Picard-lindelf Theorem
In
mathematics
, the
Picard-Lindelf theorem
on
existence
and
uniqueness
of solutions of
differential equations
(
Picard
1890
,
Lindelf
1894
) states that an
initial value problem
y'(t)=f(t,y(t)),\quad y(t_0)=y_0
has exactly one solution if
f
is
Lipschitz continuous
and
bounded
. A simple
proof
is successive approximation: (also called
Picard iteration
) Set
\varphi_0(t)=y_0 \,\!
and
\varphi_i(t)=y_0+\int_{t_0}^{t}f(s,\varphi_{i-1}(s))\,ds.
It can then be shown rather easily that the sequence of the
\varphi_i \,\!
(called the Picard iterates) is
convergent
and that the
limit
is a solution to the problem.
See also
Frobenius theorem
Integrability conditions for differential systems
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