Smooth dependence of ODE solutions on parameters

The field $G$ is jointly smooth on the open time-state domain $D$. Its last $m$ equations are ${\lambda }^{\mathrm{\prime }}=0$, so the parameter is constant along each augmented solution. Thus projection to the first $n$ coordinates identifies augmented solutions through $(y,\lambda )$ with solutions of the displayed original IVP, in both directions.

Choose $a,r>0$ so $[t_0-a,t_0+a]\times \bar{B}(z_0,2r)\subseteq D$. By [F1], $G$ and $D_{z}G$ have bounds $M\ge 1$ and $L\ge 1$ on this cylinder, and the mean-value inequality on each open state ball gives the same state-Lipschitz bound $L$. Choose $H>0$ with $H<a$, $HM<r$ and $HL<1$. For every initial state $z\in B(z_0,r)$, the radius-$r$ cylinder centred at $z$ lies in the larger cylinder. Hence [F2] gives a unique augmented solution $Z(t,z)$ on $J=[t_0-H,t_0+H]$, all of whose graphs lie in that one compact cylinder. This establishes the common interval needed below.

Now [L1] applies to the jointly smooth field $G$ on the common interval $J$ of step 2.1, and gives smooth dependence of $Z(t,z)$ on $z$ near $z_0$. We record why this dependence has the joint regularity needed for time derivatives. At first order, [F4] gives $X_1=D_{z}Z$ and $X_1^{\mathrm{\prime }}=D_{z}G(t,Z)X_1$, $X_1(t_0)=I_d$. The coefficient is jointly continuous; the uniform-in-time estimates in [F4], combined with continuity of the solution of this linear equation in time, make $X_1$ jointly continuous. The solution $Z$ itself is jointly continuous by the same initial-data estimates and its continuity in time.

The higher-order argument does not require a new existence assumption. For $r\ge 1$, use the finite-dimensional jet variables $X_0,\dots ,X_r$, where $X_j$ is a $j$-linear map and $X_0\in {\mathbb{R}}^d$. Repeated chain rules in [F3] give $X_0^{\mathrm{\prime }}=G(t,X_0)$ and $X_j^{\mathrm{\prime }}=D_{z}G(t,X_0)X_j+P_j(t,X_0,\dots ,X_{j-1})$ for $j\ge 1$, with $P_1=0$. Here $P_j$ is the finite sum over partitions of the $j$ differentiation directions into at least two nonempty blocks, applying the corresponding higher derivative of $G$ to the lower jets indexed by those blocks. Initial values are $X_0(t_0)=z$, $X_1(t_0)=I_d$ and $X_j(t_0)=0$ for $j\ge 2$. This jet field is jointly smooth. For nearby full jet initial values, $X_0$ exists on $J$ by step 2.1; successively each higher component is affine linear with bounded continuous coefficients and forcing on $J$, so [F5] gives its evolution there. The same linear integral estimates bound each component uniformly for nearby initial values, giving a common compact jet cylinder for each finite order. Thus [F4] legitimately applies to this enlarged system on $J$. Starting with its first-order variational equation, differentiation of the Volterra identities and uniqueness in [F5] identify the prescribed jet with $(Z,D_{z}Z,\dots ,D_z^{r}Z)$. Its highest component is $C^1$ in the initial state, so these identities upgrade $C^r$ to $C^{r+1}$; this uses the highest component, not merely the first one. Induction, with the uniform estimates of [F4], gives joint continuity of all state derivatives.

The equation ${\mathrm{\partial }}_{t}Z=G(t,Z)$ now supplies all mixed derivatives: differentiating it in state variables gives the chain-rule expressions in the jointly continuous state jets, and their displayed jet equations permit subsequent time differentiation. Induction on the time order, using joint smoothness of $G$, proves joint smoothness on the open time interval $(t_0-H,t_0+H)$ and the initial-state neighbourhood. Put $I=[t_0-H\mathrm{/}2,t_0+H\mathrm{/}2]$; the solution map on $I$ therefore extends smoothly to an open time neighbourhood. Projection as in step 1.1 gives the claimed common local solutions and smooth dependence on $(y,\lambda )$, including the endpoints of $I$. All solutions and all jet evolutions are uniquely determined by their IVPs; the Picard sequences use prescribed iteration and the argument makes no family selection or additional Choice assumption.