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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

  

Source: Anderson, Douglas R. - Department of Mathematics and Computer Science, Concordia College

 

Collections: Mathematics