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

Abstract

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.

1. Introduction

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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.

2. Preliminaries

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

(2.1)

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

(2.2)

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

(2.3)

So we have

(2.4)

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

(2.5)

or

(2.6)

Proof.

First, by integrating (1.1) on , we have

(2.7)

then

(2.8)

Thus

(2.9)

or

(2.10)

According to the boundary condition, we have

(2.11)

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

Now we define an operator by

(2.12)

Lemma 2.5.

is completely continuous.

Proof.

Let ; then from the definition of , we have

(2.13)

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

(2.14)

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 ,

(2.15)

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

(2.16)

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

(2.17)

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:

(2.18)

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 ,

(2.19)

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 ,

.

3. The Existence of One or Two Positive Solutions

For convenience, we denote

(3.1)

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,

(3.2)

This implies that

(3.3)

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

(3.4)

From (2.13), we have

(3.5)

This implies that

(3.6)

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

  

  (3.7)

Choosing such that

(3.8)

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

Case 1 ().

In this case,

(3.9)

Case 2 ().

In this case,

(3.10)

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

Next, turning to , there exists such that

(3.11)

Let . For every , we have . So

(3.12)

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

(3.13)

(II), , and there exists such that

(3.14)

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

4. The Existence of Three 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

(4.1)

By Lemmas 2.1 and 2.2, the functionals defined above satisfy

(4.2)

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

(4.3)

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

(4.4)

This proves that .

To check condition () of Theorem 2.9, we choose

(4.5)

Let

(4.6)

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

(4.7)

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

(4.8)

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

Secondly, for with , we have

(4.9)

Thus condition () of Theorem 2.9 holds.

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

(4.10)

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

(4.11)

Hence condition () of Theorem 2.9 is also satisfied.

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

(4.12)

5. Examples

In this section, we give three examples as applications.

Example 5.1.

Let , . Now we consider the BVP

(5.1)

where for .

Let , . Choosing . By calculations we obtain

(5.2)

For ,

(5.3)

for ,

(5.4)

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

Example 5.2.

Let , . Consider the BVP

(5.5)

where for .

Let , . Then . It easy to see

(5.6)

Choosing , for , .

(5.7)

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

Example 5.3.

Let , ; consider the boundary value problem

(5.8)

where

(5.9)

Choosing , , , , then by calculations we obtain that

(5.10)

It is easy to check that

(5.11)

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

(5.12)