Parameterized shadowing for nonautonomous dynamics

TL;DR

This paper establishes conditions for shadowing and stability in nonautonomous, nonlinear differential and difference equations depending on parameters, including cases with non-hyperbolic linear parts.

Contribution

It provides new sufficient conditions for $C^k$ shadowing and Hyers-Ulam stability in parameter-dependent nonautonomous systems, extending previous results to non-hyperbolic cases.

Findings

Conditions for $C^k$ shadowing in nonhyperbolic systems

Results on parameterized Hyers-Ulam stability for hyperbolic systems

Applicability to a broad class of nonlinear, nonautonomous equations

Abstract

For nonautonomous and nonlinear differential and difference equations depending on a parameter, we formulate sufficient conditions under which they exhibit $C^k$, $k\in \N$ shadowing with respect to a parameter. Our results are applicable to situations when the linear part is not hyperbolic. In the case when the linear part is hyperbolic, we obtain results dealing with parameterized Hyers-Ulam stability.

Equations

$$x^{\prime }=A(t)+f_{\lambda }(t,x),t\in \mathbb{R}.$$

$$x^{\prime }=A(t)x,t\in \mathbb{R}.$$

$$\mathcal{G}(t,s)=\{\begin{array}{ll}T(t,s)P(s)& t\ge s;\\ -T(t,s)(\text{Id}-P(s))& t<s,\end{array}$$

$$|f_{\lambda }(t,x)-f_{\lambda }(t,y)|\le c(t)|x-y|,$$

$$x^{\prime }=A(t)x+f_{\lambda }(t,x),t\in \mathbb{R}.$$

$$q:=\underset{t\in \mathbb{R}}{\sup }{\int }_{-\infty }^{\infty }c(s)\Vert \mathcal{G}(t,s)\Vert ds<1.$$

$$L:=\underset{t\in \mathbb{R}}{\sup }{\int }_{-\infty }^{\infty }\epsilon (s)\Vert \mathcal{G}(t,s)\Vert ds<+\infty ,$$

$$|y_{\lambda }^{\prime }(t)-A(t)y_{\lambda }(t)-f_{\lambda }(t,y_{\lambda }(t))|\le \epsilon (t)\text{for t∈R.}$$

$$\underset{t\in \mathbb{R}}{\sup }|x_{\lambda }(t)-y_{\lambda }(t)|\le \frac{L}{1-q}.$$

$$\Vert z{\Vert }_{\infty }:=\underset{t\in \mathbb{R}}{\sup }|z(t)|<+\infty .$$

$$({\mathcal{T}}_{\lambda }z)(t):={\int }_{-\infty }^{\infty }\mathcal{G}(t,s)(A(s)y_{\lambda }(s)+f_{\lambda }(s,y_{\lambda }(s)+z(s))-y_{\lambda }^{\prime }(s))ds,$$

$$\begin{array}{rl}& |A(s)y_{\lambda }(s)+f_{\lambda }(s,y_{\lambda }(s)+z(s))-y_{\lambda }^{\prime }(s)|\\ & \le |A(s)y_{\lambda }(s)+f_{\lambda }(s,y_{\lambda }(s))-y_{\lambda }^{\prime }(s)|+|f_{\lambda }(s,y_{\lambda }(s)+z(s))-f_{\lambda }(s,y_{\lambda }(s))|\\ & \le \epsilon (s)+c(s)|z(s)|,\end{array}$$

$$|({\mathcal{T}}_{\lambda }z)(t)|\le {\int }_{-\infty }^{\infty }\Vert \mathcal{G}(t,s)\Vert (\epsilon (s)+c(s)|z(s)|)ds,$$

$$\Vert {\mathcal{T}}_{\lambda }z{\Vert }_{\infty }\le L+q\Vert z{\Vert }_{\infty },z\in \mathcal{Y}.$$

$$\Vert {\mathcal{T}}_{\lambda }0{\Vert }_{\infty }\le L.$$

$$|f_{\lambda }(s,y_{\lambda }(s)+z_1(s))-f_{\lambda }(s,y_{\lambda }(s)+z_2(s))|\le c(s)|z_1(s)-z_2(s)|,$$

$$|({\mathcal{T}}_{\lambda }{z}_1)(t)-({\mathcal{T}}_{\lambda }{z}_2)(t)|\le {\int }_{-\infty }^{\infty }c(s)\Vert \mathcal{G}(t,s)\Vert |z_1(s)-z_2(s)|ds,$$

$$\Vert {\mathcal{T}}_{\lambda }{z}_1-{\mathcal{T}}_{\lambda }{z}_2{\Vert }_{\infty }\le q\Vert z_1-z_2{\Vert }_{\infty }.$$

$$\mathcal{D}:=\bigg{\{}z\in\mathcal{Y}:\|z\|_{\infty}\leq\frac{L}{1-q}\bigg{\}}.$$

$$\Vert {\mathcal{T}}_{\lambda }z{\Vert }_{\infty }\le \Vert {\mathcal{T}}_{\lambda }0{\Vert }_{\infty }+\Vert {\mathcal{T}}_{\lambda }z-{\mathcal{T}}_{\lambda }0{\Vert }_{\infty }\le L+\frac{qL}{1-q}=\frac{L}{1-q}.$$

$$\underset{\lambda \in \Sigma }{\sup }\underset{x\in X}{\sup }\Vert \frac{{\partial }^i}{\partial {\lambda }^{i-j}\partial x^j}{f}_{\lambda }(t,x)\Vert \le C\epsilon (t);$$

$$\bigg{|}\frac{\partial^{i}}{\partial\lambda^{i}}y_{\lambda}^{\prime}(t)-A(t)\frac{\partial^{i}}{\partial\lambda^{i}}y_{\lambda}(t)-\frac{d^{i}}{d\lambda^{i}}f_{\lambda}(t,y_{\lambda}(t))\bigg{|}\leq C\varepsilon(t)$$

$$\max_{1\leq i\leq k+1}\sup_{t\in\mathbb{R}}\bigg{\|}\frac{\partial^{i}}{\partial\lambda^{i}}y_{\lambda}(t)\bigg{\|}\leq M_{\lambda_{0}}\quad\text{for each}\quad\lambda\in\Sigma_{0}.$$

$$\begin{array}{rl}\Vert z_{\lambda +h}-z_{\lambda }{\Vert }_{\infty }& =\Vert {\mathcal{T}}_{\lambda +h}{z}_{\lambda +h}-{\mathcal{T}}_{\lambda }{z}_{\lambda }{\Vert }_{\infty }\\ & \le \Vert {\mathcal{T}}_{\lambda +h}{z}_{\lambda +h}-{\mathcal{T}}_{\lambda +h}{z}_{\lambda }{\Vert }_{\infty }+\Vert {\mathcal{T}}_{\lambda +h}{z}_{\lambda }-{\mathcal{T}}_{\lambda }{z}_{\lambda }{\Vert }_{\infty }\\ & \le q\Vert z_{\lambda +h}-z_{\lambda }{\Vert }_{\infty }+\Vert {\mathcal{T}}_{\lambda +h}{z}_{\lambda }-{\mathcal{T}}_{\lambda }{z}_{\lambda }{\Vert }_{\infty }.\end{array}$$

$$\begin{array}{rl}\Vert z_{\lambda +h}-z_{\lambda }{\Vert }_{\infty }& \le \frac{1}{1-q}\Vert {\mathcal{T}}_{\lambda +h}{z}_{\lambda }-{\mathcal{T}}_{\lambda }{z}_{\lambda }{\Vert }_{\infty }.\end{array}$$

$$\frac{{\partial }^i}{\partial {\lambda }^i}{z}_{\lambda }={\left(\text{Id}-\frac{\partial }{\partial z}{\mathcal{T}}_{\lambda }{z}_{\lambda }\right)}^{-1}\mathcal{F}(\lambda ),$$

$${\mathcal{F}}(\lambda):=\sum\left(\frac{\partial^{\ell}}{\partial\lambda^{\ell-j}\partial z^{j}}\mathcal{T}_{\lambda}z_{\lambda}\right)\bigg{(}\frac{\partial^{\ell_{1}}}{\partial\lambda^{\ell_{1}}}z_{\lambda},\ldots,\frac{\partial^{\ell_{j}}}{\partial\lambda^{\ell_{j}}}z_{\lambda}\bigg{)},$$

$$z_{\lambda +h}-z_{\lambda }$$

$$={\mathcal{T}}_{\lambda +h}{z}_{\lambda +h}-{\mathcal{T}}_{\lambda +h}{z}_{\lambda }+{\mathcal{T}}_{\lambda +h}{z}_{\lambda }-{\mathcal{T}}_{\lambda }{z}_{\lambda }$$

$$={\int }_0^1\left(\frac{\partial }{\partial z}{\mathcal{T}}_{\lambda +h}(z_{\lambda }+\theta (z_{\lambda +h}-z_{\lambda }))\right)(z_{\lambda +h}-z_{\lambda })d\theta$$

$$+\left(\frac{\partial }{\partial \lambda }{\mathcal{T}}_{\lambda }{z}_{\lambda }\right)h+\mathbf{o}(h),$$

$$z_{\lambda +h}-z_{\lambda }$$

Full text

1. Introduction

In the present paper, we consider nonautonomous and nonlinear differential equations of the form

[TABLE]

Here, $A(t)$, $t\in\mathbb{R}$ are linear operators acting on a Banach space $X=(X,|\cdot|)$ and $f_{\lambda}\colon\mathbb{R}\times X\to X$ is a nonlinear map for each $\lambda\in\Sigma$, where $\Sigma$ is an open subset of some Banach space. In [8], the authors have formulated very general conditions under which (1) admits a shadowing property which guarantees that in a neighborhood of each approximate solution of (1) we can construct its exact solution. An important feature of the results established in [8] is that they do not require any hyperbolicity conditions for the linear part of (1). In the setting when the linear part of (1) is hyperbolic (i.e. admits an exponential dichotomy or trichotomy), the results of [8] essentially reduce to the previously known results devoted to Hyers-Ulam stability for (1) (see [4]).

We stress that the literature devoted to Hyers-Ulam stability for differential and difference equations is vast and contains many interesting contributions. In particular, for some recent results devoted to the relationship between Hyers-Ulam stability and hyperbolicity, we refer to [2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 19] and references therein. In addition, we mention the works of Anderson and Onitsuka [1], Fukutaka and Onitsuka [14, 15, 16], Popa and Raşa [21, 22] and Wang et. al [23, 24] among others. For the description of the shadowing theory in the context of smooth dynamical systems, we refer to [17, 18, 20].

In order to describe the results of the present paper, suppose that for each $\lambda\in\Sigma$ we have an approximate solution $y_{\lambda}$ of (1) which is shadowed by an unique exact solution $x_{\lambda}$. Our main objective is to study the dependence of $x_{\lambda}$ on the parameter $\lambda$. More precisely, given a $k\in\mathbb{N}$, in Theorem 2 we formulate sufficient conditions under which the map $\lambda\mapsto x_{\lambda}(t)$ is $C^{k}$ for each $t\in\mathbb{R}$. In the particular case when the linear part of (1) admits an exponential dichotomy, our result simplifies and yields a parameterized Hyers-Ulam stability result (see Corollary 1). To the best of our knowledge the parameterized version of Hyers-Ulam stability has not been studied earlier and even our Corollary 1 is a completely new result.

Our techniques rely on those developed in [8]. More precisely, $y_{\lambda}-x_{\lambda}$ can be obtained as a fixed point of an operator $\mathcal{T}_{\lambda}$ which is a contraction on a closed ball around origin in a suitable Banach space $\mathcal{Y}$. Thus, it remains to study the regularity of the fixed point of $\mathcal{T}_{\lambda}$ with respect to the parameter $\lambda$.

The paper is organized as follows: in Section 2 we recall some preliminary material from [8]. In Section 3 we establish a parameterized shadowing result for (1). Finally, in Section 4 we discuss the parameterized shadowing for a discrete-time counterpart of (1).

2. Preliminaries

Let $X=(X,|\cdot|)$ be an arbitrary Banach space. By $\mathcal{B}(X)$ we will denote the space of all bounded linear operators on $X$ equipped with the operator norm $\|\cdot\|$. Let $A\colon\mathbb{R}\to\mathcal{B}(X)$ be a continuous map. We consider the associated linear differential equation

[TABLE]

By $T(t,s)$ we will denote the evolution family corresponding to (2). Assume that $P\colon\mathbb{R}\to\mathcal{B}(X)$ is a continuous map and set

[TABLE]

where Id denotes the identity operator on $X$.

Let $\Sigma=(\Sigma,|\cdot|)$ be (an open subset of) another Banach space. Although the norms on $X$ and $\Sigma$ are denoted by the same symbol this will not cause confusion. Suppose that for each $\lambda\in\Sigma$, we have continuous map $f_{\lambda}\colon\mathbb{R}\times X\to X$ with the property that there exists a continuous map $c\colon\mathbb{R}\to[0,\infty)$ such that

[TABLE]

for $t\in\mathbb{R}$, $\lambda\in\Sigma$ and $x,y\in X$.

For each $\lambda\in\Sigma$, we consider the nonlinear differential equation given by

[TABLE]

The following result is essentially a consequence of [8, Theorem 1]. We include it primarily because many arguments in the following section are based on its proof.

Theorem 1**.**

Suppose that (2) does not have nonzero bounded solutions, and assume that

[TABLE]

Furthermore, let $\varepsilon\colon\mathbb{R}\to(0,\infty)$ be a continuous map such that

[TABLE]

and, given $\lambda\in\Sigma$, let $y_{\lambda}\colon\mathbb{R}\to X$ be a continuously differentiable map satisfying

[TABLE]

Then, for each $\lambda\in\Sigma$ there exists a unique solution $x_{\lambda}\colon\mathbb{R}\to X$ of (4) such that

[TABLE]

Proof.

Let $\mathcal{Y}$ denote the space of all continuous maps $z\colon\mathbb{R}\to X$ such that

[TABLE]

Then, $(\mathcal{Y},\|\cdot\|_{\infty})$ is a Banach space. Take a fixed $\lambda\in\Sigma$ and set

[TABLE]

for $t\in\mathbb{R}$ and $z\in\mathcal{Y}$. Note that it follows from (3) and (7) that

[TABLE]

for $s\in\mathbb{R}$. Hence,

[TABLE]

for $t\in\mathbb{R}$. This together with (5) and (6) implies that

[TABLE]

In particular, $\mathcal{T}_{\lambda}\colon\mathcal{Y}\to\mathcal{Y}$ is well-defined. Moreover, setting $z=0$ in (10) we have that

[TABLE]

Take now $z_{1},z_{2}\in\mathcal{Y}$. Observe that (3) implies that

[TABLE]

for $s\in\mathbb{R}$. Consequently,

[TABLE]

for $t\in\mathbb{R}$. This together with (5) gives that

[TABLE]

Set

[TABLE]

It is apparent that $\mathcal{D}$ is a non-empty closed subset of $\mathcal{Y}$ and it is therefore a complete metric space with the distance $\|\cdot\|_{\infty}$. For $z\in\mathcal{D}$, it follows from (11) and (12) that

[TABLE]

Thus, $\mathcal{T}_{\lambda}(\mathcal{D})\subset\mathcal{D}$. By (12), we have that $\mathcal{T}_{\lambda}$ is a contraction on $\mathcal{D}$, and therefore it has a unique fixed point $z_{\lambda}\in\mathcal{D}$. It is straightforward to verify that $x_{\lambda}:=y_{\lambda}+z_{\lambda}$ is a solution of (4). Moreover, since $z_{\lambda}\in\mathcal{D}$, we have that (8) holds. The uniqueness of $x_{\lambda}$ can be established by repeating the arguments in the proof of [8, Theorem 1]. The proof of the theorem is completed. ∎

3. Regularity with respect to a parameter

We are now interested in formulating sufficient conditions under which the map $\lambda\mapsto x_{\lambda}(t)$ is of class $C^{k}$ with $k\in\mathbb{N}$, for each $t\in\mathbb{R}$.

3.1. $C^{k}$ regularity without exponential dichotomy

First, we recall some notions for clarity. Let $\phi:Y_{1}\times Y_{2}\to Z$ be a map, where $Y_{1}$, $Y_{2}$ and $Z$ are three Banach spaces. We say that the map $y_{2}\mapsto\phi(y_{1},y_{2})$ is $C^{k}$ on an open set $\mathcal{U}\subset Y_{1}\times Y_{2}$ if $(y_{1},y_{2})\mapsto\phi(y_{1},y_{2})$ is $k$-times differentiable with respect to $y_{2}$ on $\mathcal{U}$ and $\frac{\partial^{i}}{\partial{y_{2}}^{i}}\phi$, $i$-th partial derivative of $\phi$ with respect to $y_{2}$, is continuous on $\mathcal{U}$ for $1\leq i\leq k$. By $\frac{\partial^{i}}{\partial{y_{2}}^{i}}\phi(y_{1},y_{2})$ we denote the value of $\frac{\partial^{i}}{\partial{y_{2}}^{i}}\phi$ at the point $(y_{1},y_{2})\in Y_{1}\times Y_{2}$. For a map with more variables, we can have analogous notions.

Theorem 2**.**

For each $\lambda\in\Sigma$ let $y_{\lambda}:\mathbb{R}\to X$ be a continuously differentiable map satisfying (7) and suppose that the assumptions of Theorem 1 hold. Let $x_{\lambda}:\mathbb{R}\to X$ be the map associated to $y_{\lambda}$ by Theorem 1 and take $k\in\mathbb{N}$. In addition, suppose there exists $C>0$ such that the following conditions hold:

• •

the map $(\lambda,x)\mapsto f_{\lambda}(t,x)$ is $C^{k+1}$ and for all $2\leq i\leq k+1$ and $0\leq j\leq i$,

[TABLE]

• •

the maps $\lambda\mapsto y_{\lambda}(t)$ and $\lambda\mapsto y_{\lambda}^{\prime}(t)$ both are $C^{k+1}$ such that

[TABLE]

for all $1\leq i\leq k+1$, all $t\in\mathbb{R}$ and all $\lambda\in\Sigma$. Furthermore, for every $\lambda_{0}\in\Sigma$ there exist $M_{\lambda_{0}}>1$ and a neighborhood $\Sigma_{0}$ of $\lambda_{0}$ such that

[TABLE]

Then, the map $\lambda\mapsto x_{\lambda}(t)$ is $C^{k}$ for each $t\in\mathbb{R}$.

Remark 1**.**

Observe that if the map $\lambda\mapsto y_{\lambda}$ is constant and condition (13) holds then conditions (14) and (15) are automatically satisfied.

In order to establish the statement of Theorem 2, we need to set up several auxiliary results. We first observe that since $x_{\lambda}=y_{\lambda}+z_{\lambda}$, it is sufficient to prove that $\lambda\mapsto z_{\lambda}(t)$ is $C^{k}$ for each $t\in\mathbb{R}$. In fact, we will prove that the map $\lambda\mapsto z_{\lambda}$ is $C^{k}$ as a map from $\Sigma$ to $\mathcal{Y}$, which immediately implies the desired conclusion.

Let $\mathcal{T}_{\lambda}$ be the operator constructed in the proof of Theorem 1 (see (9)).

Lemma 1**.**

Suppose $\Sigma\ni\lambda\mapsto\mathcal{T}_{\lambda}z_{\lambda_{0}}$ is continuous at $\lambda_{0}$ for every $\lambda_{0}\in\Sigma$. Then, $\lambda\mapsto z_{\lambda}$ is continuous.

Proof.

Fix an arbitrary $\lambda\in\Sigma$. Then, for $h\in\Sigma$ we have (see (12)) that

[TABLE]

Since $q<1$, we see that

[TABLE]

According to our assumption, the right hand side of (16) goes to [math] as $|h|\to 0$, which results in the continuity as claimed. The proof is completed. ∎

Lemma 2**.**

Let $k\in\mathbb{N}$. Suppose that the map $\Sigma\times\mathcal{Y}\ni(\lambda,z)\mapsto\mathcal{T}_{\lambda}z$ is $C^{k}$ on an open set $\mathcal{W}$ containing the set ${\mathcal{S}}:=\{(\lambda,z_{\bar{\lambda}}):\lambda,\bar{\lambda}\in\Sigma\}$. Then, $\Sigma\ni\lambda\mapsto z_{\lambda}$ is also $C^{k}$.

Proof.

We utilize induction to prove that the $i$-th derivative of the map $\lambda\mapsto z_{\lambda}$ for $1\leq i\leq k$ is continuous and has the form

[TABLE]

where

[TABLE]

where the sum is taken over $1\leq\ell\leq i$, $0\leq j\leq\ell$ and some nonnegative integers $\ell_{1},\ldots,\ell_{j}$ satisfying $\ell_{1}+\ldots+\ell_{j}=i+j-\ell$.

We start by observing that, since $(\lambda,z)\mapsto\mathcal{T}_{\lambda}z$ is $C^{1}$ on the open set $\mathcal{W}$ containing ${\mathcal{S}}$, we have that

[TABLE]

where $\mathbf{o}(h)$ means that $\lim_{h\to 0}\frac{\|\mathbf{o}(h)\|_{\infty}}{|h|}=0$. Since the derivative of $z\mapsto\mathcal{T}_{\lambda}z$ is continuous at $(\lambda,z_{\lambda})$, it follows from (3.1) and the conclusion of Lemma 1 that

[TABLE]

where $\alpha$ is a linear operator such that $\|\alpha\|\to 0$ as $|h|\to 0$. Moreover, by (16), given $\varepsilon>0$, whenever $|h|$ is sufficiently small we have that

[TABLE]

In particular, when $|h|$ is sufficiently small, we have that

[TABLE]

Therefore, since $\|\alpha\|\to 0$ when $|h|\to 0$, it follows that $\alpha(z_{\lambda+h}-z_{\lambda})=\mathbf{o}(h)$. Plugging this information into (3.1) we obtain that

[TABLE]

Now, observing that (12) implies

[TABLE]

it follows that the map $\text{\rm Id}-\frac{\partial}{\partial z}\mathcal{T}_{\lambda}z_{\lambda}$ is invertible. Consequently, using (20) we obtain that

[TABLE]

This proves that $\lambda\mapsto z_{\lambda}$ is differentiable and, moreover, that its derivative is given by

[TABLE]

which is of the form in (17) with $i=1$. Due to Lemma 1 and the assumptions in Lemma 2, it is easy to see that the derivative map $\lambda\mapsto\frac{\partial}{\partial\lambda}z_{\lambda}$ is continuous. Hence, it is proved that $\lambda\mapsto z_{\lambda}$ is $C^{1}$.

Assume that (17) holds and that $\frac{\partial^{i}}{\partial\lambda^{i}}z_{\lambda}$ is continuous for all $1\leq i\leq m$, $1\leq m\leq k-1$. Before proving that (17) holds for $i=m+1$ and that $\frac{\partial^{m+1}}{\partial\lambda^{m+1}}z_{\lambda}$ is continuous, let us discuss the differentiability of $\left(\text{\rm Id}-\frac{\partial}{\partial z}\mathcal{T}_{\lambda}z_{\lambda}\right)^{-1}$. By our assumptions, for sufficiently small $h\in\Sigma$, we have that

[TABLE]

and similarly

[TABLE]

It follows that $\left(\text{\rm Id}-\frac{\partial}{\partial z}\mathcal{T}_{\lambda}z_{\lambda}\right)^{-1}$ is differentiable with respect to $\lambda$ and that its derivative is given by

[TABLE]

for $h\in\Sigma$. It follows that $\frac{\partial^{m}}{\partial\lambda^{m}}z_{\lambda}$ is differentiable. By left-multiplying (17) (with $i=m$) by ${\rm Id}-\frac{\partial}{\partial z}\mathcal{T}_{\lambda}z_{\lambda}$ and differentiating both sides, we obtain that

[TABLE]

which gives that

[TABLE]

Clearly, $\frac{\partial^{m+1}}{\partial\lambda^{m+1}}z_{\lambda}$ is of the form (17) and is continuous according to the inductive assumption and $C^{k}$ regularity of the map $(\lambda,z)\mapsto\mathcal{T}_{\lambda}z$. By induction, (17) is proved and $\frac{\partial^{i}}{\partial\lambda^{i}}z_{\lambda}$ is continuous for all $1\leq i\leq k$. Therefore, the map $\lambda\to z_{\lambda}$ is $C^{k}$. The proof of the lemma is completed. ∎

Remark 2**.**

Observe that Lemmas 1 and 2 are general results about fixed points of contractions on Banach spaces. In fact, in the proof of these results we have only exploited the fact that $z_{\lambda}\in\mathcal{D}$ is a fixed point of a contraction map $\mathcal{T}_{\lambda}$ acting on a Banach space and not the particular form of the operator $\mathcal{T}_{\lambda}$. In particular, these results hold true for any map $\lambda\to z_{\lambda}$ where $z_{\lambda}$ is a fixed point of a contraction $\mathcal{T}_{\lambda}$ on a Banach space.

In order to conclude the proof of Theorem 2 what remains to be done is to show that the hypothesis given in its statement imply that the assumptions of Lemma 2 are satisfied. This is accomplished in Lemmas 3 and 4 below.

Lemma 3**.**

Suppose that the assumptions of Theorem 2 hold. Then, $\mathcal{Y}\ni z\mapsto\mathcal{T}_{\lambda}z$ is $C^{k}$ on an open set $\mathcal{W}$ containing the set ${\mathcal{S}}$, where ${\mathcal{S}}$ is defined in Lemma 2.

Proof.

It suffices to prove that, for any $(\lambda_{0},z_{\bar{\lambda}_{0}})\in{\mathcal{S}}$, there exists an open set $\mathcal{N}\ni(\lambda_{0},z_{\bar{\lambda}_{0}})$ on which the map $z\mapsto\mathcal{T}_{\lambda}z$ is $C^{k}$. We use induction to prove that on $\mathcal{N}$, the $i$-th derivative of $z\mapsto\mathcal{T}_{\lambda}z$ is given by

[TABLE]

for $1\leq i\leq k$, where $\eta_{j}\in\mathcal{Y}$ for $1\leq j\leq i$. By arguing as in (21), it follows from (3) that

[TABLE]

We first show that (3.1) holds at the point $(\lambda_{0},z_{\bar{\lambda}_{0}})$. Using the definition of $\mathcal{T}_{\lambda}$ and the first assumption of Theorem 2, we get that for $\eta\in\mathcal{Y}$ with sufficiently small $\|\eta\|_{\infty}$,

[TABLE]

where

[TABLE]

By the first assumption of Theorem 2 again, $|\alpha_{1}(\tau)|$ can be estimated as follows:

[TABLE]

Due to (5), (6), (23) and (3.1), all the integrals above converge. In addition, by (6) and (3.1), we have that

[TABLE]

for every $t\in\mathbb{R}$, and consequently

[TABLE]

This fact combined with (3.1) implies that $z\mapsto\mathcal{T}_{\lambda_{0}}z$ is differentiable at $z_{\bar{\lambda}_{0}}$ and that its derivative is given by

[TABLE]

for $t\in\mathbb{R}$ and $\eta\in\mathcal{Y}$.

Similarly, one can show that $z\mapsto\mathcal{T}_{\lambda}z$ is also differentiable at every point in the neighborhood $\mathcal{N}$ of $(\lambda_{0},z_{\bar{\lambda}_{0}})$ and that its derivative has the same form as in (26).

Assume that (3.1) holds for $i=j$ with $1\leq j\leq k-1$. Then, by the inductive assumption and the first hypothesis of Theorem 2, for $\eta\in\mathcal{Y}$ with $\|\eta\|_{\infty}$ sufficiently small, we have that

[TABLE]

where $\eta_{\ell}\in\mathcal{Y}$ for all $1\leq\ell\leq j$ and

[TABLE]

According to the first assumption of Theorem 2,

[TABLE]

Then, we have that

[TABLE]

This together with (3.1) implies that (3.1) holds at point $(\lambda_{0},z_{\bar{\lambda}_{0}})$ for $i=j+1$.

Similarly, one can show that $z\mapsto\mathcal{T}_{\lambda}z$ is also differentiable of order $j+1$ at every point in the neighborhood $\mathcal{N}$ of $(\lambda_{0},z_{\bar{\lambda}_{0}})$ and that the $(j+1)$-th derivative has the same form as in (3.1). Thus, by induction, (3.1) is proved.

Now, we show that $\frac{\partial^{i}}{\partial z^{i}}\mathcal{T}$ is continuous on $\mathcal{N}$ for all $1\leq i\leq k$. Without loss of generality, we only show the continuity at $(\lambda_{0},z_{\bar{\lambda}_{0}})$. In fact, fixing any $1\leq i\leq k$, using the first assumption of Theorem 2, for every $\eta\in{\mathcal{Y}},\mu\in\Sigma$ sufficiently small, by (6), (13) and (15) we have

[TABLE]

and

[TABLE]

Hence, it follows that $\frac{\partial^{i}}{\partial z^{i}}\mathcal{T}$ is continuous at $(\lambda_{0},z_{\bar{\lambda}_{0}})$ and it is proved that the map $z\mapsto{\mathcal{T}}_{\lambda}z$ is $C^{k}$ on $\mathcal{N}$. This completes the proof of the lemma. ∎

Lemma 4**.**

Suppose we are in the hypotheses of Theorem 2. Then, $\Sigma\ni\lambda\mapsto\mathcal{T}_{\lambda}z$ is $C^{k}$ on an open set $\mathcal{W}$ containing the set ${\mathcal{S}}$, where ${\mathcal{S}}$ is defined in Lemma 2.

In order to prove Lemma 4, we need the following auxiliary result.

Lemma 5**.**

Fix $\lambda_{*}\in\Sigma$ and $2\leq i\leq k+1$ and suppose that conditions (13) and (15) are satisfied. Then, there exists $N>0$ such that for every $\mu\in\Sigma$ with sufficiently small $|\mu|$ and for all $\tau\in\mathbb{R}$,

[TABLE]

where $z\in\mathcal{Y}$ and $M_{*}>1$ is a constant such that

[TABLE]

for every $\mu\in\Sigma$ with sufficiently small $|\mu|$ whose existence is given by (15).

Proof.

By the chain rule, differentiating $f_{\lambda}(\tau,y_{\lambda}(\tau)+z(\tau))$ with respect to $\lambda$ recursively we get that $\frac{d^{i}}{d\lambda^{i}}f_{\lambda}(\tau,y_{\lambda}(\tau)+z(\tau))$ is a sum of factors of the form

[TABLE]

where $1\leq m\leq i$ and $j_{1}+\cdots+j_{\ell}=i-m+\ell$. Similarly, $\frac{d^{i}}{d\lambda^{i}}f_{\lambda}(\tau,y_{\lambda}(\tau))$ is also a sum of factors of the form

[TABLE]

with $1\leq m\leq i$ and $j_{1}+\cdots+j_{\ell}=i-m+\ell$. By (13) and (15), we derive that for every $\mu\in\Sigma$ with sufficiently small $|\mu|$,

[TABLE]

where $M_{*}$ is a constant satisfying (29). Finally, since $\frac{d^{i}}{d\lambda^{i}}f_{\lambda}(\tau,y_{\lambda}(\tau)+z(\tau))$ and $\frac{d^{i}}{d\lambda^{i}}f_{\lambda}(\tau,y_{\lambda}(\tau))$) have a finite number of factors (depending only on $i$), there exists a sufficiently large integer $N=N(i)>0$ such that (28) holds. The proof is completed. ∎

We now continue with the proof of Lemma 4.

Proof of Lemma 4.

The proof is similar to that of Lemma 3. Fixing a point $(\lambda_{0},z_{\bar{\lambda}_{0}})\in{\mathcal{S}}$ arbitrarily, we use induction to prove that there exists an open neighborhood $\mathcal{V}$ of $(\lambda_{0},z_{\bar{\lambda}_{0}})$ on which the $i$-th derivative of $\lambda\mapsto\mathcal{T}_{\lambda}z$ is given by

[TABLE]

for $1\leq i\leq k$, where $\mu_{j}\in\Sigma$ for $1\leq j\leq i$ and $\frac{d^{i}}{d\lambda^{i}}$ denotes $i$-th derivative with respect to $\lambda$.

We first show that (3.1) holds at $(\lambda_{0},z_{\bar{\lambda}_{0}})$. In order to simplify notation, we introduce

[TABLE]

Using the definition of $\mathcal{T}_{\lambda}$ and the assumptions of Theorem 2, we get that for $\mu\in\Sigma$ with sufficiently small $|\mu|$,

[TABLE]

where

[TABLE]

In order to prove that (3.1) holds at $(\lambda_{0},z_{\bar{\lambda}_{0}})$ for $i=1$, it suffices to show

[TABLE]

By the assumptions of Theorem 2 (in particular, assumptions (14) and (15)) and Lemma 5, we have that

[TABLE]

Therefore, by (6) we have that

[TABLE]

for every $t\in\mathbb{R}$ and thus (32) holds. Consequently, since all the integrals in (31) converge, it follows that $\lambda\mapsto\mathcal{T}_{\lambda}z_{\bar{\lambda}_{0}}$ is differentiable at $\lambda_{0}$ and that its derivative is given by

[TABLE]

for $t\in\mathbb{R}$ and $\mu\in\Sigma$.

Similarly, we can show that $\lambda\mapsto\mathcal{T}_{\lambda}z$ is also differentiable at every point in the neighborhood $\mathcal{V}$ of $(\lambda_{0},z_{\bar{\lambda}_{0}})$ and the derivative has the same form as in (33).

Assume that (3.1) holds at the point $(\lambda_{0},z_{\bar{\lambda}_{0}})$ for $i=j$ with $1\leq j\leq k-1$. Then, by the inductive assumption and the hypotheses of Theorem 2, for $\mu\in\Sigma$ with sufficiently small $|\mu|$ we get that

[TABLE]

where

[TABLE]

It follows from (14), (15) and Lemma 5 that

[TABLE]

Then, we have that

[TABLE]

This together with (3.1) results in that (3.1) holds at point $(\lambda_{0},z_{\bar{\lambda}_{0}})$ for $i=j+1$.

Similarly, by induction one can show that $\lambda\mapsto\mathcal{T}_{\lambda}z$ is also differentiable of order $j+1$ at every point in the neighborhood $\mathcal{V}$ of $(\lambda_{0},z_{\bar{\lambda}_{0}})$ and that the $(j+1)$-th derivative has the same form as in (3.1). Hence, by induction, (3.1) is proved.

Now, we show that $\frac{\partial^{i}}{\partial\lambda^{i}}\mathcal{T}$ is continuous on $\mathcal{V}$ for all $1\leq i\leq k$. Without loss of generality, we only show the continuity at $(\lambda_{0},z_{\bar{\lambda}_{0}})$. In fact, fixing any $1\leq i\leq k$, for every $\mu\in\Sigma$ sufficiently small, using (6), the assumptions of Theorem 2 and Lemma 5, we have

[TABLE]

Similarly, we can prove that for $\eta\in\mathcal{Y}$ sufficiently small

[TABLE]

Hence, it is shown that $\frac{\partial^{i}}{\partial\lambda^{i}}\mathcal{T}$ is continuous at $(\lambda_{0},z_{\bar{\lambda}_{0}})$. Therefore, it is proved that the map $\lambda\mapsto{\mathcal{T}}_{\lambda}z$ is $C^{i}$ at $(\lambda_{0},z_{\bar{\lambda}_{0}})$ for all $1\leq i\leq k$. The proof is completed. ∎

Finally, we observe that Theorem 2 follows readily from Lemmas 2, 3 and 4. $\square$

3.2. Parameterized Hyers-Ulam stability

In this subsection, we discuss an important consequence of Theorem 2. We first recall the notion of exponential dichotomy.

Definition 1**.**

The equation (2) is said to admit an exponential dichotomy if there exist a family $P(t)$, $t\in\mathbb{R}$, of projections on $X$ and constants $D,\rho>0$ such that:

• •

for $t,s\in\mathbb{R}$, $T(t,s)P(s)=P(t)T(t,s)$;

• •

for $t,s\in\mathbb{R}$, $\|\mathcal{G}(t,s)\|\leq De^{-\rho|t-s|}$, where

[TABLE]

The following result is a straightforward consequence of Theorem 1.

Corollary 1**.**

Suppose that (2) admits an exponential dichotomy and that (3) holds with $c(t)=c$ for all $t\in\mathbb{R}$, where $c\geq 0$. Finally, assume that

[TABLE]

Given $\varepsilon>0$, for each $\lambda\in\Sigma$ let $y_{\lambda}\colon\mathbb{R}\to X$ be a differentiable map satisfying

[TABLE]

Then, for each $\lambda\in\Sigma$, there exists a unique solution $x_{\lambda}\colon\mathbb{R}\to X$ of (4) such that

[TABLE]

where $\tilde{L}=\frac{2\varepsilon D}{\rho}$.

Proof.

The desired conclusion follows readily from Theorem 1 applied to the case when $\varepsilon(t)=\varepsilon$ and $c(t)=c$ for all $t\in\mathbb{R}$. ∎

The following result is a direct consequence of Theorem 2.

Corollary 2**.**

Suppose that the assumptions of Corollary 1 hold. Take $\varepsilon>0$ and for each $\lambda\in\Sigma$ let $y_{\lambda}\colon\mathbb{R}\to X$ be a differentiable map satisfying (36). In addition, suppose there exists $C>0$ such that the following conditions hold:

• •

the map $(\lambda,x)\mapsto f_{\lambda}(t,x)$ is $C^{k+1}$ and for all $2\leq i\leq k+1$ and $0\leq j\leq i$,

[TABLE]

• •

the maps $\lambda\mapsto y_{\lambda}(t)$ and $\lambda\mapsto y_{\lambda}^{\prime}(t)$ both are $C^{k+1}$ such that

[TABLE]

for all $1\leq i\leq k+1$, all $t\in\mathbb{R}$ and all $\lambda\in\Sigma$. Furthermore, for every $\lambda_{0}\in\Sigma$ there exist $M_{\lambda_{0}}>1$ and a neighborhood $\Sigma_{0}$ of $\lambda_{0}$ satisfying (15).

Then, $\lambda\mapsto x_{\lambda}(t)$ is $C^{k}$ for each $t\in\mathbb{R}$.

Let us discuss a simple concrete example to which our results are applicable.

Example 1**.**

Take $X=\mathbb{R}$ and $A(t)=-1$ for $t\in\mathbb{R}$. Then, (2) admits an exponential dichotomy with $P(t)=\text{\rm Id}$, $D=\rho=1$. Consider $\Sigma=(0,1)\subset\mathbb{R}$ and set

[TABLE]

It is straightforward to verify that $f_{\lambda}(t,x)$ satisfies all the assumptions of Corollary 2 for every $k\in\mathbb{N}$. Fix $\varepsilon>0$ and for each $\lambda\in\Sigma$ set

[TABLE]

Note that

[TABLE]

In particular,

[TABLE]

that is, (36) is satisfied. Moreover,

[TABLE]

and clearly the derivatives of order $\geq 2$ with respect to $\lambda$ of each term on the left-hand side of (37) are all zero. Furthermore, one can also easily show that

[TABLE]

for every $i\geq 2$. Therefore, all the hypotheses of Corollary 2 in this example are satisfied. This allows us to conclude that if $x_{\lambda}$ is the map associated to $y_{\lambda}$ by Corollary 1 then the map $\lambda\to x_{\lambda}(t)$ is $C^{k}$ for each $t\in\mathbb{R}$.

3.3. Beyond exponential dichotomy

The purpose of this subsection is to illustrate that Theorem 2 is applicable to situations when (2) does not admit an exponential dichotomy. We consider a particular example.

Example 2**.**

Suppose that $\rho\colon\mathbb{R}\to(0,\infty)$ is a continuously differentiable function such that $\rho(0)=1$ and that $\rho$ is increasing. Finally, we assume that $\lim_{t\to\infty}\rho(t)=\infty$ and $\lim_{t\to-\infty}\rho(t)=0$. We take $X=\mathbb{R}$, $\Sigma=(-1,1)\subset\mathbb{R}$ and set

[TABLE]

In particular,

[TABLE]

Moreover, (2) has no nonzero bounded solution. Taking $P(t)=\text{\rm Id}$ for $t\in\mathbb{R}$, we have that

[TABLE]

In addition, we choose $\tilde{c}\colon\mathbb{R}\to\mathbb{R}$ of the form $\tilde{c}(t)=e^{-a|t|}$, $t\in\mathbb{R}$ where $a>2$. Furthermore, we now choose a continuous function $\varepsilon\colon\mathbb{R}\to\mathbb{R}$ with the property that there exists $C>1$ such that $\tilde{c}(t)\leq\varepsilon(t)\leq C\tilde{c}(t)$ for $t\in\mathbb{R}$. Moreover, for $\lambda\in\Sigma$, we set

[TABLE]

Hence, (3) is satisfied with $c(t)=\tilde{c}(t)\rho(-|t|)$ and

[TABLE]

In particular, (5) holds. Similarly we have that (6) is valid. For $\lambda\in\Sigma$, we define $y_{\lambda}\colon\mathbb{R}\to\mathbb{R}$ by

[TABLE]

Then,

[TABLE]

Thus, (7) is satisfied. It is straightforward to verify that (13), (14) and (15) are satisfied for each $k\in\mathbb{N}$. Hence, Theorem 2 implies that for each $t\in\mathbb{R}$, map $\lambda\mapsto x_{\lambda}(t)$ is $C^{k}$ for each $k\in\mathbb{N}$. Finally, we observe that by choosing an appropriate function $\rho$, for instance,

[TABLE]

equation (2) does not admit an exponential dichotomy.

4. The discrete time case

In this section we present a discrete time version of Theorem 2. Let $X=(X,|\cdot|)$, $\Sigma=(\Sigma,|\cdot|)$ and $(\mathcal{B}(X),\|\cdot\|)$ be as in Section 2. Given a sequence $(A_{n})_{n\in\mathbb{Z}}$ of invertible operators in $\mathcal{B}(X)$, let us consider the associated linear difference equation

[TABLE]

For $m,n\in\mathbb{Z}$, set

[TABLE]

Let $(P_{n})_{n\in\mathbb{Z}}$ be a sequence in $\mathcal{B}(X)$ and define

[TABLE]

Moreover, for each $\lambda\in\Sigma$ suppose we have a measurable map $f_{\lambda}\colon\mathbb{Z}\times X\to X$ with the property that there exists a sequence $(c_{n})_{n}$ in $[0,\infty)$ such that

[TABLE]

for every $n\in\mathbb{Z}$, $\lambda\in\Sigma$ and $x,y\in X$. Finally, associated to these choices, for each $\lambda\in\Sigma$ we consider the semilinear difference equation given by

[TABLE]

The following result is established in [8, Theorem 3].

Theorem 3**.**

Assume that (38) admits no non-trivial bounded solution and

[TABLE]

Moreover, let $(\varepsilon_{n})_{n\in\mathbb{Z}}$ be a sequence in $(0,+\infty)$ such that

[TABLE]

Then, for each $\lambda\in\Sigma$ and for each sequence $\textbf{y}^{\lambda}:=(y^{\lambda}_{n})_{n\in\mathbb{Z}}\subset X$ satisfying

[TABLE]

there exists a unique sequence $\textbf{x}^{\lambda}:=(x^{\lambda}_{n})_{n\in\mathbb{Z}}\subset X$ satisfying (42) such that

[TABLE]

Given a sequence $(y^{\lambda}_{n})_{n\in\mathbb{Z}}\subset X$ satisfying (45), our objective now is to formulate sufficient conditions under which the map $\lambda\mapsto x^{\lambda}_{n}$ is of class $C^{k}$ with $k\in\mathbb{N}$ for every $n\in\mathbb{Z}$. As in the previous section, by $\frac{\partial^{i}}{\partial x^{i}}f_{\lambda}(n,x)$ and $\frac{\partial^{i}}{\partial\lambda^{i}}f_{\lambda}(n,x)$ we denote the $i$-th partial derivative of $f_{\lambda}(n,x)$ with respect to $x$ and $\lambda$, respectively.

Theorem 4**.**

Let $\textbf{y}^{\lambda}:=(y^{\lambda}_{n})_{n\in\mathbb{Z}}$ be a sequence in $X$ satisfying (45) and suppose that the assumptions of Theorem 3 hold. Let $\textbf{x}^{\lambda}:=(x_{n}^{\lambda})_{n\in\mathbb{Z}}$ be the sequence associated to $\textbf{y}^{\lambda}$ by Theorem 3 and take $k\in\mathbb{N}$. Moreover, suppose there exists $C>0$ such that the following conditions hold:

• •

for every $n\in\mathbb{Z}$ the map $(\lambda,x)\mapsto f_{\lambda}(n,x)$ is $C^{k+1}$ and for all $2\leq i\leq k+1$ and $0\leq j\leq i$,

[TABLE]

• •

the map $\lambda\mapsto y^{\lambda}_{n}$ is $C^{k+1}$ for every $n\in\mathbb{Z}$ and

[TABLE]

for every $1\leq i\leq k+1$, all $n\in\mathbb{Z}$ and all $\lambda\in\Sigma$. Furthermore, for every $\lambda_{0}\in\Sigma$ there exist $\hat{M}_{\lambda_{0}}>1$ and a neighborhood $\Sigma_{0}$ of $\lambda_{0}$ such that

[TABLE]

Then, the map $\lambda\mapsto x^{\lambda}_{n}$ is of class $C^{k}$ for every $n\in\mathbb{Z}$.

The proof of this theorem is very similar to the proof of its continuous time version Theorem 2. For this reason, at some steps of the proof, we will only provide a sketch of the argument. Let us start with the proof of Theorem 4 by recalling some constructions from the proof of [8, Theorem 3].

Let

[TABLE]

Then, $(\hat{\mathcal{Y}},\|\cdot\|_{\infty})$ is a Banach space. For $\textbf{z}\in\hat{\mathcal{Y}}$, we set

[TABLE]

for $n\in\mathbb{Z}$. It is observed in [8, Theorem 3] that $\hat{\mathcal{T}}_{\lambda}$ is well-defined and, moreover, that it is a contraction on

[TABLE]

where $\hat{L}$ and $\hat{q}$ are given in (44) and (43) respectively. In particular, it has a unique fixed point $\textbf{z}^{\lambda}\in\hat{\mathcal{D}}$ such that $\hat{\mathcal{T}}_{\lambda}\textbf{z}^{\lambda}=\textbf{z}^{\lambda}$. Finally, it is observed that $\textbf{x}^{\lambda}=\textbf{y}^{\lambda}+\textbf{z}^{\lambda}$ is a solution of (42) satisfying (46). In particular, in order to prove Theorem 4, it remains to show that $\lambda\to\textbf{z}^{\lambda}$ is of class $C^{k}$. With this purpose in mind we present some auxiliary results.

Lemma 6**.**

Let $k\in\mathbb{N}$. Suppose that the map $\Sigma\times\hat{\mathcal{Y}}\ni(\lambda,\textbf{z})\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is $C^{k}$ on an open set $\hat{\mathcal{W}}$ containing the set $\hat{\mathcal{S}}:=\{(\lambda,\textbf{z}^{\bar{\lambda}}):\lambda,\bar{\lambda}\in\Sigma\}$. Then, $\Sigma\ni\lambda\mapsto\textbf{z}^{\lambda}$ is also $C^{k}$.

Proof.

This result follows from Lemma 2 and Remark 2. ∎

Lemma 7**.**

Suppose that the assumptions of Theorem 4 hold. Then, $\hat{\mathcal{Y}}\ni\textbf{z}\mapsto\mathcal{T}_{\lambda}\textbf{z}$ is $C^{k}$ on an open set $\hat{\mathcal{W}}$ containing $\hat{\mathcal{S}}$, where $\hat{\mathcal{S}}$ is as in Lemma 6.

Proof.

The proof of this result is very similar to the proof of Lemma 3 and, for this reason, we only present a sketch of the argument. It suffices to prove that, for any $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})\in\hat{\mathcal{S}}$, there exists an open set $\hat{\mathcal{N}}\ni(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ on which the map $\textbf{z}\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is $C^{k}$. We use induction to prove that on $\hat{\mathcal{N}}$, the $i$-th derivative of $\textbf{z}\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is given by

[TABLE]

for $1\leq i\leq k$, where $\boldsymbol{\eta}^{j}=(\eta^{j}_{n})_{n\in\mathbb{Z}}\in\hat{\mathcal{Y}}$ for $1\leq j\leq i$ .

By arguing as in (21), it follows from (41) that

[TABLE]

We first show that (4) holds at the point $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$. Using the definition of $\hat{\mathcal{T}}_{\lambda}$ and the first assumption of Theorem 4, we can show that for $\boldsymbol{\eta}\in\hat{\mathcal{Y}}$ with sufficiently small $\|\boldsymbol{\eta}\|_{\infty}$,

[TABLE]

where

[TABLE]

Using again the first assumption of Theorem 4, $|\alpha_{k-1}|$ can be estimated as

[TABLE]

By (44) and (53), we have that

[TABLE]

for every $n\in\mathbb{Z}$, and consequently

[TABLE]

This fact combined with (4) implies that $\textbf{z}\mapsto\hat{\mathcal{T}}_{\lambda_{0}}\textbf{z}$ is differentiable at $\textbf{z}^{\bar{\lambda}_{0}}$ and that its derivative is given by

[TABLE]

for $n\in\mathbb{Z}$ and $\boldsymbol{\eta}=(\eta_{n})_{n\in\mathbb{Z}}\in\hat{\mathcal{Y}}$.

Similarly, we can show that $\textbf{z}\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is also differentiable at every point in the neighborhood $\hat{\mathcal{N}}$ of $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ and the derivative has the same form as in (54).

Assume that (4) holds for $i=j$ with $1\leq j\leq k-1$. Then, using the inductive assumption and the first hypothesis of Theorem 4, for $\boldsymbol{\eta}\in\hat{\mathcal{Y}}$ with $\|\boldsymbol{\eta}\|_{\infty}$ sufficiently small, we can show that

[TABLE]

where $\boldsymbol{\eta}^{\ell}=(\eta^{\ell}_{n})_{n\in\mathbb{Z}}\in\hat{\mathcal{Y}}$ for all $1\leq\ell\leq j$ and

[TABLE]

Using again the first hypothesis of Theorem 4 we can show that

[TABLE]

Then, we have that

[TABLE]

This together with (4) implies that the results in (4) holds at the point $(\lambda_{0},\textbf{z}_{\bar{\lambda}_{0}})$ for $i=j+1$.

Similarly, we can show that $\textbf{z}\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is also differentiable of order $j+1$ at every point in the neighborhood $\hat{\mathcal{N}}$ of $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ and that the $j+1$-th derivative has the same form as in (4). Thus, by induction, (4) is proved.

Now, we show that $\frac{\partial^{i}}{\partial\textbf{z}^{i}}\hat{\mathcal{T}}$ is continuous on $\hat{\mathcal{N}}$ for all $1\leq i\leq k$. Without loss of generality, we only show the continuity at $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$. In fact, fixing any $1\leq i\leq k$, using the assumptions of Theorem 4 and (44) we can show that there exists $\hat{M}_{\lambda_{0}}>0$ such that for every $\boldsymbol{\eta}\in\hat{\mathcal{Y}}$ and $\mu\in\Sigma$ sufficiently small,

[TABLE]

Hence, it follows that $\frac{\partial^{i}}{\partial\textbf{z}^{i}}\hat{\mathcal{T}}$ is continuous at $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ and it is proved that the map $\textbf{z}\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is $C^{k}$ on $\hat{\mathcal{N}}$. This completes the proof of the lemma. ∎

Lemma 8**.**

Suppose we are in the hypotheses of Theorem 4. Then, $\Sigma\ni\lambda\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is $C^{k}$ on an open set $\hat{\mathcal{W}}$ containing $\hat{\mathcal{S}}$, where $\hat{\mathcal{S}}$ is as in Lemma 6.

In order to prove this lemma we need the following auxiliary result whose proof is completely analogous to the proof of Lemma 5 and therefore we omit it.

Lemma 9**.**

Fix $\lambda_{*}\in\Sigma$ and $2\leq i\leq k+1$ and suppose that conditions (47) and (49) are satisfied. Then, there exists $N>0$ such that for every $\mu\in\Sigma$ with sufficiently small $|\mu|$ and for all $n\in\mathbb{Z}$,

[TABLE]

where $\textbf{z}=(z_{n})_{n\in\mathbb{Z}}\in\hat{\mathcal{Y}}$ and $\hat{M}_{*}>1$ is a constant such that

[TABLE]

for every $\mu\in\Sigma$ with sufficiently small $|\mu|$ whose existence is given by (49).

Proof of Lemma 8.

The proof of this result is similar to that of Lemma 4 and again we provide only a sketch of the argument. For any $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})\in\hat{\mathcal{S}}$, we use induction to prove that there exists an open neighborhood $\hat{\mathcal{V}}$ of $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ on which the $i$-th derivative of $\lambda\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is given by

[TABLE]

for $1\leq i\leq k$, where $\mu_{j}\in\Sigma$ for $1\leq j\leq i$.

We first show that (4) holds at $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$. In order to simplify notations, define

[TABLE]

Using the definition of $\hat{\mathcal{T}}_{\lambda}$ and the first assumption of Theorem 4, one can show that for $\mu\in\Sigma$ with sufficiently small $|\mu|$,

[TABLE]

where

[TABLE]

By the assumptions of Theorem 4 and Lemma 9 we can get that

[TABLE]

Moreover, by (44) and (60), we have

[TABLE]

for every $n\in\mathbb{Z}$, which implies that

[TABLE]

This fact combined with (4) implies that $\lambda\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}^{\bar{\lambda}}$ is differentiable at $(\lambda_{0},\mathbf{z}^{\bar{\lambda}_{0}})$ and that its derivative is given by

[TABLE]

for $n\in\mathbb{Z}$ and $\mu\in\Sigma$.

Similarly, we can show that $\lambda\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is also differentiable at every point in the neighborhood $\hat{\mathcal{V}}$ of $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ and the derivative has the same form as in (61).

Assume that (4) holds for $i=j$ with $1\leq j\leq k-1$. Then, using the inductive assumption and the hypotheses of Theorem 4, we can show that for $\mu\in\Sigma$ with sufficiently small $|\mu|$,

[TABLE]

where

[TABLE]

Using the hypotheses of Theorem 4 and Lemma 9 we can show that

[TABLE]

Then, we have that

[TABLE]

This together with (4) implies that (4) holds at point $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ for $i=j+1$.

Similarly, by induction one can show that $\lambda\mapsto\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is also differentiable of order $j+1$ at every point in the neighborhood $\hat{\mathcal{V}}$ of $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$ and that the $(j+1)$-th derivative has the same form as in (4). Hence, by induction, (4) is proved.

Now, we show that $\frac{\partial^{i}}{\partial\lambda^{i}}\hat{\mathcal{T}}$ is continuous on $\hat{\mathcal{V}}$ for all $1\leq i\leq k$. Without loss of generality, we only show the continuity at $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$. In fact, fixing any $1\leq i\leq k$, for every $\boldsymbol{\eta}=(\eta_{n})_{n\in\mathbb{Z}}\in\hat{\mathcal{Y}}$ and $\mu\in\Sigma$ sufficiently small, using (44) and the assumptions of Theorem 4, we can show that

[TABLE]

and

[TABLE]

Hence, it is shown that $\frac{\partial^{i}}{\partial\lambda^{i}}\hat{\mathcal{T}}$ is continuous at $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})$. Therefore, it is proved that the map $\lambda\to\hat{\mathcal{T}}_{\lambda}\textbf{z}$ is $C^{i}$ at $(\lambda_{0},\textbf{z}^{\bar{\lambda}_{0}})\in\hat{\mathcal{W}}$ for all $1\leq i\leq k$. The proof is completed. ∎

Finally, Theorem 4 follows readily from Lemmas 6, 7 and 8 and its proof is then completed. $\square$

Remark 3**.**

As in Subsection 3.2, one can interpret Theorem 4 in the case when (38) admits an exponential dichotomy. In particular, it is straightforward to formulate a discrete time version of Corollary 2. Moreover, one can easily build discrete time versions of the examples given in Sections 3.2 and 3.3. Therefore, we refrain from doing so.

Acknowledgements

L. Backes was partially supported by a CNPq-Brazil PQ fellowship under Grant No. 307633/2021-7. D.D. was supported in part by Croatian Science Foundation under the Project IP-2019-04-1239 and by the University of Rijeka under the Projects uniri-prirod-18-9 and uniri-prprirod-19-16. X. Tang was supported by NSFC $\#$12001537, the Start-up Funding of Chongqing Normal University 20XLB033, and the Research Project of Chongqing Education Commission CXQT21014.