Global attracting set for a class of nonautonomous neutral functional differential equations

Abstract

In this paper, a class of nonlinear and nonautonomous neutral functional differential equations is considered. By developing a new integral inequality, we obtain sufficient conditions for the existence of a global attracting set of neutral functional differential equations with time-varying delays. The results have extended and improved the related reports in the literature.

MSC:26D10, 26D15, 34K40.

1 Introduction

The asymptotic properties of neutral functional differential equations have attracted considerable attention in the past few decades, and many significant results have been obtained [1–6]. One of the most popular ways to analyze the stability property and asymptotic behavior is the method of Lyapunov functionals [1–3]. However, to construct a suitable Lyapunov functional is not easy for certain equations. The characteristic equation is another important researching tool. It seems to work well for constant delays in neutral equations [4, 5]. Meanwhile, the known approach to the study of differential inequalities for nonautonomous neutral functional differential equations was presented in Azbelev’s book [7]. These inequality methods are based on the representation formula of a solution and the analysis of Cauchy, Green’s and fundamental matrices. In this way, various assertions about the estimates of solutions, maximum principles and stability on neutral differential equations were obtained. Important results in this direction can be found in [7–10] and in the monograph [11]. Recently, by using differential and integral inequalities, Xu et al. have studied attracting and invariant sets of functional differential systems [12, 13] and impulsive functional differential equations [14]. However, the inequalities mentioned above are ineffective in studying the attracting sets of a class of nonlinear and nonautonomous neutral functional differential equations.

Motivated by the above discussions, in this paper, we will improve the inequality established in [15] so that it is effective for neutral functional differential equations. Combining with the properties of nonnegative matrices, we obtained some sufficient conditions ensuring the global attracting set for a class of nonlinear and nonautonomous neutral differential equations with time-varying coefficients and unbounded delays. The results extend the earlier publications.

2 Preliminaries

In this section, we introduce some notations and recall some basic definitions.

E denotes the $n\times n$ identity matrix, $\mathbb{R}$ is the set of real numbers and ${\mathbb{R}}_{+}=[0,+\mathrm{\infty })$. $A\le B$ ($A<B$) means that each pair of corresponding elements of A and B satisfies the inequality “≤ (<).” Especially, A is called a nonnegative matrix if $A\ge 0$, where 0 denotes the $n\times n$ zero matrix.

$C[X,Y]$ denotes the space of continuous mappings from the topological space X to the topological space Y. Especially, let $C\stackrel{\mathrm{\Delta }}{=}C[(-\mathrm{\infty },t_0],{\mathbb{R}}^n]$ denote the family of all bounded continuous ${\mathbb{R}}^n$-valued functions ϕ on $(-\mathrm{\infty },t_0]$, here $t_0\ge 0$.

For $x\in {\mathbb{R}}^n$, $A\in {\mathbb{R}}^{n\times n}$, $\phi \in C$, $\tau \in C[\mathbb{R},{\mathbb{R}}_{+}]$, we define ${[x]}^{+}={(|x_1|,|x_2|,\dots ,|x_n|)}^T$, ${[A]}^{+}={(|a_{ij}|)}_{n\times n}$, ${[\phi (t)]}_{\tau (t)}^{+}={({\parallel {\phi }_1(t)\parallel }_{\tau (t)},{\parallel {\phi }_2(t)\parallel }_{\tau (t)},\dots ,{\parallel {\phi }_n(t)\parallel }_{\tau (t)})}^T$, ${\parallel {\phi }_i(t)\parallel }_{\tau (t)}={sup}_{0\le s\le \tau (t)}|{\phi }_i(t-s)|$, $i=1,2,\dots ,n$.

We consider the following differential equation

$$\{\begin{array}{cc}\frac{d}{dt}[x(t)-H(t,x(t-\tau (t)))]=A(t)x(t)+F(t,x(t-\tau (t))),\hfill & t\ge t_0,\hfill \\ x(t)=\phi (t),-\mathrm{\infty }<t\le t_0,\hfill \end{array}$$

(1)

where $x={(x_1,\dots ,x_n)}^T\in {\mathbb{R}}^n$, $H,F\in C[\mathbb{R}\times {\mathbb{R}}^n,{\mathbb{R}}^n]$, $F(t,0)\equiv 0$, $\tau \in C[\mathbb{R},{\mathbb{R}}_{+}]$, ${lim}_{t\to +\mathrm{\infty }}(t-\tau (t))=+\mathrm{\infty }$. We always assume that for any $\phi \in C$, the system (1) has at least one solution through $(t_0,\phi )$ denoted by $x(t,t_0,\phi )$ or simply $x(t)$ if no confusion should occur.

Definition 2.1 (Xu [16])

$f\in U{C}_t$ means that $f\in C[{\mathbb{R}}_{+}\times \mathbb{R},{\mathbb{R}}_{+}]$ and for any given α and any $\epsilon >0$ there exist positive numbers B, T and A satisfying

$${\int }_{\alpha }^{t}f(t,s)ds\le B,{\int }_{\alpha }^{t-T}f(t,s)ds<\epsilon ,\mathrm{\forall }t\ge A.$$

(2)

Especially, $f\in U{C}_t$ if $f(t,s)=f(t-s)$ and ${\int }_0^{\mathrm{\infty }}f(u)du<\mathrm{\infty }$.

Definition 2.2 The set $S\subset {\mathbb{R}}^n$ is called a global attracting set of (1), if for any initial value $\phi \in C$, the solution $x(t)\stackrel{\mathrm{\Delta }}{=}x(t,t_0,\phi )$ converges to S as $t\to +\mathrm{\infty }$. That is,

$$dist(x(t),S)\to 0\text{as}t\to +\mathrm{\infty },$$

where dist$(x,S)=inf\{|x-a|:a\in S\}$ and $|\cdot |$ is a norm of ${\mathbb{R}}^n$.

For a nonnegative matrix $A\in {\mathbb{R}}^{n\times n}$, let $\rho (A)$ denote the spectral radius of A. Then $\rho (A)$ is an eigenvalue of A and its eigenspace is denoted by

$${\mathrm{\Omega }}_{\rho }(A)\stackrel{\mathrm{\Delta }}{=}\{z\in {\mathbb{R}}^n|Az=\rho (A)z\},$$

which includes all positive eigenvectors of A provided that the nonnegative matrix A has at least one positive eigenvector (see Refs. [17]).

Lemma 2.1 (Lasalle [18])

If$A\ge 0$and$\rho (A)<1$, then${(E-A)}^{-1}\ge 0$.

3 Main results

Theorem 3.1 Let $y\in C[\mathbb{R},{\mathbb{R}}_{+}^n]$ be a solution of the delay integral inequality

(3)

(4)

where$G\in C[{\mathbb{R}}_{+}\times {\mathbb{R}}_{+},{\mathbb{R}}_{+}^n]$, $B\in {\mathbb{R}}_{+}^{n\times n}$, $Q\in C[{\mathbb{R}}_{+}\times \mathbb{R},{\mathbb{R}}_{+}^{n\times n}]$, $J={(j_1,\dots ,j_n)}^T\ge 0$, $\phi \in C[(-\mathrm{\infty },t_0],{\mathbb{R}}_{+}^n]$, ${lim}_{t\to +\mathrm{\infty }}(t-\tau (t))=+\mathrm{\infty }$. Assume that the following conditions are satisfied: (${\mathbb{A}}_1$) = $G(t,t_0)\to 0$as$t\to +\mathrm{\infty }$, and there exists a constant matrix${\mathrm{\Pi }}_1\ge 0$such that

$${\int }_{t_0}^{t}Q(t,s)ds\le {\mathrm{\Pi }}_1,\mathrm{\forall }t\ge t_0.$$

(5)

; (${\mathbb{A}}_2$) = Let$\mathrm{\Pi }={\mathrm{\Pi }}_1+B$, $\rho (\mathrm{\Pi })<1$.Then there exist$z>0$, $z\in {\mathrm{\Omega }}_{\rho }(\mathrm{\Pi })$and a constant$k>0$, such that

$$y(t)<kz+{(E-\mathrm{\Pi })}^{-1}J,\text{for}t\ge t_0.$$

(6)

.

Proof By the condition $\mathrm{\Pi }\ge 0$ and the properties of nonnegative matrices, there exists a positive vector z such that $\mathrm{\Pi }z=\rho (\mathrm{\Pi })z$. Together with $\rho (\mathrm{\Pi })<1$ and Lemma 2.1, this implies that ${(E-\mathrm{\Pi })}^{-1}$ exists and ${(E-\mathrm{\Pi })}^{-1}\ge 0$.

For the initial conditions $y(t)\le \phi (t)$, $-\mathrm{\infty }<t\le t_0$, we have

$$y(t)\le k_{0}z+{(E-\mathrm{\Pi })}^{-1}J,-\mathrm{\infty }<t\le t_0,$$

(7)

where $z\in {\mathrm{\Omega }}_{\rho }(\mathrm{\Pi })$, $z>0$, $k_0=\frac{\parallel \phi \parallel }{{min}_{1\le i\le n}{z}_i}\ge 0$, $\parallel \phi \parallel ={max}_{1\le i\le n}\{{sup}_{-\mathrm{\infty }<t\le t_0}|{\phi }_i(t)|\}$.

From ${lim}_{t\to +\mathrm{\infty }}G(t,t_0)=0$, there must be a constant $T>0$ such that

$$G(t,t_0)\le \frac{1-\rho (\mathrm{\Pi })}{2}{k}_{0}z,\text{for}t>t_0+T.$$

(8)

By the continuity of $y(t)$, together with (4) and (7), there exists a constant $k>k_0$ such that

$$y(t)<kz+{(E-\mathrm{\Pi })}^{-1}J,\text{for}t\in (-\mathrm{\infty },t_0+T].$$

(9)

In the following, we shall prove that

$$y(t)<kz+{(E-\mathrm{\Pi })}^{-1}J,\text{for}t>t_0+T.$$

(10)

If this is not true, from (9) and the continuity of $y(t)$, then there must be a constant $t_1>t_0+T$ and some integer i such that

(11)

(12)

where ${\{\cdot \}}_i$ denotes the i th component of vector $\{\cdot \}$.

Using (3), (5), (8), (12) and $\rho (\mathrm{\Pi })<1$, we obtain that

$$\begin{array}{rl}{y}_i(t_1)& \le {\{G(t_1,t_0)+B{[y(t_1)]}_{\tau (t_1)}^{+}+{\int }_{t_0}^{t_1}Q(t_1,s){[y(s)]}_{\tau (s)}^{+}ds+J\}}_i\\ \le {\{\frac{1-\rho (\mathrm{\Pi })}{2}{k}_{0}z+[B+{\int }_{t_0}^{t_1}Q(t_1,s)ds][kz+{(E-\mathrm{\Pi })}^{-1}J]+J\}}_i\\ \le {\{\frac{1-\rho (\mathrm{\Pi })}{2}kz+\mathrm{\Pi }[kz+{(E-\mathrm{\Pi })}^{-1}J]+(E-\mathrm{\Pi }){(E-\mathrm{\Pi })}^{-1}J\}}_i\\ \le {\{\frac{1-\rho (\mathrm{\Pi })}{2}kz+\rho (\mathrm{\Pi })kz+(\mathrm{\Pi }+E-\mathrm{\Pi }){(E-\mathrm{\Pi })}^{-1}J\}}_i\\ \le {\{\frac{1+\rho (\mathrm{\Pi })}{2}kz+{(E-\mathrm{\Pi })}^{-1}J\}}_i\\ <{\{kz+{(E-\mathrm{\Pi })}^{-1}J\}}_i.\end{array}$$

(13)

This contradicts the equality in (11), and so (10) holds. The proof is complete. □

In order to study the attracting set, we rewrite Eq. (1) as

$$\{\begin{array}{c}x(t)=\mathrm{\Phi }(t,t_0)[\phi (t_0)-H(t_0,\phi (t_0-\tau (t_0)))]+H(t,x(t-\tau (t)))\hfill \\ \phantom{x(t)=}+{\int }_{t_0}^t\mathrm{\Phi }(t,s)[A(s)H(s,x(s-\tau (s)))+F(s,x(s-\tau (s)))]ds,\hfill & t\ge t_0,\hfill \\ x(t)=\phi (t),-\mathrm{\infty }<t\le t_0,\hfill \end{array}$$

(14)

where $\mathrm{\Phi }(t,t_0)$ is the fundamental matrix of the linear equation $\dot{x}(t)=A(t)x(t)$.

For (1), we suppose the following: (${\mathbb{B}}_1$): = ${[F(t,x(t-\tau (t)))]}^{+}\le Q(t){[x(t)]}_{\tau (t)}^{+}+J(t)$, ${[H(t,x(t-\tau (t)))]}^{+}\le \hat{H}{[x(t)]}_{\tau (t)}^{+}$, where $Q\in C[\mathbb{R},{\mathbb{R}}_{+}^{n\times n}]$, $\hat{H}\in {\mathbb{R}}_{+}^{n\times n}$, $J(t)={(J_1(t),\dots ,J_n(t))}^T\ge 0$, $J_i\in C[\mathbb{R},{\mathbb{R}}_{+}]$, $i=1,2,\dots ,n$.; (${\mathbb{B}}_2$): = ${[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}={(w_{ij}(t,s))}_{n\times n}$, $w_{ij}\in U{C}_t$, ${lim}_{t\to +\mathrm{\infty }}\mathrm{\Phi }(t,t_0)=0$. For $\mathrm{\forall }t\ge t_0$, there exist a constant matrix ${\mathrm{\Pi }}_1\ge 0$ and a vector $\hat{J}\ge 0$ such that

$${\int }_{t_0}^t{[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}ds\le {\mathrm{\Pi }}_1,{\int }_{t_0}^t{[\mathrm{\Phi }(t,s)]}^{+}J(s)ds\le \hat{J}.$$

; (${\mathbb{B}}_3$): = Let $\mathrm{\Pi }={\mathrm{\Pi }}_1+\hat{H}$, $\rho (\mathrm{\Pi })<1$..

Theorem 3.2 Assume that (${\mathbb{B}}_1$)-(${\mathbb{B}}_3$) hold. Then$S=\{x\in {\mathbb{R}}^n|{[x]}^{+}\le {(E-\mathrm{\Pi })}^{-1}\hat{J}\}$is a global attracting set of (1).

Proof From (14) and (${\mathbb{B}}_1$), we can get for $\mathrm{\forall }t\ge t_0$,

$$\begin{array}{rl}{[x(t)]}^{+}\le & {[\mathrm{\Phi }(t,t_0)]}^{+}{[\phi (t_0)-H(t_0,\phi (t_0-\tau (t_0)))]}^{+}+\hat{H}{[x(t)]}_{\tau (t)}^{+}\\ +{\int }_{t_0}^t{[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}{[x(s)]}_{\tau (s)}^{+}ds+{\int }_{t_0}^t{[\mathrm{\Phi }(t,s)]}^{+}J(s)ds.\end{array}$$

(15)

By (${\mathbb{B}}_2$)-(${\mathbb{B}}_3$) and Theorem 3.1, there exists a constant $k>0$ such that

$${[x(t)]}^{+}<kz+{(E-\mathrm{\Pi })}^{-1}\hat{J},\text{for}t\ge t_0,$$

(16)

where $z\in {\mathrm{\Omega }}_{\rho }(\mathrm{\Pi })$, $z>0$, $k>\frac{\parallel \phi \parallel }{{min}_{1\le i\le n}{z}_i}$, $\parallel \phi \parallel ={max}_{1\le i\le n}\{{sup}_{-\mathrm{\infty }<t\le t_0}|{\phi }_i(t)|\}$.

From (16), there must be a constant vector $\sigma \ge 0$ such that

$$\overline{\underset{t\to +\mathrm{\infty }}{lim}}{[x(t)]}^{+}=\sigma \le kz+{(E-\mathrm{\Pi })}^{-1}\hat{J}.$$

(17)

Next, we will show that $\sigma \in S$. From ${lim}_{t\to +\mathrm{\infty }}\mathrm{\Phi }(t,t_0)=0$, $w_{ij}\in U{C}_t$, for any $\epsilon >0$ and $e={(1,1,\dots ,1)}^T\in {\mathbb{R}}_{+}^n$, there exists a positive number $T_1>t_0$ such that for all $t>T_1$

(18)

(19)

According to the definition of superior limit and ${lim}_{t\to +\mathrm{\infty }}(t-\tau (t))=+\mathrm{\infty }$, there exists sufficiently large $T_2\ge T_1$ such that

$${[x(t)]}_{r(t)}^{+}<\sigma +\epsilon e,t\ge T_2,$$

(20)

where $r(t)=T_1+{sup}_{t-T_1\le s\le t}\tau (s)$. Therefore, from (${\mathbb{B}}_2$), (15) and (18)-(19), when $t\ge T_2$, we obtain

$$\begin{array}{rl}{[x(t)]}^{+}\le & \frac{\epsilon e}{2}+\hat{H}{[x(t)]}_{\tau (t)}^{+}+{\int }_{t_0}^{t-T_1}{[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}{[x(s)]}_{\tau (s)}^{+}ds\\ +{\int }_{t-T_1}^t{[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}{[x(s)]}_{\tau (s)}^{+}ds+\hat{J}\\ \le & \frac{\epsilon e}{2}+\hat{H}{[x(t)]}_{\tau (t)}^{+}\\ +{\int }_{t_0}^{t-T_1}{[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}[kz+{(E-\mathrm{\Pi })}^{-1}\hat{J}]ds\\ +{\int }_{t-T_1}^t{[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}{[x(s)]}_{\tau (s)}^{+}ds+\hat{J}\\ \le & \epsilon e+\hat{H}(\sigma +\epsilon e)+{\int }_{t-T_1}^t{[\mathrm{\Phi }(t,s)]}^{+}\{{[A(s)]}^{+}\hat{H}+Q(s)\}(\sigma +\epsilon e)ds+\hat{J}\\ \le & \epsilon e+\mathrm{\Pi }(\sigma +\epsilon e)+\hat{J}.\end{array}$$

(21)

Due to (17) and the definition of superior limit, there exists $T_3\ge T_2$ such that ${[x(T_3)]}^{+}>\sigma -\epsilon e$. Combining with (21), we get

$$\sigma -\epsilon e<{[x(T_3)]}^{+}\le \epsilon e+\mathrm{\Pi }(\sigma +\epsilon e)+\hat{J}.$$

(22)

Letting $\epsilon \to 0$, we have $\sigma \le {(E-\mathrm{\Pi })}^{-1}\hat{J}$, that is $\sigma \in S$, and the proof is completed. □

Remark 3.1 Theorem 3.2 is a generalization of the results in [12, 13] as $H\equiv 0$ in (1) without the boundedness of $\tau (t)$.

Corollary 3.1 Suppose that the conditions of Theorem 3.2 hold and$J(t)\equiv 0$. If$x(t)=0$is an equilibrium point of System (1), then the equilibrium point$x(t)=0$is globally asymptotically stable.

4 Example

Consider the following scalar equation

$$\frac{d}{dt}[x(t)-\frac{1}{4}x(t-\tau (t))]=-atx(t)+btx(t-\tau (t))+J(t),t\ge 0,$$

(23)

where $a>0$, b are constants, $J(t)=sint$ and $\tau (t)=\frac{1}{2}(t+sint)$.

We easily verify that $|\mathrm{\Phi }(t,0)|=e^{-a{\int }_0^{t}vdv}$, ${lim}_{t\to +\mathrm{\infty }}\mathrm{\Phi }(t,0)=0$. For any $t\ge 0$, we get

If $\frac{|b|}{a}<\frac{1}{2}$, we can get $S=\{x\in {\mathbb{R}}^n||x|\le \frac{2}{a-2|b|}\}$ is the global attracting set for (23).

Remark 4.1 If $J(t)\equiv 0$ and $\frac{|b|}{a}<\frac{1}{2}$, then every solution of (23) tends to zero at ∞. However, the methods in [4, 6] are inefficient for (23) because the variable coefficients $A(t)=-at$ and $Q(t)=bt$ are unbounded for $t\ge 0$.