Hi
I know nothing of delay differential equation, but there is a theorem that ensures the existence and uniqueness of a solution : Picard?Lindelöf theorem - Wikipedia, the free encyclopedia
But it requires some conditions.
Dear friends, I need some help with the existence and/or uniqueness of global solutions to first-order linear differential equations.
For instance, let and and consider the following differential equation
Which theorem ensures existence of global solutions to this initial value problem?
I am actually wondering to find this result for delay differential equations
where , , for all and .
Thanks for your help.
Hi
I know nothing of delay differential equation, but there is a theorem that ensures the existence and uniqueness of a solution : Picard?Lindelöf theorem - Wikipedia, the free encyclopedia
But it requires some conditions.
Consider the following initial value problem
where , and satisfies for all .
Set , where .
Theorem 1. Let for some . Assume that there exist and such that for all and for all . Then (1) has a unique solution on , where .
Proof. The proof can be given by following exactly the same arguments in the proof of Picard-Lidelof theorem, and thus omitted here.
Now, consider the following initial value problem
where , and the other arguments are same to that of (1).
Corollary 1. (2) admits a unique solution on .
Proof. Let for , and satisfy and for all (since are continuous, we may always find such constants). The Lipschitz condition holds on with the Lipschitz constant . Let satisfy for . For convenience in the notation define and . We may pick such that , we see that for all . Applying Theorem 1, we see that
admits a unique solution on since . Next, let . We may find such that . And we have for all . Applying Theorem 1, we see that
admits a unique solution on . Repeating in this manner, we obtain the unique solution of (2) on .
Remark 1. If we need to obtain the unique global solution to
where in addition is assumed to hold, we may pick an increasing divergent sequence with the convenience and such that for all , and apply Corollary 1 successively to obtain the unique solution on each of the intervals for by assuming the solution obtained in the previous step as the initial function on the current interval with the convenience . Then, letting for for , we obtain the unique global solution to (3).
Appendix. It is clear that the function as for any fixed .
Remark. The proof is very simple in the case .
proof by bkarpuz