- Research Article
- Open Access
Multiple Positive Solutions of -Point BVPs for Third-Order -Laplacian Dynamic Equations on Time Scales
Advances in Difference Equations volume 2009, Article number: 262857 (2009)
This paper is concerned with the existence of multiple positive solutions for the third-order -Laplacian dynamic equation with the multipoint boundary conditions , where with . Using the fixed point theorem due to Avery and Peterson, we establish the existence criteria of at least three positive solutions to the problem. As an application, an example is given to illustrate the result. The interesting points are that not only do we consider third-order -Laplacian dynamic equation but also the nonlinear term is involved with the first-order delta derivative of the unknown function.
The theory of dynamic equations on time scales was introduced by Stefan Hilger in 1988 . This theory has attracted many researchers' attention and interest since it cannot only unify differential and difference equations but also provides accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. In addition, time-scale calculus would allow exploration of a variety of situations in economic, biological, heat transfer, stock market, and epidemic models [2, 3], and so forth.
Recently, there has been much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales; see [4–10] and the references therein. On the one hand, higher-order nonlinear boundary value problems have been studied extensively; see [11–14] and the references therein. On the other hand, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [15–17]. However, very little work has been done to the third-order -Laplacian dynamic equations on time scales [18, 19].
For convenience, throughout this paper, we denote as -Laplacian operator, that is, for with and We also assume that is a closed subset of with ; an interval always means . Other types of intervals are defined similarly.
For example, Sun and Li  studied the two-point boundary value problem:
In , Su et al. investigated the existence of positive solutions for the following singular -Laplacian -point boundary value problem on time scales:
The main techniques are Schauder fixed point theorem and upper and lower solutions method.
In , Han and Kang considered the following third-order -Laplacian dynamic equation on time scales:
By using fixed point theorems in cones, the existence criteria of multiple positive solutions are established.
In , Zhao and Sun studied the following second-order nonlinear three-point boundary value problem on time scales:
They gave sufficient condition for the existence of three positive solutions by using a fixed point theorem due to Avery and Peterson .
subject to the boundary condition
where , for . By using fixed point theorem due to Avery and Peterson , we prove that the boundary value problems (1.5) and (1.6) have at least three positive solutions under suitable assumptions. The interesting points are that not only do we consider third-order -Laplacian dynamic equation on time scales but also the nonlinear term is involved with the first-order delta derivative of the unknown function.
Throughout this paper, it is assumed that
and , both and do not vanish identically on any closed subinterval of , and there exists such that hold;
there exist nonnegative constants and satisfying for
To prove the main results in this paper, we will employ several lemmas. And the following lemma is based on the linear BVP:
If , then the problems (2.1) and (2.2) have the unique nonnegative solution:
For any , suppose that is a solution of the BVPs (2.1) and (2.2). By integrating (2.1) from to , and combining the boundary condition, it follows that
Using (2.2), we can easily obtain
Then it is easy to see that
So . On the other hand, it is easy to verify that if is as in (2.3), then is a solution of (2.1) and (2.2). Thus in (2.3) is the unique solution of (2.1) and (2.2).
Let = be endowed with the norm
It follows that is a Banach space. Define the cone by
If , then there exists a constant such that
For , implies that
In addition, since
then we have
Therefore, We can choose and the proof is complete.
If , then for .
If , then is decreasing and , and thus and are decreasing. So we have
By the concavity of , for , there is
Then we have
The proof is complete.
Let and be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on . Then for positive real numbers and , we define the following convex sets:
and a closed set
The following fixed point theorem due to Avery and Peterson is fundamental in the proof of our main results.
Lemma 2.4 (see ).
Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on , let be a nonnegative continuous concave functional on , and let be a nonnegative continuous functional on satisfying for , such that for some positive numbers and ,
for all . Suppose that is completely continuous and there exist positive numbers , and with such that
and for ;
for with ;
and for with .
Then has at least three fixed points , such that
3. Existence Results
In this section, by using the Avery-Peterson fixed point theorem, we shall give the sufficient conditions for the existence of at least three positive solutions to the BVPs (1.5) and (1.6).
Firstly, we define the nonnegative continuous concave functional , the nonnegative continuous convex functionals , and the nonnegative continuous functional on , respectively, by
For notation convenience, we denote
Now we state and prove our main result.
Let and suppose that satisfies the following conditions:
Then problems (1.5) and (1.6) have at least three positive solutions , and such that
Define an integral operator by
for . It is easy to obtain that is a completely continuous operator and every fixed point of is a solution of (1.5) and (1.6).
Thus we set out to verify that the operator satisfies Avery-Peterson fixed point theorem which will prove the existence of three fixed points of . Now the proof is divided into some steps.
By virtue of , , and Lemma 2.2 we know that there exists a constant such that
We first show that implies that
In fact, for , , by Lemma 2.2, there is It follows from that
Thus (3.6) holds.
Next we show that condition in Lemma 2.4 holds. Let Then it is easy to see that , , and for , so . Also, we have
So . Hence
If , then for It follows from condition that
Therefore we have
That is, condition in Lemma 2.4 is satisfied.
We now prove that in Lemma 2.4 holds. In fact, since then with Lemma 2.3 it follows that
for with . Hence condition in Lemma 2.4 is satisfied.
Finally, we assert that in Lemma 2.4 also holds.
Observe that , so Suppose with Then, by hypothesis we have
Thus condition in Lemma 2.4 holds.
Therefore an application of Lemma 2.4 implies that the BVPs (1.5) and (1.6) have at least three positive solutions , and such that (3.3) holds.
In this section, we present an example to explain our result.
Let , , , and , , , , . We consider the following boundary value problem:
Choosing , , , , direct calculation shows that
Then all conditions of Theorem 3.1 hold. Thus with Theorem 3.1, the BVP (4.1) has at least three positive solutions , , and such that
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Supported by the NNSF of China (10801065) and NSF of Gansu Province of China (0803RJZA096).