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# Multiple positive solutions for a fourth-order integral boundary value problem on time scales

Yongkun Li* and Yanshou Dong

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Department of Mathematics, Yunnan University Kunming, Yunnan 650091 People's Republic of China

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Boundary Value Problems 2011, 2011:59  doi:10.1186/1687-2770-2011-59

 Received: 9 November 2011 Accepted: 29 December 2011 Published: 29 December 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel'skill. To demonstrate the applications of our results, an example is also given in the article.

##### Keywords:
positive solutions; fixed points; integral boundary conditions; time scales

### 1 Introduction

Boundary value problem (BVP) for ordinary differential equations arise in different areas of applied mathematics and physics and so on, the existence and multiplicity of positive solutions for such problems have become an important area of investigation in recent years, lots of significant results have been established by using upper and lower solution arguments, fixed point indexes, fixed point theorems and so on (see [1-8] and the references therein). Especially, the existence of positive solutions of nonlinear BVP with integral boundary conditions has been extensively studied by many authors (see [9-18] and the references therein).

However, the corresponding results for BVP with integral boundary conditions on time scales are still very few [19-21]. In this article, we discuss the multiple positive solutions for the following fourth-order system of integral BVP with a parameter on time scales

(1.1)

where ai, bi, ci, di ≥ 0, and ρi = aiciσ(T) + aidi + bici > 0(i = 1, 2), 0 < λ, μ < +∞, f, , ℝ+ = [0, +∞), Ai and Bi are nonnegative and rd-continuous on .

The main purpose of this article is to establish some sufficient conditions for the existence of at least two positive solutions for system (1.1) by using the fixed point theorem of cone expansion and compression type. This article is organized as follows. In Section 2, some useful lemmas are established. In Section 3, by using the fixed point theorem of cone expansion and compression type, we establish sufficient conditions for the existence of at least two positive solutions for system (1.1). An illustrative example is given in Section 4.

### 2 Preliminaries

In this section, we will provide several foundational definitions and results from the calculus on time scales and give some lemmas which are used in the proof of our main results.

A time scale is a nonempty closed subset of the real numbers ℝ.

Definition 2.1. [22]For , we define the forward jump operator by , while the backward jump operator by .

In this definition, we put and , where ∅, denotes the empty set. If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Also, if and σ(t) = t, then t is called right-dense, and if and ρ(t) = t, then t is called left-dense. We also need, below, the set , which is derived from the time scale as follows: if has a left-scattered maximum m, then . Otherwise, .

Definition 2.2. [22]Assume that is a function and let . Then x is called differentiable at if there exists a θ ∈ ℝ such that for any given ε > 0, there is an open neighborhood U of t such that

In this case, xΔ(t) is called the delta derivative of x at t. The second derivative of x(t) is defined by xΔΔ(t) = (xΔ)Δ(t).

In a similar way, we can obtain the fourth-order derivative of x(t) is defined by x(4Δ)(t) = (((xΔ)Δ)Δ)Δ(t).

Definition 2.3. [22]A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist at left-dense points in . The set of rd-continuous functions will be denoted by .

Definition 2.4. [22]A function is called a delta-antiderivative of provide FΔ(t) = f(t) holds for all . In this case we define the integral of f by

For convenience, we denote , and for i = 1, 2, we set

where

To establish the existence of multiple positive solutions of system (1.1), let us list the following assumptions:

In order to overcome the difficulty due to the dependence of f, g on derivatives, we first consider the following second-order nonlinear system

(2.1)

where A0 is the identity operator, and

(2.2)

For the proof of our main results, we will make use of the following lemmas.

Lemma 2.1. The fourth-order system (1.1) has a solution (x, y) if and only if the nonlinear system (2.1) has a solution (u, v).

Proof. If (x, y) is a solution of the fourth-order system (1.1), let u(t) = xΔΔ(t), v(t) = yΔΔ(t), then it follows from the boundary conditions of system (1.1) that

Thus (u, v) = (xΔΔ(t), yΔΔ(t)) is a solution of the nonlinear system (2.1).

Conversely, if (u, v) is a solution of the nonlinear system (2.1), let x(t) = A2u(t), y(t) = A2v(t), then we have

which imply that

Consequently, (x, y) = (A2u(t), A2v(t)) is a solution of the fourth-order system (1.1). This completes the proof.

Lemma 2.2. Assume that D11D21 ≠ 1 holds. Then for any h1 C(I', ℝ+), the following BVP

(2.3)

has a solution

where

Proof. First suppose that u is a solution of system (2.3). It is easy to see by integration of BVP(2.3) that

(2.4)

Integrating again, we can obtain

(2.5)

Let t = σ(T) in (2.4) and (2.5), we obtain

(2.6)

(2.7)

Substituting (2.6) and (2.7) into the second boundary value condition of system (2.3), we obtain

(2.8)

From (2.8) and the first boundary value condition of system (2.3), we have

(2.9)

(2.10)

Substituting (2.9) and (2.10) into (2.5), we have

(2.11)

By (2.11), we get

(2.12)

(2.13)

By (2.12) and (2.13), we get

(2.14)

(2.15)

Substituting (2.14) and (2.15) into (2.11), we have

(2.16)

Conversely, suppose , then

(2.17)

Direct differentiation of (2.17) implies

and

and it is easy to verify that

This completes the proof.

Lemma 2.3. Assume that D12D22 ≠ 1 holds. Then for any h2 C(I', ℝ+), the following BVP

has a solution

where

Proof. The proof is similar to that of Lemma 2.2 and will omit it here.

Lemma 2.4. Suppose that (H1) is satisfied, for all t, s I and i = 1, 2, we have

(i) Gi(t, s) > 0, Hi(t, s) > 0,

(ii) LimiGi(σ(s), s) ≤ Hi(t, s) ≤ MiGi(σ(s), s),

(iii) mGi(σ(s), s) ≤ Hi(t, s) ≤ MGi(σ(s), s),

where

Proof. It is easy to verify that Gi(t, s) > 0, Hi(t, s) > 0 and Gi(t, s) ≤ Gi(σ(s), s), for all t, s I. Since

Thus Gi(t, s)/Gi(σ(s), s) ≥ Li and we have

On the one hand, from the definition of Li and mi, for all t, s I, we have

and on the other hand, we obtain easily that from the definition of Mi, for all t, s I,

Finally, it is easy to verify that mGi(σ(s), s) ≤ Hi(t, s) ≤ MGi(σ(s), s). This completes the proof.

Lemma 2.5. [23]Let E be a Banach space and P be a cone in E. Assume that Ω1 and Ω2 are bounded open subsets of E, such that 0 ∈ Ω1, , and let be a completely continuous operator such that either

(i) ||Tu|| ≤ ||u||, ∀u P ∩ ∂Ω1 and ||Tu|| ≥ ||u||, ∀u P ∩ ∂Ω2, or

(ii) ||Tu|| ≥ ||u||, ∀u P ∩ ∂Ω1 and ||Tu|| ≤ ||u||, ∀u P ∩ ∂Ω2

holds. Then T has a fixed point in .

To obtain the existence of positive solutions for system (2.1), we construct a cone P in the Banach space Q = C(I, ℝ+) × C(I, ℝ+) equipped with the norm by

It is easy to see that P is a cone in Q.

Define two operators Tλ, Tμ : P C(I, ℝ+) by

Then we can define an operator T : P C(I, ℝ+) by

Lemma 2.6. Let (H1) hold. Then T : P P is completely continuous.

Proof. Firstly, we prove that T : P P. In fact, for all (u, v) ∈ P and t I, by Lemma 2.4(i) and (H1), it is obvious that Tλ(u, v)(t) > 0, Tμ(u, v)(t) > 0. In addition, we have

(2.18)

which implies . And we have

In a similar way,

Therefore,

This shows that T : P P.

Secondly, we prove that T is continuous and compact, respectively. Let {(uk, vk)} ∈ P be any sequence of functions with ,

from the continuity of f, we know that ||Tλ(uk, vk) - Tλ(u, v)|| → 0 as k → ∞. Hence Tλ is continuous.

Tλ is compact provided that it maps bounded sets into relatively compact sets. Let , and let Ω be any bounded subset of P, then there exists r > 0 such that ||(u, v)|| ≤ r for all (u, v) ∈ Ω. Obviously, from (2.16), we know that

so, TλΩ is bounded for all (u, v) ∈ Ω. Moreover, let

We have

Thus, for any (u, v) ∈ Ω and ∀ε > 0, let , then for t1, t2 I, |t1 - t2| < δ, we have

So, for all (u, v) ∈ Ω, TλΩ is equicontinuous. By Ascoli-Arzela theorem, we obtain that Tλ : P P is completely continuous. In a similar way, we can prove that Tμ : P P is completely continuous. Therefore, T : P P is completely continuous. This completes the proof.

### 3 Main results

In this section, we will give our main results on multiplicity of positive solutions of system (1.1). In the following, for convenience, we set

where qi(t), qj(t) ∈ Crd(I', ℝ+) satisfy

Theorem 3.1. Assume that (H1) holds. Assume further that

(H2) there exist a constant R > 0, and two functions pi(t) ∈ Crd(I, R+) satisfying such that

and one of the folloeing conditions is satisfied

(E1) , ,

(E2) , ,

(E3) , μ ∈ (0, N4),

(E4) λ ∈ (0, M4), ,

where

O1, O2 satisfy . Then system (1.1) has at least two positive solutions.

Proof. We only prove the case in which (H2) and (E1) hold, the other case can be proved similarly. Firstly, from (2.2), we have

(3.1)

Take , and let Ω1 = {(u, v) ∈ Q; ||(u, v)|| < R1}. For any t I, (u, v) ∈ ∂Ω1 P, it follows from λ < M4, μ < N4 and (H2) that

and

Consequently, for any (u, v) ∈ ∂Ω1 P, we have

(3.2)

Second, from , we can choose ε1 > 0 such that λf0 > M3 + ε1, then there exists 0 < l1 < NR1 such that for any and t I,

And since

(3.3)

Take

For all (u, v) ∈ Ω2 P, where Ω2 = {(u, v) ∈ Q; ||(u, v)|| < R2}, we have

Thus, for all (u, v) ∈ Ω2 P, we have

Consequently, for all (u, v) ∈ Ω2 P, we have

(3.4)

Finally, from μ > N3/g, we can choose ε2 > 0 such that μg> N3 + ε2. then, there exists such that for any and t I,

Take

For all (u, v) ∈ Ω3 P, where Ω3 = {(u, v) ∈ Q; ||(u, v)|| < R3}, from (3.3), we have

(3.5)

Thus, for all (u, v) ∈ Ω3 P, we have

Consequently, for all (u, v) ∈ Ω3 P, we have

(3.6)

From (3.2), (3.4), and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u1, v1) ∈ P with R2 ≤ ||(u1, v1)|| ≤ R1. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x1, y1). In the same way, from (3.2), (3.6), and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u2, v2) ∈ P with R1 ≤ ||(u2, v2)|| ≤ R3. Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x2, y2). Above all, system (1.1) has at least two positive solutions. This completes the proof.

Theorem 3.2. Assume that (H1) holds. Suppose further that

(H3) there exist a constant R0 > 0, and two functions wi(t) ∈ Crd(I, R+) (i = 1, 2) satisfying such that

(3.7)

or

(3.8)

Then system (1.1) has at least two positive solutions for each and , where

Proof. We only prove the case in which (3.7) holds. The other case in which (3.8) holds can be proved similarly.

Take

and let . For any t I, (u, v) ∈ ∂Ω4 P, it follows from λ > M5 and (H3) that

Consequently, for any (u, v) ∈ ∂Ω4 P, we have

(3.9)

From , , we know that , , we can choose ε3 > 0 such that M6 - ε3 > 0, N6 - ε3 > 0 and λf0 < M6 - ε3, μg0 < N6 - ε3. Then there exists such that for any and t I,

Take and . Then, for any (u, v) ∈ Ω5 P, from(3.1), we have

and

Consequently, for any (u, v) ∈ ∂Ω5 P, we have

(3.10)

From , we know that , , we can choose ε4 > 0 such that M6 - ε4 > 0, N6 - ε4 > 0 and λf< M6 - ε4, μg< N6 - ε4. Then there exists such that for any and t I,

Take and let . Then, for any (u, v) ∈ Ω6 P, we have

and

Consequently, for any (u, v) ∈ ∂Ω6 P, we have

(3.11)

From (3.9), (3.10) and (i) of Lemma 2.5, it follows that system (2.1) has one positive solution (u1, v1) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x1, y1). In the same way, from (3.9), (3.11) and (ii) of Lemma 2.5, it follows that system (2.1) has one positive solution (u2, v2) ∈ P with . Therefore, from Lemma 2.1, it follows that system (1.1) has one positive solution (x2, y2). Above all, system (1.1) has at least two positive solutions. This completes the proof.

### 4 An example

Consider the following BVP with integral boundary conditions:

(4.1)

where A1(t) = B1(t) = t, A2(t) = B2(t) = t/2 and

we choose O1 = 2, O2 = 4, R = 1, p1(t) = 2t, , q1(t) = q3(t) = 1. It is easy to check that f0 = g= ∞, (H1), (H2) and (E1) are satisfied. Therefore, by Theorem 3.1, system (4.1) has at least two positive solutions for each λ ∈ (0, M4), μ ∈ (0, N4).

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally to the manuscript and typed, read and approved the final manuscript.

### Acknowledgements

This study was supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.

### References

1. Yuan, C, Jiang, D, O'Regan, D: Existence and uniqueness of positive solutions for fourth-order nonlinear singular continuous and discrete boundary value problems. Appl Math Comput. 203, 194–201 (2008). Publisher Full Text

2. Yang, Z: Existence and uniqueness of positive solutions for a higher order boundary value problem. Comput Math Appl. 54, 220–228 (2007). Publisher Full Text

3. Zhang, X, Feng, M: Positive solutions for a class of 2nth-order singular boundary value problems. Nonlinear Anal. 69, 1287–1298 (2008). Publisher Full Text

4. Zhao, J, Ge, W: A necessary and sufficient condition for the existence of positive solutions to a kind of singular three-point boundary value problem. Nonlinear Anal. 71, 3973–3980 (2009). Publisher Full Text

5. Lian, H, Pang, H, Ge, W: Triple positive solutions for boundary value problems on infinite intervals. Nonlinear Anal. 67, 2199–2207 (2007). Publisher Full Text

6. Wei, Z, Zhang, M: Positive solutions of singular sub-linear boundary value problems for fourth-order and second-order differential equation systems. Appl Math Comput. 197, 135–148 (2008). Publisher Full Text

7. Bai, Z: Positive solutions of some nonlocal fourth-order boundary value problem. Appl Math Comput. 215, 4191–4197 (2010). Publisher Full Text

8. Zhu, F, Liu, L, Wu, Y: Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems. Appl Math Comput. 216, 448–457 (2010). Publisher Full Text

9. Kang, P, Wei, Z, Xu, J: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Appl Math Comput. 206, 245–256 (2008). Publisher Full Text

10. Feng, M: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl Math Lett. 24, 1419–1427 (2011). Publisher Full Text

11. Zhang, X, Ge, W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Comput Math Appl. 58, 203–215 (2009). Publisher Full Text

12. Feng, M, Ji, D, Ge, W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. J Comput Appl Math. 222, 351–363 (2008). Publisher Full Text

13. Zhao, J, Wang, P, Ge, W: Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces. Commun Nonlinear Sci Numer Simulat. 16, 402–413 (2011). Publisher Full Text

14. Yang, Z: Existence and uniqueness of positive solutions for an integral boundary value problem. Nonlinear Anal. 69, 3910–3918 (2008). Publisher Full Text

15. Kong, L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 72, 2628–2638 (2010). Publisher Full Text

16. Jankowski, T: Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions. Nonlinear Anal. 73, 1289–1299 (2010). Publisher Full Text

17. Boucherif, A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 70, 364–371 (2009). Publisher Full Text

18. Zhang, X, Feng, M, Ge, W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Anal. 69, 3310–3321 (2008). Publisher Full Text

19. Anderson, DR, Tisdell, CC: Third-order nonlocal problems with sign-changing nonlinearity on time scales. Electron J Diff Equ. 2007(19), 1–12 (2007)

20. Li, YK, Shu, J: Multiple positive solutions for first-order impulsive integral boundary value problems on time scales. Bound Value Probl. 2011, 12 (2011). BioMed Central Full Text

21. Li, YK, Shu, J: Solvability of boundary value problems with Riemann-Stieltjes -integral conditions for second-order dynamic equations on time scales at resonance. Adv Diff Equ. 2011, 42 (2011). BioMed Central Full Text

22. Bohner, M, Peterson, A: Dynamic Equations on Time Scales, An introduction with Applications. Birkhauser, Boston (2001)

23. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New york (1988)