Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions

We establish new existence results for multiple positive solutions of fourth order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many nonlocal boundary conditions, with a unified approach. Our method is to show that each boundary value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is \emph{not} assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution, for multiple solutions we seek optimal values of some of the constants that occur in the theory which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our nonlocal boundary conditions contain multi-point problems as special cases and, for the first time in fourth order problems, we allow coefficients of both signs.