Splitting the Fucík Spectrum and the Number of Solutions to a Quasilinear ODE

For φ an increasing homeomorphism from R onto R , and f ∈ C ( R ) , we consider the problem ( φ ( u ′ )) ′ + f ( u ) = 0 , t ∈ (0 ,L ) , u (0) = 0 = u ( L ) . The aim is to study multiplicity of solutions by means of some gen- eralized Pseudo Fuˇc ́ık spectrum (at infinity, or at zero). N ew insights that lead to a very precise counting of solutions are obtaine d by split- ting these spectra into two parts, called Positive Pseudo Fu ˇc ́ık Spectrum (PPFS) and Negative Pseudo Fuˇc ́ık Spectrum (NPFS) (at infin ity, or at zero, respectively), in this form we can discuss separate ly the two cases u ′ (0) > 0 and u ′ (0) < 0 .