Existence and Iteration of Positive Solutions for One-Dimensional -Laplacian Boundary Value Problems with Dependence on the First-Order Derivative

Abstract

This paper deals with the existence and iteration of positive solutions for the following one-dimensional -Laplacian boundary value problems: , , subject to some boundary conditions. By making use of monotone iterative technique, not only we obtain the existence of positive solutions for the problems, but also we establish iterative schemes for approximating the solutions.

1. Introduction

In this paper, we are concerned with the existence and iteration of positive solutions for the following one-dimensional -Laplacian boundary value problems:

(1.1)

subject to one of the following boundary conditions:

(1.2)

or

(1.3)

where with and satisfy the following:

(H₁) is continuous;

() is continuous;

(H₂) is nondecreasing in for all , that is, for all ;

() is nondecreasing in for all , that is, for all ;

(H₃) is measurable, and is not identically zero on any compact subinterval of . Furthermore, satisfies .

Here, a positive solution of (1.1), (1.2) or (1.1), (1.3) means a solution of (1.1), (1.2) or (1.1), (1.3) satisfying .

The boundary value problems (1.1), (1.2) and (1.1), (1.3) deserve a special mention because these forms occur in the study of the -dimensional -Laplacian equation, non-Newtonian fluid theory and turbulent flow of a gas in a porous medium [1].

A consistent account on the existing literature on equation

(1.4)

is provided and it emphasizes the use of upper and lower solution technique and the fixed point theory, for instance, Krasnoselskii fixed point theorem, the fixed point index of a completely continuous operator with respect to a cone in a Banach space, one may see [2–5] and the references therein. In [6], by using the monotone iterative technique, Ma et al. obtained the existence of monotone positive solution and established the corresponding iterative schemes of (1.4) under the multipoint boundary value condition. However, in their discussion, the nonlinear term is not involved with the first-order derivative .

Recently, there is much attention focused on the study of the boundary value problems like (1.1) which the nonlinear term is involved with the first-order derivative explicitly. In [7], Bai et al. considered the boundary value problems (1.1), (1.2) and (1.1), (1.3) and they proved that problems (1.1), (1.2) and (1.1), (1.3) possessed at least three positive solutions by applying a fixed point theorem due to Avery and Peterson [8]. In [9], the authors also deal with the boundary value problem (1.1), (1.2) via Krasnoselskii fixed point theorem. Here, we should mention that Sun and Ge [10] have got the positive solution of the boundary value problem (1.1), (1.2) by making use of monotone iterative technique.

On the other hand, when is involved with the first-order derivative explicitly, we can see easily that the results obtained in [1, 7, 9] are only the existence of positive solutions under some suitable conditions. Seeing such a fact, it is an interesting problem which shows how to find these solutions since they exist definitely. Motivated by the above-mentioned results, by making use of the classical monotone iterative technique, we will investigate not only the existence of positive solutions for the boundary value problems (1.1), (1.2) and (1.1), (1.3), but also give iterative schemes for approximating the solutions. Unlike the idea of [10], we will construct a special subset (see Section 3) and look at or as a unit to overcome difficulties when depends on both and . It is worth starting that the first term of our iterative schemes are simple functions which are determined with some linear ordinary equations and cone (see Remark 3.2). Therefore, the iterative schemes are significant and feasible. At the same time, we will correct some mistakes in [11–13] (see Remark 3.5).

This paper is organized as follows. After this section, some definitions and lemmas will be established in Section 2. In Section 3, we will give our main results Theorems 3.1 and 3.4. Finally, an example is also presented to illustrate our results in Section 4.

2. Preliminaries

In this section, we provide some background material from the theory of cones in Banach spaces. We also state some lemmas which are important to proof our main results.

Definition 2.1.

Let be a real Banach space. A nonempty closed set is called a cone, if it satisfies the following two conditions:

(i) for all and all ;

(ii) implies .

Definition 2.2.

A map is said to be concave on , if

(2.1)

for all and

Consider the Banach space equipped with the norm

(2.2)

and define the cone by

(2.3)

Lemma 2.3 (see {[3]}).

If , then , where .

Define the operator as follows:

(2.4)

Lemma 2.4.

Assume (), () hold, then is completely continuous.

Proof.

From (), (), it is obviously that . Since , we can see that is continuous and nonincreasing on , that is, is concave on , so is well defined. The continuity of is clear because of the continuity of and . Now, we will prove that is compact. Let be a bounded set, then there exists , such that . For any , we have , then we have

(2.5)

The Arzela-Ascoli theorem guarantees that is relatively compact, which means that is compact. Then is completely continuous.

Lemma 2.5.

Assume (), () and () hold. If such that , then .

Proof.

Noticing that is nondecreasing in , the proof is simple, here we omit it.

3. Main Results

For convenience, we denote

(3.1)

where . It is easy to see that .

Theorem 3.1.

Assume (), (), and () hold. Moreover, suppose that there exist six constants , with and , such that

(H₄);

(H₅).

Then the boundary value problem (1.1), (1.2) has at least two nonincreasing positive solutions and with

(3.2)

and , where

(3.3)

and are arbitrary constants which satisfy .

Proof.

We denote a set by

(3.4)

Based on the preceding preliminaries, we can divide our proof into three steps.

Step 1.

We first prove . Let , note that is nonincreasing on , then

(3.5)

By Lemma 2.3, we have

(3.6)

Since is nondecreasing in , and by assumptions (H₄) and (H₅), we obtain

(3.7)

which imply that

(3.8)

On the other hand, for , we have

(3.9)

In virtue of (3.8)-(3.9), .

Step 2.

Let , is an arbitrary constant which satisfies , then . Hence, . Let , next we claim that . Indeed, it is easy to check that

(3.10)

Using assumptions and , we have , these imply that

(3.11)

Therefore, . We denote

(3.12)

Since From Lemma 2.4, is compact, we assert that has a convergent subsequence and there exists , such that . Now, since , for , we have

(3.13)

This combined with Lemma 2.5 gives

(3.14)

so

(3.15)

By induction,

(3.16)

Hence, we assert that . Let in (3.12) to obtain since is continuous. Since and is a nonnegative concave function on , we conclude that . It is well known that the fixed point of operator is the solution of the boundary value problem (1.1), (1.2). Therefore, is a positive, nonincreasing solution of the boundary value problem (1.1), (1.2).

Step 3.

Put , is an arbitrary constant satisfying , then . Hence, . From the definition of and , we derive that

(3.17)

Setting , in what follows, we will prove that . In fact, similar to (3.10)-(3.11), combined with the above inequalities, one has

(3.18)

We deduce from (3.18) that . Denote

(3.19)

Since , from Lemma 2.4, we have

(3.20)

By Lemma 2.5, we get

(3.21)

so . By induction, Hence, we assert that , and . Therefore, is a positive, nonincreasing solution of the boundary value problem (1.1), (1.2).

Remark 3.2.

• (i) We can easily get that and are the maximal and minimal solutions of the boundary value problem (1.1), (1.2) in . Of course and may coincide and then the boundary value problem (1.1), (1.2) has only one solution in .

• (ii) It is worth pointing out that . In fact, , which are determined with some linear ordinary equations and the cone , this is different from the results in [6, 10], where

  

  (3.22)

Corollary 3.3.

Assume (H₁), (H₂) and (H₃) hold, suppose that

(H₆);

(H₇).

(Particularly, ). Then there exist four constants , with , such that the boundary value problem (1.1), (1.2) has at least two nonincreasing positive solutions and with

(3.23)

and , where

(3.24)

and are arbitrary constants which satisfy .

Proof.

It is very easy to verify the conditions (H₄) and (H₅) can be obtained from (H₆) and (H₇), so we omit the proof.

Obviously, though the similar arguments of Theorem 3.1, we could get the following theorem.

Theorem 3.4.

Assume (), () and () hold, suppose that there exist six positive constants , with and , such that

(H₈);

(H₉).

Then the boundary value problem (1.1), (1.3) has at least two nondecreasing positive solutions and with

(3.25)

and , where

(3.26)

and are arbitrary constants which satisfy .

Remark 3.5.

In [11], Liu and Zhang studied the following boundary value problem:

(3.27)

where and is an increasing positive homomorphism and homeomorphism with (for more details, see [11–13]). In that paper, the authors claimed that is a solution if and only if is a solution of the following integral equation:

(3.28)

where , if ; , if , otherwise is a solution of the equation

(3.29)

where

(3.30)

Unfortunately, such a claim is incorrect since

(3.31)

Similar reason, the results in [12, 13] are also incorrect, as they deal with the nonlinear systems. Under the boundary condition (1.3), we could avoid (3.31) to occur. With the similar argument of Theorem 3.4, we could have the similar theorem to Theorem 3.4 for the following boundary value problem:

(3.32)

Here, we omit the proofs.

4. Example

In this section, we will give an example to illustrate our results.

Example 4.1.

Consider the boundary value problem

(4.1)

where with , take ,

(4.2)

we have and . Take , we get . Choose , and , so satisfies the following:

(1) is continuous;

(2) is nondecreasing in for all ;

(3);

(4).

Therefore, by Theorem 3.1, the boundary value problem (4.1) has at least two nonincreasing positive solutions and , such that

(4.3)

and , where

(4.4)

and are arbitrary constants which satisfy .

For the two iterative schemes are , is arbitrary constant with

(4.5)

, is arbitrary constant with

(4.6)

Remark 4.2.

The nonlinear term in is nonincreasing, so the results in [10] do not hold.