 
Summary: SOLUTIONS TO SECONDORDER THREEPOINT PROBLEMS ON TIME
SCALES
DOUGLAS R. ANDERSON
Abstract. In the first part of the paper we establish the existence of multiple positive solutions
to the nonlinear secondorder threepoint boundary value problem on time scales,
u
(t) + f(t, u(t)) = 0, u(0) = 0, u() = u(T)
for t [0, T] T, where T is a time scale, > 0, (0, (T)) T, and < T. We employ
the LeggettWilliams fixedpoint theorem in an appropriate cone to guarantee the existence of
at least three positive solutions to this nonlinear problem. In the second part we establish the
existence of at least one positive solution to the related problem
u
(t) + a(t)f(u(t)) = 0, u(0) = 0, u() = u(T),
using Krasnoselskii's fixedpoint theorem of cone expansion and compression of norm type.
1. preliminaries about time scales
The following definitions, that can be found in Atici and Guseinov [4] and Bohner and
Peterson [7], lay out the terms and notation needed later in the discussion. A time scale T is
any nonempty closed subset of R. It follows that the jump operators , : T T
(t) = inf{s T : s > t} and (t) = sup{s T : s < t}
(supplemented by inf := sup T and sup := inf T) are well defined. The point t T is
