: neighborhood of existence, local existence — solution exists locally.
interval of existence.
If the solution exists in the interval , then we say that the solution exists non-locally.
If and are intervals of existence, then is also an interval of existence.
is a collection of intervals of existence, then is called the maximal interval of existence.
Example.
Interval of existence:
Theorem
Suppose be a bounded continuous function defined on a domain and is Lipschitz in on the domain.
Then, the largest open interval over which the solution with of the IVP
is defined in any one of the following two types:
(i) where both are finite, or either or is finite.
(ii) the entire axis, .
Proof
Let .
Given that,
We have and by Picard’s theorem, there exists a solution. Let it be .
Let and .
Clearly .
Reapply Picard’s theorem to
There exists a solution, say , such that in , .
Define
Claim is a solution.
Clearly, is continuous.
In , we have
In , we have
i.e.,
But,
Continue the process, we can extend the solution to , .
Apply the above process to the left side of the interval , we get an interval such that,
Let and whether limit exists finitely or infinitely.
In either case, we can extend the interval to an where is of the following form: , , , or depends on the limit.
Theorem
Let be a real continuous function on the strip,
and satisfy the Lipschitz condition with constant . Then the successive approximation of ; exists on the existence interval and converges to a solution of the IVP.
Proof
We note that the region is a strip and need not be bounded on . Since is not bounded, the procedure that we followed in Picard’s theorem to prove the convergence is not applicable.
But is the partial sum of the series,
Since is not bounded, we follow a different approach to prove that the series converges.
Let and .
Claim and are well defined.
Since is given to us, is well defined. For a fixed , is continuous on .
Hence,
is continuous on .
attains maximum on .
is well defined.
Let .
Then,
In general,
By applying Weierstrass method similar to Picard’s theorem, we see that the same converges to .