EXISTENCE OF A PERIODIC SOLUTION FOR CONTINUOUS AND DISCRETE PERIODIC SECOND-ORDER EQUATIONS WITH VARIABLE POTENTIALS Summary: EXISTENCE OF A PERIODIC SOLUTION FOR CONTINUOUS AND DISCRETE PERIODIC SECOND-ORDER EQUATIONS WITH VARIABLE POTENTIALS DOUGLAS R. ANDERSON AND RICHARD I. AVERY Abstract. Green's functions for new second-order 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 second-order periodic boundary value problems with periodic coefficient functions. A new version of the Leggett-Williams fixed point theorem is employed. 1. Introduction to the Continuous Periodic Problem Consider the new second-order 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 second-order 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 Collections: Mathematics