problem at the equilibrium function ${\varphi }_e$ of the nonlinear problem, with impulses defined by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dy(t)}{dt}}& =& f_{{\varphi }_e}^{\prime }(x_t)y_t,\mathrm{a}.\mathrm{e}.t\in [0,T]\hfill \\ \hfill y(j{T}^{+})& =& {\displaystyle \frac{\partial h_j(x)}{\partial x}}(y(jT)),y(jT)=y(j{T}^{-})\forall j\ge 1,\hfill \\ \hfill y_0& =& \varphi \in X,y(0^{+})=\varphi (0^{+})\in {\mathbb{R}}^n,\hfill \end{array}$$

where h satisfies condition $(H_3)$; then, ${\displaystyle \frac{\partial h_j(x)}{\partial x}}(y(jT))$ exists for any $j\in \mathbb{N}$. Furthermore, the map $f_{{\varphi }_e}^{\prime }:\mathbb{R}\times \mathcal{R}\to {\mathbb{R}}^n$ satisfies

$(H_4)$

For each regulated map $x:[a,b]\to {\mathbb{R}}^n$, with $b-a>r$, let us assume that the map $f_{{\varphi }_e}^{\prime }(x_t)$ is measurable over $[a+r,b]$.

By restriction in ${\mathcal{R}}^{+}$, we can consider the following problem on interval $[-r,T]$:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dy(t)}{dt}}& =& f_{{\varphi }_e}^{\prime }(x_t)y_t,\mathrm{a}.\mathrm{e}.t\in [0,T]\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill y(T^{+})& =& {\displaystyle \frac{\partial h(x)}{\partial x}}(y(T)),y(T)=y(T^{-})\in {\mathbb{R}}^n,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill y_0& =& {\varphi }_e\in \mathcal{R},y(0^{+})={\varphi }_e(0^{+})\in {\mathbb{R}}^n,\hfill \end{array}$$

(12)

We now consider the Poincaré operator $U_t$ defined by

$$\begin{array}{ccc}\hfill U_t:{\mathcal{R}}^{+}& \to & {\mathcal{R}}^{+}\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \Phi & \to & y_t(\Phi )\hfill \end{array}$$

(14)

where ${\Phi }_e=({\varphi }_e,{\varphi }_e(0^{+}))\in {\mathcal{R}}^{+}=\mathcal{R}\times {\mathbb{R}}^n$, and $y_t({\Phi }_e)=(y_{{\varphi }_e}(t),y({\varphi }_e)(t^{+}))\in {\mathcal{R}}^{+}$ is the solution of the linear delay differential Equations (10)–(12). I want to prove that the Fréchet derivative of $J_T$ at ${\varphi }_e$ is $J_T^{\prime }({\Phi }_e)=U_T$.

Theorem 4. 

Let ${\varphi }_e\in \mathcal{R}$ be the equilibrium function of the nonlinear problem with impulses (4)–(6), and suppose that the map f satisfies $(H_3)$ and $(H_4)$. Then, the Fréchet derivative of the Poincaré operator $J_T$ at ${\varphi }_e$, $J_T^{\prime }({\Phi }_e)=U_T$, where $U_T$ is the Poincaré operator of (10)–(12), defined by (13) and (14).

Proof. 

We have, from Equations (7) and (8),

$$\begin{array}{cc}\hfill J_T(\Phi )(\theta )& =J_T(\varphi (\theta ),\varphi (0))\hfill \\ \hfill & =(x_T(\varphi )(\theta ),x_T(\varphi )(0^{+}))\hfill \\ \hfill & =\left(\varphi (0^{+})+{\int }_0^{T+\theta }f(x_s(\varphi ))ds,h(x_T(\varphi )(0))\right).\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill U_T(\Phi )(\theta )& =U_T(\varphi (\theta ),\varphi (0^{+}))\hfill \\ \hfill & =(y_T(\varphi )(\theta ),y_T(0^{+}))\hfill \\ \hfill & =\left(y_T(0^{+})+{\int }_0^{t+\theta }{f}_{{\varphi }_e}^{\prime }(x_s(\varphi ))(y_s(\varphi ))ds,h^{\prime }(x_T(\varphi )(0))\right).\hfill \end{array}$$

If

$$\begin{array}{cc}\hfill D_T({\Phi }_e)(\Phi )& =J_T({\Phi }_e+\Phi )-J_T({\Phi }_e)-U_T(\Phi )\hfill \\ \hfill & =\left(x_T({\varphi }_e+\varphi )-x_T({\varphi }_e)-y_T(\varphi ),x_T({\varphi }_e+\varphi )(0^{+})-x_T({\varphi }_e)(0^{+})-y_T(\varphi )(0^{+})\right)\hfill \\ \hfill & =\left(D_T(\varphi ),D_T(\varphi )(0^{+})\right)\hfill \end{array}$$

then, we have

$$\begin{array}{ccc}\hfill D_T({\varphi }_e)(\varphi (\theta ))& =& {\int }_0^{T+\theta }f(x_s({\varphi }_e+\varphi ))-f(x_s({\varphi }_e))-f_{{\varphi }_e}(y_s(\varphi ))(y_s(\varphi ))ds\hfill \\ & =& {\int }_0^{T+\theta }{f}_{\varphi }(x_s(\varphi ))\left(x_s({\varphi }_e+\varphi )-x_s(\varphi )\right)+O\left(x_s({\varphi }_e+\varphi )-x_s(\varphi )\right)-\hfill \\ & & f_{{\varphi }_e}(x_s(\varphi ))(y_s(\varphi ))ds.\hfill \end{array}$$

Then, by Gronwall’s lemma, we have

$$\begin{array}{cc}\hfill \parallel D_T({\varphi }_e)(\varphi )\parallel & \le {\int }_0^{T}M\parallel D_s(\varphi )\parallel ds+ϵT\parallel \varphi \parallel \hfill \\ \hfill & \le ϵT\parallel \varphi \parallel e^{MT}.\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill D_T({\varphi }_e)(\varphi (0^{+}))& =& h(x_T({\varphi }_e+\varphi )(0))-h(x_T(\varphi )(0))-h^{\prime }(x_T(\varphi )(0))(x_T(\varphi )(0))\hfill \\ & =& h^{\prime }((x_T(\varphi ))(0))((x_T({\varphi }_e+\varphi ))(0)-(x_T(\varphi ))(0))-(y_T(\varphi ))(0))+\hfill \\ & & O\left((x_T({\varphi }_e+\varphi ))(0)-(x_T(\varphi ))(0))\right)\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \parallel D_T({\varphi }_e)(\varphi (0^{+}))\parallel & \le \parallel h^{\prime }((x_T(\varphi )))(0)\parallel \parallel D_T(\varphi )(0)\parallel +ϵ\parallel \varphi \parallel \hfill \\ \hfill & \le o(\parallel \varphi \parallel )(1+T{e}^{MT}\parallel h^{\prime }((x_T(\varphi ))(0))\parallel ).\hfill \end{array}$$

Since we can define the Fréchet derivative of the Poincaré operator $J_T$ at the equilibrium function ${\varphi }_e$, we have $J_T^{\prime }({\varphi }_e)=U_T$. □

Now, let us consider the operator S defined by

$$\begin{array}{cc}\hfill S_T:{\mathcal{R}}^{+}& \to {\mathcal{R}}^{+}\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill (y_0,y(0^{+}))& \to \left(y_T,y(T)\right)\hfill \end{array}$$

(16)

and the operator $K_T$ defined by

$$\begin{array}{cc}\hfill K_T:{\mathcal{R}}^{+}& \to {\mathcal{R}}^{+}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill (y_0,y(0^{+}))& \to \left(0,{\displaystyle \frac{\partial h(x)}{\partial x}}(y(T))\right)\hfill \end{array}$$

(18)

We have

$$\begin{array}{cc}\hfill U_T(y_0,y(0^{+}))& =\left(y_T,{\displaystyle \frac{\partial h(x)}{\partial x}}(y(T))\right)\hfill \\ \hfill & =S_T(y_0,y(0^{+}))+K_T(y_0,y(0^{+}))\hfill \end{array}$$

where $y_T$ is the solution of the delayed Equations (13) and (14).

Lemma 6. 

Under the assumptions $(H_1)$–$(H_3)$, and $T>r$, the operators $U_T,S_T:{\mathcal{R}}^{+}\to {\mathcal{R}}^{+}$ are completely continuous.

Proof. 

Similar to Theorem 4.4 in [16], the result follows by induction. For the operator $U_T$, it is easy to see that the operator $K_T$ is compact. Therefore, the result follows. □

Let $\sigma (S)$ be the spectrum of a closed operator S containing a bounded part ${\sigma }^{\prime }$ separated from the rest ${\sigma }^{″}$ in such a way that a rectifiable, simple closed curve $\Gamma$ (or more generally, a finite number of such curves) can be drawn to enclose an open set containing ${\sigma }^{\prime }$ in its interior and ${\sigma }^{″}$ in its exterior. S is said to be decomposed according to a Banach space $X=M^{\prime }\oplus M^{″}$ if

$$PD(S)\in D(S),S{M}^{\prime }\subset M^{\prime },S{M}^{″}\subset M^{″},$$

where $D(S)$ is the domain of S, and P in $\mathcal{B}(X)$ is the projection onto M along N, i.e., $P^2=P$, where $M^{\prime }=PX$ and $M^{″}=(I{d}_{|X}-P)X$. It should be added that $M^{\prime }$ and $M^{″}$ are closed linear manifolds of X.

Refer to comprehensive classic books on functional analysis by [17,18,19,20]. These books are known for their clear explanation and rigorous treatment of the subject, including spectral analysis and decomposition theorem. Under such circumstances, we have the following decomposition theorem:

Theorem 5. 

Let $\sigma (S)$ be separated into parts ${\sigma }^{\prime }$ and ${\sigma }^{″}$ as described above. Then, there exists a decomposition $X=M^{\prime }\oplus M^{″}$ such that for the operator S, $S_M^{\prime }=PS$ and $S_M^{″}=(I{d}_{|X}-P)S$, $\sigma (S_M^{\prime })={\sigma }^{\prime }$ and $\sigma (S_M^{″})={\sigma }^{″}$, $S_M^{\prime }\in \mathcal{B}(M^{\prime })$ and $S_M^{″}\in \mathcal{B}(M^{″})$.

Proof. 

Let $\sigma (S)$ be separated into parts ${\sigma }^{\prime }$ and ${\sigma }^{″}$ as described. Define $M^{\prime }$ and $M^{″}$ as the ranges of the Riesz projections corresponding to ${\sigma }^{\prime }$ and ${\sigma }^{″}$, respectively. Then,

$$X=M^{\prime }\oplus M^{″}.$$

Now, let P be the projection onto $M^{\prime }$ along $M^{″}$. Then, P is a bounded linear operator on X such that $P^2=P$.

Consider $S_M^{\prime }=PS$. It is easy to verify that $S_M^{\prime }\in \mathcal{B}(M^{\prime })$. Since $S_M^{\prime }=PS$, and P is the projection onto $M^{\prime }$, it follows that the spectrum of $S_M^{\prime }$ coincides with ${\sigma }^{\prime }$, i.e., $\sigma (S_M^{\prime })={\sigma }^{\prime }.$

Similarly, consider $S_M^{″}=(I{d}_{|X}-P)S$. It is easy to verify that $S_M^{″}\in \mathcal{B}(M^{″})$. Since $S_M^{″}=(I{d}_{|X}-P)S$, and $I{d}_{|X}-P$ is the projection onto $M^{″}$, it follows that the spectrum of $S_M^{″}$ coincides with ${\sigma }^{″}$, i.e., $\sigma (S_M^{″})={\sigma }^{″}.$

Thus, S is decomposed according to the decomposition $X=M^{\prime }\oplus M^{″}$ such that the spectra of the parts $S_M^{\prime }$ and $S_M^{″}$ coincide with ${\sigma }^{\prime }$ and ${\sigma }^{″}$, respectively; $S_M^{\prime }\in \mathcal{B}(M^{\prime })$; and $S_M^{″}\in \mathcal{B}(M^{″})$. □

Theorem 6. 

Let $\varphi \in \mathcal{R}$ be the equilibrium function of the nonlinear problem with impulses (4)–(6). Suppose that ${\displaystyle \frac{\partial f}{\partial x}}\in \mathcal{R}$, $\parallel f^{\prime }(\varphi )\parallel \le M$ for any $\varphi \in \mathcal{R}$, and ${\displaystyle \frac{\partial h}{\partial x}}(x(T))$ exists, where $T\ge r$, and $x_{\varphi }(t)$ is defined by (9). Then, the Fréchet derivative operator $J_T$ at ϕ, $J_T^{\prime }(\varphi )=U_T$, where $U_T$ is the Poincaré operator of (10)–(12), defined by (13) and (14). There exist a constant $ϵ>0$ and sequences $({\varphi }_n)$ converging to ${\varphi }_0$ and $(t_n)$ of positive real numbers such that $\parallel J_{t_n}{\varphi }_n-{\varphi }_0\parallel \ge ϵ$.

Proof. 

We have, from Lemma 6, that $U_T$ is compact. Then, by using the properties of the compact linear operator $U_T$, we can conclude that the spectrum of $U_T$, denoted by $\sigma (U_T)$, is countable, with 0 being the only possible point of accumulation [17]. Therefore, there exists a circle $C=\{\lambda \mid |\lambda |=\rho \}$ with $1<\rho <{\lambda }_0$ such that $\sigma (U_T)\cap C=\varnothing$. We have from Theorem 5 that ${\mathcal{R}}^{+}={\mathcal{R}}_1^{+}+{\mathcal{R}}_2^{+}$ with

$$\sigma (U_{|{\mathcal{R}}_1^{+}})={\sigma }^{\prime }=\{\lambda \in \sigma (U_T)\mid \left|\lambda \right|>\rho \},$$

and

$$\sigma (U_T{|}_{{\mathcal{R}}_2^{+}})={\sigma }^{″}=\{\lambda \in \sigma (U_T)\mid \left|\lambda \right|<\rho \}.$$

To prove that ${\mathcal{R}}_1^{+}$ is finite-dimensional, let us assume the opposite, that ${\mathcal{R}}_1^{+}$ is infinite-dimensional. Since $U_T$ is compact, it maps bounded sets to sets that are relatively compact. Given that $0\notin \sigma (U_T{|}_{{\mathcal{R}}_1^{+}})$, $U_T$ is invertible on ${\mathcal{R}}_1^{+}$, meaning its inverse ${(U_T)}^{-1}$ exists and is bounded. If ${(U_T)}^{-1}$ is compact, it would also map bounded sets to relatively compact sets. However, this creates a contradiction because a compact operator cannot have a bounded inverse in an infinite-dimensional space (see Section 8.3 in [17] for more on the spectral properties of compact linear operators). Therefore, ${\mathcal{R}}_1^{+}$ must be finite-dimensional.

We have

$$\sigma (U_{|{\mathcal{R}}_2^{+}})=\underset{k\to \infty }{lim}\sqrt[k]{U{(kT)}_{|{\mathcal{R}}_2^{+}}}<\rho ,$$

which uses the fact that the spectral radius of a compact operator on an infinite-dimensional space is less than $\rho$. This leads to growth rate

$$\omega =\underset{k\to \infty }{lim}{\displaystyle \frac{1}{kT}}ln\parallel U{(kT)}_{|{\mathcal{R}}_2^{+}}\parallel =\underset{k\to \infty }{lim}{\displaystyle \frac{1}{T}}ln\sqrt[k]{U{(kT)}_{|{\mathcal{R}}_2^{+}}}<{\displaystyle \frac{1}{T}}ln\rho .$$

Consequently, for $\lambda \in \sigma (U{(kT)}_{|{\mathcal{R}}_1^{+}})=\sigma (U{(T)}_{|{\mathcal{R}}_1^{+}}^k)$, we can write $\lambda ={\nu }^k$, where $\nu \in \sigma (U{(T)}_{|{\mathcal{R}}_1^{+}}^k)$, and it follows that $\left|\lambda \right|>{\rho }^k$. Finally,

$$\mu =inf\left\{\right|\mu |\mid \mu \in \sigma (U{(kT)}_{|{\mathcal{R}}_1^{+}})\}>e^{kln(\rho )}>e^{\omega kT}.$$

Then, the result follows from Theorem 3. Therefore, the Fréchet derivative operator $J_T$ at $\varphi$ is $U_T$, the Poincaré operator, and there exist a constant $ϵ>0$ and sequences $({\varphi }_n)$ converging to ${\varphi }_0$ and $(t_n)$ of positive real numbers such that $\parallel J_{t_n}{\varphi }_n-{\varphi }_0\parallel \ge ϵ$. □

4. Example and Result

Consider the following nonlinear delay differential equation with impulses:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dx(t)}{dt}}& =f(x(t-1)),\mathrm{a}.\mathrm{e}.t>0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill x_0(t)& =\varphi (t),-1\le t\le 0,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill x(j{T}^{+})-x(jT)& =g_j(x(jT)),x(jT)=x(j{T}^{-}),\hfill \end{array}$$

(21)

where $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ and $g_j:{\mathbb{R}}^2\to {\mathbb{R}}^2$ are nonlinear functions satisfying conditions $(H_1)$–$(H_4)$.

To linearize this system around a steady-state solution $x^{*}$, where $x^{*}$ satisfies $f(x^{*})=0$, we define $x(t)=x^{*}+u(t)$. Substituting into the original equations and assuming $u(t)$ is small, perform a Taylor series expansion around $x^{*}$ and keep only the linear terms. This results in a linearized system:

$$\begin{array}{cc}\hfill {\displaystyle \frac{du(t)}{dt}}& ={\displaystyle \frac{\partial f}{\partial x}}(x^{*})u(t-1),\mathrm{a}.\mathrm{e}.t>0,\hfill \\ \hfill u_0(t)& =\varphi (t)-x^{*},-1\le t\le 0,\hfill \\ \hfill u(j{T}^{+})-u(jT)& ={\displaystyle \frac{\partial g_j}{\partial x}}(x^{*})u(jT),u(jT)=u(j{T}^{-}).\hfill \end{array}$$

Let $N={\displaystyle \frac{\partial f}{\partial x}}(x^{*})=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, where $a,b,c,d\in \mathbb{R}$.

Then, the linearized system simplifies to

$$\begin{array}{cc}\hfill {\displaystyle \frac{du(t)}{dt}}& =Nu(t-1),\mathrm{a}.\mathrm{e}.t>0,u\in {\mathbb{R}}^2,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill u_0(t)& =\varphi (t)-x^{*},-1\le t\le 0,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill u(j{T}^{+})-u(jT)& ={\displaystyle \frac{\partial g_j}{\partial x}}(x^{*})u(jT),u(jT)=u(j{T}^{-}).\hfill \end{array}$$

(24)

Following the assumption in [16], we consider that (22) is asymptotically stable; that is, all the solutions tend to zero as t tends to $+\infty$. This implies that the exponential growth rate of the fundamental solution $v(t)$ of (22) and (23) is negative [21]. More precisely, there exist constants $M\ge 1$ and $\omega >0$ such that

$$\parallel v(t)\parallel \le Mexp(-\omega (t-\sigma )),t\ge \sigma \ge 0.$$

Then, for any $\varphi \in \mathcal{R}$, the solution $u(\varphi )(t)$ of the initial value problem (22) and (23) without impulses tends to zero as $t\to +\infty$. To study the relationship between the eigenvalues and the system’s parameters in (22), let us examine the characteristic equation of (22), which is given by

$$P(\lambda )=det(\lambda I-N{e}^{-\lambda })=det(\lambda e^{\lambda }I-N)=0$$

and we have

$$y^2-(a+d)y+ad-cb=0,\mathrm{where}y=\lambda e^{\lambda }.$$

Then,

$$y=\lambda e^{\lambda }={\displaystyle \frac{(a+d)\pm \sqrt{{(a+d)}^2-4(ad-cb)}}{2}},\mathrm{with}{(a+d)}^2-4(ad-cb)\ge 0.$$

It is clear that $\lambda =-1$ is a root of the characteristic equation if the following relation is true:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{(a+d)\pm \sqrt{{(a+d)}^2-4(ad-cb)}}{2}}& =& -e^{-1},\hfill \end{array}$$

(25)

and the fundamental solution is given by [16]

$$v(t)=M_{t}w{e}^{-t},\mathrm{where}{M}_t=2(t+{\displaystyle \frac{1}{3}}),w=\left[\begin{array}{cc}1+{\displaystyle \frac{de}{2+(a+d)e}}& -{\displaystyle \frac{be}{2+(a+d)e}}\\ {\displaystyle \frac{ce}{2+(a+d)e}}& 1+{\displaystyle \frac{ae}{2+(a+d)e}}\end{array}\right].$$

Moreover, if

$$\underset{t\to +\infty }{lim}supB(t,0)<\infty \mathrm{and}\underset{k\to +\infty }{lim}supB(kT,0)<1,$$

where

$$B(t,0)=\sum _{\sigma \le jT<t}\parallel v(t-jT)\parallel {∥{\displaystyle \frac{\partial g_j}{\partial x}}∥}_{\infty },$$

and additionally to (25), we also suppose

$$\underset{j\to +\infty }{lim}sup{∥{\displaystyle \frac{\partial g_j}{\partial x}}∥}_{\infty }<{\displaystyle \frac{1}{M_T\parallel w\parallel }}(exp(T)-1),M_T\mathrm{is}\mathrm{a}\mathrm{constant}\mathrm{and}T>1.$$

Then, by using Theorem 7 in [16] and Theorems 3 and 4, we have that for any $\varphi \in \mathcal{R}$, the solution $x(\varphi )(t)$ of the initial value problem (19)–(21) tends to zero as $t\to +\infty$.

5. Discussion

In this study, I investigated the stability of semigroups in delay differential equations with impulses using regulated spaces. I extended the classical results on the existence of periodic solutions for ordinary differential equations to delay differential equations. Additionally, I expanded the classical results on the linearized stability for nonlinear semigroups to periodic delay differential equations with impulses.

My study shows that the stability properties of delay differential equations with impulses can be analyzed effectively by considering the Fréchet derivative of the Poincaré operator. I established conditions under which the Fréchet derivative of the Poincaré operator at the equilibrium function is equal to the Poincaré operator of the linearized problem. This provides valuable insights into the behavior of the equilibrium under small perturbations.

I also decomposed the Poincaré operator into closed linear manifolds and analyzed the spectral properties of these decomposed operators. This decomposition sheds light on the stability and long-term behavior of equilibria in delay differential equations with impulses. My detailed spectral analysis allowed the identification of the conditions under which the equilibrium solution may lose stability under small perturbations.

The example provided demonstrated the practical application of my theoretical results. By examining the stability of an equilibrium solution in a specific delay differential equation with impulses, I showed how my methods can be used to determine stability conditions based on the parameters of the system.

Future research could explore the application of my results to more complex systems and investigate the effects of different types of impulses and delays. Additionally, further work could focus on numerical methods for solving delay differential equations with impulses and verifying the theoretical results presented in this paper.

Overall, my research contributes to the understanding of the stability of delay differential equations with impulses and provides a solid foundation for further investigations in this area.