On eigenvalue intervals and twin eigenfunctions of higher-order boundary value problems

Abstract

In this paper we shall consider the boundary value problem y(n) + λQ(t, y, y1, . . . , y(n-2)) = λP(t, y, y1, . . . , y(n-2)), n ≥ 2, t ∈ (0, 1), y(i)(0) = 0, 0 ≤ i ≤ n - 3, αy(n-2)(0) - βy(n-2)(0) = 0, γy(n-2)(1) + δy(n-1)(1) = 0, where λ > 0, α, α, γ and δ are constants satisfying αγ + αδ + βγ > 0, β, δ ≥ 0, β + α > 0 and δ + γ > 0. Intervals of λ are determined to ensure the existence of a positive solution of the boundary value problem. For λ = 1, we shall also offer criteria for the existence of two positive solutions of the boundary value problem. In addition, upper and lower bounds for these positive solutions are obtainsd for special cases. Several examples are included to dwell upon the importance of the results obtained. © 1998 Elsevier Science B.V. All rights reserved.