Positive solutions, existence of smallest eigenvalues, and comparison of smallest eigenvalues of a fourth order three point boundary value problem

The existence of smallest positive eigenvalues is established for the linear differential equations u(4) + lambda1q(t)u = 0 and u(4) + lambda2r(t)u = 0, 0 ≤ t ≤ 1, with each satisfying the boundary conditions u(0) = u'(p) = u''(1) = u'''(1) = 0 where 1-33 ≤ p < 1. A comparison theorem for smallest positive eigenvalues is then obtained. Using the same theorems, we will extend the problem to the fifth order via the Green's Function and again via Substitution. Applying the comparison theorems and the properties of u0-positive operators to determine the existence of smallest eigenvalues. The existence of these smallest eigenvalues is then applied to characterize extremal points of the differential equationu(4)+q(t)u = 0 satisfying boundary conditions u(0) = u'(p) = u''(b) = u'''(b) = 0 where 1-33 ≤ p ≤ b ≤ 1. These results are applied to show the existence of a positive solution to a nonlinear boundary value problem.