Positive Solutions for Fourth-Order Singular -Laplacian Differential Equations with Integral Boundary Conditions

Abstract

By employing upper and lower solutions method together with maximal principle, we establish a necessary and sufficient condition for the existence of pseudo- as well as positive solutions for fourth-order singular -Laplacian differential equations with integral boundary conditions. Our nonlinearity may be singular at , , and . The dual results for the other integral boundary condition are also given.

1. Introduction

In this paper, we consider the existence of positive solutions for the following nonlinear fourth-order singular -Laplacian differential equations with integral boundary conditions:

(1.1)

where , , , , , , , , and is nonnegative. Let , . Throughout this paper, we always assume that , and nonlinear term satisfies the following hypothesis:

(H) is continuous, nondecreasing on and nonincreasing on for each fixed , and there exists a real number such that, for any ,

(1.2)

there exists a function , and is integrable on such that

(1.3)

Remark 1.1.

Condition is used to discuss the existence and uniqueness of smooth positive solutions in [1].

(i)Inequality (1.2) implies that

(1.4)

Conversely, (1.4) implies (1.2).

(ii)Inequality (1.3) implies that

(1.5)

Conversely, (1.5) implies (1.3).

Remark 1.2.

Typical functions that satisfy condition are those taking the form = , where , , ; .

Remark 1.3.

It follows from (1.2) and (1.3) that

(1.6)

Boundary value problems with integral boundary conditions arise in variety of different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. They include two point, three point, and nonlocal boundary value problems (see [2–5]) as special cases and have attracted much attention of many researchers, such as Gallardo, Karakostas, Tsamatos, Lomtatidze, Malaguti, Yang, Zhang, and Feng (see [6–13], e.g.). For more information about the general theory of integral equations and their relation to boundary value problems, the reader is referred to the book by Corduneanu [14] and Agarwal and O'Regan [15].

Recently, Zhang et al. [13] studied the existence and nonexistence of symmetric positive solutions for the following nonlinear fourth-order boundary value problems:

(1.7)

where , , , , is nonnegative, symmetric on the interval , is continuous, and are nonnegative, symmetric on .

To seek necessary and sufficient conditions for the existence of solutions to the ordinary differential equations is important and interesting, but difficult. Professors Wei [16, 17], Du and Zhao [18], Graef and Kong [19], Zhang and Liu [20], and others have done much excellent work under some suitable conditions in this direction. To the author's knowledge, there are no necessary and sufficient conditions available in the literature for the existence of solutions for integral boundary value problem (1.1). Motivated by above papers, the purpose of this paper is to fill this gap. It is worth pointing out that the nonlinearity permits singularity not only at but also at . By singularity, we mean that the function is allowed to be unbounded at the points and .

2. Preliminaries and Several Lemmas

A function and is called a (positive) solution of BVP (1.1) if it satisfies (1.1) ( for ). A (positive) solution of (1.1) is called a psuedo- (positive) solution if , for . Denote that

(2.1)

Definition 2.1.

A function is called a lower solution of BVP (1.1) if satisfies

(2.2)

Definition 2.2.

A function is called an upper solution of BVP (1.1) if satisfies

(2.3)

Suppose that , and

(2.4)

To prove the main results, we need the following maximum principle.

Lemma 2.3 (Maximum principle).

If , such that , , then , ,

Proof.

Set

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

then , , , and

(2.10)

Let

(2.11)

then

(2.12)

(2.13)

By integration of (2.12), we have

(2.14)

Integrating again, we get

(2.15)

Let in (2.15), we obtain that

(2.16)

Substituting (2.13) and (2.16) into (2.15), we obtain that

(2.17)

where

(2.18)

Notice that

(2.19)

therefore,

(2.20)

Substituting (2.20) into (2.17), we have

(2.21)

where

(2.22)

Obviously, , , . From (2.21), it is easily seen that for By (2.11), we know that that is, Thus, we have proved that , . Similarly, the solution of (2.5) and (2.7) can be expressed by

(2.23)

where

(2.24)

By (2.23), we can get that ,

Lemma 2.4.

Suppose that holds. Let be a positive solution of BVP (1.1). Then there exist two constants such that

(2.25)

Proof.

Assume that is a positive solution of BVP (1.1). Then can be stated as

(2.26)

where

(2.27)

It is easy to see that

(2.28)

By (2.26), for , we have that

(2.29)

From (2.26) and (2.27), we get that

(2.30)

Setting

(2.31)

then from (2.29) and (2.30), we have (2.25).

Lemma 2.5.

Suppose that holds. And assume that there exist lower and upper solutions of BVP (1.1), respectively, and , such that , for . Then BVP (1.1) has at least one positive solution such that , . If, in addition, there exists such that

(2.32)

then the solution of BVP (1.1) is a pseudo- positive solution.

Proof.

For each , for all , , we defined an auxiliary function

(2.33)

By condition , we have that is continuous.

Let be sequences satisfying , and as and let , , be sequences satisfying

(2.34)

For each , consider the following nonsingular problem:

(2.35)

For convenience, we define linear operators as follows:

(2.36)

By the proof of Lemma 2.3, is a solution of problem (2.35) if and only if it is the fixed point of the following operator equation:

(2.37)

By (2.33), it is easy to verify that is continuous and is a bounded set. Moreover, by the continuity of , we can show that is a compact operator and is a relatively compact set. So, is a completely continuous operator. In addition, is a solution of (2.35) if and only if is a fixed point of operator . Using the Shauder's fixed point theorem, we assert that has at least one fixed point , by , we can get

We claim that

(2.38)

From this it follows that

(2.39)

Indeed, suppose by contradiction that on . By the definition of , we have

(2.40)

Therefore,

(2.41)

On the other hand, since is an upper solution of (1.1), we also have

(2.42)

Then setting

(2.43)

By (2.41) and (2.42), we obtain that

(2.44)

By Lemma 2.3, we can conclude that

(2.45)

Hence,

(2.46)

Set

(2.47)

Then

(2.48)

By Lemma 2.3, we can conclude that

(2.49)

which contradicts the assumption that Therefore, is impossible.

Similarly, we can show that So, we have shown that (2.38) holds.

Using the method of [21] and Theorem .2 in [22], we can obtain that there is a positive solution of (1.1) such that and a subsequence of converging to on any compact subintervals of .

In addition, if (2.32) holds, then . Hence, is absolutely integrable on . This implies that is a pseudo- positive solution of (1.1).

3. The Main Results

Theorem 3.1.

Suppose that holds, then a necessary and sufficient condition for BVP (1.1) to have a pseudo- positive solution is that the following integral condition holds:

(3.1)

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

Suppose that is a pseudo- positive solution of (1.1). Then both and exist. By Lemma 2.4, there exist two constants such that

(3.2)

Without loss of generality, we may assume that . This together with condition implies that

(3.3)

On the other hand, since is a pseudo- positive solution of (1.1), we have

(3.4)

Otherwise, let . By the proof of Lemma 2.3, we have that , , that is, which contradicts that is a pseudo- positive solution. Therefore, there exists a positive such that . Obviously, . By (1.6) we have

(3.5)

Consequently, , which implies that

(3.6)

It follows from (3.3) and (3.6) that

(3.7)

which is the desired inequality.

Sufficiency.

First, we prove the existence of a pair of upper and lower solutions. Since is integrable on , we have

(3.8)

Otherwise, if , then there exists a real number such that when , which contradicts the condition that is integrable on . In view of condition and (3.8), we obtain that

(3.9)

(3.10)

where .

Suppose that (3.1) holds. Firstly, we define the linear operators and as follows:

(3.11)

(3.12)

where is given by (2.27). Let

(3.13)

It is easy to know from (3.11) and (3.12) that By Lemma 2.4, we know that there exists a positive number such that

(3.14)

Take sufficiently small, then by (3.10), we get that , that is,

(3.15)

Let

(3.16)

Thus, from (3.14) and (3.16), we have

(3.17)

Considering , it follows from (3.15), (3.17), and condition that

(3.18)

From (3.13) and (3.16), it follows that

(3.19)

Thus, we have shown that and are lower and upper solutions of BVP (1.1), respectively.

Additionally, when , , by (3.17) and condition , we have

(3.20)

From (3.1), we have So, it follows from Lemma 2.5 that BVP (1.1) admits a pseudo- positive solution such that

Remark 3.2.

Lin et al. [23, 24] considered the existence and uniqueness of solutions for some fourth-order and conjugate boundary value problems when , where

(3.21)

under the following condition:

for and , there exists such that

(3.22)

Lei et al. [25] and Liu and Yu [26] investigated the existence and uniqueness of positive solutions to singular boundary value problems under the following condition:

for all , where and is nondecreasing on and nonincreasing on .

Obviously, (3.21)-(3.22) imply condition and condition implies condition . So, condition is weaker than conditions and . Thus, functions considered in this paper are wider than those in [23–26].

In the following, when admits the form , that is, nonlinear term is not mixed monotone on , but monotone with respect , BVP (1.1) becomes

(3.23)

If satisfies one of the following:

is continuous, nondecreasing on , for each fixed , there exists a function , and is integrable on such that

(3.24)

Theorem 3.3.

Suppose that holds, then a necessary and sufficient condition for BVP (3.23) to have a pseudo- positive solution is that the following integral condition holds

(3.25)

Proof.

The proof is similar to that of Theorem 3.1; we omit the details.

Theorem 3.4.

Suppose that holds, then a necessary and sufficient condition for problem (3.23) to have a positive solution is that the following integral condition holds

(3.26)

Proof.

The proof is divided into two parts, necessity and suffeciency.

Necessity.

Assume that is a positive solution of BVP (3.23). By Lemma 2.4, there exist two constants and , , such that

(3.27)

Let be a constant such that . By condition , we have

(3.28)

By virtue of (3.28), we obtain that

(3.29)

By boundary value condition, we know that there exists a such that

(3.30)

For by integration of (3.29), we get

(3.31)

Integrating (3.31), we have

(3.32)

Exchanging the order of integration, we obtain that

(3.33)

Similarly, by integration of (3.29), we get

(3.34)

Equations (3.33) and (3.34) imply that

(3.35)

Since is a positive solution of BVP (1.1), there exists a positive such that . Obviously, . On the other hand, choose , then . By condition , we have

(3.36)

Consequently, , which implies that

(3.37)

It follows from (3.35) and (3.37) that

(3.38)

which is the desired inequality.

Sufficiency.

Suppose that (3.26) holds. Let

(3.39)

It is easy to know, from (3.11) and (3.26), that

(3.40)

Thus, (3.12), (3.39), and (3.40) imply that By Lemma 2.4, we know that there exists a positive number such that

(3.41)

Take sufficiently small, then by (3.10), we get that , that is,

(3.42)

Let

(3.43)

Thus, from (3.41) and (3.43), we have

(3.44)

Notice that , it follows from (3.42)–(3.44) and condition that

(3.45)

From (3.39) and (3.43), it follows that

(3.46)

Thus, we have shown that and are lower and upper solutions of BVP (1.1), respectively.

From the first conclusion of Lemma 2.5, we conclude that problem (1.1) has at least one positive solution .

4. Dual Results

Consider the fourth-order singular -Laplacian differential equations with integral conditions:

(4.1)

(4.2)

Firstly, we define the linear operator as follows:

(4.3)

where is given by (2.27).

By analogous methods, we have the following results.

Assume that is a positive solution of problem (4.1). Then can be expressed by

(4.4)

Theorem 4.1.

Suppose that holds, then a necessary and sufficient condition for (4.1) to have a pseudo- positive solution is that the following integral condition holds:

(4.5)

Theorem 4.2.

Suppose that holds, then a necessary and sufficient condition for problem (4.2) to have a pseudo- positive solution is that the following integral condition holds:

(4.6)

Theorem 4.3.

Suppose that holds, then a necessary and sufficient condition for problem (4.2) to have a positive solution is that the following integral condition holds:

(4.7)

Consider the fourth-order singular -Laplacian differential equations with integral conditions:

(4.8)

(4.9)

Define the linear operator as follows:

(4.10)

If is a positive solution of problem (4.8). Then can be expressed by

(4.11)

Theorem 4.4.

Suppose that holds, then a necessary and sufficient condition for problem (4.8) to have a pseudo- positive solution is that the following integral condition holds:

(4.12)

Theorem 4.5.

Suppose that holds, then a necessary and sufficient condition for problem (4.9) to have a pseudo- positive solution is that the following integral condition holds:

(4.13)

Theorem 4.6.

Suppose that holds, then a necessary and sufficient condition for problem (4.9) to have a positive solution is that the following integral condition holds:

(4.14)

Consider the fourth-order singular -Laplacian differential equations with integral conditions:

(4.15)

(4.16)

Define the linear operator as follows:

(4.17)

If is a positive solution of problem (4.15). Then can be expressed by

(4.18)

Theorem 4.7.

Suppose that holds, then a necessary and sufficient condition for problem (4.15) to have a pseudo- positive solution is that the following integral condition holds:

(4.19)

Theorem 4.8.

Suppose that holds, then a necessary and sufficient condition for problem (4.16) to have a pseudo- positive solution is that the following integral condition holds:

(4.20)

Theorem 4.9.

Suppose that holds, then a necessary and sufficient condition for problem (4.16) to have a positive solution is that the following integral condition holds:

(4.21)