A Note on the Eigenvalue Criteria for Positive Solutions of a Cantilever Beam Equation with Free End

In this paper, we study the existence of at least one positive solution to the fourth-order two-point boundary value problem (BVP) {u (t)=λq(t) f (t, u(t)), 0 t 1, u (0) = u (0) = u (1)= u (1) = 0,which models a cantilever beam equation, where one end is kept free. Here f∈C([0,1]×R+,R+), g∈C([0,1],R+) and λ is a positive parameter. The sufficient conditions are interesting, new and easy to verify. We have used some inequalities on the nonlinear function f and eigenvalues of a linear integral operator as bounds for the parameter λ to obtain our results. Our approach is based on a revised version of a fixed point theorem due to Gustafson and Schmitt.