Constant sign solutions of two-point fourth order problems

In this paper we characterize the sign of the Green's function related to the fourth order linear operator u((4)) + Mu coupled with the two point boundary conditions u(1) = u(0) = u'(0) = u ''(0) = 0. We obtain the exact values on the real parameter M for which the related Green's function is negative in (0, 1) (0, 1). Such property is equivalent to the fact that the operator satisfies a maximum principle in the space of functions that fulfil the homogeneous boundary conditions. When M > 0 the best estimate follows from spectral theory. When M < 0, we obtain an estimation by studying the disconjugacy properties of the solutions of the homogeneous equation u((4)) + Mu = 0. The optimal value is attained by studying the exact expression of the Green's function. Such study allow us to ensure that there is no real parameter M for which the Green's function is positive on (0, 1) x (0, 1). Moreover, we obtain maximum principles of this operator when the solutions verify suitable non-homogeneous boundary conditions. We apply the obtained results, by means of the method of lower and upper solutions, to nonlinear problems coupled with these boundary conditions. (C) 2015 Elsevier Inc. All rights reserved.