Research Article | Open Access

You-Hui Su, Weili Wu, Xingjie Yan, "Existence Theory for Pseudo-Symmetric Solution to -Laplacian Differential Equations Involving Derivative", *Abstract and Applied Analysis*, vol. 2011, Article ID 182831, 19 pages, 2011. https://doi.org/10.1155/2011/182831

# Existence Theory for Pseudo-Symmetric Solution to -Laplacian Differential Equations Involving Derivative

**Academic Editor:**Yuri V. Rogovchenko

#### Abstract

We all-sidedly consider a three-point boundary value problem for -Laplacian differential equation with nonlinear term involving derivative. Some new sufficient conditions are obtained for the existence of at least one, triple, or arbitrary odd positive pseudosymmetric solutions by using pseudosymmetric technique and fixed-point theory in cone. As an application, two examples are given to illustrate the main results.

#### 1. Introduction

Recent research results indicate that considerable achievement was made in the existence of positive solutions to dynamic equations; for details, please see [1–6] and the references therein. In particular, the existence of positive pseudosymmetric solutions to -Laplacian difference and differential equations attract many researchers' attention, such as [7–11]. The reason is that the pseudosymmetry problem not only has theoretical value, such as in the study of metric manifolds [12], but also has practical value itself; for example, we can apply this characteristic into studying the chemistry structure [13]. On another hand, there are much attention paid to the positive solutions of boundary value problems (BVPs) for differential equation with the nonlinear term involved with the derivative explicitly [14–18]. Hence, it is natural to continue study pseudosymmetric solutions to -Laplacian differential equations with the nonlinear term involved with the first-order derivative explicitly.

First, let us recall some relevant results about BVPs with -Laplacian, We would like to mention the results of Avery and Henderson [7, 8], Ma and Ge [11] and Sun and Ge [16]. Throughout this paper, we denote the -Laplacian operator by ; that is, for with and .

For the three-point BVPs with -Laplacian
here, is constant, by using the five functionals fixed point theorem in a cone [19], Avery and Henderson [8] established the existence of at least *three* positive pseudosymmetric solutions to BVPs (1.1). The authors also obtained the similar results in their paper [7] for the discrete case. In addition, Ma and Ge [11] developed the existence of at least *two* positive pseudosymmetric solutions to BVPs (1.1) by using the monotone iterative technique.

For the three-point -Laplacian BVPs with dependence on the first-order derivative
Sun and Ge [16] obtained the existence of at least *two* positive pseudosymmetric solutions to BVPs (1.2) via the monotone iterative technique again. However, it is worth mentioning that the above-mentioned papers [7, 8, 10, 11, 16], the authors only considered results on the existence of positive pseudosymmetric solutions partly, they failed to further provide comprehensive results on the existence of positive pseudosymmetric solutions to -Laplacian. Naturally, in this paper, we consider the existence of positive pseudosymmetric solutions for -Laplacian differential equations in all respects.

Motivated by the references [7, 8, 10, 11, 16, 18], in present paper, we consider all-sidedly -Laplacian BVPs (1.2), using the compression and expansion fixed point theorem [20] and Avery-Peterson fixed point theorem [21]. We obtain that there exist at least *one, triple or arbitrary odd *positive pseudosymmetric solutions to problem (1.2). In particular, we not only get some local properties of pseudosymmetric solutions, but also obtain that the position of pseudosymmetric solutions is determined under some conditions, which is much better than the results in papers [8, 11, 16]. Correspondingly, we generalize and improve the results in papers Avery and Henderson [8]. From the view of applications, two examples are given to illustrate the main results.

Throughout this paper, we assume that(S1) is continuous, does not vanish identically on interval , and is pseudosymmetric about on ,(S2) is pseudosymmetric about on , and does not vanish identically on any closed subinterval of . Furthermore, .

#### 2. Preliminaries

In the preceding of this section, we state the definition of cone and several fixed point theorems needed later [20, 22]. In the rest of this section, we will prove that solving BVPs (1.2) is equivalent to finding the fixed points of a completely continuous operator.

We first list the definition of cone and the compression and expansion fixed point theorem [20, 22].

*Definition 2.1. * Let be a real Banach space. A nonempty, closed, convex set is said to be a cone provided the following conditions are satisfied:(i)if and , then ,(ii)if and , then .

Lemma 2.2 (see [20, 22]). *Let be a cone in a Banach space . Assume that are open bounded subsets of with . If is a completely continuous operator such that either *(i)*, or*(ii)*. **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 .

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

Next, we list the fixed point theorem due to Avery-Peterson [21].

Lemma 2.3 (see [21]). *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
*

Now, let . Then, is a Banach space with norm Define a cone by

The following lemma can be founded in [11], which is necessary to prove our result.

Lemma 2.4 (see [11]). *If , then the following statements are true: *(i)* for , here , *(ii)* for ,*(iii)*. *

Lemma 2.5. *If , then the following statements are true: *(i)*, *(ii)*,*(iii)*. *

*Proof. * (i) Since
which reduces to

(ii) By using for , we have is monotone decreasing function on . Moreover,
which implies that , so, for and for .

Now, we define the operator by here, .

Lemma 2.6. * is a completely continuous operator.*

*Proof. *In fact, for , and .

It is easy to see that the operator is pseudosymmetric about on .

In fact, for , we have , and according to the integral transform, one has
here, . Hence,

For , we note that , by using the integral transform, one has
where . Thus,

Hence, is pseudosymmetric about on .

In addition,
is continuous and nonincreasing in ; moreover, is a monotone increasing continuously differentiable function
it is easy to obtain . By using the similar way, we can deduce . So, . It is easy to obtain that is completely continuous.

Hence, the solutions of BVPs (1.2) are fixed points of the completely continuous operator .

#### 3. One Solutions

In this section, we will study the existence of one positive pseudosymmetric solution to problem (1.2) by Krasnosel'skii's fixed point theorem in a cone.

Motivated by the notations in reference [23], for , let

In the following, we discuss the problem (1.2) under the following four possible cases.

Theorem 3.1. *If and , problem (1.2) has at least one positive pseudosymmetric solution .*

*Proof. *In view of , there exists an such that
here, and satisfies

If with , by Lemma 2.5, we have
hence,
If set , one has .

According to , there exists an such that
where , and satisfies
Set

For , we have since .

If with , Lemmas 2.4 and 2.5 reduce to
For , according to (3.6), (3.7) and (3.9), we get
Thus, by (i) of Lemma 2.2, the problem (1.2) has at least one positive pseudosymmetric solution in .

Theorem 3.2. *If and , problem (1.2) has at least one positive pseudosymmetric solution . *

*Proof. *Since , there exists an such that
here, and is such that
If with , Lemma 2.5 implies that
now, by (3.11), (3.12), and (3.13), we have
If let , one has .

Now, we consider .

Suppose that is bounded, for some constant , then
Pick
here, is an arbitrary positive constant and satisfy the (3.21). Let

If , one has , then (3.15) and (3.16) imply that

Suppose that is unbounded.

By definition of , there exists such that
where and satisfies

From , we have
here, is an arbitrary positive constant.

Then, for , we have

If , one has , which reduces to
Consequently, for any cases, if we take , we have . Thus, the condition (ii) of Lemma 2.2 is satisfied.

Consequently, the problem (1.2) has at least one positive pseudosymmetric solution

Theorem 3.3. * Suppose that the following conditions hold: *(i)*there exist nonzero finite constants and such that and ,*(ii)*there exist nonzero finite constants and such that and .**Then, problem (1.2) has at least one positive pseudosymmetric solution .*

*Proof. *(i) In view of , there exists an such that
here, and satisfies

If with , by Lemma 2.5, we have
hence,
If set , one has .

According to , there exists an such that
where , and satisfies
Set
If with , Lemmas 2.4 and 2.5 reduce to
For , according to (3.29), (3.30) and (3.32), we get
Thus, by (i) of Lemma 2.2, the problem (1.2) has at least one positive pseudosymmetric solution in .

(ii) By using the similar way as to Theorem 3.2, we can prove to it.

#### 4. Triple Solutions

In the previous section, some results on the existence of at least one positive pseudosymmetric solutions to problem (1.2) are obtained. In this section, we will further discuss the existence criteria for at least *three* and arbitrary odd positive pseudosymmetric solutions of problems (1.2) by using the Avery-Peterson fixed point theorem [21].

Choose a , for the notational convenience, we denote

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

Now, we state and prove the results in this section.

Theorem 4.1. *Suppose that there exist constants , and such that . In addition, satisfies the following conditions: *(i)* for ,*(ii)* for ,*(iii)* for .**Then, problem (1.2) has at least three positive pseudosymmetric solutions , such that
*

*Proof. *According to the definition of completely continuous operator and its properties, we need to show that all the conditions of Lemma 2.3 hold with respect to .

It is obvious that

Firstly, we show that .

For any , we have
hence, the assumption (i) implies that
From the above analysis, it remains to show that (i)–(iii) of Lemma 2.3 hold.

Secondly, we verify that condition (i) of Lemma 2.3 holds; let , and it is easy to see that
in addition, we have , since . Thus
For any
one has
it follows from the assumption (ii) that

Thirdly, we prove that the condition (ii) of Lemma 2.3 holds. In fact,
For any , we have

Finally, we check condition (iii) of Lemma 2.3.

Clearly, since , we have . If
then
Hence, by assumption (iii), we have
Consequently, from above, all the conditions of Lemma 2.3 are satisfied. The proof is completed.

Corollary 4.2. * If the condition in Theorem 4.1 is replaced by the following condition (i′):*(i′)* ** then the conclusion of Theorem 4.1 also holds.*

*Proof. *From Theorem 4.1, we only need to prove that (i′) implies that (i) holds. That is, assume that (i′) holds, then there exists a number such that

Suppose on the contrary that for any , there exists such that
Hence, if we choose , then there exist such that
and so
Since the condition (i′) holds, there exists satisfying
Hence, we have
Otherwise, if
it follows from (4.21) that
which contradicts (4.19).

Let
then
which also contradicts (4.20).

Theorem 4.3. * Suppose that there exist constants such that
**
here, and . In addition, suppose that satisfies the following conditions:*(i)* for ,*(ii)* for ,*(iii)* for .**Then, problem (1.2) has at least positive pseudosymmetric solutions. *

*Proof. * When , it is immediate from condition (i) that
It follows from the Schauder fixed point theorem that has at least one fixed point
which means that

When , it is clear that Theorem 4.1 holds (with . Then, there exists at least three positive pseudosymmetric solutions , and such that
Following this way, we finish the proof by induction. The proof is complete.

#### 5. Examples

In this section, we present two simple examples to illustrate our results.

*Example 5.1. *Consider the following BVPs:

Note that
Hence, Theorem 3.2 implies that the BVPs in (5.1) have at least one pseudosymmetric solution .

*Example 5.2. *Consider the following BVPs with :
where and

Note that , then a direct calculation shows that
If we take , then holds; furthermore,