Multiple Solutions Of Periodic Boundary Value Problems Of Second Order Ordinary Differential

In this paper, we study the existence of multiple nontrivial solutions for a kind of ordinary differential equation of second order with periodic boundary value by applying minimax methods and Morse theory.Consider the boundary value problemwhere f : [0, 2Ï€] × R â†' R is C¹ function , f(t, 0) = 0 and satisfiesBecause the eigenvalue of linear eigenvalue problemis m², m = 0,1, 2, 3 .... and the multiplicity of m² is 2 when m≥ 1. (1.1) implies (P) is resonant between two consecutive eigenvalues. In this paper, we study the existence of nontrivial 2π—period solutions. In Sobolev spacewe definethen H is a Hilbert space. Definewhere F(t,x) = ∫₀^x ∫(t,s) ds. Since f∈C¹. J∈ C² (H, R), and its Frechet drivative...