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Discrete Dynamics in Nature and Society
Volume 2009, Article ID 141929, 27 pages
http://dx.doi.org/10.1155/2009/141929
Research Article

Multiple Positive Symmetric Solutions to -Laplacian Dynamic Equations on Time Scales

1School of Mathematics and Physical Sciences, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221008, China
2Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received 1 July 2009; Accepted 18 November 2009

Academic Editor: Leonid Shaikhet

Copyright © 2009 You-Hui Su and Can-Yun Huang. 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.

Abstract

This paper makes a study on the existence of positive solution to -Laplacian dynamic equations on time scales . Some new sufficient conditions are obtained for the existence of at least single or twin positive solutions by using Krasnosel'skii's fixed point theorem and new sufficient conditions are also obtained for the existence of at least triple or arbitrary odd number positive solutions by using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem. As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations, as well as in the general time-scale setting.

1. Introduction

Initiated by Hilger in his Ph.D. thesis [1] in 1988, the theory of time scales has been improved ever since, especially in the unification of the theory of differential equations in the continuous case and the theory of difference equations in the discrete case. For the time being, it remains a field of vitality and attracts attention of many distinguished scholars worldwide. In particular, the theory is also widely applied to biology, heat transfer, stock market, wound healing, epidemic models [25], and so forth.

Recent research results indicate that considerable achievement has been made in the existence problems of positive solutions to dynamic equations on time scales. For details, please see [613] and the references therein. Symmetry and pseudosymmetry have been widely used in science and engineering [14]. The reason is that symmetry and pseudosymmetry are not only of its theoretical value in studying the metric manifolds [15] and symmetric graph [16, 17], and so forth, but also of its practical value, for example, we can apply this characteristic to study graph structure [18, 19] and chemistry structure [20]. Yet, few literature resource [21, 22] is available concerning the characteristics of positive solutions to -Laplacian dynamic equations on time scales.

Throughout this paper, we denote the -Laplacian operator by , that is, for with and

For convenience, we think of the blanket as an assumption that are points in for an interval we always mean Other type of intervals is defined similarly.

We would like to mention the results of Sun and Li [11, 12]. In [12], Sun and Li considered the two-point BVP

and established the existence theory for positive solutions of the above problem. They [11] also considered the -point boundary value problem with -Laplacian

and gave the existence of single or multiple positive solutions to the above problem. The main tools used in these two papers are some fixed-point theorems [2325].

It is also noted that the researchers mentioned above [11, 12] only considered the existence of positive solutions. As a results, they failed to further provide characteristics of solutions, such as, symmetry. Naturally, it is quite necessary to consider the characteristics of solutions to -Laplacian dynamic equations on time scales.

Let be a symmetric time scale such that . we consider the following -Laplacian boundary value problem on time scales of the form:

By using symmetric technique, the Krasnosel'skii's fixed point theorem [24], the generalized Avery-Henderson fixed point theorem [26], and Avery-Peterson fixed point theorem [27], we obtain the existence of at least single, twin, triple, or arbitrary odd positive symmetric solutions of problem (1.3). As applications, two examples are given to illustrate the main results and their differences. These results are even new for the special cases of continuous and discrete equations as well as in the general time-scale setting.

The rest of the paper is organized as follows. In Section 2, we present several fixed point results. In Section 3, by using Krasnosel'skii's fixed point theorem, we obtain the existence of at least single or twin positive symmetric solutions to problem (1.3). In Section 4, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to problem (1.3) are established. In Section 5, we present two simple examples to illustrate our results.

For convenience, we now give some symmetric definitions.

Definition 1.1. The interval is said to be symmetric if any given we have

We note that such a symmetric time scale exists. For example, let

It is obvious that is a symmetric time scale.

Definition 1.2. A function is said to be symmetric if is symmetric over the interval That is, , for any given

Definition 1.3. We say is a symmetric solution to problem (1.3) on provided that is a solution to boundary value problem (1.3) and is symmetric over the interval

Basic definitions on time scale can be found in [6, 7, 28]. Another excellent sources on dynamical systems on measure chains are the book in [29].

Throughout this paper, it is assumed that

(H1) is continuous, and does not vanish identically;(H2) is symmetric over the interval and does not vanish identically on any closed subinterval of where denotes the set of all left dense continuous functions from to

2. Preliminaries

Let and equip norm

then is a Banach space. Define a cone by

Assume that with . By using the symmetric and concave characters of and , it is easy to obtain the following results.

Lemma 2.1. Assume that with . If then (i)(ii)

From the previous lemma we know that for

The operator is defined by

It is obvious that is completely continuous operator and all the fixed points of are the solutions to the boundary value problem (1.3).

In addition, it is easy to see that the operator is symmetric. In fact, for , we have , by using the integral transform, we have

Hence, is symmetric.

Now, we provide some background material from the theory of cones in Banach spaces [24, 26, 27, 30], and then state several fixed point theorems needed later.

Firstly, we list the Krasnosel'skii's fixed point theorem [24].

Lemma 2.2 (see [24]). Let be a cone in a Banach space Assume that , are open subsets of with , If is a completely continuous operator such that either(i), and , or(ii), and , then has a fixed point in

Given a nonnegative continuous functional on a cone of a real Banach space we define, for each the set

Secondly, we state the generalized Avery-Henderson fixed point theorem [26].

Lemma 2.3 (see [26]). Let be a cone in a real Banach space Let , , and be increasing, nonnegative continuous functional on such that for some and and for all Suppose that there exist positive numbers and with and is a completely continuous operator such that (i) for all (ii) for all (iii) and for ,then has at least three fixed points , , and belonging to such that

The following lemma can be found in [21].

Lemma 2.4 (see [21]). Let be a cone in a real Banach space Let , , and be increasing, nonnegative continuous functional on such that for some and and for all Suppose that there exist positive numbers and with and is a completely continuous operator such that:(i) for all (ii) for all (iii) and for ,then has at least three fixed points , and belonging to such that

Let and be nonnegative continuous convex functionals on , is a nonnegative continuous concave functional on , and is a nonnegative continuous functional, respectively on We define the following convex sets:

and a closed set

Finally, we list the fixed point theorem due to Avery-Peterson [27].

Lemma 2.5 (see [27]). Let be a cone in a real Banach space and defined as above, moreover, satisfies for such that, for some positive numbers and , for all Suppose that is completely continuous and there exist positive real numbers , , , with such that (i) and for (ii) for with (iii) and for all with then has at least three fixed points such that

3. Single or Twin Solutions

Let

We define number of zeros in the set and number of infinities in the set . Clearly, , or and there exist six possible cases: (i) and ; (ii) and ; (iii) and ; (iv) and ; (v) and ; (vi) and . In the following, by using Krasnosel'skii's fixed point theorem in a cone, we study the existence of positive symmetric solutions to problem (1.3) under the above six possible cases.

3.1. For the Case and

In this subsection, we discuss the existence of single positive symmetric solution of the problem (1.3) under and .

Theorem 3.1. Problem (1.3) has at least one positive symmetric solution in the case and

Proof. We divide the proof into two cases.
Case 1 ( and ). In view of , there exists an such that for , where arbitrary small and satisfies .
If with then We let then for
From there exists an such that for , where and satisfies the following inequality: Set If with then, by Lemma 2.1, one has For in terms of (3.3) and (3.5), we get Thus, by (i) of Lemma 2.2, problem (1.3) has at least single positive symmetric solution in with
Case 2 ( and ). Since there exists an such that for where is such that If with then, by (3.7), one has If we let then for
Now, we consider By definition, there exists such that where satisfies
Suppose that is bounded, then for all and some constant Pick If with then

Suppose that is unbounded. From we have for arbitrary here and are arbitrary positive constants. This implies that if . Hence, it is easy to know that there exists such that for If with then by using (3.9) and (3.10), we have Consequently, in either case, if we take then, for we have . Thus, condition (ii) of Lemma 2.2 is satisfied. Consequently, problem (1.3) has at least single positive symmetric solution in with The proof is complete.

3.2. For the Case and

In this subsection, we discuss the existence of positive symmetric solutions to problems (1.3) under and .

First, we will state and prove the following main result of problem (1.3).

Theorem 3.2. Suppose that the following conditions hold:(i)there exists constant such that for where (ii)there exists constant such that for where furthermore, then problem (1.3) has at least one positive symmetric solution such that lies between and

Proof. Without loss of generality, we may assume that
Let For any in view of condition (i), we have which yields
Now, set For Lemma 2.1 implies that Hence, by condition (ii) we get So, if we take then Consequently, in view of , (3.15) and (3.18), it follows from Lemma 2.2 that problem (1.3) has a positive symmetric solution in The proof is complete.

3.3. For the Case and or and

In this subsection, under the conditions and or and we discuss the existence of positive symmetric solutions to problem (1.3).

Theorem 3.3. Suppose that and hold. Then problem (1.3) has at least one positive symmetric solution.

Proof. It is easy to see that under the assumptions, conditions (i) and (ii) in Theorem 3.2 are satisfied. So the proof is easy and we omit it here.

Theorem 3.4. Suppose that and hold, then problem (1.3) has at least one positive symmetric solution.

Proof. Firstly, let there exists a sufficiently small that satisfies Thus, we have which implies that condition (ii) in Theorem 3.2 holds.
Nextly, for there exists a sufficiently large such that We consider two cases.
Case 1. Assume that is bounded, that is, here some constant. If we take sufficiently large such that then Consequently, from the above inequality, condition (i) of Theorem 3.2 is true.Case 2. Assume that is unbounded.
From there exists such that Since , by (3.21), we get hence Thus, condition (i) of Theorem 3.2 is fulfilled.
Consequently, Theorem 3.2 implies that the conclusion of this theorem holds.

From the proof of Theorems 3.1 and 3.2, respectively, we have the following two results.

Corollary 3.5. Suppose that and condition (ii) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

Corollary 3.6. Suppose that and condition (ii) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

Theorem 3.7. Suppose that and hold, then problem (1.3) has at least one positive symmetric solution.

Proof. First, in view of then by inequality (3.7), we have for Next, by for there exists a sufficiently small such that which implies that (i) of Theorem 3.2 holds, that is, (3.14) is true. Hence, we obtain for The result is obtained and the proof is complete.

Theorem 3.8. Suppose that and hold, then problem (1.3) has at least one positive symmetric solution.

Proof. On one hand, since by inequality (3.9), one gets , On the other hand, since from the technique similar to the second part proof in Theorem 3.4, one obtains that condition (i) of Theorem 3.2 is satisfied, that is, inequality (3.14) holds, one has , where Hence, problem (1.3) has at least one positive symmetric solution. The proof is complete.

From Theorems 3.7 and 3.8, respectively, it is easy to obtain the following two corollaries.

Corollary 3.9. Assume that and condition (i) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

Corollary 3.10. Assume that and condition (i) in Theorem 3.2 hold, then problem (1.3) has at least one positive symmetric solution.

3.4. For the Case and or and

In this subsection, under and or and we study the existence of multiple positive solutions to problems (1.3).

Combining the proofs of Theorems 3.1 and 3.2, it is easy to prove the following two theorems.

Theorem 3.11. Suppose that and and condition (i) of Theorem 3.2 hold, then problem (1.3) has at least two positive solutions such that

Theorem 3.12. Suppose that and and condition (ii) of Theorem 3.2 hold, then problem (1.3) has at least two positive solutions such that

4. Triple Solutions

In the previous section, we have obtained some results on the existence of at least single or twin positive symmetric solutions to problem (1.3). In this section, we will further discuss the existence of positive symmetric solutions to problem (1.3) by using two different methods. And the conclusions we will arrive at are different with their own distinctive advantages.

Based on the obtained symmetric solution position and local properties, we can only get some local properties of solutions by using method one; however, the position of solutions is not determined. In contrast, by means of method two, we cannot only get some local properties of solutions but also give the position of all solutions, with regard to some subsets of the cone, which has to meet some conditions which are stronger than those of method one. Obviously, the local properties of obtained solutions are different by using the two different methods. Hence, it is convenient for us to comprehensively comprehend the solutions of the models by using the two different techniques.

In Section 5, two examples are given to illustrate the differences of the results obtained by the two different methods.

For the notational convenience, we denote

4.1. Result 1

In this subsection, in view of the generalized Avery-Henderson fixed-point theorem [26], the existence criteria for at least triple and arbitrary odd positive symmetric solutions to problems (1.3) are established.

For we define the nonnegative, increasing, continuous functionals , and by

It is obvious that for each By Lemma 2.1, one obtains for all , here .

We now present the results in this subsection.

Theorem 4.1. If there are positive numbers , , such that In addition, satisfies the following conditions:(i) for (ii) for (iii) for Then problem (1.3) has at least three positive symmetric solutions , , and such that

Proof. By the definition of completely continuous operator and its properties, it has to be demonstrated that all the conditions of Lemma 2.3 hold with respect to It is easy to obtain that
Firstly, we verify that if , then .
If then Lemma 2.1 implies that we have Thus, by condition (i), one has
Secondly, we show that for
If we choose then In view of Lemma 2.1, we have So Using condition (ii), we get
Finally, we prove that and for all
In fact, the constant function Moreover, for we have which implies for Hence, Therefore By using assumption (iii), one has Thus, all the conditions in Lemma 2.3 are satisfied. From (H1) and (H2), we have that the solutions to problem (1.3) do not vanish identically on any closed subinterval of . Consequently, problem (1.3) has at least three positive symmetric solutions , , and belonging to and satisfying (4.3). The proof is complete.

From Theorem 4.1, we see that, when assumptions as (i), (ii), and (iii) are imposed appropriately on we can establish the existence of an arbitrary odd number of positive symmetric solutions to problem (1.3).

Theorem 4.2. Let Suppose that there exist positive numbers , , such that In addition, satisfies the following conditions:(i) for (ii) for (iii) for Then problem (1.3) has at least positive symmetric solutions.

Proof. When it is clear that Theorem 4.1 holds. Then we can obtain at least three positive symmetric solutions , and satisfying Following this way, we finish the proof by induction. The proof is complete.

Using Lemma 2.4, it is easy to have the following results.

Theorem 4.3. Suppose that there are positive numbers , , such that In addition, satisfies the following conditions:(i) for (ii) for (iii) for Then problem (1.3) has at least three positive symmetric solutions , , and such that

From Theorem 4.3, we can obtain Theorem 4.4 and Corollary 4.5.

Theorem 4.4. Let Suppose that there existence positive numbers , , such that In addition, satisfies the following conditions:(i) for (ii) for (iii) for Then problem (1.3) has at least positive symmetric solutions.

Corollary 4.5. Assume that satisfies the following conditions:(i)(ii)there exists such that for then problem (1.3) has at least three positive symmetric solutions.

Proof. First, by condition (ii), let one gets which implies that (ii) of Theorem 4.3 holds.
Second, choose sufficiently large to satisfy Since there exists sufficiently small such that Without loss of generality, suppose Choose such that For we have and Thus, by (4.18) and (4.19), we have this implies that (iii) of Theorem 4.3 is true.
Third, choose sufficiently large such that Since there exists sufficiently large such that Without loss of generality, suppose Choose Then which means that (i) of Theorem 4.3 holds.
From above analysis, we get then, all conditions in Theorem 4.3 are satisfied. Hence, problem (1.3) has at least three positive symmetric solutions.

In terms of Theorem 4.1, we also have the following corollary.

Corollary 4.6. Assume that satisfies conditions(i)(ii)there exists such that for then problem (1.3) has at least three positive symmetric solutions.

4.2. Result 2

In this subsection, the existence criteria for at least triple positive or arbitrary odd positive symmetric solutions to problems (1.3) are established by using the Avery-Peterson fixed point theorem [27].

Define the nonnegative continuous convex functionals and , nonnegative continuous concave functional and nonnegative continuous functional , respectively, on by

Now, we list and prove the results in this subsection.

Theorem 4.7. Suppose that there exist constants , , such that In addition, suppose that holds, satisfies the following conditions:(i) for (ii) for (iii) for then problem (1.3) has at least three positive symmetric solutions such that

Proof. By the definition of completely continuous operator and its properties, it suffices to show that all the conditions of Lemma 2.5 hold with respect to
For all and . Hence, condition (2.8) is satisfied.
Firstly, we show that
For any in view of and assumption (i), one has From the above analysis, it remains to show that (i)–(iii) of Lemma 2.5 hold.
Secondly, we verify that condition (i) of Lemma 2.5 holds, let with From the definitions of , and , respectively, it is easy to see that and . In addition, in view of , we have Thus For any then we get for all . Hence, by assumption (ii), we have
Thirdly, we prove that condition (ii) of Lemma 2.5 holds. For any with that is, So, in view of , and (4.30), one has
Finally, we check condition (iii) of Lemma 2.5. Clearly, since we have If with then Lemma 2.1 implies that This yields for all Hence, by assumption (iii), we have Consequently, all conditions of Lemma 2.5 are satisfied. The proof is completed.

We remark that condition (i) in Theorem 4.7 can be replaced by the following condition ():

which is a special case of (i).

Corollary 4.8. If condition (i) in Theorem 4.7 is replaced by (), then the conclusion of Theorem 4.7 also holds.

Proof. By Theorem 4.7, we only need to prove that (i') implies that (i) holds, that is, if (i') holds, then there is a number such that for
Suppose on the contrary that for any there exists such that Hence, if we choose with then there exist such that and so Since condition () holds, then there exists such that Hence, we have Otherwise, if then it follows from (4.38) that which contradicts (4.36).
Let then which also contradicts (4.37). The proof is complete.

Theorem 4.9. Let Suppose that there exist constants , , such that In addition, suppose that holds, then satisfies the following conditions:(i) for (ii) for