Multiple Positive Solutions of -Point BVPs for Third-Order -Laplacian Dynamic Equations on Time Scales
© 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
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
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).
Therefore, We can choose and the proof is complete.
If , then for .
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 ).
for all . Suppose that is completely continuous and there exist positive numbers , and with such that
and for ;
for with ;
and for with .
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
Now we state and prove our main result.
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.
Thus (3.6) holds.
So . Hence
That is, condition in Lemma 2.4 is satisfied.
for with . Hence condition in Lemma 2.4 is satisfied.
Finally, we assert that in Lemma 2.4 also holds.
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.
Supported by the NNSF of China (10801065) and NSF of Gansu Province of China (0803RJZA096).
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