Asymptotic problems for fourth-order nonlinear differential equations

By a solution of () we mean a function x ∈ C[Tx,∞), Tx ≥ , which satisfies () on [Tx,∞). A solution is said to be nonoscillatory if x(t) =  for large t; otherwise, it is said to be oscillatory. Observe that if λ≥ , according to [, Theorem .], all nontrivial solutions of () satisfy sup{|x(t)| : t ≥ T} >  for T ≥ Tx, on the contrary to the case λ < , when nontrivial solutions satisfying x(t)≡  for large t may exist. Fourth-order differential equations have been investigated in detail during the last years. The periodic boundary value problem for the superlinear equation x() = g(x) + e(t) has been studied in []. In [], the fourth-order linear eigenvalue problem, together with the nonlinear boundary value problem x() – f (t,x) = , has been investigated. Oscillatory properties of solutions for self-adjoint linear differential equations can be found in []. Equation () with q(t)≡  can be viewed as a prototype of even-order two-term differential equations, which are the main object of monographs [, , ].