# Mixed-endpoint boundary value problem

Consider the following mixed-endpoint boundary-value problem:

$$- \displaystyle \frac{d}{dt} \left (p(t) \frac{du}{dt} \right ) + q(t)u = \lambda u, \ \ \ \ u(0) = u(T), \ u'(0) = u'(T)$$

here the coefficients $p$ and $q$ are defined and continuous for all real $t$ and periodic with period $T: p(t + T) = p(t)$ and $q (t + T) = q (t)$; in addition assume that $p$ is positive and $C^1$. Show that any solution satisfying the given data is likewise periodic with period $T$.

We have just started studying boundary-value problems. Could anybody please give me any help at all on this problem? Thank you very much in advance!

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You want to prove something about $u(t+T)$. Does it satisfy some differential equation, by any chance? Maybe even with the same initial conditions as $u$? –  user53153 Mar 7 '13 at 4:53
I am very sorry but I am failing to understand whether this is a hint or a question. –  user44069 Mar 7 '13 at 5:00
It was meant as a hint. To make it more explicit and more actionable: prove that $u(t+T)$ is also a solution of your IVP. –  user53153 Mar 7 '13 at 5:03
Thank you very much for the hint! –  user44069 Mar 7 '13 at 5:04
Check Wikipedia on Sturm–Liouville theory –  Kaster Mar 7 '13 at 6:48