Research Article | Open Access
Existence of Positive Solutions for -Laplacian Dynamic Equations with Derivative on Time Scales
We consider the existence of positive solutions of nonlinear p-Laplacian dynamic equations with derivative on time scales. Applying the Avery-Peterson fixed point theorem, we obtain at least three positive solutions to the problem. An example is also presented to illustrate the applications of the obtained results.
In this paper we consider the existence of positive solutions of nonlinear p-Laplacian dynamic equations with derivative on time scales where is a time scale, is a delta derivative, is a nabla derivative, , , , , , and other types of intervals are defined similarly. Throughout this paper, we denote the p-Laplacian operator by ; that is, ,, and .
The theory of dynamic equations on time scales was introduced firstly by Stefan Hilger in 1988. Since then, more and more scholars are interested in this area. The main reason is that the time scales theory can not only unite continuous and discrete dynamic equations but also have important applications, for example, in the study of insect population models, neural networks, heat transfer, economic, stock market, and epidemic models.
In recent years, there were a lot of attention focused on the study of boundary value problems of nonlinear dynamic equations. Hong in  studied the existence of positive solutions for the boundary value problem of p-Laplacian dynamic equations Yaslan in  considered nonlinear second-order three-point boundary value problems for dynamic equations on time scales P. G. Wang and Y. Wang in  studied the following boundary value problem: Yaslan in  considered the existence of solutions of the following m-point boundary value problem on time scales They obtained the existence of at least three nonnegative solutions by using fixed point theorems in cones. The authors in [5–10] dealt with some other classes of differential equations on time scales.
However, the nonlinear terms in the above mentioned literatures [1–4] are not involving a derivative. As far as we know, there is no paper to study the existence of positive solutions for second-order p-Laplacian dynamic equations with derivative on time scales. This paper is to fill this gap. By constructing a special cone and using the Avery-Peterson fixed point theorem, we obtain the existence of at least three nonnegative solutions.
The rest of this paper is organized as follows. In Section 2, we prove several preliminary results which are needed later. In Section 3, conditions for the existence of at least three nonnegative solutions to the BVP (1) are discussed by using the Avery-Peterson fixed point theorem. In Section 4, we give one example to illustrate our results.
2. Preliminaries and Some Lemmas
In this section, we first introduce some concepts on time scales and several lemmas.
Let be a time scale (an arbitrary nonempty closed subset of the real numbers ). For we define the forward jump operator by while the backward jump operator is defined by We set , if has a maximum , and , if has a minimum . If , we say that is right scattered, while we say that is left scattered, if . Also, if and , then is called right dense, and if and , then is called left dense. If has a right-scattered minimum m, we define ; otherwise set . If has a left-scattered maximum M, we define ; otherwise set .
For and , we define the delta derivative of , to be the number (provided it exists) with the property that, for any , there is a neighborhood of such that for all . For and , we define the nabla derivative of , to be the number (provided it exists) with the property that, for any , there is a neighborhood of such that for all .
A function is left-dense continuous or ld-continuous provided it is continuous at left-dense points in , and its right-sided limits exist (finite) at right-dense points in .
For more details about the subject of concepts and calculations on time scales, please see the book by Bohner and Peterson .
Consider the Banach space ; that is,
is differential on , and is ld-continuous on with the norm Let
It is easy to see that is a cone in .
For convenience, we list the following assumptions. does not vanish identically on any closed subinterval of . The function is left-dense continuous, and .
To obtain our main results, we make use of the following lemmas.
Lemma 1. Assume that and are satisfied. Then is the solution of the BVP (1) if and only if
Proof. Firstly, we prove the necessity. Let be the solution of the BVP (1). -integrating on (1) from to , we have
By using the second boundary condition, we get
∆-integrating from to , we get
Submitting into the first boundary condition of the problem (1), we obtain Thus,
Secondly, we prove the sufficiency. Let be the solution of the problem (12). Then, Taking the -differentiating for (20), we obtain that is, By (12), we have Hence,
The proof is complete.
Lemma 2. Assume that and hold. Then is a completely continuous operator.
Proof. Firstly we prove .
For , by Lemma 1, , and , we obtain which implies .
Secondly, maps a bounded set into a bounded set. Let , where . Because is continuous, there exists a such that Hence, for , we have Consequently, is bounded.
For , we get Thus is relatively compact by using the Arzela-Ascoli theorem.
Finally we prove that is continuous. Let satisfying This means that Note that Using the Lebesgue dominated convergence theorem on time scales, this together with (28) implies that From the definition of , we know that Therefore is continuous on . The proof is complete.
Lemma 3. If , then(i), ;(ii), and .
Proof. First, prove (i). Let . From , , and , we have .
Second, prove (ii). Since is a concave function, for , we have
From , it follows that . The proof is complete.
Lemma 4. Let . Then there exists a real number such that , where .
Proof. Since , , and , we have By Lemma 3, we obtain The proof is complete.
In the rest of this section, we state the main tool used in this paper.
Let be functionals on cone in a real Banach space, where are nonnegative continuous convex functionals, is a nonnegative continuous concave functional, is a nonnegative continuous functional, and ; define the following sets:
Lemma 5 (see , the Avery-Peterson fixed point theorem). Let be a cone in a real Banach space and , and defined as above; moreover, satisfies for , and
for some positive numbers and . Suppose that is a completely continuous operator and that there exist positive real numbers , and with such that the following conditions are satisfied:(i), and for ;(ii) for with ;(iii) and for with .
Then has at least three fixed points , and such that
3. Main Results
Define the nonnegative continuous convex functionals , nonnegative continuous concave functional , and nonnegative continuous functional as follows: By Lemma 3, we have
Theorem 6. Assume that and hold. In addition, suppose that there exist constants such that and the function satisfies the following conditions:;;.Then BVP (1) has at least three nonnegative solutions , and satisfying(i);(ii), , and .
Proof. It is sufficient to show that has at least three fixed points. To this purpose, we show that all conditions of Lemma 5 are fulfilled. Now we divide this proof into three steps.
Step 1. Consider, .
For , we have . By Lemma 4, we get . From , it follows that
Applying this and Lemma 2, we have that is completely continuous.
Step 2. (i) and (ii) in Lemma 5 hold.
Let . Clearly , and So .
we have , where .
By , for , we know
So (i) in Lemma 5 is satisfied.
For with , since , by (ii) in Lemma 3, we can get So
Then (ii) in Lemma 5 holds.
Step 3. (iii) in Lemma 5 holds.
Clearly , so . If with , by , we obtain
From above, the hypotheses of the Avery-Peterson theorem are satisfied. Therefore the problem (1) has at least three nonnegative solutions , and satisfying(i);(ii), . The proof is complete.
Let . We consider the following BVP: where , and
By a simple calculation, we obtain . Choose . Thus satisfies(1), (2), (3).
The authors express their heartful thanks to reviewer(s) comments. This research is supported by Innovation Project for Graduate Education of Shandong Province (SDYY10058).
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Copyright © 2013 Jinjun Fan and Liqing Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.