Eigenvalue theorems for discrete multipoint conjugate boundary value problems

Abstract

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, Σi=1 r ni = 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 λto 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. © 2000 Elsevier Science B.V. All rights reserved.