The Slef-adjointness And Eigenvalue Of A Class Of4-order Discontinuous Differential Operators With Indefinite Weight Function

In this paper, we study a class of4-order discontinuous differentialoperators with indefinite weight function and eigenparameter-dependentboundary conditions. Their eigenvalue problems and self-adjointness arediscussed. Firstly, a Krein space K and a new operator A related to theboundary value problem are constructed in order to the eigenvalues for Aare identical with the the boundary value problems, then, we prove theoperator A is self-adjointness in the space K. Secondly, we alsoconstruct a Hilbert space H%associated with the space K andself-adjoint operator S in the space H%, by using spectrum theory ofself-adjoint operator in Krein space and the spectral properties of operatorS, we get the conclusion that all of eigenvaules of A are real and they areunbounded from below and from above. An example is given to show thereal and the distribution of eigenvalues. Moreover, applying spectrumtheory of differential operator, the-dependent solutions satisfying theinitial conditions is given, and the necessary and sufficient condition foreigenvalues is obtained, i.e., the problem of eigenvalues is converted tothat of the zeros of the characteristic function. Furthermore, we constructthe Green’s function of the problem.