The Spectral Properties Of A Class Of Fourth-order Differential Operators

We have obtained some characterizations about the left-definite Sturm - Liouville problems and inequalities of the eigenvalue with different boundary conditions, similar-ly, we have gotten the eigenvalue interlacing for the second order left-definite differential operators. But it is not deep to study the characterizations of the eigenvalue for the fourth order left-definite differential operators, especially, there are no better conclu-sions about the comparison of eigenvalues for the fourth order differential operators with different potential functions.Inspired by [1,3,4,6,7,27,28], we put the characteris-tic of eigenvalues comparison for second order Sturm - Liouville problems into the spectrum problems of fourth order differential operators. And by the comparison the-orem of fourth order differential operators, we can get the comparison of eigenvalues for the fourth order right-definite or left-definite differential operators with different potential functionsIn this thesis, we firstly combine the self-adjoint conditions with the fourth order differential equation to construct fourth order differential operators, by the definition of left-definite fourth order differential operators, and contacting the right-definite prob-lem, we can obtain the determination about left-definite problem. According to the knowledge of the operator theory about fourth right-definite order differential opera-tors, we use the the characteristic of right-definite problem to study the inequality of eigenvalues for the left-definite problem which has separate self-adjoint condition or couple self-adjoint condition respectively. Finally, we adopt the comparison theorem of the fourth order differential equation, combining the self-adjoint conditions, and get the comparison of eigenvalues for the fourth order right-definite or left-definite differen-tial operators with different potential functions The thesis is divided into four chapters according to contents.Chapter1 Introduction simply summarize the phylogeny of the problem and some results people have studies, then we further research the conclusions combing above,that is the main work of present paper.Chapter 2 Judgement of the fourth order left-definite differential operator.we shall consider the fourth order differential operator consisting of:?p₁y"?" +py = ?ry, on I = [a, b],and the self-adjoint boundary conditions CY?a? + DY?b? =0,where p₁-1,p, r?L1?I, R?,on I=[a,b],p₁ > 0 a.e. and the weight function r changes sign on I.Combing the fourth order differential equation, on I?[a,b],We can get the judgement of the fourth order left-definite differential operator.Chapter 3 Using the characterization of the fourth order right-definite differential operator to research the fourth order left-definite differential operatorIn this section, we introduce a new fourth order right-definite differential equation, on I = [a, b],where ? is the spectral parameter.Firstly, we study the characterization of this fourth order right-definite differential operator,after we obtain some results about the fourth order left-definite differential equation by the proof.Chapter 4 The comparison of eigenvalues for the fourth order differential operator with different potential functions???In first part, we firstly introduce the Sturm comparison theorem about two or-der differential equation which we have learn before, it is sufficient to provide basis for the following theorems.???In second part, we construct the Sturm comparison theorem about the fourth order homogeneous differential equation,in the same way, where q?t? ? C[a, b].According to the different conditions which are defined, we study the nature of the zero point about the solutions of the fourth order homogeneous differential equation with different potential functions.???In third part, we extend the Sturm comparison theorem about the fourth order homogeleous differential equation and compare the characteristic of the zero point about the solution of the fourth order differential equation?p₁?t?x"?t??"+q1?t?x?t?=0,?P2?t?y"?t??" + q2?t?y?t??t? =0.when p₁?t?,p2?t? and q1?t?,q2?t? are different.???In fourth part, we consider the fourth order differential operator, which are constructed by the fourth order differential equation?p₁?t?y" ?t??"+p?t?y?t? = ?r?t?y?t?, ?I1=?a,b??,and the self-adjoint separate condition:When the fourth order differential operator are left-definite or right-definite respec-tively, we can obtain the comparison of eigenvalues, for these fourth order differential operators under the different potential functions.