- Research Article
- Open Access

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

- Zhiyong Wang
^{1}Email author and - Jihui Zhang
^{1}

**2008**:860414

https://doi.org/10.1155/2008/860414

© Z. Wang and J. Zhang. 2008

**Received:**7 April 2008**Accepted:**2 May 2008**Published:**18 May 2008

## 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.

## Keywords

- Banach Space
- Nonlinear Term
- Arbitrary Constant
- Fixed Point Theorem
- Iterative Scheme

## 1. Introduction

where with and satisfy the following:

(H_{1})
is continuous;

( ) is continuous;

(H_{2})
is nondecreasing in
for all
, that is,
for all
;

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

(H_{3})
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].

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.

for all and

Lemma 2.3 (see {[3]}).

If , then , where .

Lemma 2.4.

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

Proof.

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

where . It is easy to see that .

Theorem 3.1.

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

(H_{4})
;

(H_{5})
.

and are arbitrary constants which satisfy .

Proof.

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

Step 1.

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

Step 2.

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.

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

- (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)

Corollary 3.3.

Assume (H_{1}), (H_{2}) and (H_{3}) hold, suppose that

(H_{6})
;

(H_{7})
.

and are arbitrary constants which satisfy .

Proof.

It is very easy to verify the conditions (H_{4}) and (H_{5}) can be obtained from (H_{6}) and (H_{7}), 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

(H_{8})
;

(H_{9})
.

and are arbitrary constants which satisfy .

Remark 3.5.

Here, we omit the proofs.

## 4. Example

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

Example 4.1.

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

(1) is continuous;

(2) is nondecreasing in for all ;

(3) ;

(4) .

and are arbitrary constants which satisfy .

Remark 4.2.

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

## Declarations

### Acknowledgments

The project is supported by Foundation of Major Project of Science and Technology of Chinese Education Ministry, SRFDP of Higher Education, NSF of Education Committee of Jiangsu Province, and Graduate Innovation Foundation of Jiangsu Province (1612005022).

## Authors’ Affiliations

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