Define the seminorm $$[u]_{\alpha} = \sup_{(x,t), (y,s) \in Q} \frac{|u(x,t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}},$$ and the norm $$\lVert u\rVert_{{C}^{k, \alpha}(\overline{Q})} = \sum_{i+2j \leq k} \left\lVert \frac{\partial^{i+j}u}{\partial x^i \partial t^j}\right\rVert_{C(\overline{Q})} + \sum_{i+2j = k} \bigg[\frac{\partial^{i+j}u}{\partial x^i \partial t^j}\bigg]_\alpha$$ composed of the seminorm above and the usual sup norm on continuous functions.
I want to show that the map $f:C^{k+2, \alpha} \to C^{k, \alpha}$ defined by $$f(u) = u_t - g(x,t,u,u_x),$$ where $g$ is smooth in all its arguments, is Frechet differentiable at the point $v = g(x,t,0,0)t$. I know that the directional derivative of $f$ at $v$ is $$df(v)h = h_t - (g_z|_{v}h + g_p|_{v}h_x)$$ where $g(x,t,z,p)$ are the arguments.
So I need to prove that $$\lim_{h \to 0}\frac{\lVert f(v+h)-f(v) - h_t - (g_z|_{v}h + g_p|_{v}h_x) \rVert_{C^{k,\alpha}}}{\lVert h\rVert_{C^{k+2,\alpha}}} = 0.$$ The LHS is $$\lim_{h \to 0}\frac{ \lVert g(x,t,v, v_x) - g(x,t, v+ h, v_x + h_x) + g_z|_v h + g_p|_vh_x \rVert_{C^{k,\alpha}}}{\lVert h\rVert_{C^{k+2,\alpha}}} $$
which is a horrible thing to simplify. How can I prove that the limit is 0? I need serious help please. I will add a bounty when to a good answer. Thanks.