Nodal Solutions for a Class of Fourth-Order Two-Point Boundary Value Problems

We consider the fourth-order two-point boundary value problem u′′′′ Mu λh t f u , 0 < t < 1, u 0 u 1 u′ 0 u′ 1 0, where λ ∈ R is a parameter, M ∈ −π4, π4/64 is given constant, h ∈ C 0, 1 , 0,∞ with h t /≡ 0 on any subinterval of 0, 1 , f ∈ C R,R satisfies f u u > 0 for all u/ 0, and limu→−∞f u /u 0, limu→ ∞f u /u f ∞, limu→ 0f u /u f0 for some f ∞, f0 ∈ 0, ∞ . By using disconjugate operator theory and bifurcation techniques, we establish existence and multiplicity results of nodal solutions for the above problem.