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)= λω(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).λ is the spectral parameter,i2 =-1,-π<α ≤π.p:[0,N-1]z → R and p(-1)= p(N-1),g:[0,N-1]z→[0,+∞).(?)ω:[O.N-1」z → R and ω(n)≠ 0.i.e.,there exists a proper subset,N+ of[0,N-1]z,such thatω(n)>0;for n N+,ω(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].