# Positive Solutions of Singular Complementary Lidstone Boundary Value Problems

- RaviP Agarwal
^{1}Email author, - Donal O'Regan
^{2}and - Svatoslav Staněk
^{3}

**2010**:368169

**DOI: **10.1155/2010/368169

© Ravi P. Agarwal et al. 2010

**Received: **7 October 2010

**Accepted: **21 November 2010

**Published: **2 December 2010

## Abstract

We investigate the existence of positive solutions of singular problem , , , . Here, and the Carathéodory function may be singular in all its space variables . The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used.

## 1. Introduction

The function is positive and may be singular at the value zero of all its space variables .

*singular at the value zero of its space variable*if for a.e. and all , , such that , the relation

holds.

A function
(i.e.,
has absolutely continuous
th derivative on
) is a *positive solution of problem* (1.1), (1.2) if
for
,
satisfies the boundary conditions (1.2) and (1.1) holds a.e. on
.

was discussed in [1]. Here, is continuous at least in the interior of the domain of interest. Existence and uniqueness criteria for problem (1.5) are proved by the complementary Lidstone interpolating polynomial of degree . No contributions exist, as far as we know, concerning the existence of positive solutions of singular complementary Lidstone problems.

where the differential equation and derivatives in the boundary conditions are even orders. For ( ), regular Lidstone problems were discussed in [2–9], while singular ones in [10–15].

The aim of this paper is to give the conditions on the function in (1.1) which guarantee that the singular problem (1.1), (1.2) has a solution. The existence results are proved by regularization and sequential techniques, and in limit processes, the Vitali convergence theorem [16, 17] is applied.

Throughout the paper, and , stands for the norm in and , respectively. denotes the set of functions (Lebesgue) integrable on and meas the Lebesgue measure of .

We work with the following conditions on the function in (1.1).

*H*

_{1}) and there exists such that

for a.e. and each .

*H*

_{2}) For a.e. and for all , the inequality

The paper is organized as follows. In Section 2, we construct a sequence of auxiliary regular differential equations associated with (1.1). Section 3 is devoted to the study of auxiliary regular complementary Lidstone problems. We show that the solvability of these problems is reduced to the existence of a fixed point of an operator . The existence of a fixed point of is proved by a fixed point theorem of cone compression type according to Guo-Krasnosel'skii [18, 19]. The properties of solutions to auxiliary problems are also investigated here. In Section 4, applying the results of Section 3, the existence of a positive solution of the singular problem (1.1), (1.2) is proved.

## 2. Regularization

If a function satisfies (2.8) for a.e. , then is called a solution of (2.8).

## 3. Auxiliary Regular Problems

Lemma 3.1 (see [10, Lemmas 2.1 and 2.3]).

hold.

We prove the existence of a fixed point of by the following fixed point result of cone compression type according to Guo-Krasnosel'skii (see, e.g., [18, 19]).

Lemma 3.2.

holds. Then, has a fixed point in .

We are now in the position to prove that problem (2.8), (1.2) has a solution.

Lemma 3.3.

Let ( ) and ( ) hold. Then, problem (2.8), (1.2) has a solution.

Proof.

Then, is a cone in and since for by (3.4) and satisfies (2.5), we see that . The fact that is a completely continuous operator follows from , from Lebesgue dominated convergence theorem, and from the Arzelà-Ascoli theorem.

The conclusion now follows from Lemma 3.2 (for and ).

The properties of solutions to problem (2.8), (1.2) are collected in the following lemma.

Lemma 3.4.

Let ( ) and ( ) be satisfied. Let be a solution of problem (2.8), (1.2). Then, for all , the following assertions hold:

(i) for , , and for a.e. ,

(ii) is increasing on , and for , is decreasing on , and there is a unique such that ,

for ,

(iv) the sequence is bounded in .

Proof.

holds for and . Now, using (1.2), (3.4), (3.30), and (3.31), we see that assertion (i) is true. Hence, is decreasing on for and is increasing on this interval. Due to for , there exists a unique such that for . Consequently, assertion (ii) holds.

Then estimate (3.29) follows from relations (3.32)–(3.37).

is fulfilled. The last inequality together with estimate (3.46) gives for . Consequently, for , , and assertion (iv) follows.

The following result gives the important property of for applying the Vitali convergent theorem in the proof of Theorem 4.1.

Lemma 3.5.

Proof.

are uniformly integrable on . Due to and for by ( ), this fact follows from [13, Criterion 11.10 (with and )].

## 4. The Main Result

The following theorem is the existence result for the singular problem (1.1), (1.2).

Theorem 4.1.

Proof.

As a result, and is a solution of (1.1). Consequently, is a positive solution of problem (1.1), (1.2) and inequality (4.1) follows from (4.3).

Example 4.2.

on , where , (that is, is essentially bounded and measurable on ) are nonnegative, for a.e. . If for and , for , then, by Theorem 4.1, the problem has a positive solution satisfying inequality (4.1).

## Declarations

### Acknowledgment

This work was supported by the Council of Czech Government MSM no. 6198959214.

## Authors’ Affiliations

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