Estimation Of Non - Real Eigenvalue Of Differential Spectral Problem Of Differential Operator Under Different Conditions

As we all know, the regular indefinite Sturm-Liouville problems may possess non-real eigenvalues. In general, determining the exact num-ber of non-real eigenvalues in term of the parameters of the problem is a difficult and interesting open problem in Sturm-Liouville theory. (see [12,13]). With the extensive research of the differential of oper-ator theory, the contents of spectrum operators have greatly enriched, especially on the eigenvalues of spectrum have achieved a major break-through. More and more researchers have paid close attention and re-search on the left-definite, indefinite Sturm-Liouville problems and the estimate on the non-real eigenvalues. The various discussions and appli-cations of this kind of problem are found in many literature [4,5],[7-12] and [3,14,15,25,27,28]. Particularly, most researchers only estimated the eigenvalues of second order Sturm-Liouville problems. So that the re-search value and practical have been greatly limited. And the estimation in most articles were in the classical boundary conditions (such as Dirich-let boundary conditions, separation boundary conditions, etc.). There are few people research the estimation of eigenvalues in couple bound-ary conditions. Inspired by [14,15,26,27,28], the main objective of this paper is to given a special weight function with different limit condition-s, put the estimation of non-real eigenvalues of second order indefinite Sturm-Liouville problems into fourth order differential equation, mean-while we investigate the real and imaginary parts of the regular indefinite Sturm-Liouville problems with couple boundary conditions.By using the classical analysis techniques and spectral theory of linear operator, we obtain the estimate on the non-real eigenvalues of fourth or-der regular indefinite differential equation problems which depend on the turning point, symmetry conditions on weight function and the limita-tion of absolutely continuous functions in different circumstances. Mean-while by using couple boundary conditions, obtain the estimate on the non-real eigenvalues of regular indefinite Sturm-Liouville problems.The thesis is divided into four chapters according to contents.Chapter 1 Preference,simply summarize the phylogeny of Sturm-Liouville theory and introduce the work of present paper.Chapter 2 Fourth order indefinite differential equation problems. We shall consider the left-definite and right-definite problems of the fourth order indefinite differential expression: where r-1,p,q,w∈L1[a,b],r>0 a.e.in[a,b],and the weight function w changes sign.Chapter 3 The estimate on the non-real eigenvalues of the fourth order regular indefinite differential equation problems. We shall consider the fourth order regular indefinite Sturm-Liouville differential expression with weight function witch obtain the estimate on the real and imaginary parts of these eigenvalues,where q and w are real-valued functions satisfying q,w∈ L1[-1,1],w(x)≠0 a.e.on[-1,1]and w(x)changes sign on[-1,1].Chapter 4 The estimate on the non-real eigenvalues of the regular indefinite Sturm-Liouville problems with couple boundary conditions. We shall consider the regular indefinite Sturm-Liouville problems.-(py’)’+qy=λwy, with couple boundary conditions where weight function w and the real coefficients p-1,q∈L|w|1[-1,1].