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


Introduction
We are concerned with the following even-order three-point boundary value problem on time scales T: n ≥ 1, a < η < σ b ; we assume that σ b is right dense so that σ j b σ b for j ≥ 1 and that for each 1 ≤ i ≤ n, α i , β i , γ i coefficients satisfy the following condition: 1.4 They have studied the existence of at least one positive solution to the BVP 1.4 using the functional-type cone expansion-compression fixed point theorem.
In 18 , Anderson and Karaca were concerned with the dynamic three-point boundary value problem 1.2 and the eigenvalue problem −1 n y Δ 2n t λf t, y σ t with the same boundary conditions where λ is a positive parameter.Existence results of bounded solutions of a noneigenvalue problem were first established as a result of the Schauder fixed point theorem.Second, the monotone method was discussed to ensure the existence of solutions of the BVP 1.2 .Third, they established criteria for the existence of at least one positive solution of the eigenvalue problem by using the Krasnosel'skii fixed point theorem.Later, they investigated the existence of at least two positive solutions of the BVP 1.2 by using the Avery-Henderson fixed point theorem.
In 19 , Han and Liu studied the existence and uniqueness of nontrivial solution for the following third-order p-Laplacian m-point eigenvalue problems on time scales: where They obtained several sufficient conditions of the existence and uniqueness of nontrivial solution of the BVP 1.5 when λ is in some interval.Their approach was based on the Leray-Schauder nonlinear alternative.
Very recently, Yaslan 20 investigated the existence of solutions to the nonlinear evenorder three-point boundary value problem on time scales T: 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 Krasnose Ískȋ and Zabreȋko 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 .

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 G i t, s be Green's function for the following boundary value problem: where

2.5
Lemma 2.4 see 18 .Assume that condition H is satisfied.For G as in 2.2 , take H 1 t, s : G 1 t, s and recursively define for 2 ≤ j ≤ n.Then H n t, s is Green's function for the homogeneous problem:

2.7
Lemma 2.5 see 18 .Assume that H holds.If one defines K n−1 j 1 K j , then the Green function H n t, s in Lemma 2.4 satisfies the following inequalities: where Lemma 2.6 see 22 .LetX be a Banach space and let F : X → X be completely continuous.If there exists a bounded and linear operator A : X → X such that 1 is not an eigenvalue of A and then F has a fixed point in X.
Lemma 2.7 see 25 .Let X be a real Banach space, let Ω be a bounded open subset of X, 0 ∈ Ω, and let F : Ω → X be a completely continuous operator.Then either there exist x ∈ ∂Ω, λ > 1 such that F x λx or there exists a fixed point x * ∈ Ω.
Suppose that B denotes the Banach space C a, σ b with the norm y sup t∈ a,σ b |y t |.

Existence Results
In this section, we apply Lemmas 2.6 and 2.7 to establish some existence criteria for 1.1 and 1.2 .From the proof of Theorem 3.1 of 18 , we can know that F : B → B is completely continuous.In order to apply Lemma 2.6, we consider the following BVP:

3.11
Case 2 Y ≥ Y 1 .In this case, when s ∈ a, σ b , from 3.7 , we see that f y σ s − my σ s < y σ s ≤ y .

3.12
Thus, we can deduce from 3.11 and 3.12 that for any y ∈ B and y > Y f y σ s − my σ s < y , ∀s ∈ a, σ b .

3.13
From 3.13 , we have 3.14 that is to say, Then, it follows from Lemma 2.6 that F has a fixed point y * ∈ B. In other words, y * is a solution of the BVP 1.1 .Moreover, we can assert that y * is nontrivial when f 0 / 0. In fact, if f 0 / 0, then that is, 0 is not a solution of the BVP 1.1 .

Corollary 3.2.
Assume that condition H holds, and f : 0, ∞ → 0, ∞ is continuous with lim s → ∞ f s /s 0. Then the BVP 1.1 has a nonnegative solution. Proof.Let 3.17 then f * : R → 0, ∞ is continuous, and from lim s → ∞ f s /s 0, we know that lim Consider the following BVP:

3.19
It follows from Theorem 3.1 that the BVP 3.19 has a solution y * , that is,

3.20
Since H n t, s and f * are nonnegative, we can get that y * ≥ 0 on a, σ b .Consequently, from the definition of f * , we have

3.21
It follows from the boundary conditions of 3.20 and 3.21 that y * is a nonnegative solution of the BVP 1.1 .
Remark 3.3.In Corollary 3.2, we only need that lim s → ∞ f s /s 0. Thus, f may not be sublinear.where K and G n •, s are defined in Lemmas 2.5 and 2.1, respectively.Then the BVP 1.2 has a nonnegative solution y 1 with y 1 < r.

Theorem 3.4. Assume that condition H holds, and
Proof.We consider the following boundary value problem:

3.26
Let y be any solution of 3.

3.31
Since y / r, any solution y ∈ ∂U of y λNy with 0 < λ < 1 cannot occur.Lemma 2.7 guarantees that N has a fixed point y 1 in U.In other words, the BVP 1.2 has a solution y 1 ∈ B with y 1 < r.Theorem 3.5.Assume that condition H is satisfied.Suppose that f t, 0 / ≡ 0, t ∈ a, σ b , f : a, σ b × R → R is continuous, and there exist nonnegative integrable functions k, h such that

3.32
Then, the BVP 1.2 has at least one nontrivial solution y * ∈ B.

3.33
By hypothesis B < 1.Since f t, 0 / ≡ 0, there exists m, n ⊂ a, σ b such that min t∈ m,n |f t, 0 | > 0. On the other hand, from the condition h t ≥ |f t, 0 |, a.e.t ∈ a, σ b , we know that A > 0. Let

3.35
Therefore which contradicts λ > 1.By Lemma 2.7, T has a fixed point y * ∈ Ω d .Noting f t, 0 / ≡ 0, the BVP 1.2 has at least one nontrivial solution y * ∈ B. This completes the proof.
Corollary 3.6.Assume that condition H is satisfied.Suppose that f t, 0 / ≡ 0, t ∈ a, σ b , f : a, σ b × R → R is continuous, and there exist nonnegative integrable functions k, h such that , t ∈ a, σ b .

3.37
Then, the BVP 1.2 has at least one nontrivial solution y * ∈ B.
Proof.In this case, we have By Theorem 3.5, this completes the proof.
Corollary 3.7.Assume that condition H is satisfied.Suppose that f t, 0 / ≡ 0, t ∈ a, σ b , f : a, σ b × R → R is continuous, and there exist nonnegative integrable functions k, h such that

3.39
Then, the BVP 1.2 has at least one nontrivial solution y * ∈ B.
then, there exists c > 0 such that and it follows from 3.41 and 3.42 that

3.43
By Corollary 3.6, we can deduce that Corollary 3.7 is true.

Two Examples
In the section, we present two examples to explain our results.

4.4
From the proof of Lemma 2.5 in 18 , we can get that where

, 3 . 1
then the BVP 1.1 has a solution y * , and y * / 0 when f 0 / 0. Proof.Define the integral operator F : B → B by Fy t σ b a H n t, s f y σ s Δs 3.2 for t ∈ a, σ b .Obviously, the solutions of the BVP 1.1 are the fixed points of operator F.
with g ≥ 0 continuous, and nondecreasing on 0, ∞ and ϕ : a, σ b → 0, ∞ continuous 3.23 ∃r > 0 with r > g r σ b a ϕ s K G n •, s Δs, 3.24 Then for any y ∈ B and y > Y Y > 0 , we distinguish the following two cases.
A : B → B by First, we claim that 1 is not an eigenvalue of A. In fact, if m 0, then it is obvious that the BVP 3.3 has no nontrivial solution.If m / 0 and the BVP 3.3 has a nontrivial solution y, then y > 0, and so In fact, for any > 0, since lim s → ∞ f s /s m, there must exist a number Y 1 > 0 such thatf s − ms < |s|, |s| > Y 1 .Thus, when s ∈ a, σ b and |y σ s | ≤ Y 1 , we have f y σ s − my σ s ≤ f y σ s |m| y σ s ≤ M |m|Y 1 < Y < y , Suppose y ∈ ∂Ω d , λ > 1 such that Ty λy, then d A 1 − B −1 , Ω d {y ∈ B : y < d}.For t ∈ a, σ b , the operator T is defined by from the proof of Theorems 3.1 and 3.4, we have known that T : B → B is a completely continuous operator, and the BVP 1.2 has at least one nontrivial solution y * ∈ B if and only if y * is a fixed point of T in B.