On the existence of eigenvalues of differential operators dependent on a parameter

Abstract:

In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on $\lambda$, \[ \begin {array}{*{20}{c}} {{{u’}_k}(x) = \sum \limits _{j = 1}^n {{b_{kj}}(x, \lambda ){u_j}(x),} } & {\sum \limits _{i = 0}^p {\sum \limits _{j = 1}^N {a_{kj}^i{u_j}({x_i}) = 0,} } } \\ \end {array} \] , $k = 1,..., N$. In order to get these results we prove that this system can be reduced to a standard system of the form \[ \begin {array}{*{20}{c}} {{{y’}_k}(x) = \sum \limits _{j = 1}^n {{a_{kj}}(x, \lambda ) {y_j}(x)} ,} & {{y_k}(0) = {a_k}(\lambda ),} & {{y_n}(1) = 0,} \\ \end {array} \] $k = 1,..., n$.