Second-order n-point eigenvalue problems on time scales

Abstract

We discuss conditions for the existence of at least one positive solution to a nonlinear second-order Sturm-Liouville-type multipoint eigenvalue problem on time scales. The results extend previous work on both the continuous case and more general time scales, and are based on the Guo-Krasnosel'skiȋ fixed point theorem.

Structure of Eigenvalues of Multi-Point Boundary Value Problems

“…Theorem 1.4. Given q ∈ L 1 R , there exists some constant A q > 0, depending on the norm q only, such that if α ∈ R m satisfies α ≤ A q , then one has Σ q α,η ⊂ R.…”

“…Notice that the constant A q,η in 4.44 is constructed from the implicit function theorem. Generally speaking, A q,η depends on η ∈ 0, 1 and all information of the potential q ∈ L 1 R . However, during the application of the implicit function theorem to 4.37 , the derivatives of ∂ α λ n α can be well controlled using estimates in 11 , like 3.8 -3.11 .…”

“…We end the paper with an open problem. Given α, η ∈ R m × Δ m , for any q ∈ L 1 R , due to Theorem 1.2, problem M q has always a sequence of real eigenvalues λ n q λ n,α,η q which tends to ∞. In applications of eigenvalues to nonlinear problems, the smallest real eigenvalues λ 1 q are of great importance.…”

“…In this paper, we will establish some fundamental results for eigenvalue theory of multi-point boundary value problems. Precisely, for a real potential q ∈ L 1 R : L 1 0, 1 , R , we consider the eigenvalue problem −y q x y λy, x ∈ 0, 1 , When q x ≡ 0, 1.1 is −y λy, x ∈ 0, 1 .…”

Positive solutions for second-order semipositone problems on time scales

“…In related papers, Anderson [14] and Anderson and Ma [15] consider the second-order multi-point problem…”