Path-Dependent SDEs in Hilbert Spaces

Abstract

We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove the Gâteaux differentiability of generic order n of mild solutions with respect to the starting point and the continuity of the Gâteaux derivatives with respect to all the data.

This research has been partially supported by the ERC 321111 Rofirm.

4 Appendix

Proof of Proposition 2.1 Suppose that the derivatives $\partial ^j_{x_1\ldots x_j}f(u)$ exists for all $u\in U$, $x_1,\ldots ,x_j\in X_0$, $j=1,\ldots ,n$, separately continuous in $u,x_1,\ldots , x_j$. We want to show that $f\in \mathscr {G}^{n}(U,Y;X_0)$.

We proceed by induction on n. Let $n=1$. Since $\partial _xf(u)$ is continuous in u, for all $x\in X_0$, we have that $X_0\rightarrow Y,\ x\mapsto \partial _xf(u)$ is linear ([11, Lemma 4.1.5]). By assumption, it is also continuous. Hence $x \mapsto \partial _x f(u)\in L(X_0,Y)$ for all $u\in U$. This shows the existence of $\partial _{X_0}f$. The continuity of $U\rightarrow L_s(X_0,Y),\ u\mapsto \partial _{X_0} f(u)$, comes from the separate continuity of (2.1) and from the definition of the locally convex topology on $L_s(X_0,Y)$. This shows that $f\in \mathscr {G}^{1}(U,Y;X_0)$.

Let now $n>1$. By inductive hypothesis, we may assume that $f\in \mathscr {G}^{n-1}(U,Y;X_0)$ and

$$ \partial _{X_0 }^{j}f(u).(x_1,\ldots ,x_j)=\partial _{x_1\ldots x_j}^jf(u)\qquad \forall u\in U, \ \forall j=1,\ldots ,n-1, \ \forall (x_1,\ldots , x_j)\in X_0^j. $$

Let $x_n\in X_0$. The limit

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{ \partial ^{n-1}_{X_0 }f(u+tx_n)- \partial ^{n-1}_{X_0 }f(u)}{t}=\Lambda \end{aligned}$$

(4.1)

exists in $L_s^{(n-1)}(X_0^{n-1},Y)$ if and only if $\Lambda \in L_s^{(n-1)}(X_0^{n-1},Y)$ and, for all $x_1,\ldots ,x_{n-1}\in X_0$, the limit

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{\partial ^{n-1}_{x_1\ldots x_{n-1}}f(u+tx_n)-\partial ^{n-1}_{x_1\ldots x_{n-1}}f(u)}{t}=\Lambda (x_1,\ldots ,x_{n-1}) \end{aligned}$$

(4.2)

holds in Y. By assumption, the limit (4.2) is equal to $\partial ^n_{x_1\ldots x_{n-1}x_n}f(u)$, for all $x_1,\ldots ,x_{n-1}$. Since, by assumption, $\partial ^n_{x_1\ldots x_{n-1}x_n}f(u)$ is separately continuous in $u,x_1,\ldots ,x_{n-1},x_n$, we have that the limit (4.1) exists in $L_s^{(n-1)}(X_0^{n-1},Y)$ and is given by

$$ \partial _{x_n} \partial ^{n-1} _{X_0}f(u).(x_1,\ldots ,x_{n-1})=\Lambda (x_1,\ldots ,x_{n-1}) =\partial ^n_{x_1\ldots x_{n-1}x_n}f(u)\qquad \forall x_1,\ldots ,x_{n-1}\in X_0. $$

Since u and $x_n$ were arbitrary, we have proved that $\partial _{x_n} \partial ^{n-1}_{X_0}f(u)$ exists for all u, $x_n$. Moreover, for all $x_1,\ldots ,x_n\in X_0$, the function

$$ U\rightarrow Y,\ u\mapsto \partial _{x_n} \partial ^{n-1}_{X_0}f(u).(x_1,\ldots ,x_{n-1}) =\partial _{x_n}\partial ^{n}_{x_1\ldots x_{n-1}}f(u) $$

is continuous, by separate continuity of (2.1). Then $\partial ^{n}_{x_1\ldots x_{n-1}x_n}f(u)$ is linear in $x_n$. The continuity of

$$\begin{aligned} X_0\rightarrow L_s^{(n-1)}(X_0^{n-1},Y),\ x\mapsto \partial _x \partial _{X_0 }^{n-1}f(u) \end{aligned}$$

(4.3)

comes from the continuity of $\partial ^{n}_{x_1\ldots x_{n-1}x}f(u)$ in each variable, separately. Hence (4.3) belongs to $L_s(X_0,L_s^{n-1}(X_0^{n-1},Y))$ for all $u\in U$. This shows that $\partial _{X_0 }^{n-1}f$ is Gâteaux differentiable with respect to $X_0$ and that

$$\begin{aligned} \partial _{X_0}^nf(u).(x_1,\ldots ,x_n) = \partial ^n_{x_1\ldots x_n}f(u) \qquad \forall u\in U, \ \forall x_1,\ldots ,x_n\in X_0, \end{aligned}$$

and shows also the continuity of

$$ U\rightarrow L^{(n)}_s(X_0^n,Y),\ u \mapsto \partial _{X_0}^nf(u), $$

due to the continuity of the derivatives of f, separately in each direction. Then we have proved that $f\in \mathscr {G}^{n}(U,Y;X_0)$ and that (2.2) holds.

Now suppose that $f\in \mathscr {G}^{n}(U,Y;X_0)$. By the very definition of $\partial _{X_0 }f$, $\partial _xf(u)$ exists for all $x\in X_0$ and $u\in U$, it is separately continuous in u, x, and coincides with $\partial _{X_0 }f(u).x$. By induction, assume that $\partial ^{n-1}_{x_1\ldots x_{n-1}}f(u)$ exists and that

$$\begin{aligned} \partial ^{n-1}_{X_0}f(u).(x_1,\ldots ,x_{n-1})=\partial ^{n-1}_{x_1\ldots x_{n-1}}f(u)\qquad \forall u\in U,\ \forall x_1,\ldots ,x_{n-1}\in X_0. \end{aligned}$$

(4.4)

Since $\partial _{X_0 }^{n-1}f(u)$ is Gâteaux differentiable, the directional derivative $\partial _{x_n} \partial ^{n-1}_{X_0 }f(u)$ exists. Hence, by (4.4), the derivative $\partial ^n_{x_1\ldots x_{n-1}x_n}f(u)$ exists for all $x_1,\ldots ,x_{n-1},x_n\in X_0$. The continuity of $\partial ^n_{x_1\ldots x_{n-1}x_n}f(u)$ with respect to u comes from the continuity of $\partial _{X_0 }^nf$. The continuity of $\partial ^n_{x_1\ldots x_{j}\ldots x_n}f(u)$ with respect to $x_j$ comes from the fact that, for all $x_{j+1},\ldots ,x_n\in X_0$, $u\in U$,

$$ X_0^j\rightarrow Y,\ (x_1',\ldots ,x_j') \mapsto \partial ^n_{X_0 }f(u).( x_1',\ldots ,x_j',x_{j+1},\ldots ,x_{n}) $$

belongs to $L^{(j)}_s(X_0^j,Y)$. $\blacksquare$

Proof of Theorem 2.9 The proof is by induction on n. The case $n=1$ is provided by Proposition 2.7.

Let $n\ge 2$. Clearly, it is sufficient to prove that $\varphi \in \mathscr {G}^{n}(U,Y_n)$ and that (2.11) holds true for $j=n$. Since we are assuming that the theorem holds true for $n-1$, we can apply it with the data

$$ \widetilde{h}_1:U\times \widetilde{Y}_2\rightarrow \widetilde{Y}_2,\ \ldots , \widetilde{h}_{n-1}:U\times \widetilde{Y}_n\rightarrow \widetilde{Y}_n, $$

where $\widetilde{h}_k:=h_{k+1}$, $\widetilde{Y}_k:=Y_{k+1}$, for $k=1,\ldots ,n-1$. According to the claim, the fixed-point function $\widetilde{\varphi }$ of $\widetilde{h}_1$ belongs to $\mathscr {G}^{j}(U,\widetilde{Y}_{(n-1)-j+1})$, for $j=1,\ldots ,n-1$, and formula (2.11) holds true for $\widetilde{\varphi }$ and $j=1,\ldots ,n-1$. Since $\varphi (u)=(i_{2,1}\circ \widetilde{\varphi }) (u)$, for $u\in U$, we have $\varphi \in \mathscr {G}^{j}(U,\widetilde{Y}_{n-j})=\mathscr {G}^{j}(U,Y_{n-j+1})$, for $j=1,\ldots ,n-1$, and

$$ \partial ^j_{x_1\ldots x_j}\varphi (u)=\partial ^j_{x_1\ldots x_j}\widetilde{\varphi }(u)\in \widetilde{Y}_{n-j}=Y_{n-j+1}, \qquad \forall u\in U, \ \forall x_1,\ldots ,x_j\in X. $$

Then (2.11) holds true for $\varphi$ up to order $j=n-1$. In particular $\varphi \in \mathscr {G}^{n-1}(U,Y_2)$, hence, for $x_1,\ldots ,x_n\in X$, $\epsilon >0$, we can write

$$\begin{aligned} \begin{aligned}&\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n) - \partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u) \\&= \left( \partial _{Y_1}h_1(u+\varepsilon x_n,\varphi (u+\varepsilon x_n)).\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)- \partial _{Y_1}h_1 (u,\varphi (u)).\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)\right) \\&{=:} + \left( \mathscr {S}(u+\varepsilon x_n)- \mathscr {S}(u)\right) \\&=:\mathbf {I}+\mathbf {II}, \end{aligned} \end{aligned}$$

(4.5)

where $\mathscr {S}(\cdot )$ denotes the sum

$$ \mathscr {S}(v):=\partial ^{n-1}_{x_1\ldots x_{n-1}}h_1(v,\varphi (v)) + \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_{n-1}\}} \\ \mathbf {x}\ne \emptyset \end{array}}\sum _{i=\max \{1,2-(n-1)+|\mathbf {x}|\}}^{|\mathbf {x}|}\sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_{1},\ldots ,\mathbf {p}_i) \end{array}} \partial ^{n-1} [\mathbf {x}^c,\mathbf {p}]h_1(v,\varphi (v)), $$

for $v\in U$. By recalling that $\varphi \in \mathscr {G}^{j}(U,Y_{n-j+1})$, $j=1,\ldots ,n-1$, hence by taking into account with respect to which space the derivatives of $\varphi$ are continuous, we write

$$\begin{aligned} \mathbf {I}=&\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)}h_1(u+\varepsilon x_n,\varphi (u+\varepsilon x_n))- \partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)) \nonumber \\ =&\int _0^1 \partial _{x_n}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)}h_1(u+\theta \varepsilon x_n,\varphi (u+\varepsilon x_n))\varepsilon {d} \theta \nonumber \\&+\int _0^1 \partial _{\frac{\varphi (u+\varepsilon x_n)-\varphi (u)}{\varepsilon }}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)}h_1(u,\varphi (u)+\theta (\varphi (u+\varepsilon x_n)-\varphi (u)))\varepsilon {d} \theta \nonumber \\&+ \partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)- {\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)} }h_1(u,\varphi (u)) \nonumber \\ =&\mathbf {I_1}+\mathbf {I_2}+ \partial _{Y_1}h_1(u,\varphi (u)).\left( \partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)- {\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}\right) ,\nonumber \\ \end{aligned}$$

(4.6)

with (⁵)

$$ \lim _{\varepsilon \rightarrow 0}\frac{\mathbf {I_1}}{\varepsilon }=\partial _{x_n}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)) \qquad \text { and } \qquad \lim _{\varepsilon \rightarrow 0}\frac{\mathbf {I_2}}{\varepsilon }=\partial _{\partial _{x_n}\varphi (u)}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)) . $$

In a similar way,

$$\begin{aligned} \begin{aligned} \lim _{\varepsilon \rightarrow 0}&\frac{\mathbf {II}}{\varepsilon }= \partial _{x_n}\partial ^{n-1}_{x_1\ldots x_{n-1}}h_1(u,\varphi (u)) +\partial _{\partial _{x_n}\varphi (u)}\partial ^{n-1}_{x_1 \ldots x_{n-1}}h_1(u,\varphi (u))\\&+\sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_{n-1}\}}\\ \mathbf {x}\ne \emptyset \end{array}} \sum _{i=\max \{1,2-(n-1)+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i) \end{array}} \partial _{x_n} \partial ^{n-1} [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u))\\&+ \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_{n-1}\}}\\ \mathbf {x}\ne \emptyset \end{array}} \sum _{i=\max \{1,2-(n-1)+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {p})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i) \end{array}} \Bigg( \partial _{\partial _{x_n}\varphi (u)} \partial ^{n-1} [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u)) \\&+ \sum _{j=1}^i\partial _{\mathbf {x} ^c}^{|\mathbf {x}^c|}\partial _{\partial ^{| \mathbf {p}_1|}_{\mathbf {p}_1}\varphi (u)} \ldots \partial _{\partial ^{|\mathbf {p}_{j-1} |}_{\mathbf {p}_{j-1}}\varphi (u)} \partial _{\partial _{x_n}\partial ^{| \mathbf {p}_j |}_{ \mathbf {p}_j }\varphi (u)} \partial _{\partial ^{|\mathbf {p}_{j+1}|}_{\mathbf {p} _{j+1}}\varphi (u)} \ldots \partial _{\partial ^{|\mathbf {p}_i|}_{\mathbf {p}_i }\varphi (u)} h_1(u,\varphi (u)) \Bigg) . \end{aligned} \end{aligned}$$

(4.7)

Notice that

$$\begin{aligned} \begin{aligned}&\sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_{n-1}\}} \mathbf {x}\ne \emptyset \end{array}} \sum _{i=\max \{1,2-(n-1)+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} p_\pi \in P^i(\mathbf {x})\\ \mathbf {p}= ( \mathbf {p}_1,\ldots ,\mathbf {p}_i ) \end{array}} \partial _{x_n} \partial ^{n-1} [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u))\\&= \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_n\}}\\ \mathbf {x}\ne \emptyset \\ x_n\not \in \mathbf {x} \end{array}} \sum _{i=\max \{1,2-n+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i) \end{array}} \partial ^n [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u)) -\partial _{x_n} \partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)) \end{aligned} \end{aligned}$$

(4.8)

and

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_{n-1}\}}\\ \mathbf {x}\ne \emptyset \end{array}}&\sum _{i=\max \{1,2-(n-1)+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i) \end{array}} \partial _{\partial _{x_n}\varphi (u)}\partial ^{n-1} [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u))\\&= \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_n\}} \\ x_n\in \mathbf {x}\\ \mathbf {x}\ne \{x_n\} \end{array}} \sum _{i=\max \{1,2-n+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i)\\ \{x_n\}\in \mathbf {p} \end{array}} \partial ^n [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u))\\&{=} -\partial _{\partial _{x_n}\varphi (u)}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)) \end{aligned} \end{aligned}$$

(4.9)

and

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_{n-1}\}}\\ \mathbf {x} \ne \emptyset \end{array}}&\sum _{i=\max \{1,2-(n-1)+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i ) \end{array}} \sum _{j=1}^i L(\mathbf {p},j;u) \\&= \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_n\}}\\ x_n\in \mathbf {x}\\ \mathbf {x}\ne \{x_n\} \end{array}} \sum _{i=\max \{1,2-n+|\mathbf {x}|\}}^{|\mathbf {x}|} \sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i)\\ \{x_n\}\not \in \mathbf {p} \end{array}} \partial ^n [\mathbf {x}^c,\mathbf {p} ]h_1(u,\varphi (u)) \end{aligned} \end{aligned}$$

(4.10)

where

$$ L(\mathbf {p},j;u) :=\partial _{\mathbf {x} ^c}^{|\mathbf {x}^c|} \partial ^{|\mathbf {x}|} _{ \partial ^{|\mathbf {p}_1|}_{\mathbf {p} _1}\varphi (u) \ldots \partial ^{|\mathbf {p}_{j-1}|}_{ \mathbf {p}_{j-1} \varphi (u)} \partial _{x_n}\partial ^{|\mathbf {p}_j |}_{\mathbf {p}_j}\varphi (u) \partial ^{|\mathbf {p}_{j+1} |}_{\mathbf {p}_{j+1}}\varphi (u) \ldots \partial ^{|\mathbf {p}_i|}_{\mathbf {p}_i }\varphi (u)} h_1(u,\varphi (u)). $$

By collecting (4.7), (4.8), (4.9), (4.10), we obtain

$$\begin{aligned} \begin{aligned} \lim _{\varepsilon \rightarrow 0}\frac{\mathbf {II}}{\varepsilon }=&\, \partial _{\partial _{x_n}\varphi (u)}\partial ^{n-1}_{x_1\ldots x_{n-1}}h_1(u,\varphi (u)) +\partial ^n_{x_1\ldots x_{n}}h_1(u,\varphi (u)) -\partial _{\partial _{x_n}\varphi (u)}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)) \\&+ \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1\ldots x_n\}} \\ \mathbf {x}\ne \emptyset \\ \mathbf {x}\ne \{x_n\} \end{array}}\sum _{i=\max \{1,2-n+|\mathbf {x}|\}}^{|\mathbf {x}|}\sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i) \end{array}} \partial ^n [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u)) -\partial _{x_n}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u))\\ =&\partial ^n_{x_1\ldots x_{n}}h_1(u,\varphi (u)) -\partial _{\partial _{x_n}\varphi (u)}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)) \\&+\sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_n\}} \\ \mathbf {x}\ne \emptyset \end{array}}\sum _{i=\max \{1,2-n+|\mathbf {x}|\}}^{|\mathbf {x}|}\sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i) \end{array}} \partial ^n [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u)) -\partial _{x_n}\partial _{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)}h_1(u,\varphi (u)). \end{aligned} \end{aligned}$$

Hence

$$ \lim _{\varepsilon \rightarrow 0}\left( \frac{\mathbf {I_1}}{\varepsilon }+ \frac{\mathbf {I_2}}{\varepsilon }+ \frac{\mathbf {II}}{\varepsilon }\right) = \sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_n\}} \\ \mathbf {x}\ne \emptyset \end{array}}\sum _{i=\max \{1,2-n+|\mathbf {x}|\}}^{|\mathbf {x}|}\sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots ,\mathbf {p}_i) \end{array}} \partial ^n [\mathbf {x}^c,\mathbf {p}] h_1(u,\varphi (u)) +\partial ^n_{x_1\ldots x_{n}}h_1(u,\varphi (u)), $$

and, by recalling (4.5), (4.6), we obtain

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0} \left( I- \partial _{Y_1}h_1(u,\varphi (u))\right) .\frac{\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u+\varepsilon x_n)- {\partial ^{n-1}_{x_1\ldots x_{n-1}}\varphi (u)} }{\varepsilon } \\&\qquad \qquad =\sum _{\begin{array}{c} \mathbf {x}\in 2^{\{x_1,\ldots ,x_n\}} \\ \mathbf {x}\ne \emptyset \end{array}}\sum _{i=\max \{1,2-n+|\mathbf {x}|\}}^{| \mathbf {x}|}\sum _{\begin{array}{c} \mathbf {p}\in P^i(\mathbf {x})\\ \mathbf {p}=(\mathbf {p}_1,\ldots , \mathbf {p}_i) \end{array}} \partial ^n [\mathbf {x}^c,\mathbf {p}]h_1(u,\varphi (u)) +\partial ^n_{x_1\ldots x_{n}}h_1(u,\varphi (u)). \end{aligned}$$

Finally, we can conclude the proof by recalling that $I- \partial _{Y_1}h_1(u,\varphi (u))$ is invertible with strongly continuous inverse. $\blacksquare$