Asymptotic Distribution Of Eigenvalue For A Class Of Regular Non-local Sturm-Liouville Problems

The theory of ordinary differential operators originated from the mathematical treat-ment of heat conduction model by Fourier.In 1830,Sturm and Liouville discussed the Fourier's method in general cases when studying the solution of the vibrating string equa-tion,the results is the theoretical basis for solving a large class of mathematical physics equations,their research results were promoted the development of differential equation-s,and they had done a lot of research in this field,they had been published a series of articles,eventually established Sturm-Liouville operator represented by differential opera-tor theory with the development of mathematics step by step.In 1910,H.Weyl extended the classical S-L problem to the singular S-L problem which is defined in the infinite interval and pioneered the theory of the singular S-L problem.The important,theory of quantum mechanics was established in early 20th century,quantum mechanics is a theory that describes microscopic materials,it is considered the two basic theories of modern physics with relativity theory.The theory of singular operators has played an important role in the initial development of quantum mechanics,so more and more scholars had started to study the theory of differential operators.The main contents of the study on the theory of differential operators are includ-ing spectrum analysis,defect index,inverse spectrum problems.Spectral analysis is also called eigenvalue analysis,it is the study of eigenvalue related properties and distribution and other problems.In 1950,Titchmarch,Everitt and other mathematicians had started to study eigenvalue properties,spectrum of operators,eigenfunction expansion and other problems,they had made important research results.In 1960,because of the developmen-t of modern physics and modern science,the researchers are no longer confined to the study of classical differential equations,they started to study the definite solutions of nonlocal differential equations,non-local differential equations come from a wide range of sources,for example,reaction diffusion theory?quantum mechanical point interference problem?electrical power system of Voltage-Driven,etc.It is necessary to solve these non-local differential problems.In 1987,Cao zhijiang obtained the asymptotic estimation formula of the eigenvalue of S-L problem with respect to n with the method of theory of functions.In 1996,Kong Q and Zettl A proved that the coefficients,weight functions and boundary conditions of the equation were all related to the eigenvalues.Since then,many scholars has started to esti-mate the eigenvalues estimated asymptotically with different research methods,the results were got more and more accurate.Many researchers have also done a lot of research on this problem under special circumstances for the spectrum of non-local problems,however,the study of spectrum theory of non-local differential equations is still at an initial stage,there are not many results of asymptotic analysis of eigenvalues.This dissertation mainly considers asymptotic distribution of eigenvalues for the regular nonlocal Sturm-Liouville problem,first,we obtain the analytic form for the so-lution of Cauchy problem,secondly,we obtained the asymptotic expression for the so-lution of Cauchy problem by iterative method and infinitesimal analysis,then we got the asymptotic distribution formula for the eigenvalues in the case of q?0 by Rouche theorem,furthermore,we obtained the asymptotic distribution formula of eigenvalues by Frechet derivative.The main research work are arranged as following:Chapter 1,prolegomenon,we mainly introduces the domestic and foreign background of the research problem,the source of the problem and the current research status in this chapter.Chapter 2,we give preliminary knowledge of theoretical knowledge for the regular Sturm-Liouville problem and the relevant conclusions of this paper.Chapter 3,we mainly research the asymptotic estimation of eigenvalues for regu-lar nonlocal problems,at first,we obtain the analytic form for the solution of Cauchy problem.Secondly,we obtained the asymptotic formula for the solution of Cauchy prob-lem by using iteration method and infinitesimal analysis.Then,we got the asymptotic distribution formula of the eigenvalue of the local potential function q?0 by Rouche theorem.Finally,we obtained the asymptotic distribution formula of the eigenvalues of the nonlocal Sturm-Liouville problem by Frechet derivative.