The proof constructs the center manifold as a fixed point of a contraction mapping on a space of graphs. The strategy is: (1) cut off the nonlinearity outside a small ball to make it globally Lipschitz, (2) reformulate the invariance condition as an [integral](/page/Integral) equation using the variation-of-constants formula, (3) show the integral operator is a contraction on a suitable [Banach space](/page/Banach%20Space) of [functions](/page/Function), and (4) verify that the fixed point is a $C^k$ manifold tangent to $E^c$. **Step 1: Cutoff and global extension.** Let $\chi: \mathbb{R}^n \to [0, 1]$ be a smooth cutoff with $\chi(x) = 1$ for $|x| \le \delta$ and $\chi(x) = 0$ for $|x| \ge 2\delta$, where $\delta > 0$ is small. Replace the nonlinearities $f, g$ by $\tilde{f}(x, y) = \chi(x, y) f(x, y)$ and $\tilde{g}(x, y) = \chi(x, y) g(x, y)$. The modified system agrees with the original on $B(0, \delta)$, and $\tilde{f}, \tilde{g}$ are globally bounded with Lipschitz constant $L(\delta) \to 0$ as $\delta \to 0$. **Step 2: The graph space.** Consider the Banach space $\mathcal{X}_\eta := \{h: \mathbb{R}^c \to \mathbb{R}^s \mid h \text{ is Lipschitz}, \, h(0) = 0, \, \|h\|_\eta < \infty\}$ equipped with the norm $\|h\|_\eta := \sup_{x \neq 0} |h(x)|/(|x| \cdot e^{\eta|x|})$ for a suitable $\eta > 0$. The condition $Dh(0) = 0$ is enforced by the vanishing at the origin and the Lipschitz constraint. **Step 3: The integral equation.** A graph $y = h(x)$ is invariant under the flow if and only if for every solution $x(t)$ of the center equation $\dot{x} = Ax + \tilde{f}(x, h(x))$, the function $y(t) = h(x(t))$ solves $\dot{y} = By + \tilde{g}(x(t), h(x(t)))$. Since $B$ has eigenvalues with strictly negative real parts, the stable component is characterised by the requirement that it remains bounded as $t \to -\infty$. The variation-of-constants formula gives: \begin{align*} h(x_0) = \int_{-\infty}^0 e^{-Bs} \tilde{g}(x(s; x_0, h), h(x(s; x_0, h))) \, d\mathcal{L}^1(s), \end{align*} where $x(s; x_0, h)$ is the solution of $\dot{x} = Ax + \tilde{f}(x, h(x))$ at time $s$ with $x(0) = x_0$. This is a fixed-point equation $h = \mathcal{T}(h)$ for the operator $\mathcal{T}$ defined by the right-hand side. **Step 4: Contraction.** [claim:$\mathcal{T}$ Is A Contraction For Small $\delta$] For $\delta > 0$ sufficiently small, $\mathcal{T}: \mathcal{X}_\eta \to \mathcal{X}_\eta$ is a contraction with Lipschitz constant $< 1$. [/claim] [proof] The key estimates are: (a) $\|e^{Bs}\| \le M e^{-\alpha s}$ for $s \le 0$ (exponential decay, since $B$ has eigenvalues with negative real parts, with $\alpha > 0$ the spectral gap); (b) $\|\tilde{g}\|_{\mathrm{Lip}} \le L(\delta) \to 0$ as $\delta \to 0$. For two graphs $h_1, h_2 \in \mathcal{X}_\eta$, the difference $\mathcal{T}(h_1)(x_0) - \mathcal{T}(h_2)(x_0)$ involves both the difference in the integrand $\tilde{g}(\cdot, h_1) - \tilde{g}(\cdot, h_2)$ and the difference in the trajectories $x(s; x_0, h_1) - x(s; x_0, h_2)$. Both are controlled by $\|h_1 - h_2\|_\eta$ via Gronwall-type estimates, and the smallness of $L(\delta)$ together with the exponential decay of $e^{Bs}$ ensures the integral converges with a contracting prefactor. [/proof] **Step 5: Regularity.** The Banach fixed-point theorem gives a unique $h \in \mathcal{X}_\eta$ with $h = \mathcal{T}(h)$. The graph $W^c_{\mathrm{loc}} = \{(x, h(x)) : x \in \mathbb{R}^c\}$ is Lipschitz by construction. To obtain $C^k$ regularity, one differentiates the fixed-point equation $k$ times with respect to $x_0$, producing a [sequence](/page/Sequence) of linear fixed-point problems for $D^j h$ ($j = 1, \ldots, k$) that can be solved iteratively. Alternatively, one works in the Banach space of $C^k$ graphs from the start. The condition $h(0) = 0$ and $Dh(0) = 0$ follow from the integral representation and the fact that $\tilde{f}(0,0) = 0$, $\tilde{g}(0,0) = 0$, $D\tilde{f}(0) = 0$, $D\tilde{g}(0) = 0$. **Step 6: Non-uniqueness.** The cutoff $\chi$ depends on $\delta$, so different choices of $\delta$ may produce different center manifolds outside the ball $B(0, \delta)$. Within any fixed neighbourhood, the Taylor expansion of $h$ at the origin is unique (determined by the jet of $f, g$ at the origin), but the global manifold is not. This is why the Approximation Theorem guarantees agreement of Taylor expansions without asserting uniqueness of the manifold itself.