Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales

We are concerned with the following nonlinear second-order three-point boundary value problem on time scales −xΔΔ t f t, x t , t ∈ a, b T , x a 0, x σ2 b δx η , where a, b ∈ T with a < b, η ∈ a, b T and 0 < δ < σ2 b − a / η − a . A new representation of Green’s function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method.


Introduction
Let T be a time scale, that is, T is an arbitrary nonempty closed subset of R. For each interval I of R, we define I T I ∩ T. For more details on time scales, one can refer to 1-5 . Recently, three-point boundary value problems BVPs for short for second-order dynamic equations on time scales have received much attention. For example, in 2002, Anderson 6 studied the following second-order three-point BVP on time scales: where 0, T ∈ T, η ∈ 0, ρ T T and 0 < α < T/η. Some existence results of at least one positive solution and of at least three positive solutions were established by using the well-known Krasnoselskii and Leggett-Williams fixed point theorems. In 2003, Kaufmann 7 applied the Krasnoselskii fixed point theorem to obtain the existence of multiple positive solutions to the BVP 1.1 . For some other related results, one can refer to 8-10 and references therein.
In this paper, we are concerned with the existence of at least one positive solution for the following second-order three-point BVP on time scales:

1.2
Throughout this paper, we always assume that a, b ∈ T with a < b, η ∈ a, b T , and 0 < δ < It is interesting that the method used in this paper is completely different from that in 6, 7, 9, 10 , that is, a new representation of Green's function for the corresponding linear BVP is obtained and some existence criteria of at least one positive solution to the BVP 1.2 are established by using the iterative method.
For the function f, we impose the following hypotheses: H3 there exists q ∈ 0, 1 such that

Main Results
Lemma 2.1. The BVP 1.2 is equivalent to the integral equation is called the Green's function for the corresponding linear BVP, here is the Green's function for the BVP: Proof. Let x * be a solution of the BVP: x a x σ 2 b 0.

2.5
Then, it is easy to know that Now, if x is a solution of the BVP 1.2 , then it can be expressed by which together with the boundary conditions in 1.2 and 2.6 implies that

2.8
On the other hand, if x satisfies 2.1 , then it is easy to verify that x is a solution of the BVP 1.2 . For any t, s ∈ a, σ 2

Lemma 2.2.
Proof. Since it is obvious from the expression of G t, s that we know that 2.9 is fulfilled.

Boundary Value Problems
Our main result is the following theorem.

2.12
Define an operator F : D → P :

2.13
Then it is obvious that fixed points of the operator F in D are positive solutions of the BVP 1.2 . First, in view of H2 , it is easy to know that F : D → P is increasing. Next, we may assert that F : D → D, which implies that for any x ∈ D, there exist positive constants l and L such that Fx t ≥ lx t for x ∈ a, σ 2 b T .

2.14
In fact, for any x ∈ D, there exist 0 < m x < 1 < M x such that which together with H2 , H3 , and Remark 1.1 implies that By Lemma 2.2 and 2.16 , for any t ∈ a, σ 2 b T , we have

2.17
Boundary Value Problems 5 If we let which shows that Fx ∈ D. Now, for any fixed h 0 ∈ D, we denote where

2.25
Moreover, if we let t 0 m/M, then it follows from 2.22 , 2.23 , 2.24 , and H3 by induction that

2.27
Therefore, there exists a w ∈ D such that {u n t } ∞ n 0 and {v n t } ∞ n 0 converge uniformly to w on a, σ 2 b T and u n t ≤ w t ≤ v n t , t ∈ a, σ 2 b T , n 0, 1, 2, . . . .

2.28
Since F is increasing, in view of 2.28 , we have u n 1 t Fu n t ≤ Fw t ≤ Fv n t v n 1 t , t ∈ a, σ 2 b T , n 0, 1, 2, . . . .