Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems

We consider three nonlinear eigenvalue problems that consist of -y '' + f(y(2))y = lambda y with one of the following boundary conditions: y(0) = y(1) = 0 y'(0) = p, y'(0) = y(1) = 0 y(0) = p, y'(0) = y'(1) = 0 y(0) = p, where p is a positive constant. Under smoothness and monotonicity conditions on f, we show the existence and uniqueness of a sequence of eigenvalues f {lambda(n)} and corresponding eigenfunctions {y(n)} such that y(n)(x) has precisely n roots in the interval (0, 1), where n = 0, 1, 2, .. . For the first boundary condition, we show that {y(n)} is a basis and that {y(n)/parallel to y(n)parallel to} is a Riesz basis in the space L-2(0, 1). For the second and third boundary conditions, we show that {y(n)} is a Riesz basis.