Dear Colleagues,

We continue our Colloquium.

Professor Johnny Henderson,

Department of Mathematics, Auburn University

Differential Inequality Methods for Solutions of Boundary Value Problems for
Ordinary Differential Equations

April 12, Friday, Room 207, 2:00pm.

Abstract. Employing upper and lower solutions methods to obtain solutions of
boundary value problems for ordinary differential equations enjoys quite a
history. In particular, if $\beta$ is an upper solution and $\alpha$ is a
lower solution, respectively, of $y'' = f(x,y,y')$ on $[a,b]$, (i.e.,
$\beta''(x) \leq f(x,\beta(x),\beta'(x))$ and $\alpha''(x) \geq
f(x,\alpha(x),\alpha'(x))$), with $\alpha(x) \leq \beta(x)$ on $[a,b]$,
conditions are sought for $f$ such that, for each $\alpha(a) \leq A \leq
\beta(a)$ and $\alpha(b) \leq B \leq \beta(b),$ there is a solution,$y$, of
$y'' = f(x,y,y')$ satisfying $y(a)=A,$ $y(b)=B,$ and $\alpha(x) \leq y(x)
\leq \beta(x)$ on $[a,b].$
Extensions of upper and lower solutions methods will also be discussed for
certain elliptic boundary value problems as well as for higher order
problems.