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
be a solution of the system
A solution , , is called a prolongation of the solution (1) if and for .
Suppose that the function
is defined in a domain and suppose . 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 (respectively, on the semi-axis , on the semi-axis ). The solution (1) is called extendible forwards (to the right) up to the boundary of if a prolongation , , of it exists with the following property: For any compact set there is a value , , such that the point does not belong to . Extendibility backward (to the left) up to the boundary is defined analogously. A solution that cannot be extended is called non-extendible.
If the function is continuous in , then every solution (1) of (2) can be either extended forwards (backward) or indefinitely or up to the boundary . In other words, every solution of (2) can be extended to a non-extendible solution. If the partial derivatives
are continuous in , then such a prolongation is unique.
An interval 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
with coefficients and right-hand sides , , that are continuous on an interval , the maximal interval of existence of a solution coincides with . 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
if , and
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 is continuous for , , and that it satisfies in this domain the estimate
where is a function continuous for , and for some , ,
Then every solution of (2) exists on the whole of .
This theorem also holds for . Sufficient conditions for indefinite extendibility of a solution are of great interest. E.g., if and its partial derivatives (3) are continuous for , , and if for these values of the estimates
hold, then the solution of (2) with exists for , for any .
Consider the Cauchy problem
for an autonomous system, where is continuously differentiable in a domain . If, as grows, the phase trajectory of the solution of (4) remains in a compact subset , then this solution can be extended to the semi-axis .
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|||V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian)|
|||V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)|
|||E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17|
|||P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)|
|||L. Cesari, "Asymptotic behavior and stability problems in ordinary differential equations" , Springer (1959)|
|||A. Wintner, "The non-local existence problem of ordinary differential equations" Amer. J. Math. , 67 (1945) pp. 277–284|
Instead of prolongation of solutions, continuation of solutions is nowadays mostly used.
Prolongation of solutions of differential equations. M.V. Fedoryuk (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Prolongation_of_solutions_of_differential_equations&oldid=16650