Two-parameter nonlinear eigenvalue problems Armands Gritsans * , Felix Sadyrbaev † and Natalija Sergejeva †,* * <author1 address> † <common address for author2 and author3> Abstract. We consider two types of Fuˇ cík like spectra, which have somewhat peculiar features. The first one stretches along the bysectrix of the plane of parameters, the second one may contain separated components. Both spectra are significant for existence of solutions to nonlinear boundary value problems with nonlinearities having asymmetric asymptotics. Keywords: nonlinear equations, spectra, asymptotically asymmetric nonlinearities PACS: 80 INTRODUCTION We are motivated by the Fuˇ cík problem x 00 = -λ x + + μ x - , (1) x(0)= 0, x(1)= 0, (2) where x + = max{x, 0}, x - = max{-x, 0}. The spectrum of this problem is well known and it is a set of hyperbolas looking like in Fig. 1. 50 100 150 200 250 300 Λ 50 100 150 200 250 300 Μ F - 0 F + 0 F - 1 F + 1 F - 2 F + 2 F - 3 F + 3 F - 4 F + 4 FIGURE 1. Spectrum for the problem (1), (2)