Eigenvalue theorems for discrete multipoint conjugate boundary value problems

摘要

We consider the following boundary value problem: Δ~n y(k) = λP(k,y,Δy,…,Δ~(n-1)y), k = 0,…,m, Δ~j y(k_i) = 0, j = 0,…,n_i - 1, i = 1,…,r, where r ≥ 2, n_i ≥ 1 for i = 1,…,r, ∑_(i=1)~r n_i = n and 0 = k_1 < k_1 + n_1 < k_2 < k_2 + n_2 < … < k_r ≤ k_r + n_r - 1 = m + n. Values of λ are characterized so that the boundary value problem has, in certain sense, a "positive" solution. In face, 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.