Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 112, pp. 1-34.

Title: Periodic boundary-value problems and the Dancer-Fucik
       spectrum under conditions of resonance

Authors: David A. Bliss   (Claremont Graduate Univ., Claremont, CA, USA)
         James Buerger    (Pomona College, Claremont, CA, USA)
	 Adolfo J. Rumbos (Pomona College, Claremont, CA, USA)

Abstract:
 We prove the existence of solutions to the nonlinear $2 \pi$-periodic
 problem
 $$\displaylines{
 u''(x)+\mu u^+(x)-\nu u^-(x)+g(x,u(x))=f(x)\,,\quad
 x\in (0,2\pi)\,,\cr
 u(0)-u(2\pi) =0 \,, \quad  u'(0) - u'(2\pi)=0,
 }$$
 where the point $(\mu,\nu)$ lies in the Dancer-Fucik
 spectrum, with
 $$
 0< \frac{4}{9}\mu \leqslant \nu<\mu \quad\hbox{and}\quad \mu<(m+1)^2,
 $$
 for some  natural number m, and the nonlinearity $g(x,\xi)$ is bounded
 with primitive,  $G(x,\xi)$, satisfying a
 Landesman-Lazer type  condition introduced by  Tomiczek in 2005.
 We use variational methods based on the generalization of the
 Saddle Point Theorem of Rabinowitz.

Submitted November 10, 2010. Published August 29, 2011.
Math Subject Classifications: 34B15, 34K13, 35A15.
Key Words: Resonance; jumping nonlinearities; Dancer-Fucik spectrum;
           saddle point theorem.