The Fučík spectrum for discrete systems and some nonlinear existence theorems

Abstract

In this paper, we study the existence solutions to nonlinear Fučík problems of the form(1)Ax=αx+−βx−+g(x), where A is a symmetric n×n matrix, α,β are real numbers, and g:Rn→Rn is continuous. The nonlinear problem (1) is motivated by application to nonlinear oscillating systems such as the Tacoma Narrows Bridge.The paper begins by developing a qualitative picture of Fučík spectrum associated with the matrix equationAx=αx+−βx−. In this setting, we present two characterizations: first, we show that under appropriate assumptions the Fučík spectrum consists of curves bifurcating from points (λ,λ)∈R2, where λ is an eigenvalue of A; second, we give more global variational characterization of the Fučík curves. In both cases, we present various qualitative properties of the Fučík curves. The paper finishes by presenting two existence theorems for the nonlinear Fučík problem under mild assumptions on the nonlinear term g.