Let $(\mathfrak{B}_{1},\|\cdot\|_{1})$ and $(\mathfrak{B}_{2},\|\cdot\|_{2})$ be two Banach spaces, $\|\cdot\|_{E}$ be the standard Euclidean metric on $\mathbb{R}^{n}$, $U\subset\mathbb{R}^{n}\times\mathfrak{B}_{1}$ be an open set, and $\mathfrak{F}:U\longrightarrow\mathfrak{B}_{2}$ be a continuously differentiable map. Denote the differential

$$D\mathfrak{F}(y,\sigma)(\dot{y}+\dot{\sigma})=D_{y}\mathfrak{F}(y,\sigma)\dot{y}+D_{\sigma}\mathfrak{F}(y,\sigma)\dot{\sigma},$$

for $(y,\sigma)\in U$, $\dot{y}\in\mathbb{R}^{n}$ and $\dot{\sigma}\in\mathfrak{B}_{1}$. Assume that $(0,0)\in U$ satisfies that $D_{\sigma}\mathfrak{F}(0,0):\mathfrak{B}_{1}\longrightarrow\mathfrak{B}_{2}$ has a bounded linear inverse $D_{\sigma}\mathfrak{F}(0,0)^{-1}:\mathfrak{B}_{2}\longrightarrow\mathfrak{B}_{1}$ with

$$\|D_{\sigma}\mathfrak{F}(0,0)^{-1}\|\leq\overline{C}$$

for a constant $\overline{C}>0$. Let $r>0$, $\delta_{0}>\delta>0$ be constants such that, if $\|y_{0}\|_{E}<r,$ and $\|\sigma\|_{1}\leq\delta_{0}$, then $(y_{0},\sigma)\in U$, and

$$\|D_{\sigma}\mathfrak{F}(y_{0},\sigma)-D_{\sigma}\mathfrak{F}(0,0)\|\leq\frac{1}{2\overline{C}}\ \ {\rm and}\ \ \|\mathfrak{F}(y_{0},0)\|_{2}\leq\frac{\delta}{4\overline{C}}.$$

Then, for any $\|y\|_{E}<r$, there exists a unique $\sigma(y)\in\mathfrak{B}_{1}$ such that

$$\mathfrak{F}(y,\sigma(y))=0,\ \ \ \|\sigma(y)\|_{1}\leq\delta.$$

Furthermore,

$$D\sigma(y)\dot{y}=-D_{\sigma}\mathfrak{F}(y,\sigma)^{-1}D_{y}\mathfrak{F}(y,\sigma)\dot{y}.$$