EXISTENCE OF SOLUTIONS FOR A CLASS OF FOURTH-ORDER MULTIPOINT BOUNDARY-VALUE PROBLEMS ON TIME SCALES

摘要

The paper deals with the existence of solutions for the dynamic equation on time scales u~(ΔΔΔΔ)(t) = f (t, u(σ(t)), u~(ΔΔ)(t)), t ∈ [0, 1]T, with the multipoint boundary conditions u(0) = 0, u(σ(1)) = m-2 ∑ i=1 a_iu(ξ_i), u~(ΔΔ)(0) = 0, u~(ΔΔ)(σ(1)) = n-2 ∑ j=1 b_ju~(ΔΔ)(η_j), where T is a time scale, [0, 1]_t = {t ∈ T | 0 ≤ t ≤ 1}, a_i ＞ 0, i = 1,2,… ,m - 2, b_j ＞ 0, j = = 1,2,…,n - 2, 0 ＜ ξ_1 ＜ξ_2 ＜ … ＜ ξ_(m-2) ＜ ρ(1), and 0 ＜ η_1 ＜ η_2 ＜ … ＜ η_(n-2) ＜ ρ(1). The existence result is given by using Green's function, the method of upper and lower solutions, and the monotone iterative technique. We also give an example to illustrate our result.