The Self-adjointness And Eigenvalues Of 4th Order Singular Differential Operators With Eigenparameter Dependent Boundary Conditions

In this paper, we study the self-adjointness and eigenvalue problems of discontinuous singular 4th order differential operator boundary problems with eigenparameters. The self-adjointness and eigenvalue problems are much important for linear differential operator problems. It is important while discribing the physical phenomenon that the eigenvalues are real valued for the self-adjoint operators. Many researchers contribute to the self-adjointness and eigenvalue of regular Sturm-Liouville problems with transferring conditions and eigenparameters dependent on the boundary, however, there are few results for the singular higher order cases.Firstly the self-adjointness of a class singular 4th order differential operators with eigenparameters dependent boundary conditions is considered. In the next, we discuss the self-adjointness and eigenvalue problems of a class of singular 4th order differential operators with transferring conditions and. eigenparameters dependent boundary conditions. By constructing the inner product associated with the transferring conditions and eigenparameters dependent boundary conditions, we studied this issue in the Hilbert space associated with the transferring conditions and eigenparameters dependent boundary conditions, and defined the operator T associated with transmission conditions and eigenparameters dependent boundary conditions, proved that they are conjugate each other, and gave the necessary and sufficient conditions of differential operators for the self-adjoint operators.We give the proof of the self-adjointness of the two problems. Then for the latter, that is a class of singular 4th order differential operators with transferring conditions and eigenparameters dependent boundary conditions. By constructing the fundamental solution, we get the necessary and sufficient conditions ofλto be the eigenvalue of this class of differential operators. And by constructing the whole function, we transferee this problem into the zero problem of whole function, finally we obtained that the eigenvalue are countable at most, and no limited accumulation.