Eigenvalue problems for second-order nonlinear dynamic equations on time scales

This paper is concerned with the existence and nonexistence of positive solutions of the secondorder nonlinear dynamic equation uΔΔ(t) + λa(t)f (u(σ (t))) = 0, t ∈ [0, 1], satisfying either the conjugate boundary conditions u(0) = u(σ (1)) = 0 or the right focal boundary conditions u(0) = uΔ(σ (1)) = 0, where a and f are positive. We show that there exists a λ∗ > 0 such that the above boundary value problem has at least two, one and no positive solutions for 0 < λ < λ∗, λ = λ∗ and λ > λ∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→0+ uλ =0 and limλ→+∞ uλ =+∞.  2005 Elsevier Inc. All rights reserved.