Resonance with respect to the Fučík spectrum for non-selfadjoint operators

In this paper, we consider a real linear (non-selfadjoint) operator L:Dom(L)⊂L2(Ω)→L2(Ω) and study the solvability of the problem Lu−αu++βu−−g(u)+f=0Lu−αu++βu−−g(u)+f=0 in the resonance case with respect to the Fučík spectrum Σ(L)Σ(L). We extend the standard Landesman–Lazer type conditions and compare our result with results for selfadjoint operators as well as with results for the resonance in the point spectrum (i.e., for α=β∈σ(L)α=β∈σ(L)).