Existence Of Solutions For Beam Equation With Parameters And Periodic-integrable Boundary Value Problems

The existence of solutions for beam equations and periodic-integrable boundary value problem are considered by many authors. Firstly, we consider the existence of nontrivial and multiple periodic solutions of beam equation u′′′′- pu′′- g(t, u) = e(t) by variational methods.To some extent, the known conclusions are generalized. Secondly, the existence of nontrivial solutions of boundary value problem u′′′′- ξu = f(t, u) with u(0) = u′(0) = u′′(1) =0, u′′′(1) = g(u(1)) is considered under the nonresonance conditions by using the same methods. In addition, we also consider the existence of solutions for a periodic-integrable boundary value problem u′′+ f(t, u) = 0 with u(0) = u(2π),∫2π0u(s)ds = 0.