Some variational principles for a nonlinear eigenvalue problem

where L and M are linear differential operators on a finite interval [u, 61 (cf. [I], p. 430). If one works in L,[a, b], in typical cases the problems are such that L is self-adjoint, and M is an operator of lower order, usually symmetric. This problem was examined by Shinbrot [2] who wrote it in the form Au = Au + h2Bu with h = p-l, A = L-l and B = -L-lM (ML-1 seems preferable if one wants to show equivalence with (1.1)). He then treated the problem Au = Au + kB,p with a! > 1, A = A* compact, and B, a bounded linear operator, Lipschitz continuous in h with respect to the operator-norm. Under conditions on the location of the spectrum of A and on the size of the norms 11 A 11 and 11 B, 11, he showed that the nonlinear eigenvalue problem has a sequence of eigenvalues converging to zero and a corresponding basis of eigenvectors. Here, with some conditions on L and M, we reduce (1.1) to a problem