Fixed Point Theory, Volume 8, No. 1, 2007, 69-85 http://www.math.ubbcluj.ro/ ∼ nodeacj/sfptcj.htm ON SOME GENERALIZATIONS OF THE LANDESMAN-LAZER THEOREM VALERI OBUKHOVSKII * , PIETRO ZECCA ** AND VICTOR ZVYAGIN * * Faculty of Mathematics Voronezh State University Universitetskaya pl.,1 (394006) Voronezh, Russia E-mail: valerio@math.vsu.ru; zvg@main.vsu.ru ** Dipartimento di Energetica ”S.Stecco” Universita di Firenze via S.Marta, 3 (50139) Firenze, Italia E-mail: zecca@unifi.it Abstract. We apply the coincidence degree theory for compact multivalued perturbations of Fredholm operators to obtain necessary and sufficient conditions for the existence of solutions for an equation containing a linear Fredholm operator with an one-dimensional kernel and a discontinuous nonlinearity. Further we consider the extension to the case when the kernel is multi-dimensional and the Fredholm operator is not necessarily self-adjoint. Some examples are given. Key Words and Phrases: Landesman–Laser equation, Fredholm operator, resonance, degeneracy, discontinuous nonlinearity, multivalued map, coincidence point, coincidence de- gree, topological degree. 2000 Mathematics Subject Classification: 47J05, 35J25, 35R05, 47H04, 47H11, 54H25. 1. Introduction In the work of E.M. Landesman and A.C. Lazer [14] it was observed that the boundary value problem for a nonlinear elliptic equation at the presence First and third author are supported by the Russian FBR Grants 04-01-00081 and 05- 01-00100. The second author is supported by the Italian Cofin 04-05. All authors work is partially supported by the NATO Grant ICS.NR.CLG 981757. 69