On a nonlinear eigenvalue problem

In the present paper we consider a self-adjoint and nonsmooth operator-valued function on (c, d) subset of R-1. We suppose that the equation (L(alpha)x, x) = 0, x not equal 0, has exactly one root p(x) is an element of (c, d) and the function f(alpha) = (L(alpha)x, x) is increasing at the point p(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.