Weighted shadowing for delay differential equations

Abstract

After introducing the notion of weighted shadowing, we give sufficient conditions under which certain nonlinear perturbations of nonautonomous linear delay differential equations exhibit the shadowing property with respect to a given weight function. As a corollary, it is shown that if the unperturbed equation admits a shifted exponential dichotomy and the Lipschitz constant of the nonlinear term is sufficiently small, then the perturbed system has a shadowing property with respect to an exponential weight function. An application to differential equations with small delay is given. The results are new even in the case of the standard shadowing property.

Appendix

For completeness, we prove a variant of the Moore–Osgood theorem which is needed for the proof of Lemma 3.2. For analogous results, see [15, Theorem 2.1.4.1 and Remark 2.1.4.1].

Moore-Osgood type theorem. Let $(X,\Vert \cdot \Vert _X)$ be a Banach space and $\sigma \in {\mathbb {R}}$. Suppose that $f_n:[\sigma ,\infty )\rightarrow X$, $n\in \mathbb N$, is a sequence of functions such that:

• (a) the limit $g_n:=\lim _{t\rightarrow \infty }f_n(t)$ exists uniformly for $n\in {\mathbb {N}}$,

• (b) the limit $h(t):=\lim _{n\rightarrow \infty }f_n(t)$ exists for all $t\ge \sigma$.

Then the limit

$$\begin{aligned} l:=\lim _{n\rightarrow \infty }g_n \end{aligned}$$

(A.1)

exists and

$$\begin{aligned} \lim _{t\rightarrow \infty }h(t)=l, \end{aligned}$$

(A.2)

i.e.,

$$\begin{aligned} \lim _{n\rightarrow \infty }\lim _{t\rightarrow \infty }f_n(t)=\lim _{t\rightarrow \infty }\lim _{n\rightarrow \infty }f_n(t). \end{aligned}$$

(A.3)

Proof

First we prove the existence of the limit in (A.1). Let $\epsilon >0$. Condition (a) guarantees that if we choose t large enough, then

$$\begin{aligned} \Vert f_n(t)-g_n\Vert _X<\frac{\epsilon }{4}\qquad \text { for all}\ n\in {\mathbb {N}}. \end{aligned}$$

(A.4)

Let such t be fixed. Condition (b) implies the existence of $n_0$ such that

$$\begin{aligned} \Vert f_n(t)-h(t)\Vert _X<\frac{\epsilon }{4}\qquad \text { for}\ n\ge n_0. \end{aligned}$$

(A.5)

If $n, m\ge n_0$, then (A.4), and (A.5), together with the triangle inequality, imply that

$$\begin{aligned} \Vert g_n-g_m\Vert _X&\le \Vert g_n-f_n(t)\Vert _X+\Vert f_n(t)-h(t)\Vert _X\\&\quad +\Vert h(t)-f_m(t)\Vert _X+\Vert f_m(t)-g_m\Vert _X<4\frac{\epsilon }{4}=\epsilon . \end{aligned}$$

Since $\epsilon >0$ was arbitrary, this shows that $\{g_n\}_{n=1}^\infty$ is a Cauchy sequence in X. Therefore, the limit in (A.1) exists.

Now we prove the limit relation (A.2). Let $\epsilon >0$. Condition (a) implies the existence of $\tau \ge \sigma$ such that

$$\begin{aligned} \Vert f_n(t)-g_n\Vert _X<\frac{\epsilon }{3}\qquad \text {whenever } t\ge \tau \text { and } n\in {\mathbb {N}}. \end{aligned}$$

(A.6)

Let $t\ge \tau$. Condition (b) guarantees the existence of $n_1$ such that

$$\begin{aligned} \Vert f_n(t)-h(t)\Vert _X<\frac{\epsilon }{3}\qquad \text { for}\ n\ge n_1, \end{aligned}$$

(A.7)

while (A.1) implies that there exists $n_2$ such that

$$\begin{aligned} \Vert g_n-l\Vert _X<\frac{\epsilon }{3}\qquad \text { for}\ n\ge n_2. \end{aligned}$$

(A.8)

Define $n:=\max \{n_1,n_2\}$. Then (A.6), (A.7), and (A.8), together with the triangle inequality, imply that

$$\begin{aligned} \Vert h(t)-l\Vert _X \le \Vert h(t)-f_n(t)\Vert _X+\Vert f_n(t)-g_n\Vert _X+\Vert g_n-l\Vert _X<3\frac{\epsilon }{3}=\epsilon . \end{aligned}$$

Thus, $\Vert h(t)-l\Vert _X<\epsilon$ for all $t\ge \tau$. Since $\epsilon >0$ was arbitrary, this proves (A.2).$\square$