View source for Prolongation of solutions of differential equations

The property of solutions of ordinary differential equations to be extendible to a larger interval of the independent variable. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

be a solution of the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754002.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754003.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754004.png" />, is called a prolongation of the solution (1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754005.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754006.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754007.png" />.

Suppose that the function

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is defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p0754009.png" /> and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540010.png" />. The solution (1) is called indefinitely extendible (indefinitely extendible forwards (to the right), indefinitely extendible backward (to the left)) if a prolongation of it exists defined on the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540011.png" /> (respectively, on the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540012.png" />, on the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540013.png" />). The solution (1) is called extendible forwards (to the right) up to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540015.png" /> if a prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540017.png" />, of it exists with the following property: For any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540018.png" /> there is a value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540020.png" />, such that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540021.png" /> does not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540022.png" />. Extendibility backward (to the left) up to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540023.png" /> is defined analogously. A solution that cannot be extended is called non-extendible.

If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540024.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540025.png" />, then every solution (1) of (2) can be either extended forwards (backward) or indefinitely or up to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540026.png" />. In other words, every solution of (2) can be extended to a non-extendible solution. If the partial derivatives

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540028.png" />, then such a prolongation is unique.

An interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540029.png" /> is called a maximal interval of existence of a solution of (2) if the solution cannot be extended to a larger interval. For any solution of a linear system

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with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540031.png" /> and right-hand sides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540033.png" />, that are continuous on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540034.png" />, the maximal interval of existence of a solution coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540035.png" />. For solutions of a non-linear system the maximal intervals of existence may differ for different solutions, and determining them is a difficult task. E.g. for the solution to the [[Cauchy problem|Cauchy problem]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540036.png" /></td> </tr></table>

one has

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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540038.png" />,

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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540040.png" />, and

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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540042.png" />.

A sufficient condition under which one can indicate the maximal interval of existence of a solution is, e.g., Wintner's theorem: Suppose that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540043.png" /> is continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540045.png" />, and that it satisfies in this domain the estimate

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540046.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540047.png" /> is a function continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540049.png" /> and for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540051.png" />,

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Then every solution of (2) exists on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540053.png" />.

This theorem also holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540054.png" />. Sufficient conditions for indefinite extendibility of a solution are of great interest. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540055.png" /> and its partial derivatives (3) are continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540057.png" />, and if for these values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540058.png" /> the estimates

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hold, then the solution of (2) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540060.png" /> exists for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540061.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540062.png" />.

Consider the Cauchy problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

for an [[Autonomous system|autonomous system]], where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540064.png" /> is continuously differentiable in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540065.png" />. If, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540066.png" /> grows, the phase trajectory of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540067.png" /> of (4) remains in a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540068.png" />, then this solution can be extended to the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075400/p07540069.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Arnol'd,   "Ordinary differential equations" , M.I.T.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Cesari,   "Asymptotic behavior and stability problems in ordinary differential equations" , Springer  (1959)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A. Wintner,   "The non-local existence problem of ordinary differential equations"  ''Amer. J. Math.'' , '''67'''  (1945)  pp. 277–284</TD></TR></table>



====Comments====
Instead of prolongation of solutions, continuation of solutions is nowadays mostly used.