Non-real Eigenvalues Of Linear Problems With Sign-changing Weight

In this paper,we study the non-real eigenvalues of linear eigenvalue problems with sign-changing weight.The main works are:In the first chapter,we consider the non-real eigenvalues of second-order indef-inite S-L eigenvalue problem with eigenparameter-dependent boundary conditions?y:=-(p(x)y')' + q(x)y = ??(x)y,x ?[0,1],B1y:= ?y(0)cos ?1-p(0)y'(0)sin ?1 = 0,B2y:= ?y(1)cos ?2-p(1)y'(1)sin ?2 = 0,where p,q,w is real-valued function and satisfy basic condition p(x)>0,?(x)? 0 a.e.x ?[0,1],1/p,q,? ?,L1[0,1],w(x)changes sign on[0,1],A is spectral parameter,and ?1,?2 ?[0,?).We obtain a priori bounds estimation for possible non-real eigenvalues and a sufficient condition for the existence and non-existence of non-real eigenvalues.In the second chapter,we consider the non-real eigenvalues of fourth-order in-definite p-Laplacian eigenvalues problem?y:=([y"]p-1)"+ q(x)[y]p-1=??(x)[y]p-1 x ?(0,1),y(0)= y(1)y"(0)= y"(1)=0,where p>1,[y]p-1= |y|p-2y,q and ? are real-valued functions and satisfy basic condition?(x)? 0 a.e.x ?[0,1],q,? ? L2[0,1],w(x)changes sign on[0,1],A is spectral parameter.We obtain a priori bounds estimation for possible non-real eigenvalues and a sufficient condition for the non-existence of non-real eigenvalues.In the third chapter,we consider the non-real eigenvalues of fourth-order indef-inite elliptic eigenvalue problem?y:= ?2y + ay = ?wy,y ? L|?|2(?),y|(?)? = ?|(?)?= 0,where ?(?)Rn(n ? 2)is bounded open set,with almost regular boundary and ?,? is real-valued function and satisfy basic condition?(x)? 0 a.e.x:? ?,? ? L1,(?),a ? L?(?),w(x)changes sign on ?,? is spectral parameter.We obtain a priori bounds for possible non-real eigenvalues and a sufficient condition for the existence and non-existence of non-real eigenvalues.