An even-order three-point boundary value problem on time scales

We study the even-order dynamic equation (−1)nx( ∇)n (t ) = λh(t)f (x(t )), t ∈ [a, c] satisfying the boundary conditions x( ∇)i (a) = 0 and x( ∇)i (c) = βx( ∇)i (b) for 0 i n − 1. The three points a, b, c are from a time scale T, where 0 < β(b − a) < c − a for b ∈ (a, c), β > 0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a, c] of the time scale.  2003 Elsevier Inc. All rights reserved.