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Continuous Open Set Equation Variable Differential Equation Recurrence Relation Ordered Geometry Initial Value Problem Lipschitz Condition Existence And Uniqueness Of Solution Of Ordinary Differential Equations
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Picards Theorem
Theorem 1 (Picard's theorem [KF])
Let 
be an open subset
of
and a continuous function
defined as
. If
and 
satisfies the Lipschitz condition
in the variable
in 
:
where 
is a constant. Then the ordinary differential equation
defined as
with the initial condition
has a unique solution
on some interval
.
be an open subset
of
and a continuous function
defined as
. If
and satisfies the Lipschitz condition
in the variable
in :
is a constant. Then the ordinary differential equation
defined as
on some interval
.The above theorem is also named the Picard-Lindelöf theorem and can be generalized to a system of first order ordinary differential equations
Theorem 2 (generalization of Picard's theorem [KF])
Let 
be an open subset of
and a continuous function
defined as
. If
and 
satisfies the Lipschitz condition in the variable
in 
:
where 
is a constant. Then the system of ordinary differential equation defined as
with the initial condition
has a unique solution
on some interval
.
be an open subset of
and a continuous function
defined as
. If
and satisfies the Lipschitz condition in the variable
in :
is a constant. Then the system of ordinary differential equation defined as
 |
 |
|
 |
||
 |
 |
.see also:
References
- KF
- Kolmogorov, A.N. & Fomin, S.V.: Introductory Real Analysis, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.