General Melnikov Approach to Implicit ODE’s

Abstract

Existence of solutions connecting a singularity of a perturbed implicit differential equations is studied. It is assumed that the unperturbed differential equation has a solution of the same kind. By a suitable, nonlinear, change of coordinates these kind of solutions are associated to homoclinic solutions to a fixed point of an ordinary differential equation with a one-dimensional centre manifold. Then we obtain a Melnikov condition for the persistence of homoclinic orbits which is simpler than the one obtained in Battelli and Fečkan (J Differ Equ 256:1157–1190, 2014). This difference is due to the fact that this method does not distinguish solutions according to the rate of convergence to the fixed point and then another assumption on the perturbation term is needed.

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Acknowledgements

We want to thank the referee for his/her very careful reading of the paper.

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Correspondence to Flaviano Battelli.

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Michal Fečkan: Partially supported by the Slovak Research and Development Agency under the Contract No. APVV-18-0308 and by the Slovak Grant Agency VEGA Nos. 1/0358/20 and 2/0127/20.

Appendices

Appendix A: Smoothness of the Fixed Point

In this section we will prove the smoothness properties, as stated in Theorem 2.3, of the solution \(y_+(s,\xi ,\varepsilon )\) whose existence is stated in Theorem 2.1. A similar conclusion holds for the solution \(y_-(s,\xi ,\varepsilon )\) whose existence is stated in Theorem 2.2.

We assume that the derivatives of \(g(t,y,\varepsilon )\) are bounded for \(|x|\le 1\) and \(|\varepsilon |\le \varepsilon _0\) uniformly with respect to \(t\in {\mathbb {R}}\). For definiteness we set

$$\begin{aligned} \begin{array}{l} \displaystyle N=\sup \{g(t,y_0(s)+y,\varepsilon ) \mid t\in {\mathbb {R}},\; s\ge 0, \; | \varepsilon |\le \varepsilon _0, \hbox { and }\; |y|<1\} \\ \displaystyle N_1=\sup \{g_t(t,y_0(s)+y,\varepsilon ) \mid t\in {\mathbb {R}},\; s\ge 0, \; |\varepsilon |\le \varepsilon _0, \hbox { and }\; |y|<1\} \\ \displaystyle N_2=\sup \{g_y(t,y_0(s)+y,\varepsilon ) \mid t\in {\mathbb {R}},\; s\ge 0, \; |\varepsilon |\le \varepsilon _0, \hbox { and }\; |y|<1\}. \end{array} \end{aligned}$$
(5.1)

Note that, from \((H_2){-}(H_3)\) we get, for \(|y|\le 1\):

$$\begin{aligned} |g(t,y_0(s)+y,\varepsilon )|=|g(t,y_0(s)+y,\varepsilon )-g(t,{\bar{y}}_0,\varepsilon )| \le L_g[|y_0(s)-{\bar{y}}_0|+|y|] \end{aligned}$$

so that

$$\begin{aligned} N\le L_g(1+\sup _{s\ge 0}|y_0(s)-{\bar{y}}_0|). \end{aligned}$$

First we observe that in [19] the smoothness of a fixed point of such a map as

$$\begin{aligned} {\hat{z}} = S\xi +KG(z) \end{aligned}$$

where \(G(z)(s):=g(z(s))\) and K is a linear map on a scale of Banach spaces is considered. In our case, Eq. (2.7) can be written as

$$\begin{aligned} {\hat{z}} = S\xi +K[F(z)+\varepsilon G(z,\varepsilon )] \end{aligned}$$
(5.2)

where

$$\begin{aligned} \begin{array}{l} (S\xi )(s)= X(s)\xi \\ \displaystyle (Kw)(s) = \int _0^s X(s)PX^{-1}(\tau )w(\tau )d\tau - \int _s^\infty X(s)({\mathbb {I}}-P)X^{-1}(\tau )w(\tau )d\tau \end{array} \end{aligned}$$
(5.3)

and

$$\begin{aligned} \begin{array}{l} F(z)(s) = f(y_0(s)+z(s))-f(y_0(s))-A(s)z(s) \\ \displaystyle G(z,\varepsilon )(s) = g\left( \int _0^s \omega (y_0(\tau )+z(\tau ))d\tau , y_0(s)+z(s),\varepsilon \right) . \end{array} \end{aligned}$$
(5.4)

In this section when we refer to Eq. (5.2) we will always assume that S and K are as in (5.3) and F(z) and \(G(z,\varepsilon )\) are as in (5.4).

Let \({{\mathcal {L}}}(X,Y)\) be the space of bounded linear maps from the Banach space X into the Banach space Y. We have, trivially:

$$\begin{aligned} \begin{array}{l} \displaystyle \Vert S\Vert _{{{\mathcal {L}}}({{\mathcal {R}}}P,C^0_\eta )} \le k \\ \displaystyle \Vert K\Vert _{{{\mathcal {L}}}(C^0_\eta ,C^0_\zeta )} \le k_\eta := k\left( \frac{1}{\delta +\eta }+\frac{1}{\delta -\eta }\right) \end{array} \end{aligned}$$
(5.5)

for any \(0\le \eta \le \zeta <\delta \). Note that \(k_\eta \) is an increasing function of \(\eta \) if \(0<\eta <\delta \).

More generally in Lemmas 5.2 and 5.4 we will study the problem of existence and Lipschitz continuity of the solution of such a fixed point equation as

$$\begin{aligned} {\hat{z}} = S\nu +K[F(z,\nu )+G(z,\nu )] \end{aligned}$$
(5.6)

where \(\nu \) belongs to a given ball B in a vector space V of parameters, \(S:V\rightarrow C^0_0\) and \(K: C^0_\beta \rightarrow C^0_\beta \) are bounded linear maps with norm \(\Vert S\Vert _{{{\mathcal {L}}}(V,C^0_0)}\le {\hat{k}}\) and \(\Vert K\Vert _{{{\mathcal {L}}}(C^0_\beta ,C^0_\beta )}\le {\hat{k}}_\beta \), for any \(0\le \beta <\delta \), and \(F,G:C^0_\beta \times V\rightarrow C^0_\beta \) satisfy certain Lipschitz conditions. Of course when S and K are as in (5.3) we can take \({\hat{k}}=k\) and \({\hat{k}}_\eta =k_\eta \). Next note that, for \(0\le \beta \le \eta <\delta \) we have

$$\begin{aligned} \begin{array}{l} \Vert S\Vert _{{{\mathcal {L}}}(V,C^0_\beta )} \le \Vert S\Vert _{{{\mathcal {L}}} (V,C^0_0)} \le {\hat{k}} \\ \Vert K\Vert _{{{\mathcal {L}}}(C^0_\beta ,C^0_\eta )}\le \Vert K\Vert _{\mathcal{L}(C^0_\beta ,C^0_\beta )}\le {\hat{k}}_\eta . \end{array} \end{aligned}$$

With reference to (5.2), (5.3), (5.4) we take \(\nu =(\xi ,\varepsilon )\) and \(F(z,\nu )\) as the Neminsky operator \(F(z,\nu )(s)=F(z(s))\) which is independent of \(\nu =(\xi ,\varepsilon )\). Note also that \(G(z,\nu )\) is independent of \(\xi \). The more general equation (5.6) is needed when studying the smoothness properties of the fixed point.

Because of the difference between the definition of G in (5.4) and that in [19], we cannot apply directly the results in [19]. Instead we adapt the methods given there to the present situation. Throughout this section we assume that

$$\begin{aligned} \rho <1. \end{aligned}$$

To start with, in Eq. (5.2) we replace F, G and \(\omega \) with

$$\begin{aligned} \begin{array}{l} F(z)(s) = {\tilde{f}}(s,z(s)), \quad \hbox {and} \\ \displaystyle G(z,\varepsilon )(s) = g\left( \int _0^s \omega _1(y_0(\tau ) +z(\tau ))d\tau , y_0(s)+z(s),\varepsilon \right) \chi (|z|) \\ \end{array} \end{aligned}$$
(5.7)

where

$$\begin{aligned} \begin{array}{l} {\tilde{f}}(s,z):= [f(y_0(s)+z)-f(y_0(s))-A(s)z]\chi (|z|) \\ \omega _1(z)=\omega (z)\chi (|z|) \end{array} \end{aligned}$$

and \(\chi (x)\), \(x\in {\mathbb {R}}\), is a cut-off function that is a \(C^\infty \)-function such that \(|\chi (x)|\le 1\) and

$$\begin{aligned} \chi (x) = \left\{ \begin{array}{ll} 1 &{} \quad \hbox {if } |x|\le 1 \\ 0 &{}\quad \hbox {if } |x|\ge 3 \end{array}\right. \end{aligned}$$

Since the fixed point \(z(t,\xi ,\varepsilon )\in B_\rho \), and \(\rho <1\), the study of its smoothness is not affected by this change. However in this way \({\tilde{f}}(s,z)\) is \(C^r\) with respect to z and the derivatives are bounded in \((s,z)\in {\mathbb {R}}^{n+1}\).

Note that \(\chi (x)\) can be taken so that \(|\chi '(x)|\le 1\) for any \(x\in {\mathbb {R}}\). Moreover for \(|z|\le 1\) and \(s\ge 0\) we have:

$$\begin{aligned} \begin{array}{l} |\omega _1'(y_0(s)+z)|=|\omega '(y_0(s)+z)|\le {\bar{\omega }} \\ \displaystyle \left| {\tilde{f}}_z(s,z)\right| \le \Delta (|z|). \end{array} \end{aligned}$$
(5.8)

where \({\tilde{f}}_z(t,z)=\frac{\partial {\tilde{f}}}{\partial z}(s,z)\), \({\bar{\omega }}\) has been defined in (2.8) and \(\Delta (\rho )\) in (2.1).

Our first result concerns the functions F and G given in (5.7). Arguing as in [19, Lemma 3] we see that

Lemma 5.1

Let \(\zeta ,\eta >0\) and F and G as in (5.7) Then \(F:C^0_\zeta \rightarrow C^0_\eta \) is continuous. Moreover, if \(0\le \zeta \le \eta \), then for any \(z_1,z_2\in C^0_\zeta \) we have

$$\begin{aligned} \Vert F(z_1)-F(z_2)\Vert _\eta \le \left| {\tilde{f}}_z\right| _0 \Vert z_1-z_2\Vert _\zeta \end{aligned}$$
(5.9)

where

$$\begin{aligned} \left| {\tilde{f}}_z\right| _0 =\sup \left\{ \left| {\tilde{f}}_z(s,\theta z_1(s)+(1-\theta )z_2(s)) \right| s\ge 0, 0\le \theta \le 1\right\} \end{aligned}$$
(5.10)

and

$$\begin{aligned} \Vert G(z_1,\varepsilon _1)-G(z_2,\varepsilon _2)\Vert _\eta \le L_g[(1+{\bar{\omega }}\eta ^{-1})\Vert z_1-z_2\Vert _\zeta + |\varepsilon _1-\varepsilon _2|]. \end{aligned}$$
(5.11)

Proof

The proof of (5.9) has been given in [19]. We repeat it here for completeness. We have

$$\begin{aligned}&\sup _{s\ge 0}|{\tilde{f}}(s,z_1(s))-{\tilde{f}}(s,z_2(s))|e^{-\eta s}\\&\le \max \left\{ \sup _{0\le s\le R}|{\tilde{f}}(s,z_1(s))-\tilde{f}(s,z_2(s))|e^{-\eta s}; 2|{\tilde{f}}|_0e^{-\eta R}\right\} . \end{aligned}$$

Take \(R\gg 1\) so that \(2|{\tilde{f}}|_0e^{-\eta R}<\varepsilon \). Next, the set

$$\begin{aligned} D=\{(s,z_1(s)) \mid 0\le s\le R\} \end{aligned}$$

is compact, so there exists \(\delta _1\) such that

$$\begin{aligned} |{\tilde{f}}(s,z)-{\tilde{f}}(s,z+{\tilde{z}})|<\varepsilon \end{aligned}$$

if \((s,z)\in D\) and \(|{\tilde{z}}|<\delta _1\). Take \(\delta =e^{-\zeta R}\delta _1\). If \(\Vert z_1-z_2\Vert _\zeta <\delta \) we have, for \(0\le s\le R\):

$$\begin{aligned} |z_1(s)-z_2(s)|\le \Vert z_1-z_2\Vert _\zeta e^{\zeta s}\le \delta e^{\zeta R}=\delta _1. \end{aligned}$$

Hence

$$\begin{aligned} \sup _{0\le s\le R}|{\tilde{f}}(s,z_1(s))-{\tilde{f}}(s,z_2(s))|e^{-\eta s}<\sup _{0\le s\le R}\varepsilon e^{-\eta s} <\varepsilon . \end{aligned}$$

This proves the first part of the Lemma. As for the second we have:

$$\begin{aligned} |{\tilde{f}}(s,z_1(s))-{\tilde{f}}(s,z_2(s))|e^{-\eta s}\le & {} \left| \int _0^1 {\tilde{f}}_z(s,\theta z_1(s)+(1-\theta )z_2(s)) d\theta \right| \Vert z_1-z_2\Vert _{\eta } \\\le & {} \left| {\tilde{f}}_z\right| _0|\Vert z_1-z_2\Vert _{\eta } \le \left| {\tilde{f}}_z\right| _0|\Vert z_1-z_2\Vert _{\zeta } \end{aligned}$$

We have

$$\begin{aligned}&|G(z_1,\varepsilon )(s)-G(z_2,\varepsilon )(s)|e^{-\eta s} \\&\quad \le L_g \left[ \left| \int _0^s \omega _1(y_0(\tau )+ z_1(\tau ))-\omega _1(y_0(\tau )+z_2(\tau ))d\tau \right| + |z_1(s)-z_2(s)|\right] e^{-\eta s} \\&\quad \le L_g \left[ \left| \int _0^s \int _0^1 \omega _1'(y_0(\tau )+\theta z_1(\tau )+(1-\theta )z_2(\tau ))d\theta (z_1(\tau )-z_2(\tau ))d\tau \right| + |z_1(s)-z_2(s)|\right] e^{-\eta s} \end{aligned}$$

It is clear that \(z(s)\mapsto z(s)\) is continuous as a map from \(C^0_\zeta \rightarrow C^0_\eta \). Next we have

$$\begin{aligned}&\left| \int _0^s \int _0^1 \omega _1'(y_0(\tau )+\theta z_1(\tau )+(1-\theta )z_2(\tau ))d\theta (z_1(\tau )-z_2(\tau ))d\tau \right| \\&\quad \le \int _0^s {\bar{\omega }} |z_1(\tau )-z_2(\tau )|d\tau \le {\bar{\omega }} \Vert z_1-z_2\Vert _\eta \int _0^s e^{\eta \tau } d\tau \le {\bar{\omega }}\eta ^{-1} \Vert z_1-z_2\Vert _\eta e^{\eta s}. \end{aligned}$$

So \(z\mapsto G(z,\varepsilon )\) is Lipschitz-continuous as a map from \(C^0_\zeta \rightarrow C^0_\eta \), with \(\zeta \le \eta \) and Lipschitz constant \(L_g(1+{\bar{\omega }}\eta ^{-1})\) that does not depend on \(\varepsilon \). Next, for \(z\in C^0_\zeta \) we have

$$\begin{aligned} \displaystyle |G(z,\varepsilon _1)(s)-G(z,\varepsilon _2)(s)|e^{-\eta s} \le L_g |\varepsilon _1-\varepsilon _2|e^{-\eta s} \le L_g |\varepsilon _1-\varepsilon _2| \end{aligned}$$

The proof is finished. \(\square \)

Now we go to the more general equation (5.6). Recall that \(\Vert K\Vert _{{{\mathcal {L}}}(C^0_\eta ,C^0_\eta )}\le {\hat{k}}_\eta \) and \(\Vert S\Vert _{{{\mathcal {L}}}(V,C^0_0)}\le {\hat{k}}\). We prove the following:

Lemma 5.2

Let \(0\le \beta \le \eta <\delta \), \(F,G:C^0_\gamma \times V\rightarrow C^0_\gamma \), with \(\gamma =\beta ,\eta \). Suppose there exists \(R>0\) such that the following conditions hold

  1. (i)

    \(F(0,\nu )=0\);

  2. (ii)

    there exists \(\mu _0\) such that for any \(z_1(s),z_2(s)\in C^0_\beta \) with \(\Vert z_1\Vert _\beta , \Vert z_2\Vert _\beta \le R\) it results

    $$\begin{aligned} \Vert F(z_1(\cdot ),\nu )-F(z_2(\cdot ),\nu )\Vert _\gamma \le \mu _0\Vert z_1-z_2\Vert _\gamma \end{aligned}$$

    where \(\gamma =\beta ,\eta \),

  3. (iii)

    there exist positive constants \(\mu _1\) and \(\mu _2\) such that for any \(z(s), z_1(s), z_2(s) \in C^0_\beta \) with \(\Vert z\Vert _\beta , \Vert z_i\Vert _\beta \le R\):

    $$\begin{aligned} \Vert G(z(\cdot ),\nu )\Vert _\eta \le \mu _1 \end{aligned}$$

    and

    $$\begin{aligned} \Vert G(z_1(\cdot ),\nu )-G(z_2(\cdot ),\nu )\Vert _\eta \le \mu _2\Vert z_1-z_2\Vert _\eta \end{aligned}$$

for any \(\nu \in V\). Then, if

$$\begin{aligned} \begin{array}{l} 2{\hat{k}}_\eta (\mu _0+\mu _2)<1\\ 2(\hat{k}|\nu |+{\hat{k}}_\beta \mu _1)<R \end{array} \end{aligned}$$

equation (5.6) has a unique fixed point \(z(s,\nu )\in C^0_\beta \). Moreover

$$\begin{aligned} \Vert z(s,\nu )\Vert _\beta \le 2(\hat{k}|\nu |+{\hat{k}}_\beta \mu _1)<R. \end{aligned}$$

Proof

First we observe that, from (i) and (ii) it follows easily:

$$\begin{aligned} \Vert F(z,\nu )\Vert _\beta \le \mu _0 \Vert z\Vert _\beta \end{aligned}$$

for \(\Vert z\Vert _\beta \le R\). Next, for any \(z(s)\in C^0_\beta \), with \(\Vert z\Vert _\beta \le R\), we define \({\hat{z}}(s)\) by the formula (5.6). We get

$$\begin{aligned} \Vert {\hat{z}}\Vert _\beta \le \hat{k}|\nu | + {\hat{k}}_\beta [\mu _0\Vert z\Vert _\beta + \mu _1] \end{aligned}$$
(5.12)

So, if \(2(\hat{k}|\nu |+{\hat{k}}_\beta \mu _1)< R\) and \(2{\hat{k}}_\beta \mu _0\le 1\), then \({\hat{z}}\in C^0_\beta \) and \(\Vert {\hat{z}}\Vert _\beta \le R\).

Next, for \(z_1(s),z_2(s)\in C^0_\beta \), with \(\Vert z_1\Vert _\beta , \Vert z_2\Vert _\beta \le R\), and \(\xi ,\varepsilon \) fixed and small as required above we have, for \(\eta \ge \beta \):

$$\begin{aligned} \begin{array}{l} \Vert {\hat{z}}_1(s)-{\hat{z}}_2(s)\Vert _\eta \le \\ {\hat{k}}_\eta \left[ \Vert F(z_1(\cdot ),\xi ,\varepsilon )-F(z_2(\cdot ),\xi ,\varepsilon )\Vert _\eta +\Vert G(z_1(\cdot ),\xi ,\varepsilon )-G(z_2(\cdot ),\xi ,\varepsilon ) \Vert _\eta \right] \le \\ {\hat{k}}_\eta (\mu _0+\mu _2)\Vert z_1-z_2\Vert _\eta \end{array} \end{aligned}$$

So, if \(2{\hat{k}}_\eta (\mu _0+\mu _2)<1\), we see that \(z\mapsto {\hat{z}}\) is a contraction with respect to the \(\eta \)-norm. We conclude that (5.2) has a unique fixed point \(z(\cdot ,\nu )\in C^0_\eta \) with \(\Vert z(\cdot ,\nu )\Vert _\eta <R\). However since \(z\mapsto {\hat{z}}\) is a map from the ball of radius R in \(C^0_\beta \) into itself we see that \(z(s,\nu )\) is the pointwise limit of a sequence \(z_k(s)\in C^0_\beta \) such that \(\Vert z_k\Vert _\beta <R\). For any \(s\ge 0\) we have then

$$\begin{aligned} |z(s,\nu )|e^{-\beta s}\le \lim _{k\rightarrow \infty }|z_k(s)|e^{-\beta s} \le R \end{aligned}$$

Thus \(z(\cdot ,\nu )\in C^0_\beta \) and \(\Vert z(\cdot ,\nu )\Vert _\beta \le R\). Finally, from (5.12) with \(z={\hat{z}}=z(\cdot ,\nu )\) we get \(\Vert z(\cdot ,\nu )\Vert _\beta \le 2(\hat{k}|\nu |+{\hat{k}}_\beta \mu _1)\). The proof is complete. \(\square \)

Remark 5.3

Lemma 5.2 can be used to prove the existence of a fixed point in \(C^0_0\) of Eq. (5.2). Indeed, we prove the existence of a fixed point satisfying \(\Vert z(\cdot )\Vert \le 1\), hence we can replace F and G in (5.4) with those in (5.7). Then we apply Lemma 5.2 replacing \(\beta \) with 0 and \(\eta \) with \(\beta \), \(0<\beta <\delta \). In Lemma 5.2. It is clear that \(F(0)=0\) and from Eq. (5.9) in Lemma 5.1 we have:

$$\begin{aligned} \Vert F(z_1)-F(z_2)\Vert _\gamma \le \left| {\tilde{f}}_z\right| _0\Vert z_1-z_2\Vert _\gamma \end{aligned}$$

where \(\gamma =0,\beta \). Now, when \(\Vert z_1\Vert , \Vert z_2\Vert \le \rho \le 1\) we have \(|\theta z_1(s)+(1-\theta )z_2(s)|\le \rho \le 1\) for any \(s\ge 0\) and \(0\le \theta \le 1\). Thus (see (5.8)):

$$\begin{aligned} \left| {\tilde{f}}_z\right| _0 \le \Delta (|z|) \end{aligned}$$

for \(\Vert z_i\Vert \le \rho \le 1\). This proves (ii) provided \(\Delta (\rho )\le \mu _0\). Next for \(G(z,\varepsilon )\) as in (5.7) we have

$$\begin{aligned} \Vert G(z,\varepsilon )\Vert _\eta \le \Vert G(z,\varepsilon )\Vert \le N \end{aligned}$$
(5.13)

where N has been defined in (5.1). Next, for any \(z_1(s),z_2(s)\in C^0_0\) such that \(\Vert z_i\Vert \le \rho \):

$$\begin{aligned}&|G(z_1(\cdot ),\varepsilon )(s)-G(z_2(\cdot ),\varepsilon )(s)| \\&\quad \le L_g \left[ \left| \int _0^s \omega _1(y_0(\tau )+z_1(\tau ))-\omega _1(y_0(\tau )+z_2(\tau ))d\tau \right| + |z_1(s)-z_2(s)|\right] \\&\quad \le L_g \left[ \left| \int _0^s \int _0^1 \omega _1'(y_0(\tau )+\theta z_1(\tau )+(1-\theta )z_2(\tau ))d\theta (z_1(\tau )-z_2(\tau ))d\tau \right| + |z_1(s)-z_2(s)|\right] \\&\quad \le L_g\left( {\bar{\omega }} \int _0^s |z_1(\tau )-z_2(\tau )|d\tau + |z_1(s)-z_2(s)|\right) \\&\quad \le L_g ({\bar{\omega }}\beta ^{-1}\Vert z_1-z_2\Vert _\beta + \Vert z_1-z_2\Vert _\beta )e^{\beta s} \end{aligned}$$

(see (2.8) for the definition of \({\bar{\omega }}\)). So:

$$\begin{aligned} \Vert G(z_1(\cdot ),\varepsilon )-G(z_2(\cdot ),\varepsilon )\Vert _\beta \le L_g(1+{\bar{\omega }}\beta ^{-1}) \Vert z_1-z_2\Vert _\beta \end{aligned}$$
(5.14)

Hence, from Lemma 5.2 with \(R=\rho \), we see that, for \(|\xi |\) and \(|\varepsilon |\) sufficiently small (depending on \(\rho \)), equation (5.2) has a unique fixed point \(z(s,\xi ,\varepsilon )\in C^0_0\) and \(\Vert z(\cdot ,\xi ,\varepsilon )\Vert \le \rho \).

Next we prove the following

Lemma 5.4

Suppose the same assumptions of Lemma 5.2 hold. Suppose, further that constants \(L_F\) and \(L_G\) exists such that for any \(z(s)\in C^0_\beta \) with \(\Vert z(s)\Vert _\beta \le R\), and any \(\xi _1,\xi _2,\varepsilon _1,\varepsilon _2\) with \(|\xi _i|\le {\bar{\xi }}\) and \(|\varepsilon _i|\le {\bar{\varepsilon }}\), it results:

$$\begin{aligned} \begin{array}{l} \Vert F(z(\cdot ),\nu _1)-F(z(\cdot ),\nu _2)\Vert _\eta \le L_F|\nu _1-\nu _2| \\ \Vert G(z(\cdot ),\nu _1)-G(z(\cdot ),\nu _2)\Vert _\eta \le L_G|\nu _1-\nu _2| \end{array} \end{aligned}$$
(5.15)

Then, if \(2{\hat{k}}_\eta (\mu _0+\mu _2)<1\), \(z(\cdot ,\nu )\) is Lipschitz continuous as a map into \(C^0_\eta \).

Proof

Let \(z(s):=z(s,\nu _1)-z(s,\nu _2)\). Then z(s) satisfies the equation

$$\begin{aligned} z(\cdot )= & {} S(\nu _1-\nu _2) + K[F(z(\cdot ,\nu _1),\nu _1)-F(z(\cdot ,\nu _2),\nu _2) \\&+G(z(\cdot ,\nu _1),\nu _1) - G(z(\cdot ,\nu _2),\nu _2)]. \end{aligned}$$

Then:

$$\begin{aligned} \Vert z\Vert _\eta\le & {} \hat{k}|\nu _1-\nu _2| +{\hat{k}}_\eta \Vert F(z(\cdot ,\nu _1),\nu _1)-F(z(\cdot ,\nu _2),\nu _2)\Vert _\eta \\&+{\hat{k}}_\eta \Vert G(z(\cdot ,\nu _1),\nu _1)-G(z(\cdot ,\nu _2),\nu _2) \Vert _\eta \\\le & {} \hat{k}|\nu _1-\nu _2|+{\hat{k}}_\eta \mu _0\Vert z\Vert _\eta + {\hat{k}}_\eta \Vert F(z(s,\nu _2),\nu _1)-F(z(s,\nu _2),\nu _2)\Vert _\eta \\&+ {\hat{k}}_\eta \mu _2\Vert z\Vert _\eta + {\hat{k}}_\eta \Vert G(z(s,\nu _2),\nu _1)-G(z(s,\nu _2),\nu _2) \Vert _\eta \\\le & {} \hat{k}|\nu _1-\nu _2| + {\hat{k}}_\eta (\mu _0+\mu _2)\Vert z\Vert _\eta + {\hat{k}}_\eta (L_F+L_G) |\nu _1-\nu _2|. \end{aligned}$$

Using \(2{\hat{k}}_\eta (\mu _0+\mu _2)<1\) we get

$$\begin{aligned} \Vert z\Vert _\eta \le 2[{\hat{k}}_\eta (L_F+L_G)+ k]|\nu _1-\nu _2| \end{aligned}$$

The proof is complete. \(\square \)

From Lemma 5.4 we obtain the following

Proposition 5.5

Let \(z(\cdot ,\xi ,\varepsilon )\in C^0_0\) be the fixed point of equation (5.2). Then, for any \(\eta >0\), the map \((\xi ,\varepsilon )\mapsto z(\cdot ,\xi ,\varepsilon )\in C^0_\eta \) is Lipschitz continuous.

Proof

Since F(z) is independent of \((\xi ,\varepsilon )\) we trivially have \(F(z(\cdot ),\xi _1,\varepsilon _1)-F(z(\cdot ),\xi _2,\varepsilon _2)=0\). So the first condition in (5.15) holds. Next it is easy to see that

$$\begin{aligned} \Vert G(z(\cdot ),\varepsilon _1)-G(z(\cdot ),\varepsilon _2)\Vert _\eta \le L_g|\varepsilon _1-\varepsilon _2| \end{aligned}$$
(5.16)

The conclusion follows from Lemma 5.4. \(\square \)

Now, we prove that \(z(\cdot ,\xi ,\varepsilon )\in B_\rho \subset C^0_0\) is \(C^1\) if we consider it as a map \((\xi ,\varepsilon )\mapsto z(\cdot ,\xi ,\varepsilon )\in C^0_{2\beta }\), with \(0<2\beta \le \delta \) and the derivatives take values into \(C^0_\beta \). So, we assume that f(y) and \(g(t,y,\varepsilon )\) are \(C^2\) and that the second order derivatives of \(g(t,y,\varepsilon )\) are bounded in \((t,y,\varepsilon )\) for any \(t\in {\mathbb {R}}\), \(|y|\le 1\) and \(|\varepsilon |\le \varepsilon _0\).

For uv running in the set of symbols \(\{y,t,\varepsilon \}\), let

$$\begin{aligned} \begin{array}{l} {\bar{g}}_{uv}=\sup \{g_{uv}(t,y_0(s)+z,\varepsilon ) \mid t\in {\mathbb {R}}, s\ge 0, |z|\le \rho , |\varepsilon |\le \varepsilon _0\}. \end{array} \end{aligned}$$
(5.17)

Taking formally the derivative with respect to \(\xi \) we see that \(z_\xi (\cdot ,\xi ,\varepsilon ){\hat{\xi }}\) should be a fixed point in \(C^0_\beta \) of the equation

$$\begin{aligned} {\hat{w}} = S{\hat{\xi }} +K[F'(z(\cdot ,\xi ,\varepsilon ))+\varepsilon G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )]w, \end{aligned}$$
(5.18)

where

$$\begin{aligned} {[}F'(z(\cdot ))w](s):={\tilde{f}}_z(s,z(s))w(s) \end{aligned}$$

and

$$\begin{aligned}{}[G_z(z(\cdot ),\varepsilon )w](s):= & {} g_y\left( \int _0^s \omega _1(y_0(\tau )+z(\tau ))d\tau ,y_0(s)+z(s),\varepsilon \right) w(s) \nonumber \\&+ g_t\left( \int _0^s \omega _1(y_0(\tau )+z(\tau ))d\tau ,y_0(s)+z(s),\varepsilon \right) \nonumber \\&\times \int _0^s \omega '_1(y_0(\tau )+z(\tau ))w(\tau )d\tau . \end{aligned}$$
(5.19)

Similarly, \(z_\varepsilon (\cdot ,\xi ,\varepsilon )\) should be a fixed point in \(C^0_\beta \) of the equation

$$\begin{aligned} {\hat{w}} = K\{[F'(z(\cdot ,\xi ,\varepsilon ))+\varepsilon G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )]w+[G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )+\varepsilon G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon )]\},\nonumber \\ \end{aligned}$$
(5.20)

where

$$\begin{aligned} G_\varepsilon (z(\cdot ),\varepsilon )(s)=g_\varepsilon \left( \int _0^s \omega _1(y_0(\tau )+z(\tau ))d\tau ,y_0(s)+z(s),\varepsilon \right) . \end{aligned}$$
(5.21)

First we consider Eq. (5.18) and take \(0<2\beta <\delta \). Then we prove the following

\((s_1\)):

equation (5.18) has a unique fixed point \({\tilde{z}}(s,\xi ,\varepsilon )\hat{\xi }\in C^0_\beta \);

\((s_2\)):

the map \((\xi ,\varepsilon )\mapsto {\tilde{z}}(s,\xi ,\varepsilon )\in C^0_{2\beta }\) is Lipschitz continuous;

\((s_3\)):

\({\tilde{z}}(s,\xi ,\varepsilon )\) is indeed the derivative of \(z(s,\xi ,\varepsilon )\) in \(C^0_{2\beta }\).

Proof of\(s_1)\). We apply Lemma 5.2 with \(F(z,\nu )(s)=F'(z(s,\xi ,\varepsilon ))z(s)\) and \(G(z,\nu )(s)=G_z(z(s,\xi ,\varepsilon ),\varepsilon )z(s)\) instead of F and G. As \(\Vert z(s,\xi ,\varepsilon )\Vert \le \rho \le 1\), we have

$$\begin{aligned} |[F'(z(\cdot ,\xi ,\varepsilon ))z](s)|= & {} |{\tilde{f}}_z(s,z(s,\xi ,\varepsilon ))z(s)| \nonumber \\= & {} |[f'(y_0(s)+z(s,\xi ,\varepsilon ))-f'(y_0(s))]z(s)| \le \Delta (\rho )|z(s)|\qquad \end{aligned}$$
(5.22)

So (i) trivially holds and

$$\begin{aligned} \Vert [F'(z(\cdot ,\xi ,\varepsilon ))(z_1-z_2)](s)\Vert _\beta \le \Delta (\rho )\Vert z_1-z_2\Vert _\beta \end{aligned}$$
(5.23)

for any \(z_1(s),z_2(s)\in C^0_\beta \) (no matter of \(0\le \beta <\delta \)) which proves (ii). Next, recalling (5.1), (2.8) and using (5.19) we have, for any \(u(s)\in C^0_0\) such that \(\Vert u\Vert <\rho \) and any \(z(s)\in C^0_\beta \):

$$\begin{aligned} |[G_z(u(\cdot ),\varepsilon )z](s)|\le & {} N_2|z(s)| + N_1\int _0^s|\omega '_1(y_0(\tau )+u(\tau ))z(\tau )|d\tau \nonumber \\\le & {} N_2\Vert z\Vert _\beta e^{\beta s} + N_1{\bar{\omega }}\Vert z\Vert _\beta \int _0^s e^{\beta \tau }d\tau \le (N_2+N_1{\bar{\omega }}\beta ^{-1})\Vert z\Vert _\beta e^{\beta s}.\nonumber \\ \end{aligned}$$
(5.24)

In particular:

$$\begin{aligned} \Vert G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )z(\cdot )\Vert _\beta \le (N_2+N_1{\bar{\omega }}\beta ^{-1})\Vert z\Vert _\beta \end{aligned}$$
(5.25)

for any \(\beta >0\) and \(z(\cdot )\in C^0_\beta \). Hence, being \(G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )z\) linear in z:

$$\begin{aligned} \Vert G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )z_1(\cdot )-G_z (z(\cdot ,\xi ,\varepsilon ),\varepsilon )z_2(\cdot )\Vert _\beta \le (N_2+N_1{\bar{\omega }}\beta ^{-1})\Vert z_1-z_2\Vert _\beta \end{aligned}$$

for any \(\beta >0\). We conclude that, for any \(0<\beta <\delta \), Eq. (5.18) has a unique fixed point \(\tilde{z}(s,\xi ,\varepsilon ){\hat{\xi }}\). The linearity of \({\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }}\) with respect to \({\hat{\xi }}\) easily follows from its uniqueness and the fact that the map (5.18) is linear in \((z,{\hat{\xi }})\). Finally from the fact that \({\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }}\) satisfies equation (5.18) it follows

$$\begin{aligned} \Vert {\tilde{z}}(\cdot ,\xi ,\varepsilon ){\hat{\xi }}\Vert _\beta \le 2k|{\hat{\xi }}| \end{aligned}$$
(5.26)

provided

$$\begin{aligned} 2k_\beta (\Delta (\rho )+{\bar{\varepsilon }}(N_2+N_1{\bar{\omega }}\beta ^{-1}))<1 \end{aligned}$$

Proof of\(s_2)\). We apply Lemma 5.4 with \(F'(z(\cdot ,\xi ,\varepsilon )z(\cdot )\) and \(G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )z(\cdot )\) instead of F and G. We will prove that (5.15) holds with \(\eta =2\beta >0\). For \(|z_1(s)|, |z_2(s)|\le \rho \) we have

$$\begin{aligned}&|[F'(z_1(\cdot ))-F'(z_2(\cdot ))]z](s)| = |{\tilde{f}}_z(s,z_1(s))-{\tilde{f}}_z(s,z_2(s))|\, |z(s)| \nonumber \\&\quad =|f'(y_0(s)+z_1(s))-f'(y_0(s)+z_2(s))|\, |z(s)| \le L_{f'}|z_1(s)-z_2(s)|\, |z(s)|. \end{aligned}$$
(5.27)

\(L_{f'}\) being the Lipschitz constant for \(f'(x)\) in a fixed compact set containing

$$\begin{aligned} \{y_0(s)+z \mid s\ge 0, |z|\le 1\}. \end{aligned}$$

Thus for any \(z\in C^0_\beta \) we have, because of Proposition 5.5

$$\begin{aligned} \Vert [F'(z(\cdot ,\xi _1,\varepsilon _1))-F'(z(\cdot ,\xi _2,\varepsilon _2))]z(\cdot )\Vert _{2\beta } \le L_zL_{f'}[|\xi _1-\xi _2|+|\varepsilon _1-\varepsilon _2|]\, \Vert z\Vert _\beta \end{aligned}$$
(5.28)

\(L_z\) being the Lipschitz constant of \(z(\cdot ,\xi ,\varepsilon )\) whose existence is stated in Proposition 5.5. Next, let \(z_1(s), z_2(s)\in C^0_0\) and set

$$\begin{aligned} t_i(s)=t(s,z_i(s),\varepsilon ):=\int _0^s \omega _1(y_0(\tau )+z_i(\tau ))d\tau . \end{aligned}$$
(5.29)

For any \(z(s)\in C^0_\beta \) we have

$$\begin{aligned}&|[(G_z(z_1(\cdot ),\varepsilon _1)-G_z(z_2(\cdot ),\varepsilon _2))z](s)| \\&\quad \le |g_y(t_1(s),y_0(s)+z_1(s),\varepsilon _1)-g_y(t_2(s),y_0(s)+z_2(s),\varepsilon _2)|\, |z(s)| \\&\qquad +|g_t(t_1(s),y_0(s)+z_1(s),\varepsilon _1)-g_t(t_2(s),y_0(s)+z_2(s),\varepsilon _2)| \\&\qquad \cdot \int _0^s |\omega '_1(y_0(\tau )+z_1(\tau ))z(\tau )|d\tau \\&\qquad +|g_t(t_2(s),y_0(s)+z_2(s),\varepsilon _2)| \\&\qquad \cdot \int _0^s |\omega '_1(y_0(\tau )+z_1(\tau ))-\omega '_1(y_0(\tau )+z_2(\tau ))|\, |z(\tau )|d\tau . \end{aligned}$$

Using (5.17) we get

$$\begin{aligned}&|g_y(t_1(s),y_0(s)+z_1(s),\varepsilon _1)-g_y(t_2(s),y_0(s)+z_2(s),\varepsilon _2)| \\&\quad \le {\bar{g}}_{yt}|t_1(s)-t_2(s)| + {\bar{g}}_{yy}|z_1(s)-z_2(s)| + {\bar{g}}_{y\varepsilon }|\varepsilon _1-\varepsilon _2| \end{aligned}$$

and similarly

$$\begin{aligned}&|g_t(t_1(s),y_0(s)+z_1(s),\varepsilon _1)-g_t(t_2(s),y_0(s)+z_2(s),\varepsilon _2)| \\&\quad \le {\bar{g}}_{tt}|t_1(s)-t_2(s)| + {\bar{g}}_{ty}|z_1(s)-z_2(s)| + {\bar{g}}_{t\varepsilon }|\varepsilon _1-\varepsilon _2|. \end{aligned}$$

Now, \(t(s)=t_1(s)-t_2(s)\) satisfies the differential equation

$$\begin{aligned}&t' = \omega _1(y_0(s)+z_1(s))-\omega _1(y_0(s)+z_2(s)) \\&t(0)=0 \end{aligned}$$

so

$$\begin{aligned} |t(s,z_1(s),\varepsilon ))-t(s,z_2(s),\varepsilon ))|\le & {} \int _0^s |\omega _1(y_0(\tau )+z_1(\tau ))-\omega _1(y_0(\tau )+z_2(\tau ))| d\tau \nonumber \\\le & {} \int _0^s \int _0^1 |\omega '_1(y_0(\tau )+\theta z_1(\tau )\nonumber \\&+(1-\theta )z_2(v))|d\theta e^{\beta \tau }d\tau \Vert z_1-z_2\Vert _\beta \nonumber \\\le & {} {\bar{\omega }}\beta ^{-1}\Vert z_1-z_2\Vert _\beta e^{\beta s}. \end{aligned}$$
(5.30)

As a consequence:

$$\begin{aligned}&\Vert [G_z(z_1(\cdot ),\varepsilon _1)-G_z(z_2(\cdot ),\varepsilon _2)]z\Vert _{2\beta } \nonumber \\&\quad \le \left[ ({\bar{g}}_{yt}{\bar{\omega }}\beta ^{-1} + {\bar{g}}_{yy})\Vert z_1-z_2\Vert _\beta + {\bar{g}}_{y\varepsilon }|\varepsilon _1-\varepsilon _2|\right] \Vert z\Vert _\beta \nonumber \\&\qquad + \left[ ({\bar{g}}_{tt}{\bar{\omega }}\beta ^{-1}+{\bar{g}}_{ty})\Vert z_1-z_2\Vert _\beta + {\bar{g}}_{t\varepsilon }|\varepsilon _1-\varepsilon _2|\right] {\bar{\omega }}\beta ^{-1}\Vert z\Vert _\beta \nonumber \\&\qquad + N_1L_{\omega '}(2\beta )^{-1}\Vert z_1-z_2\Vert _\beta \Vert z\Vert _\beta \le C_{1,\beta }[\Vert z_1-z_2\Vert _\beta + |\varepsilon _1-\varepsilon _2|]\Vert z\Vert _\beta \end{aligned}$$
(5.31)

where \(L_{\omega '}\) is the Lipschitz constant for \(\omega '_1(y)\) and \(C_{1,\beta }\) a suitable constant.

Putting everything together and using the Lipschitz continuity of \((\xi ,\varepsilon )\mapsto z(s,\xi ,\varepsilon )\in C^0_\beta \), we see that, for any \(z\in C^0_\beta \) we have

$$\begin{aligned} \Vert [G_z(z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _1)-G_z(z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)]z(\cdot )\Vert _{2\beta } \le C[|\xi _1-\xi _2|+|\varepsilon _1-\varepsilon _2|]\Vert z\Vert _\beta \nonumber \\ \end{aligned}$$
(5.32)

for some positive constant C can that be evaluated explicitly. Then \(s_2)\) follows from Lemma 5.4.

Proof of\(s_3)\). Let \(\xi ,\hat{\xi }\in {{\mathcal {R}}}P\) such that \(z(s,\xi ,\varepsilon )\) and \(z(s,\xi +{\hat{\xi }},\varepsilon )\) exists, and set

$$\begin{aligned} z(s):=z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )-{\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }}. \end{aligned}$$
(5.33)

Then \(z(s)\in C^0_\beta \) satisfies the equation

$$\begin{aligned} {\hat{z}}= & {} K[F(z(\cdot ,\xi +{\hat{\xi }},\varepsilon ))-F(z(\cdot ,\xi ,\varepsilon ))-F'(z(\cdot ,\xi ,\varepsilon )){\tilde{z}}(\cdot ,\xi ,\varepsilon )\hat{\xi }] \nonumber \\&+\varepsilon K[G(z(\cdot ,\xi +{\hat{\xi }},\varepsilon ),\varepsilon )-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )-G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )\tilde{z}(\cdot ,\xi ,\varepsilon )\hat{\xi }]. \end{aligned}$$
(5.34)

We want to apply Lemma 5.2 to (5.34). However, Eq. (5.34) is not in the form of a fixed point equation so, to apply Lemma 5.2, we write it in another, equivalent, form. The idea is to write the right hand side of Eq. (5.34) as the sum of a linear map of small norm with a function of \(({\hat{\xi }},\varepsilon )\) of small norm. To do so we will also need to include in the new \(F(z,\nu )\) the linear part of the function \(G(z,\varepsilon )\). This is the reason why we changed Eq. (5.2) into (5.6). Using (5.33) we write:

$$\begin{aligned}&{[}F(z(\cdot ,\xi +{\hat{\xi }},\varepsilon ))-F(z(\cdot ,\xi ,\varepsilon ))-F'(z(\cdot ,\xi ,\varepsilon )){\tilde{z}}(\cdot ,\xi ,\varepsilon ){\hat{\xi ]}}(s) \\&\quad = {\tilde{f}}(s,z(s,\xi +{\hat{\xi }},\varepsilon ))-{\tilde{f}}(s,z(s,\xi ,\varepsilon ))-{\tilde{f}}_z(s,z(s,\xi ,\varepsilon )){\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }} \; \\&\quad = \int _0^1 {\tilde{f}}_z(s,\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon ))d\theta z(s) \\&\qquad + \int _0^1 \left\{ {\tilde{f}}_z(s,\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon ))- {\tilde{f}}_z(s,z(s,\xi ,\varepsilon ))\right\} d\theta {\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }} \\&\quad = [F_1({\hat{\xi }},\varepsilon )z](s) + F_2(s,{\hat{\xi }},\varepsilon ) \tilde{z}(s,\xi ,\varepsilon ){\hat{\xi }} \end{aligned}$$

where

$$\begin{aligned}{}[F_1({\hat{\xi }},\varepsilon )z](s)= & {} \int _0^1 {\tilde{f}}_z(s,\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon )) d\theta z(s) \\ F_2(s,{\hat{\xi }},\varepsilon )= & {} \int _0^1 {\tilde{f}}_z(s,\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon ))- \tilde{f}_z(s,z(s,\xi ,\varepsilon ))d\theta \end{aligned}$$

As \(|z(s,\xi +{\hat{\xi }},\varepsilon )|,|z(s,\xi ,\varepsilon )|\le \rho \), we see that \(|\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon )|\le \rho \) and then

$$\begin{aligned} \begin{array}{l} |[F_1({\hat{\xi }},\varepsilon )z](s)| \le \Delta (\rho )|z(s)| \\ |F_2(s,{\hat{\xi }},\varepsilon )| \le \frac{1}{2}L_{f'}|z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )| \end{array} \end{aligned}$$
(5.35)

Next, using (5.19) we have, with \(t(s,\xi ,\varepsilon )=t(s,z(\cdot ,\xi ,\varepsilon ),\varepsilon )\) (see (5.29))

$$\begin{aligned} \begin{array}{l} {[}G(z(\cdot ,\xi +{\hat{\xi }},\varepsilon ),\varepsilon )-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )-G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon ){\tilde{z}}(\cdot ,\xi ,\varepsilon )\hat{\xi }](s) \\ = g(t(s,\xi +{\hat{\xi }},\varepsilon ),y_0(s)+z(s,\xi +{\hat{\xi }},\varepsilon ),\varepsilon )-g(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon ) \\ \qquad - g_y(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon ){\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }} \\ \displaystyle \qquad - g_t(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon )\int _0^s \omega '_1(y_0(\tau )+z(\tau ,\xi ,\varepsilon )){\tilde{z}}(\tau ,\xi ,\varepsilon ){\hat{\xi }} d\tau \end{array} \end{aligned}$$

Since

$$\begin{aligned} g(t_2,z_2,\varepsilon )-g(t_1,z_1,\varepsilon )=\int _0^1 \frac{d}{d\theta }[g(\theta t_2+(1-\theta )t_1,\theta z_2+(1-\theta )z_1,\varepsilon )]d\theta \end{aligned}$$

we get (see also (5.30) for the last equality):

$$\begin{aligned}&G(z(\cdot ,\xi +{\hat{\xi }},\varepsilon ),\varepsilon )(s)-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )(s) \\&\quad = \int _0^1 \frac{d}{d\theta }[g(\theta t(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )t(s,\xi ,\varepsilon ),y_0(s)+\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon ),\varepsilon )]d\theta \\&\quad = g_1(s,{\hat{\xi }},\varepsilon )[z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )] + g_2(s,{\hat{\xi }},\varepsilon )[t(s,\xi +{\hat{\xi }},\varepsilon )-t(s,\xi ,\varepsilon )] \\&\quad = g_1(s,{\hat{\xi }},\varepsilon )[z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )] + g_2(s,{\hat{\xi }},\varepsilon ) [\theta ({\hat{\xi }},\varepsilon )(z(\cdot ,\xi +{\hat{\xi }},\varepsilon )-z(\cdot ,\xi ,\varepsilon ))](s) \end{aligned}$$

where

$$\begin{aligned}&g_1(s,{\hat{\xi }},\varepsilon ) \\&\quad = \int _0^1 g_y(\theta t(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )t(s,\xi ,\varepsilon ),y_0(s)+\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon ),\varepsilon )d\theta \\&g_2(s,{\hat{\xi }},\varepsilon ) \\&\quad =\int _0^1 g_t(\theta t(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )t(s,\xi ,\varepsilon ),y_0(s)+\theta z(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon ),\varepsilon ) d\theta \\&[\theta ({\hat{\xi }},\varepsilon )z](s)=\int _0^s\int _0^1 \omega '_1(y_0(\tau )+\theta z(\tau ,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(\tau ,\xi ,\varepsilon ))d\theta z(\tau ) d\tau . \end{aligned}$$

Hence we see that

$$\begin{aligned}&[G(z(\cdot ,\xi +{\hat{\xi }},\varepsilon ),\varepsilon )-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )-G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon ){\tilde{z}}(\cdot ,\xi ,\varepsilon ){\hat{\xi }} ](s) \\&\quad = {[}G_1({\hat{\xi }},\varepsilon )z](s) + G_2(s,{\hat{\xi }},\varepsilon ) \end{aligned}$$

where

$$\begin{aligned}{}[G_1({\hat{\xi }},\varepsilon )z](s)= & {} g_1(s,{\hat{\xi }},\varepsilon )z(s)+g_2(s,{\hat{\xi }},\varepsilon )[\theta ({\hat{\xi }},\varepsilon )z](s) \\ G_2(s,{\hat{\xi }},\varepsilon )= & {} [g_1(s,{\hat{\xi }},\varepsilon )-g_y(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon )]{\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }} \\&+ g_2(s,{\hat{\xi }},\varepsilon )[\theta ({\hat{\xi }},\varepsilon ){\tilde{z}}(\cdot ,\xi ,\varepsilon ){\hat{\xi ]}}(s) \\&- g_t(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon )\int _0^s \omega '_1(y_0(\tau )+z(\tau ,\xi ,\varepsilon )){\tilde{z}}(\tau ,\xi ,\varepsilon ){\hat{\xi }} d\tau . \end{aligned}$$

So, z(s) satisfies the fixed point equation

$$\begin{aligned} z = K\left\{ [F_1({\hat{\xi }},\varepsilon )+\varepsilon G_1({\hat{\xi }},\varepsilon )]z+F_2(\cdot ,{\hat{\xi }},\varepsilon )\tilde{z}(\cdot ,\xi ,\varepsilon ){\hat{\xi }} + \varepsilon G_2(\cdot ,{\hat{\xi }},\varepsilon )\right\} . \end{aligned}$$
(5.36)

We are now in position to apply Lemma 5.2 with \([F_1({\hat{\xi }},\varepsilon )+\varepsilon G_1({\hat{\xi }},\varepsilon )]z\) instead of F and \(F_2(s,{\hat{\xi }},\varepsilon ){\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }}+ \varepsilon G_2(s,{\hat{\xi }},\varepsilon )\) instead of G.

It is clear that (i) in Lemma 5.2 is satisfied. Next

$$\begin{aligned}\begin{array}{l} |g_1(s,{\hat{\xi }},\varepsilon )z(s)| \le N_2|z(s)| \\ \displaystyle |g_2(s,{\hat{\xi }},\varepsilon )[\theta ({\hat{\xi }},\varepsilon )z](s)| \le N_1{\bar{\omega }}\int _0^s |z(\tau )|d\tau \le N_1{\bar{\omega }}\beta ^{-1}\Vert z(\cdot )\Vert _\beta e^{\beta s}. \end{array} \end{aligned}$$

So

$$\begin{aligned} \Vert \varepsilon G_1(\cdot ,{\hat{\xi }},\varepsilon )]z\Vert _\beta \le |\varepsilon |(N_2+N_1{\bar{\omega }}\beta ^{-1})\Vert z(\cdot )\Vert _\beta \end{aligned}$$

This inequality, together with (5.35), proves (ii) in Lemma 5.2 provided \(|\varepsilon |\) is sufficiently small, because the map \(z\mapsto [F_1(\cdot ,{\hat{\xi }},\varepsilon )+\varepsilon G_1(\cdot ,{\hat{\xi }},\varepsilon )]z\) is linear. Next, we consider \(F_2(s,{\hat{\xi }},\varepsilon )\tilde{z}(s,\xi ,\varepsilon ){\hat{\xi }}\) and \(G_2({\hat{\xi }},\varepsilon )\). Using (5.35) and (5.26) we get:

$$\begin{aligned}&|F_2(s,{\hat{\xi }},\varepsilon ){\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }}|\le \int _0^1 L_{f'}|z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )|\theta d\theta \, |{\tilde{z}}(s,\xi ,\varepsilon ){\hat{\xi }}| \nonumber \\&\quad \frac{1}{2}L_{f'}\Vert z(\cdot ,\xi +{\hat{\xi }},\varepsilon )-z(\cdot ,\xi ,\varepsilon )\Vert _\beta \Vert {\tilde{z}}(\cdot ,\xi ,\varepsilon ){\hat{\xi }}\Vert _\beta e^{2\beta s} \le L_zL_{f'}k||{\hat{\xi }}|^2 e^{2\beta s} \end{aligned}$$
(5.37)

where \(L_z\) is the Lipschitz constant for \(z(\cdot ,\xi ,\varepsilon )\) in the \(\beta \)-norm whose existence is stated in Proposition 5.5. Similarly:

$$\begin{aligned}&|g_1(s,{\hat{\xi }},\varepsilon )-g_y(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon )|\\&\quad \le \int _0^1 \theta {\bar{g}}_{ty}|t(s,\xi +{\hat{\xi }},\varepsilon )-t(s,\xi ,\varepsilon )| + \theta {\bar{g}}_{yy}|z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )| d\theta \\&\quad = \frac{1}{2}{\bar{g}}_{yt}|t(s,\xi +{\hat{\xi }},\varepsilon )-t(s,\xi ,\varepsilon )|+\frac{1}{2}{\bar{g}}_{yy} |z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )| \\&\quad \le \frac{1}{2}\left( {\bar{g}}_{yt}{\bar{\omega }}\beta ^{-1}+{\bar{g}}_{yy}\right) \Vert z(\cdot ,\xi +{\hat{\xi }},\varepsilon )-z(\cdot ,\xi ,\varepsilon )\Vert _\beta e^{\beta s}. \end{aligned}$$

having used (5.30). Hence

$$\begin{aligned} \Vert [g_1(\cdot ,{\hat{\xi }},\varepsilon )-g_y(t(\cdot ,\xi ,\varepsilon ),y_0(\cdot )+z(\cdot ,\xi ,\varepsilon ),\varepsilon )]\tilde{z}(\cdot ,\xi ,\varepsilon ){\hat{\xi }}\Vert _{2\beta } \le c_1|{\hat{\xi }}|^2 \end{aligned}$$

for some \(c_1>0\), follows from (5.26) and the Lipschitz continuity of \(z(s,\xi ,\varepsilon )\) in the \(\beta \)-norm. Next:

$$\begin{aligned}&\left| g_2(s,{\hat{\xi }},\varepsilon )[\theta ({\hat{\xi }},\varepsilon ){\tilde{z}}(\cdot ,{\hat{\xi }},\varepsilon )](s) - g_t(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon )\right. \\&\qquad \left. \cdot \int _0^s\omega '(y_0(\tau ) +z(\tau ,\xi ,\varepsilon ){\tilde{z}}(s,{\hat{\xi }},\varepsilon ){\hat{\xi }} d\tau \right| \\&\quad \le \int _0^1 \Big | g_t(\theta t(s,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )t(s,\xi ,\varepsilon ),y_0(s)+\theta z(s,\xi +{\hat{\xi }},\varepsilon ),\varepsilon )+(1-\theta )z(s,\xi ,\varepsilon ) \\&\qquad -g_t(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon ) \Big | d\theta \\&\qquad \cdot \int _0^s\int _0^1|\omega _1'(y_0(\tau )+\theta z(\tau ,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(\tau ,\xi ,\varepsilon )){\tilde{z}}(\tau ,\xi ,\varepsilon ){\hat{\xi }} |d\theta d\tau \\&\qquad + |g_t(t(s,\xi ,\varepsilon ),y_0(s)+z(s,\xi ,\varepsilon ),\varepsilon )| \\&\qquad \cdot \int _0^s\int _0^1 |\omega '_1(y_0(\tau )+\theta z(\tau ,\xi +{\hat{\xi }},\varepsilon )+(1-\theta )z(\tau ,\xi ,\varepsilon )) -\omega '_1(y_0(\tau )+z(\tau ,\xi ,\varepsilon ))| \, |{\tilde{z}}(\tau ,\xi ,\varepsilon ){\hat{\xi }} | d\theta \, d\tau \\&\quad \le \frac{1}{2}\left\{ {\bar{g}}_{tt}|t(s,\xi +{\hat{\xi }},\varepsilon )-t(s,\xi ,\varepsilon )| + {\bar{g}}_{ty}|z(s,\xi +{\hat{\xi }},\varepsilon )-z(s,\xi ,\varepsilon )|\right\} {\bar{\omega }}\int _0^s |{\tilde{z}}(\tau ,\xi ,\varepsilon )| d\tau \\&\qquad + \frac{1}{2}N_1\int _0^s L_{\omega '} |z(\tau ,\xi +{\hat{\xi }},\varepsilon )-z(\tau ,\xi ,\varepsilon )|\, |{\tilde{z}}(\tau ,\xi ,\varepsilon ){\hat{\xi }} |d\tau \\&\quad \le \left[ \frac{{\bar{g}}_{tt}{\bar{\omega }}}{2\beta } \Vert t(s,\xi +{\hat{\xi }},\varepsilon )-t(s,\xi ,\varepsilon )\Vert _\beta \Vert z(\tau ,\xi ,\varepsilon )\Vert _\beta \right. \\&\qquad \left. + \frac{N_1L_{\omega '}}{4\beta } \Vert z(\cdot ,\xi +{\hat{\xi }},\varepsilon )-z(\cdot ,\xi ,\varepsilon )\Vert _\beta \, \Vert {\tilde{z}}(\cdot ,\xi ,\varepsilon ){\hat{\xi }}\Vert _\beta \right] e^{2\beta } \\&\quad \le c_2|{\hat{\xi }}|^2 e^{2\beta }. \end{aligned}$$

From Lemma 5.2 we get then

$$\begin{aligned} \Vert z(\cdot ,\xi +{\hat{\xi }},\varepsilon )-z(\cdot ,\xi ,\varepsilon )-\tilde{z}(\cdot ,\xi ,\varepsilon ){\hat{\xi }}\Vert _{2\beta }\le {\tilde{C}}|{\tilde{\xi }}|^2 \end{aligned}$$

that is \(s_3)\).

To complete the proof of Theorem 2.3 when \(m=2\) we prove that \(s_1), s_2)\) and \(s_3)\) are satisfied when we replace Eq. (5.18) with (5.20). With reference to Eq. (5.6) in this case we take

$$\begin{aligned}{}[F'(z(\cdot ,\xi ,\varepsilon ))+\varepsilon G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )]z \end{aligned}$$

instead of \(F(z,\nu )\) and

$$\begin{aligned} G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )+\varepsilon G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon ) \end{aligned}$$

instead of \(G(z,\nu )\) with \(\nu =(\xi ,\varepsilon )\). We apply Lemma 5.2 with \(\eta =2\beta \). It is clear that condition (i) of Lemma 5.2 is satisfied and (ii) follows from (5.22) and (5.25), taking \(\mu _0=\Delta (\rho )+\varepsilon _0(N_2+N_1{\bar{\omega }}(2\beta )^{-1})\) provided \(\rho \) and \(\varepsilon _0\) are sufficiently small.

As for (iii), only the inequality \(\Vert G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )+\varepsilon G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon )\Vert _{2\beta }<\mu _1\) has to be checked, since \(G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )+\varepsilon G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon )\) depends only on \(\nu =(\xi ,\varepsilon )\). Now, from (5.21) we get

$$\begin{aligned} |G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon )| \le {\bar{g}}_\varepsilon := \sup \{g_\varepsilon (t,y_0(s)+z,\varepsilon )\mid t\in {\mathbb {R}}, \; |z|\le 1, |\varepsilon |\le e_0\} \end{aligned}$$
(5.38)

and then, using (5.13) we see that

$$\begin{aligned} |G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )+\varepsilon G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon )|\le N+\varepsilon _0{\bar{g}}_\varepsilon . \end{aligned}$$

So, taking \(\mu _1=N+\varepsilon _0{\bar{g}}_\varepsilon \), \(\mu _2=0\), \(k=0\) (because S is missed in the equation), \(R>2k_\beta \mu _1\) and \(\rho ,\varepsilon _0\) such that \(2k_\eta [\Delta (\rho )+\varepsilon _0(N_2+N_1{\bar{\omega }}(2\beta )^{-1})]<1\), we see that (iii) holds with \(\eta =2\beta \). As a consequence, from Lemma 5.2, Eq. (5.20) has a unique solution \(Z(s,\xi ,\varepsilon )\in C^0_\beta \) such that

$$\begin{aligned} \Vert Z(\cdot ,\xi ,\varepsilon )\Vert _\beta \le 2k_\beta (N+\varepsilon _0{\bar{g}}_\varepsilon )<R. \end{aligned}$$
(5.39)

So, \(s_1)\) holds. The Lipschitz-continuity of the map \((\xi ,\varepsilon )\mapsto Z(s,\xi ,\varepsilon )\in C^0_{2\beta }\) (i.e. \(s_2)\)) follows from Lemma 5.4 with \(\eta =2\beta \). Indeed the first inequality in (5.15) follows from (5.28), (5.32) taking \(L_F=(L_zL_{f'}+\varepsilon _0C_{1,\beta })R\). As for the second inequality in (5.15) we have, using (5.14) and (5.16):

$$\begin{aligned}&\Vert G(z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)-G(z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _1)\Vert _\beta \nonumber \\&\quad \le \Vert G(z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)-G(z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _2)\Vert _\beta \nonumber \\&\qquad + \Vert G(z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _2)-G(z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _1)\Vert _\beta \nonumber \\&\quad \le L_g(1+{\bar{\omega }}\beta ^{-1})\Vert z(\cdot ,\xi _1,\varepsilon _1)-z(\cdot ,\xi _2,\varepsilon _2)\Vert _\beta + L_g|\varepsilon _1-\varepsilon _2|. \end{aligned}$$
(5.40)

Next, using (5.38) we first get:

$$\begin{aligned}&\Vert \varepsilon _2 G_\varepsilon (z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)-\varepsilon _1 G_\varepsilon (z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _1)\Vert _\beta \nonumber \\&\quad \le |\varepsilon _2-\varepsilon _1|\Vert G_\varepsilon (z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)\Vert _\beta \nonumber \\&\qquad + |\varepsilon _1|\Vert G_\varepsilon (z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)-G_\varepsilon (z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _1)\Vert _\beta \nonumber \\&\quad \le {\bar{g}}_\varepsilon |\varepsilon _2-\varepsilon _1| + |\varepsilon _1|\Vert G_\varepsilon (z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)-G_\varepsilon (z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _1)\Vert _\beta \end{aligned}$$
(5.41)

and from (5.21) and (5.17) it follows immediately

$$\begin{aligned}&\Vert G_\varepsilon (z_1(\cdot ),\varepsilon _1)-G_\varepsilon (z_2(\cdot ),\varepsilon _2)\Vert _\beta \le {\bar{g}}_{t\varepsilon }\left\| \int _0^s {\bar{\omega }} |z_1(\tau )-z_2(\tau )| d\tau \right\| _\beta \nonumber \\&\qquad + {\bar{g}}_{\varepsilon z}\Vert z_1(\cdot )-z_2(s)\Vert _\beta +{\bar{g}}_{\varepsilon \varepsilon }|\varepsilon _1-\varepsilon _2| \nonumber \\&\quad \le ({\bar{g}}_{t\varepsilon }{\bar{\omega }}\beta ^{-1}+{\bar{g}}_{\varepsilon z})\Vert z_1(\cdot )-z_2(\cdot )\Vert _\beta +{\bar{g}}_{\varepsilon \varepsilon }|\varepsilon _1-\varepsilon _2| \end{aligned}$$
(5.42)

when \(|z_1(s)|, |z_2(s)|<\rho \) and hence, using the Lipschitz continuity of \(z(\cdot ,\xi ,\varepsilon )\):

$$\begin{aligned} \begin{array}{l} \Vert G_\varepsilon (z(\cdot ,\xi _1,\varepsilon _1),\varepsilon _1)-G_\varepsilon (z(\cdot ,\xi _2,\varepsilon _2),\varepsilon _2)\Vert _{\beta } \le \\ \le ({\bar{g}}_{t\varepsilon }{\bar{\omega }}\beta ^{-1}+{\bar{g}}_{z\varepsilon }) L_z[|\xi _1-\xi _2|+|\varepsilon _1-\varepsilon _2|]+{\bar{g}}_{\varepsilon \varepsilon }|\varepsilon _1-\varepsilon _2| \\ \le c_{2,\beta }[|\xi _1-\xi _2|+|\varepsilon _1-\varepsilon _2|] \end{array} \end{aligned}$$
(5.43)

with \(c_{2,\beta }=({\bar{g}}_{t\varepsilon }{\bar{\omega }}\beta ^{-1}+\bar{g}_{z\varepsilon })L_z+{\bar{g}}_{\varepsilon \varepsilon }\). Putting (5.40), (5.41), (5.43) together we see that \(G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )+\varepsilon G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon )\) satisfies the condition of Lemma 5.4 under the \(\beta \)-norm (and hence also under the \(2\beta \)-norm).

As a consequence the map \((\xi ,\varepsilon )\mapsto Z(s,\xi ,\varepsilon )\in C^0_{2\beta }\) is Lipschitz continuous. Finally we prove that \(s_3)\) holds using again Lemma 5.2. Let \(|\varepsilon |<\varepsilon _0\) be fixed and \({\tilde{\varepsilon }}\) be such that \(|\varepsilon +{\tilde{\varepsilon }}|<\varepsilon _0\). Then

$$\begin{aligned} w(s):=z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})-z(s,\xi ,\varepsilon )-Z(s,\xi ,\varepsilon ){\tilde{\varepsilon }} \end{aligned}$$

satisfies the equation:

$$\begin{aligned}&w=K\Big [F(z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }}))-F(z(\cdot ,\xi ,\varepsilon ))-F'(z(\cdot ,\xi ,\varepsilon ))Z(s,\xi ,\varepsilon ){\tilde{\varepsilon }} \\&\qquad +\,\varepsilon [G(z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }}),\varepsilon +{\tilde{\varepsilon }})-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon ) \\&\qquad -\,G_z(z(\cdot ,\xi ,\varepsilon ),\varepsilon )Z(s,\xi ,\varepsilon ){\tilde{\varepsilon }} - G_\varepsilon (z(\cdot ,\xi ,\varepsilon ),\varepsilon ){\tilde{\varepsilon }}] \\&\qquad +\,[G(z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }}),\varepsilon +{\tilde{\varepsilon }})-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )]{\tilde{\varepsilon }} \Big ] \\&\quad = K\Big \{ [F_1(\xi ,{\tilde{\varepsilon }})+\varepsilon G_1(\xi ,{\tilde{\varepsilon }})]w + [F_2(\cdot ,{\tilde{\varepsilon }}) +\varepsilon G_2(\cdot ,\xi ,{\tilde{\varepsilon }})]{\tilde{\varepsilon }} \\&\qquad +\,[G(z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }}),\varepsilon +{\tilde{\varepsilon }})-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )]{\tilde{\varepsilon }} \Big \} \end{aligned}$$

where now

$$\begin{aligned}&[F_1(\xi ,{\tilde{\varepsilon }})w](s)=\int _0^1 {\tilde{f}}_z(s,\theta z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})+(1-\theta )z(s,\xi ,\varepsilon ))d\theta w(s) \\&F_2(s,{\tilde{\varepsilon }})= \int _0^1 {\tilde{f}}_z(s,\theta z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})+(1-\theta )z(s,\xi ,\varepsilon ))- {\tilde{f}}_z(s,z(s,\xi ,\varepsilon ))d\theta Z(s,\xi ,\varepsilon ) \\&[G_1(\xi ,{\tilde{\varepsilon }})w](s) = \int _0^1 G_z(\theta z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})+(1-\theta )z(s,\xi ,\varepsilon ),\varepsilon )w(s)\, d\theta \\&G_2(s,\xi ,{\tilde{\varepsilon }}) = \int _0^1 [G_z(\theta z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})+(1-\theta )z(s,\xi ,\varepsilon ),\varepsilon +\theta {\tilde{\varepsilon }})- G_z(z(s,\xi ,\varepsilon ),\varepsilon )]Z(s,\xi ,\varepsilon ) d\theta \\&\qquad \qquad + \int _0^1 G_\varepsilon (\theta z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})+(1-\theta )z(s,\xi ,\varepsilon ),\varepsilon +\theta {\tilde{\varepsilon }})- G_\varepsilon (z(s,\xi ,\varepsilon ),\varepsilon ) d\theta . \end{aligned}$$

So, we apply Lemma 5.2 with \(S=0\), \([F_1(\xi ,{\tilde{\varepsilon }})+\varepsilon G_1(\xi ,{\tilde{\varepsilon }})]w\) replacing F(w) (here we use the variable w instead of z) and

$$\begin{aligned} \, [F_2(\cdot ,{\tilde{\varepsilon }}) + G_2(\cdot ,\xi ,{\tilde{\varepsilon }})+G(z(\cdot ,\xi , \varepsilon +{\tilde{\varepsilon }}),\varepsilon +{\tilde{\varepsilon }})-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )]{\tilde{\varepsilon }} \end{aligned}$$

replacing \(G(w,\nu )\) which is now independent of w.

Since \(|\theta z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})+(1-\theta )z(s,\xi ,\varepsilon )|\le \theta \rho +(1-\theta )\rho =\rho \), using (5.8) and the Lipschitz-continuity of \(f'(z)\) and \(z(\cdot ,\xi ,\varepsilon )\in C^0_\beta \) we see that:

$$\begin{aligned}\begin{array}{l} \Vert F_1(\xi ,\varepsilon )w\Vert _{2\beta } \le \Delta (\rho )\Vert w\Vert _{2\beta } \\ \Vert F_2(\cdot ,\varepsilon )\Vert _{2\beta }\le \frac{1}{2}L_{f'}L_z R|{\tilde{\varepsilon }}| \end{array} \end{aligned}$$

(recall that \( \Vert Z(\cdot ,\xi ,\varepsilon )\Vert _\beta <R\)). Next, from (5.24) with \(2\beta \) instead of \(\beta \), we get

$$\begin{aligned} \Vert G_1(\xi ,{\tilde{\varepsilon }})w(\cdot )\Vert _{2\beta }\le [N_2+N_1{\bar{\omega }}(2\beta )^{-1}] \Vert w\Vert _{2\beta } \end{aligned}$$

Similarly, from (5.31), (5.39), (5.42) and the Lipschitz continuity of \(z(\cdot ,\xi ,\varepsilon )\) with respect to the \(\beta \)-norm we get

$$\begin{aligned}&\Vert G_2(\cdot ,\xi ,\varepsilon )\Vert _{2\beta }\le \int _0^1 C_{1,\beta }\theta [\Vert z(s,\xi ,\varepsilon +{\tilde{\varepsilon }})-z(s,\xi ,\varepsilon )\Vert _\beta +|{\tilde{\varepsilon }}|] Rd\theta \\&\qquad +\int _0^1 \theta [({\bar{g}}_{t\varepsilon }{\bar{\omega }}\beta ^{-1}+{\bar{g}}_{z\varepsilon })\Vert z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }}-z(\cdot ,\xi ,\varepsilon )\Vert _\beta + {\bar{g}}_{\varepsilon \varepsilon }|{\tilde{\varepsilon }}|] d\theta R \\&\qquad \qquad \le \frac{1}{2}\left[ C_{1,\beta }(L_z+1)| + (\bar{g}_{t\varepsilon }{\bar{\omega }}\beta ^{-1}+{\bar{g}}_{z\varepsilon })L_z+{\bar{g}}_{\varepsilon \varepsilon }\right] R|{\tilde{\varepsilon }}| \end{aligned}$$

Finally, from (5.40) and the Lipschitz continuity of \(z(\cdot ,\xi ,\varepsilon )\) in the \(\beta \)-norm, we get

$$\begin{aligned}&\Vert G(z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }}),\varepsilon +{\tilde{\varepsilon }})-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )\Vert _\beta \\&\quad \le L_g(1+{\bar{\omega }}\beta ^{-1})\Vert z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }})-z(\cdot ,\xi ,\varepsilon )\Vert _\beta + L_g|{\tilde{\varepsilon }}| \le {\tilde{C}}|{\tilde{\varepsilon }}|. \end{aligned}$$

and then the same holds for \(\Vert G(z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }}),\varepsilon +{\tilde{\varepsilon }})-G(z(\cdot ,\xi ,\varepsilon ),\varepsilon )\Vert _{2\beta }\). From Lemma 5.2 we get then

$$\begin{aligned} \Vert z(\cdot ,\xi ,\varepsilon +{\tilde{\varepsilon }})-z(\cdot ,\xi ,\varepsilon )-Z(\cdot ,\xi ,\varepsilon ){\tilde{\varepsilon }}\Vert _{2\beta } \le C|{\tilde{\varepsilon }}|^2 \end{aligned}$$

for some positive constant C, provided \(\Delta (\rho )+\varepsilon _0(N_2+N_1{\bar{\omega }}(2\beta )^{-1})<1\), proving \(s_3)\).

More arguments of similar nature prove that the conclusion of Theorem 2.3 holds for any \(m\le r\).

Appendix B: Roughness of Fredholm Maps

In this “Appendix” we prove the result stated in Remark 4.3, which is in general known [8] but we present its proof for the reader convenience to get the corresponding estimates.

Lemma 6.1

Let X and Y be Banach spaces and \(A\in L(X,Y)\) be a linear Fredholm map. If \(B\in L(X,Y)\) is near to 0, then \(A+B\) is Fredholm with \({{\,\mathrm{ind}\,}}(A+B)={{\,\mathrm{ind}\,}}A\) and \(\dim {{\mathcal {N}}}A\ge \dim {{\mathcal {N}}}(A+B)\).

Proof

We know that there are closed linear subspaces \(Y_1\) and \(X_2\) such that \(X={{\mathcal {N}}}A\oplus X_2\) and \(Y=Y_1\oplus {{\mathcal {R}}}A\). Let \(S:Y\rightarrow Y\) be the projection with \({{\mathcal {R}}}S={{\mathcal {R}}}A\) and \({{\mathcal {N}}}S=Y_1\). We solve \((A+B)x=y\). We set \(x=x_1+x_2\) for \(x_1\in {{\mathcal {N}}}A\), \(x_2\in X_2\) and \(y=y_1+y_2\) for \(y_1\in Y_1\), \(y_2\in {{\mathcal {R}}}A\). Then equation \((A+B)x=y\) is equivalent to

$$\begin{aligned} (SA+SB)x_2=y_2-SBx_1 \end{aligned}$$
(6.1)

and

$$\begin{aligned} ({\mathbb {I}}-S)B(x_1+x_2)=y_1. \end{aligned}$$
(6.2)

Since \(SA:X_2\rightarrow {{\mathcal {R}}}A\) is invertible, then \((SA+SB):X_2\rightarrow {{\mathcal {R}}}A\) is invertible as well for \(\Vert B\Vert \) small, i.e., \(\Vert S\Vert \Vert B\Vert \Vert (SA/X_2)^{-1}\Vert <1\) by the Neumann lemma [8]. So (6.1) gives

$$\begin{aligned} x_2=(SA+SB)^{-1}(y_2-SBx_1). \end{aligned}$$
(6.3)

Plugging (6.3) into (6.2), we obtain

$$\begin{aligned} Cx_1=y_1-({\mathbb {I}}-S)B(SA+SB)^{-1}y_2 \end{aligned}$$
(6.4)

for \(C:{{\mathcal {N}}}A\rightarrow Y_1\) given by

$$\begin{aligned} C=({\mathbb {I}}-S)B\left( {\mathbb {I}}-(SA+SB)^{-1}SB\right) . \end{aligned}$$

Taking \(y=0\) in (6.3) and (6.4), we get

$$\begin{aligned} \dim {{\mathcal {N}}}(A+B)=\dim {{\mathcal {N}}}C\le \dim {{\mathcal {N}}}A. \end{aligned}$$

Next, (6.4) implies that \(y\in {{\mathcal {R}}}(A+B)\) if and only if

$$\begin{aligned} y_1=w+({\mathbb {I}}-S)B(SA+SB)^{-1}y_2,\quad w\in {{\mathcal {R}}}C, \end{aligned}$$

i.e.,

$$\begin{aligned} y=w+({\mathbb {I}}-S)B(SA+SB)^{-1}y_2+y_2. \end{aligned}$$

On the other hand, we see that a linear mapping

$$\begin{aligned} K:Y_1\times {{\mathcal {R}}}A\rightarrow Y_1\times {{\mathcal {R}}}A \end{aligned}$$

given by

$$\begin{aligned} K(y_1,y_2)=y_1+({\mathbb {I}}-S)B(SA+SB)^{-1}y_2+y_2, \end{aligned}$$

or as \(K:Y\rightarrow Y\)

$$\begin{aligned} K(y)=y+({\mathbb {I}}-S)B(SA+SB)^{-1}Sy \end{aligned}$$

is invertible for \(\Vert B\Vert \) small, i.e., \(\Vert ({\mathbb {I}}-S)\Vert \Vert B\Vert \Vert (SA+SB)^{-1}\Vert \Vert S\Vert <1\). Thus

$$\begin{aligned} {{\mathcal {R}}}(A+B)=K({{\mathcal {R}}}C\times {{\mathcal {R}}}A), \end{aligned}$$

and so \({{\mathcal {R}}}(A+B)\) is closed with

$$\begin{aligned} {{\,\mathrm{codim}\,}}{{\mathcal {R}}}(A+B)={{\,\mathrm{codim}\,}}{{\mathcal {R}}}C. \end{aligned}$$

Consequently, we obtain

$$\begin{aligned}&{{\,\mathrm{ind}\,}}(A+B)=\dim {{\mathcal {N}}}(A+B)-{{\,\mathrm{codim}\,}}{{\mathcal {R}}}(A+B)\\&\quad =\dim {{\mathcal {N}}}C-{{\,\mathrm{codim}\,}}{{\mathcal {R}}}C=\dim {{\mathcal {N}}}C+\dim {{\mathcal {R}}}C-{{\,\mathrm{codim}\,}}{{\mathcal {R}}}A\\&\quad =\dim {{\mathcal {N}}}A-{{\,\mathrm{codim}\,}}{{\mathcal {R}}}A={{\,\mathrm{ind}\,}}A. \end{aligned}$$

The proof is complete. \(\square \)

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Battelli, F., Fečkan, M. General Melnikov Approach to Implicit ODE’s. J Dyn Diff Equat (2020). https://doi.org/10.1007/s10884-020-09859-y

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Keywords

  • Implicit differential equations
  • Homoclinic solutions
  • Melnikov conditions
  • Persistence