Positive Solutions for Integral Boundary Value Problem with ϕ-Laplacian Operator

We consider the existence, multiplicity of positive solutions for the integral boundary value problem with -Laplacian , , , , where is an odd, increasing homeomorphism from onto . We show that it has at least one, two, or three positive solutions under some assumptions by applying fixed point theorems. The interesting point is that the nonlinear term is involved with the first-order derivative explicitly.

We are interested in the existence of positive solutions for the integral boundary value problem

where , and satisfy the following conditions.

(H1) is an odd, increasing homeomorphism from onto , and there exist two increasing homeomorphisms and of onto such that

Moreover, , where denotes the inverse of .

(H2) is continuous. are nonnegative, and , .

The assumption (H1) on the function was first introduced by Wang [1, 2], it covers two important cases: and . The existence of positive solutions for two above cases received wide attention (see [3–10]). For example, Ji and Ge [4] studied the multiplicity of positive solutions for the multipoint boundary value problem

where , . They provided sufficient conditions for the existence of at least three positive solutions by using Avery-Peterson fixed point theorem. In [5], Feng et al. researched the boundary value problem

where the nonlinear term does not depend on the first-order derivative and , . They obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of by applying Krasnoselskii fixed point theorem.

As for integral boundary value problem, when is linear, the existence of positive solutions has been obtained (see [8–10]). In [8], the author investigated the positive solutions for the integral boundary value problem

The main tools are the priori estimate method and the Leray-Schauder fixed point theorem. However, there are few papers dealing with the existence of positive solutions when satisfies (H1) and depends on both and . This paper fills this gap in the literature. The aim of this paper is to establish some simple criteria for the existence of positive solutions of BVP(1.1). To get rid of the difficulty of depending on , we will define a special norm in Banach space (in Section 2).

This paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence of one or two positive solutions for BVP(1.1) is established by applying the Krasnoselskii fixed point theorem. In Section 4, we give the existence of three positive solutions for BVP(1.1) by using a new fixed point theorem introduced by Avery and Peterson. In Section 5, we give some examples to illustrate our main results.

The basic space used in this paper is a real Banach space with norm defined by , where . Let

It is obvious that is a cone in .

Lemma 2.1 (see [7]).

Let , then , .

Lemma 2.2.

Let , then there exists a constant such that .

Proof.

The mean value theorem guarantees that there exists , such that

Moreover, the mean value theorem of differential guarantees that there exists , such that

So we have

Denote ; then the proof is complete.

Lemma 2.3.

Assume that (H1), (H2) hold. If is a solution of BVP(1.1), there exists a unique , such that and , .

Proof.

From the fact that , we know that is strictly decreasing. It follows that is also strictly decreasing. Thus, is strictly concave on [0, 1]. Without loss of generality, we assume that . By the concavity of , we know that , . So we get . By , it is obvious that . Hence, , .

On the other hand, from the concavity of , we know that there exists a unique where the maximum is attained. By the boundary conditions and , we know that or 1, that is, such that and then .

Lemma 2.4.

Assume that (H1), (H2) hold. Suppose is a solution of BVP(1.1); then

or

Proof.

First, by integrating (1.1) on , we have

then

Thus

or

According to the boundary condition, we have

By a similar argument in [5], ; then the proof is completed.

Now we define an operator by

Lemma 2.5.

is completely continuous.

Proof.

Let ; then from the definition of , we have

So is monotone decreasing continuous and . Hence, is nonnegative and concave on [0, 1]. By computation, we can get . This shows that . The continuity of is obvious since is continuous. Next, we prove that is compact on .

Let be a bounded subset of and is a constant such that for . From the definition of , for any , we get

Hence, is uniformly bounded and equicontinuous. So we have that is compact on . From (2.13), we know for , , such that when , we have . So is compact on ; it follows that is compact on . Therefore, is compact on .

Thus, is completely continuous.

It is easy to prove that each fixed point of is a solution for BVP(1.1).

Lemma 2.6 (see [1]).

Assume that (H1) holds. Then for ,

To obtain positive solution for BVP(1.1), the following definitions and fixed point theorems in a cone are very useful.

Definition 2.7.

The map is said to be a nonnegative continuous concave functional on a cone of a real Banach space provided that is continuous and

for all and . Similarly, we say the map is a nonnegative continuous convex functional on a cone of a real Banach space provided that is continuous and

for all and .

Let and be a nonnegative continuous convex functionals on , a nonnegative continuous concave functional on , and a nonnegative continuous functional on . Then for positive real number , and , we define the following convex sets:

Theorem 2.8 (see [11]).

Let be a real Banach space and a cone. Assume that and are two bounded open sets in with , . Let be completely continuous. Suppose that one of following two conditions is satisfied:

(1), , and , ;

(2), , and , .

Then has at least one fixed point in .

Theorem 2.9 (see [12]).

Let be a cone in a real Banach space . Let and be a nonnegative continuous convex functionals on , a nonnegative continuous concave functional on , and a nonnegative continuous functional on satisfying for , such that for positive number and ,

for all . Suppose is completely continuous and there exist positive numbers , and with such that

and for ;

() for with ;

() and for with .

Then has at least three fixed points , such that

for ,

,

with ,

.

For convenience, we denote

where denotes 0 or .

Theorem 3.1.

Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.

(i) There exist two constants with such that

(a) for and

(b) for ;

(ii);

(iii).

Then BVP(1.1) has at least one positive solution.

Proof.

(i) Let , .

For , we obtain and , which implies . Hence, by (2.12) and Lemma 2.6,

This implies that

Next, for , we have . Thus, by (2.12) and Lemma 2.6,

From (2.13), we have

This implies that

Therefore, by Theorem 2.8, it follows that has a fixed point in . That is BVP(1.1) has at least one positive solution such that .

(ii) Considering , there exists such that

Choosing such that

then for all , let . For every , we have . In the following, we consider two cases.

Case 1 ().

In this case,

Case 2 ().

In this case,

Then it is similar to the proof of (3.6); we have for .

Next, turning to , there exists such that

Let . For every , we have . So

Then like in the proof of (3.3), we have for . Hence, BVP(1.1) has at least one positive solution such that .

(iii) The proof is similar to the (i) and (ii); here we omit it.

In the following, we present a result for the existence of at least two positive solutions of BVP(1.1).

Theorem 3.2.

Assume that (H1) and (H2) hold. In addition, suppose that one of following conditions is satisfied.

(I), , and there exists such that

(II), , and there exists such that

Then BVP(1.1) has at least two positive solutions.

In this section, we impose growth conditions on which allow us to apply Theorem 2.9 of BVP(1.1).

Let the nonnegative continuous concave functional , the nonnegative continuous convex functionals , , and nonnegative continuous functional be defined on cone by

By Lemmas 2.1 and 2.2, the functionals defined above satisfy

for all . Therefore, the condition (2.19) of Theorem 2.9 is satisfied.

Theorem 4.1.

Assume that (H1) and (H2) hold. Let and suppose that satisfies the following conditions:

for ;

for .

for ;

Then BVP(1.1) has at least three positive solutions , and satisfying

where defined as (3.1), .

Proof.

We will show that all the conditions of Theorem 2.9 are satisfied.

If , then . With Lemma 2.2 implying , so by (), we have when . Thus

This proves that .

To check condition () of Theorem 2.9, we choose

Let

Then and , so . Hence, for , there is , when . From assumption (), we have

It is similar to the proof of assumption (i) of Theorem 3.1; we can easily get that

This shows that condition () of Theorem 2.9 is satisfied.

Secondly, for with , we have

Thus condition () of Theorem 2.9 holds.

Finally, as , there holds . Suppose that with ; then by the assumption (),

So like in the proof of assumption (i) of Theorem 3.1, we can get

Hence condition () of Theorem 2.9 is also satisfied.

Thus BVP(1.1) has at least three positive solutions , and satisfying

In this section, we give three examples as applications.

Example 5.1.

Let , . Now we consider the BVP

where for .

Let , . Choosing . By calculations we obtain

For ,

for ,

Hence, by Theorem 3.1, BVP(5.1) has at least one positive solution.

Example 5.2.

Let , . Consider the BVP

where for .

Let , . Then . It easy to see

Choosing , for , .

Hence, by Theorem 3.2, BVP(5.5) has at least two positive solutions.

Example 5.3.

Let , ; consider the boundary value problem

where

Choosing , , , , then by calculations we obtain that

It is easy to check that

Thus, according to Theorem 4.1, BVP(5.8) has at least three positive solutions , and satisfying

The research was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNU-KJCXGC-3-47.

1. Wang, H: On the number of positive solutions of nonlinear systems. Journal of Mathematical Analysis and Applications. 281(1), 287–306 (2003)

2. Wang, H: On the structure of positive radial solutions for quasilinear equations in annular domains. Advances in Differential Equations. 8(1), 111–128 (2003)

3. Wang, J: The existence of positive solutions for the one-dimensional p-Laplacian. Proceedings of the American Mathematical Society. 125(8), 2275–2283 (1997). Publisher Full Text

4. Ji, D, Ge, W: Multiple positive solutions for some p-Laplacian boundary value problems. Applied Mathematics and Computation. 187(2), 1315–1325 (2007). Publisher Full Text

5. Feng, H, Ge, W, Jiang, M: Multiple positive solutions for m-point boundary-value problems with a one-dimensional p-Laplacian. Nonlinear Analysis: Theory, Methods & Applications. 68(8), 2269–2279 (2008). PubMed Abstract | Publisher Full Text

6. Liu, B: Positive solutions of three-point boundary value problems for the one-dimensional p-Laplacian with infinitely many singularities. Applied Mathematics Letters. 17(6), 655–661 (2004). Publisher Full Text

7. Wang, Z, Zhang, J: Positive solutions for one-dimensional p-Laplacian boundary value problems with dependence on the first order derivative. Journal of Mathematical Analysis and Applications. 314(2), 618–630 (2006). Publisher Full Text

8. Yang, Z: Existence and uniqueness of positive solutions for an integral boundary value problem. Nonlinear Analysis: Theory, Methods & Applications. 69(11), 3910–3918 (2008). PubMed Abstract | Publisher Full Text

9. Kong, L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 72(5), 2628–2638 (2010). PubMed Abstract | Publisher Full Text

10. Boucherif, A: Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 70(1), 364–371 (2009). PubMed Abstract | Publisher Full Text

11. Guo, DJ, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering,p. viii+275. Academic Press, Boston, Mass, USA (1988)

12. Avery, RI, Peterson, AC: Three positive fixed points of nonlinear operators on ordered Banach spaces. Computers & Mathematics with Applications. 42(3–5), 313–322 (2001). PubMed Abstract | Publisher Full Text