The assumption (f2) characterizes problem (1.1) as double resonant between two consecutive eigenvalues at infinity.
In 2011, Zhang and Wang [15] used variational methods and Morse theory to study the multiplicity of periodic solutions for (P) with double resonance between two consecutive eigenvalues at infinity.
This results in N self-adjoint Fredholm integral equations, the non-trivial solutions of which predict eigensolutions of P. As a consequence, a periodic structure P with period N will have exactly N eigenvalues lying between two consecutive eigenvalues of the substructure S i), if the interface I(i) contains only one degree of freedom.
Hence, (1.1) and (1.2) with ;, ; has only one eigenvalue between any two consecutive eigenvalues of (1.1) with (2.7), respectively.
Hence, (1.1) and (1.2) with,,, and have only one eigenvalue between any two consecutive eigenvalues of (1.1) with (2.14), respectively.
Similarly, by Propositions 3.3 3.6, the continuity of and the intermediate value theorem, reaches,, and exactly one time, respectively, between any two consecutive eigenvalues of the separated boundary value problem (1.1) with (2.7).
Similarly, by Propositions 4.1, 4.3, and 4.10, the continuity of and the intermediate value theorem, reaches,, and exactly one time between any two consecutive eigenvalues of the separated boundary value problem (1.1) with (2.14).
Assume (2) and (3) and suppose (lambda _{k}two consecutive eigenvalues of (5).
By the fact that and are two consecutive eigenvalues of with corresponding eigenspace and, we have and then, the function is strictly decreasing on with as.
characterizes problem (BP) as double resonance between two consecutive eigenvalues at infinity.
In our parallelization we use a relationship between two consecutive subspaces which allows us to calculate eigenvalues in the subspace through an arrowhead matrix.
