Discussion On Asymptotic Behavior Of Eigenvalues Of Some Differential Operators

This paper is mainly focused on eigenvalue problems of several kinds ofdifferential operators and gives asymptotic formula of eigenvalue.Firstly, using methods of the Green-Liouville transformation and the theoryof functions, the result from discussing asymptotic formula of eigenvalueswith mixed boundary conditions and a right-definite Sturm-Liouville problemis more universal than the ones in literatures [8]. Secondly, by the methods ofthe Green-Liouville transformation and Fréchet derivative, an asymptoticformula of an eigenvalue with periodic boundary conditions and right-definiteSturm-Liouville problem is studied. And the result is more precise than theones in literature [22]. This demonstrates that the coefficient of equation andweighting function influence the result of eigenvalue through asymptoticformula.Thirdly, the eigenvalue problem of the second-difference operator has beenapplied into the fourth-difference operate in analyzing the eigenvalue`sasymptotic formula of a fourth-difference operate with separated boundaryconditions and a fourth-difference operate with periodic boundary conditions.At last, the existence, distribution and asymptotic formula of eigenvalue couldbe gotten by combining the methods of theory of functions and Rouchétheorem of complex functions.There are five parts in this paper: First part is introduction; Second part isthe analysis on the asymptotic formula of asymptotic formula of eigenvalueswhich have a right-definite Sturm-Liouville problem and with MixedBoundary Conditions; Third part is the analysis on the asymptotic formula ofasymptotic formula of a eigenvalue which has a right-definite Sturm-Liouvilleproblem and with periodic Boundary Conditions; Fourth part is the analysison the asymptotic formula of a fourth-difference operate with periodic boundary conditions; Fifth part is the analysis on the asymptotic formula of afourth-difference operate with periodic boundary conditions.