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) = lambda q(t)f(,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 is an element of C ([0, 1] x R+, R+), g is an element of C ([0, 1], R+) and lambda 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 lambda to obtain our results. Our approach is based on a revised version of a fixed point theorem due to Gustafson and Schmitt.