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We establish the existence of positive solutions for singular boundary value problems of coupled systems The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and improved.

In this paper, we consider the existence of positive solutions for coupled singular system of second order ordinary differential equations

Throughout this paper, we always suppose that

In recent years, singular boundary value problems to second ordinary differential equations have been studied extensively (see [

We consider the scalar equation

with boundary conditions

Suppose that

where

here

Lemma 2.1. Suppose that

1):

2):

3):

4):

5):

6):

7):

8): For each fixed

9):

We define the function

which is the unique solution of

Following from Lemma

Let us fix some notation to be used in the following: For a given function

1)

Theorem 3.1. We assume that there exists

If

Proof A positive solution of (1.1) is just a fixed point of the completely continuous map

By a direct application of Schauder’s fixed point theorem, the proof is finished if we prove that A maps the closed convex set defined as

into itself, where

Given

Note for every

Similarly, by the same strategy, we have

Thus

Note that

and these inequalities hold for

2)

The aim of this section is to show that the presence of a weak singular nonlinearity makes it possible to find positive solutions if

Theorem 3.2. We assume that there exists

then there exists a positive solution of (1.1).

Proof In this case, to prove that

If we fix

or equivalently

The function

Taking

Similarly,

Taking

remains to prove that

since

3)

Theorem 3.3. Assume that

where

then there exists a positive solution of (1.1).

Proof We follow the same strategy and notation as in the proof of ahead theorem. In this case, to prove that

If we fix

or equivalently

If we chose

If we fix

or equivalently

According to

we have

Then the function

Note

or equivalently

Taking

Remark 1. In theorem 3.3 the right-hand side of condition (3.4) always negative, this is equivalent to proof that

Similarly, we have the following theorem.

Theorem 3.4. Assume

where

then there exists a positive solution of (1.1).

Project supported by Heilongjiang province education department natural science research item, China (12541076).