Eigenvalues Of Discrete Second-order Linear Sign-changing Weight Problems With Coupled Boundary Conditions

We consider the second-order difference equation eigenvalue problems-?[p(n)? y(n)]+ q(n)y(n)= ??(n)y(n),n ?[0,N-1]Z with the coupled boundary condition(?).Where N ? 3 is an interger,[0,N-1]z?{0,1,…,N-1}.? is the forward difference operation:?y(n)= y(n +1)-y(n),? is the backward difference opera-tion:?y(n)= y(n)—y(n-1).? is the spectral parameter,i2 =-1,-?<? ??.p:[0,N-1]z ? R and p(-1)= p(N-1),g:[0,N-1]z?[0,+?).(?)?:[O.N-1?z ? R and ?(n)? 0.i.e.,there exists a proper subset,N+ of[0,N-1]z,such that?(n)>0;for n N+,?(n)<0 for n ?[0.N-1]z?N+.Let in Abe the number of elements in N+.Then N-m is the number of ele-ments in[0,N-1]z?N+.This paper mainly generalizes and develops the results of Sun and Shi[Linear Algebra Appl,2006]?Gao and Ma[Linear Algebra Appl,2015].