- Research Article
- Open Access
© Li-Hua Bian et al. 2009
Received: 17 August 2009
Accepted: 26 October 2009
Published: 27 October 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.
The main techniques are Schauder fixed point theorem and upper and lower solutions method.
By using fixed point theorems in cones, the existence criteria of multiple positive solutions are established.
They gave sufficient condition for the existence of three positive solutions by using a fixed point theorem due to Avery and Peterson .
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
The proof is complete.
The following fixed point theorem due to Avery and Peterson is fundamental in the proof of our main results.
Lemma 2.4 (see ).
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).
Now we state and prove our main result.
Thus (3.6) holds.
In this section, we present an example to explain our result.
Supported by the NNSF of China (10801065) and NSF of Gansu Province of China (0803RJZA096).
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