Abstract and Applied Analysis

Volume 2012 (2012), Article ID 352159, 23 pages

doi:10.1155/2012/352159

Research Article

Multiple Bounded Positive Solutions to Integral Type BVPs for Singular Second Order ODEs on the Whole Line

Yuji Liu

Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, China

Received 3 June 2012; Accepted 6 August 2012

Academic Editor: Ziemowit Popowicz

Copyright © 2012 Yuji Liu. 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 is concerned with the integral type boundary value problems of the second order differential equations with one-dimensional p-Laplacian on the whole line. By constructing a suitable Banach space and a operator equation, sufficient conditions to guarantee the existence of at least three positive solutions of the BVPs are established. An example is presented to illustrate the main results. The emphasis is put on the one-dimensional p-Laplacian term involved with the function ρ, which makes the solutions un-concave.

1. Introduction

The multipoint boundary-value problems for linear second order ordinary differential equations (ODEs for short) was initiated by Il'in and Moiseev [1]. Since then, more general nonlinear multi-point boundary-value problems (BVPs for short) were studied by several authors, see the text books [2–4] and the references cited therein.

Differential equations governed by nonlinear differential operators have been widely studied. In this setting the most investigated operator is the classical one-dimensional -Laplacian, that is, with . This operator is involved in some models, for example, in non-Newtonian fluid theory, diffusion of flows in porous media, nonlinear elasticity, and theory of capillary surfaces. The related nonlinear differential equation has the form

where with is a one dimensional -Laplacian. For a comprehensive bibliography on this subject, see, for example [5–9].

In this paper, we consider the more generalized BVP for second order differential equation on the whole line with -Laplacian coupled with the integral type BCs, that is the BVP

where is a nonnegative Caratheodory function, satisfy , the integrals in mentioned equations are meant in the sense of Riemann-Stieljes, with is called a one dimensional -Laplacian, whose inverse function is denoted by .

The purpose is to establish sufficient conditions for the existence of at least three positive solutions of BVP(1.2). The result in this paper generalizes and improves some known ones since the one-dimensional -Laplacian term involved with the function , which makes the solutions unconcave and there exists no paper concerned with the existence of at least three positive solutions of this kind of integral BVPs on the whole lines. This paper fills the gap.

The remainder of this paper is organized as follows: the main result (Theorem 2.8) is presented in Section 2, and the example to show the main result is given in Section 3.

2. Main Results

In this section, we first present some background definitions in Banach spaces and state an important three fixed point theorem. Then the main results are given and proved.

Definition 2.1. Let be a real Banach space. The nonempty convex closed subset of is called a cone in if for all and and and imply .

Definition 2.2. A map is a nonnegative continuous concave or convex functional map provided is nonnegative, continuous, and satisfies , or , for all and .

Definition 2.3. An operator is completely continuous if it is continuous and maps bounded sets into precompact sets.

Definition 2.4. Let be positive constants, be two nonnegative continuous concave functionals on the cone , be three nonnegative continuous convex functionals on the cone . Define the convex sets as follows:

Lemma 2.5 (see [10]). Let be a real Banach space, be a cone in , be two nonnegative continuous concave functionals on the cone be three nonnegative continuous convex functionals on the cone . Assume that there exists a constant such that

Furthermore, suppose that are constants with . Let be a completely continuous operator. If

(C1) and ;(C2) and ;(C3) for with ;(C4) for each with ,

then has at least three fixed points , and such that .

Let us list the assumptions

(H1) satisfy , and

(H2) for with .(H3) on any finite subinterval of for each , is a Carathédory function, that is,(i) is measurable for any ,(ii) is continuous for a.e. ,(iii) for each , there exists nonnegative function such that implies

Choose

For , define the norm of by

One can prove that is a Banach space with the norm for .

Let . Consider the following auxiliary BVP

Lemma 2.6. Suppose that (H1)–(H3) hold. If such that is a solution of BVP(2.7), then

(i) is bounded and nonnegative on ;(ii) is concave with respect to ;(iii)for , it holds that with ;(iv)there exists a unique constant such that

Proof. Since , we get

Then there exists a nonnegative function such that

Then

(i) We know there exist the limits and . We claim that there exists such that . In fact, if for all , we get for all . It follows that

Then (H1) implies , which contradicts to for all . Similarly we can prove that does not hold. Then there exists such that .

Since , we know that is decreasing on . Then for all and for all . Hence

One sees that

Since and , we see that

Then we get that

This tells us that is bounded on .

It follows from (2.7) and (2.16) that

Then

Hence

Since

we get from (2.19) that and . Hence (2.16) implies that

(ii) We prove that is concave with respect to on . It is easy to see that and

Thus

It follows that

Hence

So

Since and , we get that . Hence is concave with respect to on .

(iii) Now, we prove that

Since for all , there exists the inverse function of . Denote the inverse function of by .

It follows from (2.13) that . One sees

If , note , then for , one has

Noting that and is concave with respect to , then, for ,

Similarly, if , note , for , one has

Hence (2.27) holds.

(iv) Finally, we prove the uniqueness of . Define

Then (H1) implies that is increasing on and . Let

then . By mean value theorem, there exists an unique satisfying . Then (iv) holds. This completes the proof of the lemma.

Choose and

Define the cone by

It is easy to see that is a cone in .

Define the operator by

where satisfies

It follows from Lemma 2.6(iv) that is well defined and . It is easy to show that

It follows from Lemma 2.6(i) and (iii) that for all . Then is well defined.

Lemma 2.7. Suppose (H1)–(H3) hold. Then is completely continuous.

Proof. First, we prove that is continuous.

We claim that the function is continuous in .

Let with as in . Let be the constants determined by

Since in as , there exists an such that . The fact is a Carathédory function means that there exists a nonnegative function such that

Then

So

which means that is uniformly bounded.

Suppose that does not converge to . By the bounded property, we know that there exist two subsequences and of with and and .

By the construction of , , we have

Let , using Lebesgue's dominated convergence theorem, the above equality implies

By Lemma 2.6 (iv), we get . Similarly, . Thus , a contradiction. So, for any , one has , which means is continuous.

Since is continuous, together with the continuity of , we get that is continuous.

Second, we show that is maps bounded subsets into bounded sets.

Let be bounded. Then, there exists such that . Hence

Then there exists a nonnegative function such that

Then

Thus for all . Therefore,

On the other hand, we have

Then

So, is bounded.

Third, given a bounded set , we prove that both and are equicontinuous on each finite subinterval on .

Then, there exists such that . Hence

Then there exists a nonnegative function such that

Then

For any , since is uniformly continuous on , there exists such that

For any and with , we have

Then there exists such that

Hence imply that

It follows that is equicontinuous on each finite subinterval on .

On the other hand, we have

Then is equicontinuous on each finite subinterval on .

At last given a bounded set , we show that both and are equiconvergent at , respectively.

Then, there exists such that . Hence

Then there exists a nonnegative function such that

Then

For any , since is uniformly continuous on , there exists such that

Since

uniformly as , we get that there exists such that

Hence implies that

Furthermore, we get

Hence and are equiconvergent at .

Similarly we can how that and are equiconvergent at . We omit the details.

Therefore, is equiconvergent at . So the operator is completely continuous.

Define the functionals on by

It is easy to see that are two nonnegative continuous concave functionals on the cone , are three nonnegative continuous convex functionals on the cone and for all .

For and , define

Theorem 2.8. Suppose that (H1)–(H3) hold. Given positive constants and , let , and be as above. If

and

(A1)  for all   ;(A2)  for all  ;(A3)  for all  ;

then BVP(1.2) has at least three positive solutions such that

Proof. We prove that all conditions in Lemma 2.5 are satisfied.

(i) By the definitions, it is easy to show that are two nonnegative continuous concave functionals on the cone , are three nonnegative continuous convex functionals on the cone and for all . One sees is a positive solution of BVP (1.2) if and only if is a solution of the operator equation .

(ii) For , we have

It follows that

Then

Hence

It follows that for all .

(iii) Corresponding to Lemma 2.5,

Now, we prove that all other conditions of Lemma 2.5 hold. One sees that . The remainder is divided into five steps.

Step 1. Prove that .

For , we have . Then and for all . So (A1) implies that

We have

Similarly to (ii), we can show that

It follows that

Then .

Step 2. Prove that

and .

Choose for all . Then and

It follows that .

For , one has that

Then

Thus (A2) implies that

Similarly to Lemma 2.6(i), we know that there exists such that . Then

Since

if , we get

If , we get

This completes Step 2.

Step 3. Prove that and

Choose . Then , and

It follows that .

For , one has that

Hence we get that

Then (A3) implies that

So

This completes Step 3.

Step 4. Prove that for with ;

For with , we have that and and . Then

This completes Step 4.

Step 5. Prove that for each with .

For with , we have and and . Then

This completes the Step 5.

Then Lemma 2.5 implies that has at least three fixed points , , and such that

Hence BVP(1.2) has three decreasing positive solutions and such that (2.70) holds. The proof is complete.

3. Examples

Now, we present an example, whose three positive solutions cannot be obtained by theorems in known papers, to illustrate the main results.

Example 3.1. Consider the following BVP

Corresponding to BVP(1.2), one sees that  , , , is nonnegative and continuous and is defined by

Choose , . By direct computation, we see that , and are given by

One can show that

Suppose that

From

it is easy to show that

(A₁) for all ;(A₂) for all ;(A₃) for all ;

then Theorem 2.8 implies that BVP(3.1) has at least three positive solutions such that

Remark 3.2. Example 3.1 implies that there is a large number of functions that satisfy the conditions of Theorem 2.8. In addition, the conditions of Theorem 2.8 are also easy to check.

Acknowledgments

This work was supported by Natural Science Foundation of Guangdong provinces Grant (no S2011010001900) and the Foundation for High-level talents in Guangdong Higher Education Project.

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