Non-resumable solution

In the theory of ordinary differential equations , Peano's theorem and Picard-Lindelof's theorem lead to the existence of a local solution to a given initial value problem. It is mainly interested whether this solution continues to continue, until you a non-continuable solution (sometimes maximum solution called) arrives. In a second step one is interested in the reason for the non-continuability. This is clarified by the theorem of the maximum existence interval.

Typically, the results are applied in the following order:

• First, with Peano's theorem or Picard-Lindelöf's theorem , one shows the existence of a (possibly unique) local solution of the initial value problem.
• From this follows with the sentence given below the existence of a non-continuable solution of the initial value problem. Its uniqueness is obtained by applying Gronwall's inequality .
• With the help of the theorem of the maximum existence interval, by excluding the remaining alternatives (for example with comparison arguments), one can conclude that this non-continuable solution is global.

In the following always be . ${\displaystyle \mathbb {K} \in \{\mathbb {R} ,\mathbb {C} \}}$

Existence of a non-continuable solution

Be and steady . Next is a solution of ${\displaystyle G\subset \mathbb {R} \times \mathbb {K} ^{n}}$ ${\displaystyle F:G\rightarrow \mathbb {K} ^{n}}$ ${\displaystyle y\in C([a,b);\mathbb {K} ^{n})\cap C^{1}((a,b);\mathbb {K} ^{n})}$

${\displaystyle \ y'=F(x,y)}$

on . Then there is one and a solution of the above differential equation with the properties: ${\displaystyle (a,b)}$ ${\displaystyle x^{+}\in [b,\infty )}$ ${\displaystyle u}$ ${\displaystyle (a,x^{+})}$

• ${\displaystyle y(x)=u(x)}$on . ${\displaystyle [a,b)}$
• There is none so that a solution can be continued on. ${\displaystyle s^{+}>x^{+}}$ ${\displaystyle u}$ ${\displaystyle (a,s^{+})}$

This theorem is proved by introducing a partial ordering on the set of all solutions in such a way that maximal elements are always non-continuable solutions. Their existence is proven with the Kuratowski-Zorn lemma. Details can be found in the evidence archive. Because of this evidence, the non-continuable solution is sometimes referred to as the maximal solution . But don't confuse this with the concept of the maximum solution of a non-uniquely solvable initial value problem (for continuous ). ${\displaystyle y'=F(x,y),y(a)=y_{0}}$ ${\displaystyle F}$

The theorem of the maximum existence interval

If you have a non-continuable solution, you want to know what is happening at the edge of its domain. The exclusion of this phenomenon would then result in the global nature of this solution.

formulation

Be and steady; this is explicitly permitted. Consider the differential equation ${\displaystyle D\subset \mathbb {K} ^{n}}$ ${\displaystyle F:[a,b)\times D\rightarrow \mathbb {K} ^{n}}$ ${\displaystyle b=\infty }$

${\displaystyle y'=F(x,y)\ .}$

Then applies to any non-resumable solution ${\displaystyle y:[a,x^{+})\rightarrow D}$

• ${\displaystyle \ x^{+}=b}$ (Globality) or
• ${\displaystyle \lim _{x\nearrow x^{+}}\min \left\{{\rm {dist}}(y(x),\partial D),{\frac {1}{\|y(x)\|}}\right\}=0\ .}$

This is agreed. ${\displaystyle {\rm {dist}}(z,\emptyset ):=\infty }$

Variant for locally Lipschitz continuous differential equation

Let , be continuous and locally Lipschitz continuous in the second variable and be a non-continuable solution of . Then applies ${\displaystyle G\subset \mathbb {R} \times \mathbb {K} ^{n}}$ ${\displaystyle F:G\rightarrow \mathbb {K} ^{n}}$ ${\displaystyle y:(x^{-},x^{+})\rightarrow \mathbb {K} ^{n}}$ ${\displaystyle \ y'=F(x,y)}$

• ${\displaystyle x^{+}=\infty }$ (Globality) or
• ${\displaystyle \lim _{x\nearrow x^{+}}\|y(x)\|=\infty }$ or
• there is a consequence so that the limit exists with . ${\displaystyle x_{j}\nearrow x^{+}}$ ${\displaystyle \lim _{j\rightarrow \infty }y(x_{j})=:y^{\star }}$ ${\displaystyle (x^{+},y^{\star })\in \partial G}$

literature

• Herbert Amann: Ordinary differential equations . 2nd Edition. Gruyter - de Gruyter textbooks, Berlin / New York 1995, ISBN 3-11-014582-0 .
• Wolfgang Walter: Ordinary differential equations . 6th edition. Springer-Verlag, Berlin / Heidelberg / New York 1996, ISBN 3-540-59038-2 .

Web links

Wikibooks: Proof of the Existence of Non-Continuable Solutions