Eigenvalue theorems for discrete multipoint conjugate boundary value problems

We consider the following boundary value problem:Δny(k)=λP(k,y,Δy,…,Δn−1y), k=0,…,m,Δjy(ki)=0, j=0,…,ni−1, i=1,…,r,where r⩾2, ni⩾1 for i=1,…,r, ∑i=1rni=n and 0=k1<k1+n1<k2<k2+n2<⋯<kr⩽kr+nr−1=m+n. Values of λ are characterized so that the boundary value problem has, in certain sense, a “positive” solution. In fact, we offer criteria for values of λ to form a bounded/unbounded interval. Further, the intervals of λ are explicitly given. We also include examples to dwell upon the importance of the results obtained.