Solving a systemof eqns by given initial conditions
Which has the I.C. . So I take that it means this:
and then it is solved as any other system? We get eigenvector , but since the eigenvalues of the system are two, , we have to find the second generalized eigenvector.
So the second vector would be . Using the general solutions for the system, we get
Would this be correct, or did I misinterpret the initial conditions?
How do I get an estimate for this nonlocal ODE?
Consider the following nonlocal ODE on :
where l is a positive integer and is a real number.
Define the following norm
I want to prove the estimate:
for some constant C independent of , l and f. But I am stuck.
Here is what I tried. Multiply both sides by f and integrate by parts to get:
where I used Cauchy-Schwartz in the before last line. I am not sure how to continue and how to get rid of the f'(1) term.
Any help is appreciated.