Non-resonant boundary value problems with singular -Laplacian operators

In this paper, using Leray-Schauder degree arguments, critical point theory for lower semicontinuous functionals and the method of lower and upper solutions, we give existence results for periodic problems involving the relativistic operator u bart right arrow (u'/root 1-u'2)' + r(t)u with integral(T)(0) r dt not equal 0. In particular we show that in this case we have non-resonance, that is periodic problem (u'/root 1-u'2)' + r(t)u = e(t), u(0) - u(T) = 0 = u'(0) - u'(T), has at least one solution for any continuous function e : left perpendicular0, Tright perpendicular -> R. Then, we consider Brillouin and Mathieu-Duffing type equations for which r(t) equivalent to b(1) + b(2) cos t and b(1), b(2) is an element of R.