Ninth MSU-UAB Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 20 (2013), pp. 103-117.

Title: A Landesman-Lazer condition for the boundary-value problem
       -u''=a u^+ - b u^- +g(u) with periodic boundary conditions

Authors: Quinn A. Morris (The Univ. of North Carolina, Greensboro, NC, USA)
         Stephen B. Robinson (Wake Forest Univ., Winston-Salem, NC, USA)
	  
Abstract: 
 In this article we prove the existence of solutions for the
 boundary-value problem
 $$\displaylines{
	-u''=a u^+ - b u^- +g(u)\cr
	u(0)=u(2 \pi)\cr
	u'(0)=u'(2 \pi),
 }$$
 where $(a,b)\in \mathbb{R}^2$, $u^+ (x) = \max \{u(x),0\}$,
 $u^- (x) = \max \{-u(x),0\}$, and $g: \mathbb{R} \to \mathbb{R}$
 is a bounded, continuous function. We consider both the resonance
 and nonresonance cases relative to the Fucik Spectrum.
 For the resonance case we assume a generalized Landesman-Lazer
 condition that depends upon the average values of g at $\pm\infty$.
 Our theorems generalize the results in [1] by removing
 certain restrictions on (a,b). Our proofs are also different in
 that they rely heavily on a variational characterization of
 the Fucik Spectrum given in [3].

Published October 31, 2013.
Math Subject Classifications: 34B15.
Key Words: Fucik spectrum; resonance; Landesman-Lazer condition; 
           variational approach.