Global Bifurcation for Fourth-Order Differential Equations with Periodic Boundary-Value Conditions

Аннотация

We establish the global structure of positive solutions of fourth-order periodic boundary-value problems u⁗(t) + Mu(t) = λf (t, u(t)), t ∈ [0, T], u^k(0) = u^(k)(T), k = 0, 1, 2, 3, with M ∈ (0, 4(2πnM₄/T)⁴) and u⁽⁴⁾(t) − Mu(t) + λg(t, u(t)) = 0, t ∈ [0, T], u^k(0) = u^(k)(T), k = 0, 1, 2, 3, with M ∈ (0, (2πM₄/T)⁴);here g, f ∈ C([0,T] × [0, ∞), [0, ∞)), M is constant, and λ> 0 is a real parameter. The main results are based on a global bifurcation theorem.