Abstract

We study a system of second-order dynamic equations on time scales , , satisfying four kinds of different multipoint boundary value conditions, is continuous and semipositone. We derive an interval of such that any lying in this interval, the semipositone coupled boundary value problem has multiple positive solutions. The arguments are based upon fixed-point theorems in a cone.

1. Introduction

In this paper, we consider the following dynamic equations on time scales: satisfying one of the boundary value conditions where and is continuously and nonegative functionsquad, for ; the points for with .

In the past few years, the boundary value problems of dynamic equations on time scales have been studied by many authors (see [119] and references). Recently, multipoint boundary value problems on time scale have been studied, for instance, see [112].

In 2006, Anderson and Ma [1] studied the second-order multiple time-scale eigenvalue problem: where the functions and are continuous. The authors discuss conditions for the existence of at least one positive solution to the second-order Sturm-Liouville-type multiple eigenvalue problem on time scales.

In 2009, Feng et al. [2] studied where the functions , , , . This paper shows the existence of multiple positive solutions for the boundary value problem on time scales.

In 2009, Topal and Yantir [3] studied the second-order nonlinear -point boundary value problems where ; for  all ; , , are continuously and nonegative functions. The authors deal with determining the value of , and the existences of multiple positive solutions of the equation are obtained. In 2010, Yuan and Liu [4] also study the second-order -point boundary value problems; Yuan and Liu shows the existence of multiple positive solutions if is semipositone and superlinear.

Motivated by the above results mentioned, we study the second-order nonlinear -point boundary value problem (1) with boundary condition ( ), and nonlinear term may be singularity and semipositone.

In this paper, the nonlinear term of (1) is suit to and semipositone and the superlinear case, we shall prove our two existence results for the problem (1) with ( ) by using a nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed-point theorem. This paper is organized as follows. In Section 2, we start with some preliminary lemmas. In Section 3, we give the main result which state the sufficient conditions for (1) with -point boundary value ( ) to have existence of positive solutions ( ).

2. Preliminaries

In this section, we state the preliminary information that we need to prove the main results.

In this paper, for our constructions, we shall consider the Banach space equipped with standard norm ; for each , we write . Clearly, is a Banach space. Denote by and ( ), the solutions of the equation under the initial conditions respectively. So that and ( ) satisfy respectively. For , set , the Green's function of the corresponding homogeneous boundary value problem is defined by From Lemmas and 3.3 in [1], we have the following lemma.

Lemma 1. If , is a solution of (1) with boundary value condition ( ) if only and if where , and

For the rest of the paper, we need the following assumption:

From is nondecreasing on , is nonincreasing on (see [2, Proposition 2.3]), it is easy to verify the following inequalities:

Lemma 2. The Green's function has properties

Lemma 3. For , and , one has the conclusions and where and

Proof. From Lemma 2 and we have
For or 3, we have So, we have
Since , then we also have The proof is complete.

The following theorems will play a major role in our next analysis.

Theorem 4 (see [20]). Let be a Banach space, and closed and convex. Assume is a relatively open subset of with , and let be a compact, continuous map. Then either (1) has a fixed point in , or (2)there exists and , with .

Theorem 5 (see [21]). Let be a Banach space, and let be a cone in . Let be bounded open subsets of with , and let be a completely continuous operator such that, either (1) , , , or (2) , , . Then has a fixed point in .

3. Main Results

We make the following assumptions: , moreover there exists a function such that , for any , , .( may be singular at ; moreover, there exists a function such that , for any , .( , for    .( There exists such that .( and for any , is any constant .In fact, we only consider the system with one of the boundary value conditions where and . For , from Lemma 1, ( ) is the solution of the equation respectively, satisfying the following boundary value conditions:

We will show that there exists a solution to the boundary value problem ( ) of the system (26) with . If this is true, then is a nonnegative solution (positive on ) of the system (1) with the boundary value problem , . Since for any , from we have As a result, we will concentrate our study on (26) with the boundary value problem ( ).

Employing Lemma 1, we note that is a solution of the system (26) with boundary value if and only if

We define a cone    by It is clearly that is a cone of , . Define the integral operator , , , , by where operators are defined by where . Clearly, if is a fixed point of , then is a solution of system (26) with    .

For , from (35) and Lemma 3, we have on , for , we have then .

On the other hand, when , we have Thus, . Hence .

When , we have Thus, . Hence .

Similarly discussion, we also have , . In addition, standard arguments show that is a completely continuous operator.

For simplicity, we adopt the notation: and , then, we can write , that is, , ( ).

Theorem 6. Suppose that - hold. Then there exists a constant such that, for any , (1) with boundary value condition ( ) has at least one positive solution ( ).

Proof. Fix and ( ). From ( ) let be such that
Let , and . We have
Set , since for any , fix the , we always have Then there exists a such that
Let , and be such that , that is,    . We claim that . In fact for and , we have It follows that that is, which implies that . By the nonlinear alternative of Leray-Schauder type, has a fixed point . Moreover combining (40) and the fact that , we obtain
Then has a positive fixed point and ; that is, is a positive solution of the boundary value problem (26) with for .
Let    , then is a nonnegative solution (positive on ) of the boundary value problem (1).

Theorem 7. Suppose that and - hold. Then there exists a constant such that, for any , (1) with boundary value condition ( ) has at least one positive solution ( ).

Proof. We fix ( ). Let , where and . Choose where and .
Then for any ,    and for , we have This implies
Choose a constant such that where .
By assumption and , there exists a constant such that
Choose and let . We note that    , by Lemma 3, we have . Then for any , we have or . Without loss of generality let , so we have Thus
Now since , it follows that This implies For the Krasnosel'skii's fixed point theorem, one deduces that has a fixed point with .
Since    , then
Thus is a positive solution of the boundary value problem (26) with    for .
Let    , then is a nonnegative solution (positive on ) of the boundary value problem (1).

Since condition implies conditions and then from the proof of Theorems 6 and 7, we immediately have the following theorem.

Theorem 8. Suppose that hold. Then (1) with boundary value condition ( ) has at least two positive solutions for sufficiently small ( ).

In fact with then (1) with boundary value condition ( ) has at least two positive solutions.

Remark 9. In Theorems 68, we use the assumption condition 16. If we have not the condition 16, that is, , then the system (1) and boundary condition ( ) are From Lemma 2, an argument similar to those in Theorems 68 yields the following theorems.

Theorem 10. Suppose that and hold. Then there exists a constant such that, for any , the boundary value problem (58) has at least one positive solution.

Theorem 11. Suppose that and - hold. Then there exists a constant such that, for any , the boundary value problem (58) has at least one positive solution.

Theorem 12. Suppose that hold. Then the boundary value problem (58) has at least two positive solutions for sufficiently small.

4. Example

To illustrate the usefulness of the results, we give some examples.

Example 13. Consider the boundary value problem where . Then if is sufficiently small, (59) has a positive solution with for .

To see this, we will apply Theorem 7 with Clearly for , Now , , and   hold. We note that the boundary condition of (59) is in accord with (4), and from [1], we have Then Note . Let and we have

Also let Now, if , Theorem 7 guarantees that (59) has a positive solutions with and .

Example 14. Consider the boundary value problem: satisfying one of the boundary value conditions ( ), ( ).
Then if is sufficiently small, (66) has two solutions , with for .
To see this, we will apply Theorem 8 with Clearly, for , Now hold. Let , , and we have Furthermore let , and . Note
Let . Now, if then and Theorem 6 guarantees that (66) has positive solutions with .
Next note and let so we have
Also let Now, if , Theorem 7 guarantees that (59) has a positive solutions with , .
Thus, if , Theorem 8 guarantees that (66) has two solutions and with for .