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
![]() | (1) |
be a solution of the system
![]() | (2) |
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
![]() | (3) |
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
![]() |
one has
![]() |
if ,
![]() |
if , and
![]() |
if .
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
![]() | (4) |
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
.
References
[1] | L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian) |
[2] | V.I. Arnol'd, "Ordinary differential equations" , M.I.T. (1973) (Translated from Russian) |
[3] | V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) |
[4] | E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17 |
[5] | P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) |
[6] | L. Cesari, "Asymptotic behavior and stability problems in ordinary differential equations" , Springer (1959) |
[7] | A. Wintner, "The non-local existence problem of ordinary differential equations" Amer. J. Math. , 67 (1945) pp. 277–284 |
Comments
Instead of prolongation of solutions, continuation of solutions is nowadays mostly used.
Prolongation of solutions of differential equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prolongation_of_solutions_of_differential_equations&oldid=16650