description/proof of the \(C^k\)-ness of solution for Euclidean-normed Euclidean vectors space ODE with initial condition
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean-normed Euclidean vectors space.
- The reader knows a definition of derivative of map from arbitrary subset of normed vectors space into subset of normed vectors space at point.
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, where \(k\) excludes \(0\) and includes \(\infty\).
- The reader knows a definition of matrix norm induced by vector norms.
- The reader knows a definition of Frobenius matrix norm.
- The reader knows a definition of uniformly Cauchy sequence of maps from set into metric space.
- The reader admits the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded.
- The reader admits the proposition that for any topological space, any compact subset of the space that is contained in any subspace is compact on the subspace.
- The reader admits the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
- The reader admits the proposition that the product of any finite number of compact topological spaces is compact.
- The reader admits the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
- The reader admits the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is a compact subset of the codomain.
- The reader admits the proposition that for any metric space, any open subset, and any compact subset contained in the open subset, there is a positive radius of which the open or closed ball around each point on the compact subset is contained in the open subset.
- The reader admits the proposition that for any topological space, any compact subset of any subspace is compact on the base space.
- The reader admits the proposition that for any metric space, any bounded subset, and any positive real number, the union of the open balls around each point of the subset with the number radius is bounded.
- The reader admits the proposition that for any metric space with the induced topology, the closure of any bounded subset is bounded with the diameter of the subset.
- The reader admits the proposition that for any metric space, any subset, any subset of the subset, and any positive real number, if the open ball around each point of the subset of the subset with the number-radius is contained in the subset, the closure of the union of the open balls around each point of the subset of the subset with any smaller-radius is contained in the subset.
- The reader admits the proposition that for any continuous map with any compact topological parameter space from any subspace of any Euclidean metric space with the induced topology into any any subspace of any Euclidean metric space with the induced topology, that (the map) locally satisfies Lipschitz estimates, the restriction of the map on any compact subspace domain satisfies Lipschitz estimates.
- The reader admits the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
- The reader admits the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
- The reader admits the proposition that for any continuous real map from any non-1-point interval with any closed end, if the image of the interior is bounded, the range is correspondingly equal-or-smaller-or-larger bounded.
- The reader admits the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
- The reader admits the proposition that for any uniformly convergent sequence of continuous maps from any topological space into any metric space with the induced topology, the convergence is continuous.
- The reader admits the mean-value theorem for any differentiable map from any closed interval into the \(1\)-dimensional Euclidean \(C^\infty\) manifold.
- The reader admits the proposition that for any continuous map between any metric spaces with the domain with the induced topology, the restriction of the map on any compact domain is uniformly continuous.
- The reader admits the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions.
- The reader admits the proposition that any uniformly Cauchy sequence of maps from any set into any complete metric space converges uniformly.
- The reader admits the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
Target Context
- The reader will have a description and a proof of the \(C^k\)-ness of solution for Euclidean-normed Euclidean vectors space ordinary differential equation with initial condition.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(\mathbb{R}^d\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) as the Euclidean \(C^\infty\) manifold
\(\mathbb{R}\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) as the Euclidean \(C^\infty\) manifold
\(U\): \(\in \{\text{ the open subsets of } \mathbb{R}^d\}\)
\(J\): \(\in \{\text{ the intervals of } \mathbb{R}\}\) as the Euclidean \(C^\infty\) manifold with boundary
\(f\): \(: U \times J \to \mathbb{R}^d\), \(\in \{\text{ the } C^k \text{ maps }\}\), such that \(\forall u \in U (\exists U_u \in \{\text{ the open neighborhoods of } u \text{ on } U\} (\forall u_1, u_2 \in U_u, \forall r \in J (\Vert f (u_1, r) - f (u_2, r) \Vert \le L_u \Vert u_1 - u_2 \Vert)))\)
\(r_0\): \(\in J\)
//
Statements:
\(\exists J^` \in \{\text{ the open intervals of } J\} \text{ such that } r_0 \in J^`, \exists U^` \in \{\text{ the open subsets of } U\}, \exists x: J^` \times U^` \to U (\partial_1 x (r, p) = f (x (r, p), r) \land x (r_0, p) = p)\)
\(\implies\)
\(x \in \{\text{ the } C^k \text{ maps }\}\)
//
2: Proof
Whole Strategy: prove it inductively with respect to \(k\); Step 1: take any \((r_1, p_1) \in J^` \times U^`\), any bounded open interval around \(r_1\), \(J_{r_1}\), such that \(r_0, r_1 \in J_{r_1}\) and \(\overline{J_{r_1}} \subseteq J^`\), \(K := x (\overline{J_{r_1}} \times \{p_1\})\), and \(V := \cup_{k \in K} B_{k, \epsilon} \subseteq U\); Step 2: take Lipschitz \(f \vert_{\overline{V} \times \overline{J_{r_1}}}\) with \(L\), define \(M := Sup (\{\Vert f (p, r) \Vert \vert (p, r) \in \overline{V} \times \overline{J_{r_1}}\})\), \(R := Sup (\{\Vert r - r_0 \Vert \vert r \in \overline{J_{r_1}}\})\), and \(\delta\) such that \(2 \delta e^{L R} \lt \epsilon\) and \(B_{p_1, 2 \delta} \subseteq U^`\), and see that \(x\) maps \(\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}\) into \(V\); Step 3: prove it for \(k = 0\); Step 4: prove it for \(k = 1\); Step 5: suppose that it holds for \(0 \le k \le k' - 1\) where \(2 \le k'\), and prove it for \(k = k'\); Step 6: conclude the proposition.
Step 1:
Let \((r_1, p_1) \in J^` \times U^`\) be any.
Let \(J_{r_1}\) be any bounded open interval around \(r_1\) such that \(r_0, r_1 \in J_{r_1}\) and \(\overline{J_{r_1}} \subseteq J^`\), which is possible, because \(J^`\) is an open interval.
\(x \vert_{J^` \times \{p_1\}}: J^` \times \{p_1\} \to U\) is continuous, because \(x\) is differentiable with respect to the 1st argument.
\(\overline{J_{r_1}} \subseteq J^`\) is a compact subset, by the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded and the proposition that for any topological space, any compact subset of the space that is contained in any subspace is compact on the subspace: \(\overline{J_{r_1}}\) is compact on \(\mathbb{R}\) and is compact on \(J^`\).
\(\overline{J_{r_1}} \subseteq J^`\) is a compact subspace, by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
\(\{p_1\}\) is a compact topological space, because for any open cover of \(\{p_1\}\), there is the 1-element subcover that take any element that contains \(p_1\).
\(\overline{J_{r_1}} \times \{p_1\}\) is a compact topological space, by the proposition that the product of any finite number of compact topological spaces is compact.
\(\overline{J_{r_1}} \times \{p_1\}\) is the topological subspace of \(J^` \times \{p_1\}\), by the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces.
\(\overline{J_{r_1}} \times \{p_1\}\) is a compact subset of \(J^` \times \{p_1\}\), by the proposition that the compactness of any topological subset as a subset equals the compactness as a subspace.
\(K := x (\overline{J_{r_1}} \times \{p_1\})\) is compact on \(U\), by the proposition that for any continuous map between any topological spaces, the image of any compact subset of the domain is a compact subset of the codomain.
There is an \(\epsilon \in \mathbb{R}\) such that \(0 \lt \epsilon\) and for each \(k \in K\), \(B'_{k, 2 \epsilon} \subseteq U\), by the proposition that for any metric space, any open subset, and any compact subset contained in the open subset, there is a positive radius of which the open or closed ball around each point on the compact subset is contained in the open subset.
Let \(V := \cup_{k \in K} B_{k, \epsilon} \subseteq U\).
\(V\) is an open neighborhood of \(K\) on \(U\).
\(K\) is a compact subset of \(\mathbb{R}^d\), by the proposition that for any topological space, any compact subset of any subspace is compact on the base space, so, is bounded, by the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded.
\(V\) is bounded on \(\mathbb{R}^d\), by the proposition that for any metric space, any bounded subset, and any positive real number, the union of the open balls around each point of the subset with the number radius is bounded.
\(\overline{V}\) on \(\mathbb{R}^d\) is bounded, by the proposition that for any metric space with the induced topology, the closure of any bounded subset is bounded with the diameter of the subset.
So, \(\overline{V}\) is compact on \(\mathbb{R}^d\), by the Heine-Borel theorem: any subset of any Euclidean topological space is compact if and only if it is closed and bounded.
\(\overline{V} \subseteq U\), by the proposition that for any metric space, any subset, any subset of the subset, and any positive real number, if the open ball around each point of the subset of the subset with the number-radius is contained in the subset, the closure of the union of the open balls around each point of the subset of the subset with any smaller-radius is contained in the subset.
\(\overline{V}\) is compact on \(U\) as the topological subspace, by the proposition that for any topological space, any compact subset of the space that is contained in any subspace is compact on the subspace.
Step 2:
\(f \vert_{\overline{V} \times \overline{J_{r_1}}}\) is a Lipschitz map with \(L\), which depends on \(V\), but we denote it just as \(L\), by the proposition that for any continuous map with any compact topological parameter space from any subspace of any Euclidean metric space with the induced topology into any any subspace of any Euclidean metric space with the induced topology, that (the map) locally satisfies Lipschitz estimates, the restriction of the map on any compact subspace domain satisfies Lipschitz estimates.
\(\overline{V} \times \overline{J_{r_1}}\) is a compact topological space, by the proposition that the product of any finite number of compact topological spaces is compact.
\(\overline{V} \times \overline{J_{r_1}}\) is the topological subspace of \(U \times J\), by the proposition that for any possibly uncountable number of indexed topological spaces or any finite number of topological spaces and their subspaces, the product of the subspaces is the subspace of the product of the base spaces, so, it is a compact subspace.
\(f \vert_{\overline{V} \times \overline{J_{r_1}}}: \overline{V} \times \overline{J_{r_1}} \to \mathbb{R}^d\) is continuous, by the proposition that any restriction of any continuous map on the domain and the codomain is continuous.
Let \(M := Sup (\{\Vert f (p, r) \Vert \vert (p, r) \in \overline{V} \times \overline{J_{r_1}}\}) \lt \infty\), by the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum.
Let \(R := Sup (\{\Vert r - r_0 \Vert \vert r \in \overline{J_{r_1}}\}) \lt \infty\), because \(J_{r_1}\) is bounded.
Let \(\delta \in \mathbb{R}\) be any such that \(0 \lt \delta\), \(2 \delta e^{L R} \lt \epsilon\) (which implies \(2 \delta \lt \epsilon\)), and \(\overline{B_{p_1, 2 \delta}} \subseteq U^`\), which is possible, because \(U^`\) is an open neighborhood of \(p_1\): there is a \(B_{p_1, \delta'} \subseteq U^`\), so, take \(2 \delta \lt \delta'\), then, \(\overline{B_{p_1, 2 \delta}} \subseteq B_{p_1, \delta'} \subseteq U^`\).
For each \(p, p' \in U^`\), if there is an open interval around \(r_0\), \(J_{r_0} \subset \overline{J_{r_1}}\), such that \(x (r, p), x (r, p') \in \overline{V}\) over it, let us see that we can take \(u: J_{r_0} \to \mathbb{R}^d, r \mapsto x (r, p) - x (r, p')\), \(v: [0, \infty) \to [0, \infty), r \mapsto e^{L r} \Vert x (r_0, p) - x (r_0, p') \Vert\), and \(w: [0, \infty) \to [0, \infty), r \mapsto L r\), which satisfy the conditions for the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions for \(u, v, w\) instead of \(f, g, h\).
\(u\) is differentiable; \(v\) is differentiable and satisfies \(v (0) = \Vert x (r_0, p) - x (r_0, p') \Vert = \Vert u (r_0) \Vert\); \(w\) is a Lipschitz map with \(L\) and satisfies \(\Vert \partial_1 u (r) \Vert = \Vert f (x (r, p), r) - f (x (r, p'), r) \Vert \le L \Vert x (r, p) - x (r, p') \Vert = w (\Vert u (r) \Vert)\) and \(\partial_1 v (r) = L e^{L r} \Vert x (r_0, p) - x (r_0, p') \Vert = w (v (r))\).
So, in that case, \(\Vert u (r) \Vert = \Vert x (r, p) - x (r, p') \Vert \le v (\vert r - r_0 \vert) = e^{L \vert r - r_0 \vert} \Vert x (r_0, p) - x (r_0, p') \Vert \le e^{L R} \Vert p - p' \Vert\).
Let us see that \(x (\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}) \subseteq V\).
Let us suppose that there was a \((r_2, p_2) \in \overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}\) such that \(x (r_2, p_2) \notin V\).
\(p_2 \in V\), because \(x (r_0, p_2) = p_2 \in \overline{B_{p_1, 2 \delta}} \subseteq B_{p_1, \epsilon}\) and \(p_1 = x (r_0, p_1) \in K\).
\(r_2 \neq r_0\), because \(x (r_0, p_2) = p_2 \in V\).
Let us suppose that \(r_0 \lt r_2\).
Let \(t := Inf (\{r \in \overline{J_{r_1}} \vert r_0 \lt r \land x (r, p_2) \notin V\})\), which would exist as \(r_0 \le t\), because at least \(x (r_2, p_2) \notin V\).
\(x (t, p_2) \notin V\), because if \(t\) was the upper boundary of \(\overline{J_{r_1}}\), \(t = r_2\), which would mean that \(x (t, p_2) = x (r_2, p_2) \notin V\), and otherwise, if \(x (t, p_2) \in V\), as \(V\) was open and \(x\) was continuous with respect to \(r\), there would be a \(B_{t, \rho} \subseteq J_{r_1}\) such that \(x (B_{t, \rho}, p_2) \subseteq V\), then, \(t\) would not be the infimum, a contradiction.
So, \(t \neq r_0\), so, \(r_0 \lt t\).
For each \(r \in [r_0, t)\), \(x (r, p_2) \in V\), because otherwise, \(t\) would not be the infimum, a contradiction; also \(x (r, p_1) \in V\), because \(x (r, p_1) \in K \subseteq V\).
So, there would be an open interval, \(J_{r_0}\), that contained \([r_0, t)\) such that for each \(r \in J_{r_0}\), \(x (r, p_1), x (r, p_2) \in V\).
So, for each \(r \in J_{r_0}\), \(\Vert x (r, p_2) - x (r, p_1) \Vert \le e^{L R} \Vert p_2 - p_1 \Vert \le e^{L R} 2 \delta \lt \epsilon\).
As \(x\) was continuous with respect to \(r\), \(\Vert x (t, p_2) - x (t, p_1) \Vert \le e^{L R} 2 \delta \lt \epsilon\), by the proposition that for any continuous real map from any non-1-point interval with any closed end, if the image of the interior is bounded, the range is correspondingly equal-or-smaller-or-larger bounded, which would mean that \(x (t, p_2) \in V\), because \(x (t, p_1) \in K\), a contradiction.
So, there is no such \((r_2, p_2)\) such that \(r_0 \le r_2\).
Let us suppose that \(r_2 \lt r_0\).
Let \(t := Sup (\{r \in \overline{J_{r_1}} \vert r \lt r_0 \land x (r, p_2) \notin V\})\), which would exist as \(t \le r_0\), because at least \(x (r_2, p_2) \notin V\).
\(x (t, p_2) \notin V\), because if \(t\) was the lower boundary of \(\overline{J_{r_1}}\), \(t = r_2\), which would mean that \(x (t, p_2) = x (r_2, p_2) \notin V\), and otherwise, if \(x (t, p_2) \in V\), as \(V\) was open and \(x\) was continuous with respect to \(r\), there would be a \(B_{t, \rho} \subseteq J_{r_1}\) such that \(x (B_{t, \rho}, p_2) \subseteq V\), then, \(t\) would not be the supremum, a contradiction.
So, \(t \neq r_0\), so, \(t \lt r_0\).
For each \(r \in (t, r_0]\), \(x (r, p_2) \in V\), because otherwise, \(t\) would not be the supremum, a contradiction; also \(x (r, p_1) \in V\), because \(x (r, p_1) \in K \subseteq V\).
So, there would be a \(J_{r_0}\) that contained \((t, r_0]\) such that for each \(r \in J_{r_0}\), \(x (r, p_1), x (r, p_2) \in V\).
So, for each \(r \in J_{r_0}\), \(\Vert x (r, p_2) - x (r, p_1) \Vert \le e^{L R} \Vert p_2 - p_1 \Vert \le e^{L R} 2 \delta \lt \epsilon\).
As \(x\) was continuous with respect to \(r\), \(\Vert x (t, p_2) - x (t, p_1) \Vert \le e^{L R} 2 \delta \lt \epsilon\), by the proposition that for any continuous real map from any non-1-point interval with any closed end, if the image of the interior is bounded, the range is correspondingly equal-or-smaller-or-larger bounded, which would mean that \(x (t, p_2) \in V\), because \(x (t, p_1) \in K\), a contradiction.
So, there is no such \((r_2, p_2)\) such that \(r_2 \lt r_0\).
So, there is no such whatsoever \((r_2, p_2)\).
So, \(x (\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}) \subseteq V\).
Step 3:
Let us prove the proposition for \(k = 0\).
By Step 2, \(x\) from \(\overline{J_{r_1}} \times \overline{B_{p_1, 2 \delta}}\) is into \(V\), so, for each \(r \in J_{r_1}\) (\(J_{r_1}\) is taken as \(J_{r_0}\)) and \(p, p' \in \overline{B_{p_1, 2 \delta}}\), \(\Vert x (r, p) - x (r, p') \Vert \le e^{L R} \Vert p - p' \Vert\).
For each \(r \in J_{r_1}\) and \(p \in \overline{B_{p_1, 2 \delta}} \subseteq U^`\), \(x (r, p) = p + \int^r_{r_0} f (x (s, p), s) d s\), which has been proved in Step 1 of Proof of the local unique solution existence for a closed interval domain for a Euclidean-normed Euclidean vectors space ordinary differential equation with an initial condition with a clarification on the solution domain area.
\(\Vert x (r, p) - x (r_1, p_1) \Vert = \Vert p - p_1 + \int^r_{r_0} f (x (s, p), s) d s - \int^{r_1}_{r_0} f (x (s, p_1), s) d s \Vert = \Vert p - p_1 + \int^{r_1}_{r_0} (f (x (s, p), s) - f (x (s, p_1), s)) d s + \int^r_{r_1} f (x (s, p), s) d s \Vert \le \Vert p - p_1 \Vert + \Vert \int^{r_1}_{r_0} (f (x (s, p)) - f (x (s, p_1))) d s \Vert + \Vert \int^r_{r_1} f (x (s, p)) d s \Vert \le \Vert p - p_1 \Vert + \int^{r_1}_{r_0} \Vert (f (x (s, p)) - f (x (s, p_1))) \Vert d s + \int^r_{r_1} \Vert f (x (s, p)) \Vert d s \le \Vert p - p_1 \Vert + \int^{r_1}_{r_0} L \Vert x (s, p) - x (s, p_1) \Vert d s + \int^r_{r_1} \Vert f (x (s, p)) \Vert d s \le \Vert p - p_1 \Vert + \int^{r_1}_{r_0} L e^{L R} \Vert p - p_1 \Vert d s + \int^r_{r_1} M d s \le \Vert p - p_1 \Vert + L e^{L R} \Vert p - p_1 \Vert R + M (r - r_1)\).
That implies that \(x\) is continuous at \((r_1, p_1)\), obviously.
As \((r_1, p_1) \in J^` \times U^`\) is arbitrary, \(x\) is continuous over \(J^` \times U^`\).
So, the proposition has been proved for \(k = 0\).
Step 4:
Step 4 Strategy: Step 4-1: see that \(\partial_1 x (r, p)\) exists and is continuous; Step 4-2: see that \(\partial_{l + 1} x (r, p)\) exists and is continuous.
Let us prove the proposition for \(k = 1\).
Step 4-1:
For each \((r, p) \in J^` \times U^`\), \(\partial_1 x (r, p) = f (x (r, p), r)\), so, \(\partial_1 x (r, p)\) exists, and as \(x\) is continuous at \((r, p)\) by Step 3 and \(f\) is continuous, \(\partial_1 x\) is continuous at \((r, p)\).
Step 4-2:
Step 4-2 Strategy: Step 4-2-1: define \({g_h}^j_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}, (r, p) \mapsto (x^j (r, p + h e_l) - x^j (r, p)) / h\); Step 4-2-2: define \(s: \mathbb{N} \to \mathbb{R}, n \mapsto \delta (1 / 2)^{n + 1}\) and \(s'_l: \mathbb{N} \to \{: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\}, n \mapsto {g_{s (n)}}_l\); Step 4-2-3: see that \(s'_l\) is a uniformly Cauchy sequence of maps, by applying the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions for \(u: J_{r_1} \to \mathbb{R}^{d^2}, r \mapsto {g_h}_l (r, p) - {g_{h'}}_l (r, p)\), \(v: [0, \infty) \to [0, \infty), r \mapsto 2 c d \rho e^{L R} / (c d A) (e^{c d A r} - 1)\), and \(w: [0, \infty) \to [0, \infty), r \mapsto c d A r + 2 c d \rho e^{L R}\); Step 4-2-4: see that the convergence of \(s'_l\) is \(\partial_{l + 1} x (r, p)\).
Let us see that \(\partial_{l + 1} x (r, p)\) exists and is continuous at each \((r, p) \in J^` \times U^`\).
Step 4-2-1:
Let \(e_l \in \mathbb{R}^d\) be the unit vector of \(\mathbb{R}^d\) in the direction of the \(l\) component.
Let \(h \in B_{0, \delta} \subseteq \mathbb{R}\) be any.
For each \(p \in B_{p_1, \delta}\), \(p + h e_l \in B_{p_1, 2 \delta}\), because \(dist (p + h e_l, p_1) \le dist (p + h e_l, p) + dist (p, p_1) \lt \delta + \delta = 2 \delta\).
Let \({g_h}^j_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}, (r, p) \mapsto (x^j (r, p + h e_l) - x^j (r, p)) / h\).
Let \(g_h: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^{d^2}\) be the map whose \((j, l)\) component is \({g_h}^j_l\) above: \(g_h (r, p)\) is regarded to be the \(d \times d\) matrix.
Let \({g_h}_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\) be the map whose \(j\) component is \({g_h}^j_l\) above: \({g_h}_l (r, p)\) is regarded to be the \(d\) vector.
Step 4-2-2:
Let us think of the sequence, \(s: \mathbb{N} \to \mathbb{R}, n \mapsto \delta (1 / 2)^{n + 1}\).
Then, \(s'_l: \mathbb{N} \to \{: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\}, n \mapsto {g_{s (n)}}_l\) is the sequence of maps.
Step 4-2-3:
Let us see that \(s'_l\) is a uniformly Cauchy sequence of maps.
Let \(A \in \mathbb{R} := Sup (\{\vert \partial_m f^j (p, r) \vert \vert (p, r) \in \overline{V} \times \overline{J_{r_1}}, j, m \in \{1, ..., d\}\})\) or if it is \(0\), let \(A\) be any positive real number, which exists as \(\lt \infty\), by the proposition that the image of any continuous map from any compact topological space to the \(\mathbb{R}\) Euclidean topological space has the minimum and the maximum, because \(\overline{V} \times \overline{J_{r_1}}\) is a compact subspace of \(U \times J\) (as is seen in Step 2) and \(\partial_m f^j\) is continuous over \(U \times J\).
Let \(c \in \mathbb{R}\) be any such that for the \(d \times d\) matrix norm induced by vector norms, \(\Vert \bullet \Vert\), and the Frobenius \(d \times d\) matrix norm, \(\Vert \bullet \Vert_F\), \(\Vert \bullet \Vert \lt c \Vert \bullet \Vert_F\): refer to Note for the definition of Frobenius matrix norm.
Let us take \(u: J_{r_1} \to \mathbb{R}^d, r \mapsto {g_h}_l (r, p) - {g_{h'}}_l (r, p)\), \(v: [0, \infty) \to [0, \infty), r \mapsto 2 c d \rho e^{L R} / (c d A) (e^{c d A r} - 1)\), and \(w: [0, \infty) \to [0, \infty), r \mapsto c d A r + 2 c d \rho e^{L R}\).
Let us see that \(u\), \(v\), and \(w\) satisfies the conditions for the proposition that for any differentiable map from any open interval into any Euclidean \(C^\infty\) manifold with the Euclidean norm, the norm of each point image is upper-bounded by a differentiable map between the non-negative interval with an initial condition if there is a Lipschitz map between the non-negative interval that satisfies certain conditions, with \(u\), \(v\) and \(w\) instead of \(f\), \(g\), and \(h\).
\(u\) is differentiable.
\(v\) is differentiable and \(v (0) = \Vert u (r_0) \Vert\), because while \(v (0) = 0\), \(u (r_0) = {g_h}_l (r_0, p) - {g_{h'}}_l (r_0, p) = (x (r_0, p + h e_l) - x (r_0, p)) / h - (x (r_0, p + h' e_l) - x (r_0, p)) / h' = (p + h e_l - p) / h - (p + h' e_l - p) / h' = h e_l / h - h' e_l / h' = e_l - e_l = 0\), so, \(\Vert u (r_0) \Vert = 0\).
\(w\) is a Lipschitz map with \(c d A\), obviously.
Let us see that \(\Vert \partial_1 u (r) \Vert \le w (\Vert u (r) \Vert)\).
\(\partial_1 u (r) = \partial_1 ({g_h}_l (r, p) - {g_{h'}}_l (r, p))\).
\(\partial_1 {g_h}_l (r, p) = (\partial_1 x (r, p + h e_l) - \partial_1 x (r, p)) / h = (f (x (r, p + h e_l), r) - f (x (r, p), r)) / h\).
Let us think of \({y_h}_l (r, p): [0, 1] \to \mathbb{R}^d, s \mapsto f ((1 - s) x (r, p) + s x (r, p + h e_l), r)\), which is valid, because the line segment, \(\overline{x (r, p) x (r, p + h e_l)}\), is contained in \(V\), because \(\Vert x (r, p) - x (r, p_1) \Vert \le e^{L R} \Vert p - p_1 \Vert \lt e^{L R} \delta \lt \epsilon / 2\) while \(k := x (r, p_1) \in K\), which means that \(x (r, p) \in B_{k, \epsilon / 2}\), and \(\Vert x (r, p) - x (r, p + h e_l) \Vert \le e^{L R} \Vert p + h e_l - p \Vert = e^{L R} \vert h \vert \lt e^{L R} \delta \lt \epsilon / 2\), so, \(\Vert x (r, p + h e_l) - k \Vert \le \Vert x (r, p + h e_l) - x (r, p) \Vert + \Vert x (r, p) - k \Vert \lt \epsilon / 2 + \epsilon / 2 = \epsilon\), \(x (r, p), x (r, p + h e_l) \in B_{k, \epsilon}\).
\({y_h}_l (r, p)\) is differentiable, because \(f\) is differentiable and \((1 - s) x (r, p) + s x (r, p + h e_l)\) is differentiable with respect to \(s\).
By the mean-value theorem for any differentiable map from any closed interval into the \(1\)-dimensional Euclidean \(C^\infty\) manifold, there is an \(t_j \in (0, 1)\) such that \({y_h}^j_l (r, p) (1) - {y_h}^j_l (r, p) (0) = \partial_1 {y_h}^j_l (r, p) (t_j)\).
Let \({p_2}_{l, h, j} (r, p) := (1 - t_j) x (r, p) + t_j x (r, p + h e_l)\).
\(f^j (x (r, p + h e_l), r) - f^j (x (r, p), r) = {y_h}^j_l (r, p) (1) - {y_h}^j_l (r, p) (0) = \partial_1 {y_h}^j_l (r, p) (t_j) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) (x^m (r, p + h e_l) - x^m (r, p)) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_h}^m_l (r, p) h\).
So, \(\partial_1 {g_h}^j_l (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_h}^m_l (r, p)\).
\(\partial_1 ({g_h}^j_l (r, p) - {g_{h'}}^j_l (r, p)) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_h}^m_l (r, p) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) {g_{h'}}^m_l (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) ({g_h}^m_l (r, p) - {g_{h'}}^m_l (r, p)) + \partial_m f^j ({p_2}_{l, h, j} (r, p), r) {g_{h'}}^m_l (r, p) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) {g_{h'}}^m_l (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r) ({g_h}^m_l (r, p) - {g_{h'}}^m_l (r, p)) + (\partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r)) {g_{h'}}^m_l (r, p)\).
So, \(\partial_1 ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) = M_{l, h} (r, p) ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) + (M_{l, h} (r, p) - M_{l, h'} (r, p)) {g_{h'}}_l (r, p)\), where \(M_{l, h} (r, p)\) is the matrix whose \((j, m)\) component is \({M_{l, h}}^j_m (r, p) = \partial_m f^j ({p_2}_{l, h, j} (r, p), r)\): note that \(M_{l, h} (r, p)\) is not exactly the Jacobian at any point, because \({p_2}_{l, h, j} (r, p)\) depends on \(j\), which does not matter.
\(\Vert \partial_1 ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) \Vert = \Vert M_{l, h} (r, p) ({g_h}_l (r, p) - {g_{h'}}_l (r, p)) + (M_{l, h} (r, p) - M_{l, h'} (r, p)) {g_{h'}}_l (r, p) \Vert \le \Vert M_{l, h} (r, p) \Vert \Vert {g_h}_l (r, p) - {g_{h'}}_l (r, p) \Vert + \Vert M_{l, h} (r, p) - M_{l, h'} (r, p) \Vert \Vert {g_{h'}}_l (r, p) \Vert\), where the matrix norms are the norms induced by vector norms, \(\le c \Vert M_{l, h} (r, p) \Vert_F \Vert {g_h}_l (r, p) - {g_{h'}}_l (r, p) \Vert + c \Vert M_{l, h} (r, p) - M_{l, h'} (r, p) \Vert_F \Vert {g_{h'}}_l (r, p) \Vert\), where \(\Vert \bullet \Vert_F\) is the Frobenius matrix norm.
As \(\vert {M_{l, h} (r, p)}^j_m \vert \le A\), \(\Vert M_{l, h} (r, p) \Vert_F \le d A\).
\(\Vert {g_h}_l (r, p) \Vert = \Vert x (r, p + h e_l) - x (r, p) \Vert / \vert h \vert \le e^{L R} \Vert p + h e_l - p \Vert / \vert h \vert = e^{L R} \Vert h e_l \Vert / \vert h \vert = e^{L R}\).
\(\Vert {g_{h'}}_l (r, p) \Vert \le e^{L R}\), likewise.
\(\partial_m f^j\) is continuous over \(U \times J\), so, \(\partial_m f^j\) is uniformly continuous over \(\overline{V} \times \overline{J_{r_1}}\), by the proposition that for any continuous map between any metric spaces with the domain with the induced topology, the restriction of the map on any compact domain is uniformly continuous.
So, for each \(\rho \in \mathbb{R}\) such that \(0 \lt \rho\), there is a \(\lambda \in \mathbb{R}\) such that \(0 \lt \lambda\) and for each \((p, r), (p', r') \in \overline{V} \times \overline{J_{r_1}}\) such that \(\sqrt{\Vert p' - p \Vert^2 + \Vert r' - r \Vert^2} \lt \lambda\), \(\Vert \partial_m f^j (p', r') - \partial_m f^j (p, r) \Vert \lt \rho\).
Let \(\vert h \vert, \vert h' \vert \lt \lambda e^{- L R}\).
Then, \(\Vert {p_2}_{l, h, j} (r, p) - x (r, p) \Vert = \Vert (1 - t_j) x (r, p) + t_j x (r, p + h e_l) - x (r, p) \Vert = \Vert - t_j x (r, p) + t_j x (r, p + h e_l) \Vert = \Vert t_j (x (r, p + h e_l) - x (r, p)) \Vert = t_j \Vert x (r, p + h e_l) - x (r, p) \Vert = t_j \Vert {g_h}_l (r, p) h \Vert \le t_j \vert h \vert e^{L R} \lt \lambda e^{- L R} e^{L R} = \lambda\).
\(\Vert {p_2}_{l, h', j} (r, p) - x (r, p) \Vert \lt \lambda\), likewise.
\(\vert (M_{l, h} (r, p) - M_{l, h'} (r, p))^j_m \vert = \vert \partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) \vert = \vert \partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j (x (r, p), r) + \partial_m f^j (x (r, p), r) - \partial_m f^j ({p_2}_{l, h', j} (r, p), r) \vert \le \vert \partial_m f^j ({p_2}_{l, h, j} (r, p), r) - \partial_m f^j (x (r, p), r) \vert + \vert \partial_m f^j ({p_2}_{l, h', j} (r, p), r) - \partial_m f^j (x (r, p), r) \vert \lt \rho + \rho = 2 \rho\), because \(\sqrt{\Vert \Vert {p_2}_{l, h, j} (r, p) - x (r, p) \Vert \Vert^2 + \Vert r - r \Vert^2} \lt \lambda\) and \(\sqrt{\Vert \Vert {p_2}_{l, h', j} (r, p) - x (r, p) \Vert \Vert^2 + \Vert r - r \Vert^2} \lt \lambda\).
So, \(\Vert M_{l, h} (r, p) - M_{l, h'} (r, p) \Vert_F \lt 2 \rho d\).
So, \(\Vert \partial_1 u (r) \Vert \lt c d A \Vert u (r) \Vert + 2 c d \rho e^{L R} = w (\Vert u (r) \Vert)\).
\(\partial_1 v (r) = 2 c d \rho e^{L R} e^{c d A r}\) while \(w (v (r)) = c d A (2 c d \rho e^{L R} / (c d A) (e^{c d A r} - 1)) + 2 c d \rho e^{L R} = 2 c d \rho e^{L R} (e^{c d A r} - 1) + 2 c d \rho e^{L R} = 2 c d \rho e^{L R} e^{c d A r}\), so, \(\partial_1 v (r) = w (v (r))\).
So, \(u, v, w\) satisfies the conditions, so, \(\Vert u (r) \Vert \le v (\vert r - r_0 \vert)\), so, \(\Vert {g_h}_l (r, p) - {g_{h'}}_l (r, p) \Vert \le 2 c d \rho e^{L R} / (c d A) (e^{c d A \vert r - r_0 \vert} - 1) \le 2 c d \rho e^{L R} / (c d A) (e^{c d A R} - 1)\).
So, for any \(\alpha \in \mathbb{R}\) such that \(0 \lt \alpha\), \(\rho\) can be chosen such that \(2 c d \rho e^{L R} / (c d A) (e^{c d A R} - 1) \lt \alpha\), then, \(\lambda\) can be chosen accordingly, and \(N \in \mathbb{N}\) can be chosen such that \(s (N) \lt \lambda e^{- L R}\), then for each \(n, n' \in \mathbb{N}\) such that \(N \lt n, n'\), \(s (n), s (n') \lt \lambda e^{- L R}\), so, \(\Vert s'_l (n) (r, p) - s'_l (n') (r, p) \Vert = \Vert {g_{s (n)}}_l (r, p) - {g_{s (n')}}_l (r, p) \Vert \lt \alpha\).
As \(N\) is chosen independent of \((r, p)\), \(s'_l\) is a uniformly Cauchy sequence.
So, \(s'_l\), converges uniformly to the continuous map, \(z_l: J_{r_1} \times B_{p_1, \delta} \to \mathbb{R}^d\), by the proposition that any uniformly Cauchy sequence of maps from any set into any complete metric space converges uniformly and the proposition that for any uniformly convergent sequence of continuous maps from any topological space into any metric space with the induced topology, the convergence is continuous.
\({z_l}^j\) is indeed the derivative, \(lim_{h \to 0} (x^j (r, p + h e_l) - x^j (r, p)) / h\), because \(\vert (x^j (r, p + h e_l) - x^j (r, p)) / h - {z_l}^j (r, p) \vert = \vert {g_h}^j_l (r, p) - {z_l}^j (r, p) + {g_{s (n)}}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert \le \vert {g_h}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert + \vert {z_l}^j (r, p) - {g_{s (n)}}^j_l (r, p) \vert\), and choosing \(n\) such that \(\vert {z_l}^j (r, p) - {g_{s (n)}}^j_l (r, p) \vert \lt \alpha / 2\) and \(s (n) \lt \lambda e^{- L R}\) for \(\alpha / 2\), \(\lt \vert {g_h}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert + \alpha / 2\), and for each \(\vert h \vert \lt \lambda e^{- L R}\) for \(\alpha / 2\), \(\vert {g_h}^j_l (r, p) - {g_{s (n)}}^j_l (r, p) \vert \lt \alpha / 2\), so, \(\vert (x^j (r, p + h e_l) - x^j (r, p)) / h - {z_l}^j (r, p) \vert \lt \alpha\).
As \(z_l\) exists and is continuous around each \((r_1, p_1) \in J^` \times U^`\), \(z_l\) exists and is continuous over whole \(J^` \times U^`\).
So, the proposition holds for \(k = 1\).
Step 5:
Let us suppose that the proposition holds for \(0 \le k \le k' - 1\) where \(2 \le k'\).
Let us prove that the proposition holds for \(k = k'\).
As \(f\) is \(C^{k' - 1}\), \(x\) is \(C^{k' - 1}\), by the induction hypothesis.
\(\partial_1 x^j = f^j (x (r, p), r)\), which is \(C^{k - 1}\), by the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point.
As \(x^j (r, p) = p^j + \int^r_{r_0} f^j (x (s, p), s) d s\), \(\partial_{l + 1} x^j (r, p) = \delta^j_l + \int^r_{r_0} \partial_{l + 1} \widetilde{f}^j (s, p) d s\), where \(\widetilde{f} (s, p)\) is \(f (x (s, p), s)\) regarded as the map from \(J^` \times U^`\), \(C^1\), by the proposition that for any maps between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at corresponding points, where \(k\) includes \(\infty\), the composition is \(C^k\) at the point, because \(\widetilde{f} = f \circ (x, id) \circ \iota\), where \(\iota: J^` \times U^` \to (J^` \times U^`) \times J^`, (r, p) \mapsto ((r, p), r)\), while the derivation under the integral is possible, because \(\widetilde{f}\) is \(C^1\).
\(\partial_{l + 1} \widetilde{f}^j (s, p) = \partial_m f^j (x (s, p), s) \partial_{l + 1} x^m (s, p)\), by the chain rule.
So, \(\partial_{1 + l} x^j (r, p) = \delta^j_l + \int^r_{r_0} \partial_m f^j (x (s, p), s) \partial_{l + 1} x^m (s, p) d s\).
So, \(\partial_1 \partial_{1 + l} x^j (r, p) = \partial_m f^j (x (r, p), r) \partial_{l + 1} x^m (r, p)\).
Let us think of \((\partial_{1 + l} x, x): J^` \times U^` \to \mathbb{R}^{2 d}\) and \(\partial_1 \partial_{1 + l} x (r, p) = \partial_m f (x (r, p), r) \partial_{l + 1} x^m (r, p)\) and \(\partial_1 x (r, p) = f (x (r, p), r)\) combined, which is an ordinal differential equation with initial condition, \((\partial_{1 + l} x, x) (r_0) = ((\delta^j_l), p)\) with \(f'\) instead of \(f\) such that \(f' ((\partial_{1 + l} x, x), r) = (\partial_m f (x, r) (\partial_{l + 1} x)^m, f (x, r))\).
As \(f'\) is \(C^{k - 1}\), the solution, \((\partial_{1 + l} x, x)\) is \(C^{k - 1}\), by the induction hypothesis.
So, especially, \(\partial_{1 + l} x\) is \(C^{k - 1}\).
As \(\partial_1 x\) and \(\partial_{l + 1} x\) s are \(C^{k - 1}\), \(x\) is \(C^k\).
Step 6:
By the induction principle, the proposition holds for each \(k \in \mathbb{N}\).
