- Research Article
- Open Access

# Positive Solutions of Singular Initial-Boundary Value Problems to Second-Order Functional Differential Equations

- Fengfei Jin
^{1}and - Baoqiang Yan
^{1}Email author

**Received: **23 August 2007

**Accepted: **5 August 2008

**Published: **26 August 2008

## Abstract

## Keywords

## 1. Introduction

where and the existence of positive solutions to (1.1) is obtained. When in (1.1), Agarwal and O'Regan in [5], Lin and Xu in [6] discussed the existence of positive solutions to (1.1) also. We notice that the nonlinearities in all the above-mentioned references depend on .

The more difficult case is that the term depends on for second-order functional differential equations with delay. When has no singularity at and , there are many results on the following (1.2) (see [7–9] and references therein). Up to now, to our knowledge, there are fewer results on (1.2) when the term is allowed to possess singularity for the term at and , which is of more actual significance.

where . By Leray-Schauder fixed-point theorem, the existence of positive solutions to (1.2) is obtained when is singular at and .

For and , let and . Then, and are Banach spaces. Let and . Obviously, and are cones in and respectively. Now, we give a new definition.

Deffinition.

is said to be singular at for when satisfies for and is said to be singular at for when satisfies for .

And one defines some functions which one has to use in this paper.

where is a Green's function. It is clear that for and on

We now introduce the definition of a solution to IBVP(1.2).

Deffinition.

A function is said to be a solution to IBVP(1.2) if it satisfies the following conditions:

(1) is continuous and nonnegative on ;

(4) is Lebesgue integrable on ;

Furthermore, a solution is said to be positive if on .

for all solutions, , to IBVP(1.2), where . For , let on throughout this paper. Obviously, and for all .

Throughout this paper, we assume the following hypotheses hold.

Lemma 1.3.

for all solutions, , to (1.2).

Proof.

which contradicts the assumption that and hence the claim is true provided is suitably small.

Remark 1.4.

holds provided that is sufficiently small, where is in Lemma 1.3.

Lemma 1.5 (see [7]).

Let
be the Banach space and let *X* be any nonempty, convex, closed, and bounded subset of
. If
is a continuous mapping of
into itself and
is relatively compact, then the mapping
has at least one fixed point (i.e., there exists an
with
) .

Using Lemma 1.5, we present the existence of at least one positive solution to (1.2) when is singular at and (notice the new Definition 1.1). To some extent, our paper complements and generalizes these in [1–6, 8–10].

## 2. Main Results

Theorem 2.1.

*Assume that (H*
_{
1
}
*)–(H*
_{
3
}
*) hold. Then, the IBVP( 1.2 ) has at least one positive solution.*

Proof.

Since , we can choose an such that

It is obvious that satisfies the hypotheses and .

Since on , there are the following three cases.

Case 1.

which contradicts (2.7).

Case 2.

There exists a such that and .

which contradicts (2.7).

Case 3.

There exists a such that and .

which contradicts (2.7).

To prove the existence of positive solutions to IBVP(2.5), we seek to transform (2.5) into an integral equation via the use of Green's function and then find a positive solution by using Lemma 1.5.

Together with the definition of , we get .

which implies that is integrable on .

Since on , the above inequality holds for .

Thus, is a relative compact subset of . That is, is a compact operator.

We are now going to prove that the mapping is continuous on .

that is, the claim is true.

which shows that the mapping is continuous on .

Thus, the solution of IBVP(2.5) is also the one of (1.2). The proof is complete.

## 3. Application

## 4. Conclusion

Equation (3.1) has at least one positive solution.

Now, we will check that hold in (3.1).

It is obvious that is nonincreasing and is nondecreasing.

Now, we will prove that there exists such that is decreasing on .

From the continuity of , we can find such that on . Then, on . That is, is decreasing on .

So, from Theorem 2.1, IBVP(3.1) has at least one positive solution.

## Declarations

### Acknowledgments

The research was supported by NNSF of China (10571111) and the fund of Shandong Education Committee (J07WH08).

## Authors’ Affiliations

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