 
Summary: EXISTENCE OF A PERIODIC SOLUTION FOR CONTINUOUS AND DISCRETE
PERIODIC SECONDORDER EQUATIONS WITH VARIABLE POTENTIALS
DOUGLAS R. ANDERSON AND RICHARD I. AVERY
Abstract. Green's functions for new secondorder periodic differential and difference equations with variable
potentials are found, then used as kernels in integral operators to guarantee the existence of a positive periodic
solution to continuous and discrete secondorder periodic boundary value problems with periodic coefficient
functions. A new version of the LeggettWilliams fixed point theorem is employed.
1. Introduction to the Continuous Periodic Problem
Consider the new secondorder differential equation with variable potentials given by
(1.1) Ly(t) = (py ) (t)  r(t)p(t)y (t) + q(t)y(t) = f(t, y(t)), t [0, ),
with the periodic boundary conditions
(1.2) y(t) = y(t + ), y (t) = y (t + ), t [0, ),
where > 0 is the period, and we assume the hypotheses
(H1) p, q, r C(R) with p(t), q(t) > 0 and r(t) 0 for all t [0, );
(H2) p(t) = p(t + ), q(t) = q(t + ), r(t) = r(t + ), for all t [0, );
(H3) f : [0, ) × [0, ) [0, ) is continuous with f(t, z) = f(t + , z).
There are several recent results on secondorder periodic problems. Liu, Ge, and Gui [13] (see also [4]) consider
problem (1.1), (1.2) with r(t) 0. Graef, Kong, Wang [7] give an extensive analysis for a constant coefficient
case,
y (t)  2
