Abstract |
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For p > 1, and phi (p)(s) = s (p-2)s, we consider the equation
(phi (p)(x'))' + alpha phi (p)(x(+)) - X phi (p)(x(-)) = f(t, x),
where x(+) max(x, 0); x(-) = max(-x,0), in a situation of resonance or near resonance for the period T, i.e. when alpha,beta satisfy exactly or approximately the resonance for the period T, i.e. equation
pi (p)/alpha (1/p) + pi (p)/beta (1/p) = T/n,
for some integer n. We assume that f is continuous, locally Lipschitzian in x, T-periodic in t, bounded on R-2, and having limits f(-/+)(t) for x --> +/- infinity, the limits being uniform in t. Denoting by v a solution of the homogeneous equation
(phi (p)(x'))' + alpha phi (p)(x(+)) - beta phi (p)(x(-)) = 0
we study the existence of T-periodic solutions by means of the function
Z(theta) = integral ((t is an element ofI v theta (t)>0)) f (+)(t)v(t + theta) dt + integral ((t is an element ofI v theta (t)<0)) f(-)(t)v(t + <theta>) dt,
where I-(def) double under bar [0,T]. In particular, we prove the existence of T-periodic solutions at resonance when Z has 2z zeros in the interval (0,T/n), all zeros being simple, and z being different from 1. |