On Fučík type spectrum for problem with integral nonlocal boundary condition

Abstract

The Fučík equation x' '= -μ x+λ x- with two types of nonlocal boundary value conditions are considered. The Fučík type spectrum for both problems are constructed. The visualization of the spectrum for some values of parameter γ is provided.

Localization of Fučík curves for the second order discrete Dirichlet operator

“…As in the previous case, using (39), we simplify G β (Q α (μ α,β )) as V β κ β +1 /V β κ β , which justifies (70). Finally, if we combine (69) and (70) and use (27) then we obtain (71).…”

“…Finally, let us note that asymmetric nonlinearities also surprisingly appear in the study of competing systems of species with large interactions in biology (see [4,6,22]) and the Fučík spectrum of the Dirichlet Laplacian (the Laplace operator u → −Δu with zero Dirichlet boundary conditions) is needed (see [6] for details). Nowadays, there are a number of papers in which authors study the structure of the Fučík spectrum for particular linear differential operators, let us mention here only some of them: [1,2,7,9,14,23,24] for the Dirichlet Laplacian on bounded domains, [3,13,15,16,26,27] for the ordinary differential operators with various boundary conditions (Dirichlet, Neumann, Robin, Navier, periodic, multipoint, integral type).…”