description/proof of the local uniqueness 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 map.
- 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 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 topological space, the union of any finite compact subsets is compact.
- 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 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.
Target Context
- The reader will have a description and a proof of the local uniqueness 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:
\(\mathbb{R}^d\): \(= \text{ the Euclidean-normed Euclidean vectors space }\) with the topology induced by the metric induced by the norm
\(\mathbb{R}\): \(= \text{ the Euclidean-normed Euclidean vectors space }\)
\(U\): \(\in \{\text{ the open subsets of } \mathbb{R}^d\}\)
\(J\): \(\in \{\text{ the intervals of } \mathbb{R}\}\)
\(f\): \(: U \times J \to \mathbb{R}^d\), \(\in \{\text{ the } C^0 \text{ maps }\}\), such that \(\forall x_0 \in U (\exists U_{x_0} \in \{\text{ the open neighborhoods of } x_0 \text{ on } U\} (\forall x_1, x_2 \in U_{x_0}, \forall r \in J (\Vert f (x_1, r) - f (x_2, r) \Vert \le L_{x_0} \Vert x_1 - x_2 \Vert)))\)
\(r_0\): \(\in J\)
\(x'_0\): \(\in U\)
//
Statements:
\(\exists J^` \in \{\text{ the open intervals of } J\} \text{ such that } r_0 \in J^`, \exists x: J^` \to U (d x / d r = f (x, r) \land x (r_0) = x'_0)\)
\(\implies\)
\(x\) is the unique solution over \(J^`\)
//
2: Note
The merit of this proposition against 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 is that this is on any solution over any \(J^` \subseteq J\) while 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 is on the solution over any more restricted domain specified in the proposition.
3: Proof
Whole Strategy: Step 1: suppose that there were some 2 solutions, \(x_1 (t), x_2 (t)\), over \(J^`\); Step 2: for each \(r_1 \in J^`\), take a bounded open interval, \(J^{``} \subseteq J^`\), such that \(r_0, r_1 \in J^{``}\) and \(\overline{J^{``}} \subseteq J^`\), and see that \(f \vert_{(x_1 (\overline{J^{``}}) \cup x_2 (\overline{J^{``}})) \times \overline{J^{``}}}\) satisfies a Lipschitz estimate; Step 3: see that over \(J^{``}\), \(\Vert x_1 (r) - x_2 (r) \Vert \le e^{L_{J^{``}} \vert r - r_0 \vert} \Vert x_1 (r_0) - x_2 (r_0) \Vert\); Step 4: conclude the proposition.
Step 1:
Let us suppose that there were some 2 solutions, \(x_1 (r), x_2 (r)\), over \(J^`\).
Step 2:
For each \(r_1 \in J^`\), there is a bounded open interval, \(J^{``} \subseteq J^`\), such that \(r_0, r_1 \in J^{``}\) and \(\overline{J^{``}} \subseteq J^`\).
\(\overline{J^{``}}\) is compact on \(J^`\), 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.
\(x_j (\overline{J^{``}})\) 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, and \(x_1 (\overline{J^{``}}) \cup x_2 (\overline{J^{``}})\) is compact on \(U\), by the proposition that for any topological space, the union of any finite compact subsets is compact.
The restriction of \(f\) on \((x_1 (\overline{J^{``}}) \cup x_2 (\overline{J^{``}})) \times \overline{J^{``}}\) satisfies a Lipschitz estimate, 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.
So, over \(J^{``}\), \(\Vert d (x_1 - x_2) / d r \Vert = \Vert d x_1 / d r - d x_2 / d r \Vert = \Vert f (x_1 (r), r) - f (x_2 (r), r) \Vert \le L_{J^{``}} \Vert x_1 (r) - x_2 (r) \Vert\).
Step 3:
Let \(u: J^{``} \to \mathbb{R}^d = x_1 - x_2\), \(g: [0, \infty) \to [0, \infty), r \mapsto e^{L_{J^{``}} r} \Vert x_1 (r_0) - x_2 (r_0) \Vert\), and \(h: [0, \infty) \to [0, \infty), r \mapsto L_{J^{``}} r\).
\(u\) is differentiable; \(g\) is differentiable and \(g (0) = e^{L_{J^{``}} 0} \Vert x_1 (r_0) - x_2 (r_0) \Vert = \Vert x_1 (r_0) - x_2 (r_0) \Vert = \Vert u (r_0) \Vert\); \(h\) is a Lipschitz map with \(L_{J^{``}}\), \(\Vert \partial_1 u (r) \Vert \le h (\Vert u (r) \Vert)\), as has been seen above, and \(\partial_1 g (r) = h (g (r))\).
So, by 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, \(\Vert u (r) \Vert \le g (\vert r - r_0 \vert)\), which is \(\Vert x_1 (r) - x_2 (r) \Vert \le e^{L_{J^{``}} \vert r - r_0 \vert} \Vert x_1 (r_0) - x_2 (r_0) \Vert\).
But as \(x_1 (r_0) - x_2 (r_0) = 0\), \(\Vert x_1 (r_0) - x_2 (r_0) \Vert = 0\), so, \(\Vert x_1 (r) - x_2 (r) \Vert \le 0\), so, \(\Vert x_1 (r) - x_2 (r) \Vert = 0\), so, \(x_1 (r) = x_2 (r)\).
Step 4:
As \(r_1 \in J^`\) is arbitrary, such \(J^{``}\) s cover \(J^`\), and as over each \(J^{``}\), \(x_1 = x_2\), \(x_1 = x_2\) over the whole \(J^`\).
