Solvability of a Higher-Order Three-Point Boundary Value Problem on Time Scales

and Applied Analysis 3 for 0 ≤ i ≤ n − 1, where α > 0 and β > 1 are given constants. On the one hand, the author established criteria for the existence of at least one solution and of at least one positive solution for the BVP 1.6 by using the Schauder fixed point theorem and Krasnosel’skii fixed point theorem, respectively. On the other hand, the author investigated the existence of multiple positive solutions to the BVP 1.6 by using Avery-Henderson fixed point theorem and Leggett-Williams fixed point theorem. In this paper, motivated by 21 , firstly, a new existence result for 1.1 is obtained by using a fixed point theorem, which is due to KrasnoseÍskı̆ and Zabreı̆ko 22 . Particularly, f may not be sublinear. Secondly, some simple criteria for the existence of a nonnegative solution of the BVP 1.2 are established by using Leray-Schauder nonlinear alternative. Thirdly, we investigate the existence of a nontrivial solution of the BVP 1.2 ; our approach is also based on the application of Leray-Schauder nonlinear alternative. Particularly, we do not require any monotonicity and nonnegativity on f . Our conditions imposed on f are all very easy to verify; our method is motivated by 1, 21, 23, 24 . 2. Preliminaries To state and prove the main results of this paper, we need the following lemmas. Lemma 2.1 see 18 . For 1 ≤ i ≤ n, let Gi t, s be Green’s function for the following boundary value problem: −yΔ2 t 0, t ∈ a, b ⊂ T, αiy ( η ) βiy a y a , γiy ( η ) y σ b , 2.1 and let di γi − 1 a − βi 1 − αi σ b η αi − γi . Then, for 1 ≤ i ≤ n, Gi t, s ⎧ ⎨ ⎩ Gi1 t, s , a ≤ s ≤ η, Gi2 t, s , η < s ≤ b, 2.2