Existence and Uniqueness of Solutions
There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The two main theorems are
-
Theorem Assumption Conclusion Peano existence theorem F continuous local existence only Picard–Lindelöf theorem F Lipschitz continuous local existence and uniqueness
which are both local results.
Read more about this topic: Ordinary Differential Equation
Famous quotes containing the words existence and, existence, uniqueness and/or solutions:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)
“We need not only a purpose in life to give meaning to our existence but also something to give meaning to our suffering. We need as much something to suffer for as something to live for.”
—Eric Hoffer (1902–1983)
“Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.”
—Daniel J. Boorstin (b. 1914)
“Those great ideas which come to you in your sleep just before you awake in morning, those solutions to the world’s problems which, in the light of day, turn out to be duds of the puniest order, couldn’t they be put to some use, after all?”
—Robert Benchley (1889–1945)