Elastic beam problem with higher order derivatives

We apply a fixed point theorem to obtain sufficient conditions for existence of triple positive solutions of the symmetric nonlinear fourth-order boundary value problemu(4)(t)=a(t)f(t,u(t),|u′(t)|,u″(t),|u‴(t)|),t∈(-1,1),u(-1)=u′(-1)=0,u(-1)=u′(-1)=0,u(-t)=u(t),t∈[-1,1],where f satisfies certain growth conditions. In the process, we obtain nontrivial a priori bounds on the derivatives of the solution.