For each
, we write
. Let
. Clearly,
is a Banach space and
is a cone. Similarly, for each
, we write
. Clearly,
is a Banach space and
is a cone in
. For any real constant
, define
. By a positive solution of (1.3), we mean a vector
such that
satisfies (1.3) and
,
on
. The proofs of our main result (Theorem 1.1) is based on the Guo's fixed-point theorem.

Lemma 2.1 (Guo's Fixed-Point Theorem [27]).

Let
be a cone of a real Banach space
,
,
be bounded open subsets of
and
. Suppose that
is completely continuous such that one of the following condition hold:

(i)
for
and
for
;

(ii)
for
and
for
.

Then,
has a fixed point in
.

The following result can be easily verified.

Result.

Let
such that
. Let
,
and concave on
. Then,
for all
.

Choose
such that
. For fixed
and
, the linear three-point BVP

where

is the Green's function and is given by

We note that

as

, where

is the Green's function corresponding the boundary value problem

whose integral representation is given by

Lemma 2.2 (see [9]).

Let

. If

and

, then then unique solution

of the problem (2.5) satisfies

where
.

We need the following properties of the Green's function
in the sequel.

Lemma 2.3 (see [11]).

The function

can be written as

Following the idea in [10], we calculate upper bound for the Green's function
in the following lemma.

Lemma 2.4.

The function

satisfies

where

Proof.

For
, we discuss various cases.

Case 1.

,

; using (2.3), we obtain

If

, the maximum occurs at

, hence

and if

, the maximum occurs at

, hence

Case 2.

,

; using (2.3), we have

Case 3.

,

; using (2.3), we have

Case 4.

,

; using (2.3), we have

For
, the maximum occurs at
, hence

For
, the maximum occurs at
, so

Now, we consider the nonlinear nonsingular system of BVPs

We write (2.19) as an equivalent system of integral equations

By a solution of the system (2.19), we mean a solution of the corresponding system of integral equations (

2.20). Define a retraction

by

and an operator

by

where operators

are defined by

Clearly, if
is a fixed point of
, then
is a solution of the system (2.19).

Lemma 2.5.

Assume that
holds. Then
is completely continuous.

Proof.

Clearly, for any

,

. We show that the operator

is uniformly bounded. Let

be fixed and consider

Choose a constant

such that

,

,

. Then, for every

, using (2.22), Lemma 2.4,

and

, we have

that is,

is uniformly bounded. Similarly, using (2.22), Lemma 2.4,

and

, we can show that

is also uniformly bounded. Thus,

is uniformly bounded. Now we show that

is equicontinuous. Define

Let

such that

. Since

is uniformly continuous on

, for any

, there exist

such that

implies

For

, using (2.22)–(2.27), we have

which implies that
is equicontinuous. Similarly, using (2.22)–(2.27), we can show that
is also equicontinuous. Thus,
is equicontinuous. By Arzelà-Ascoli theorem,
is relatively compact. Hence,
is a compact operator.

Now we show that

is continuous. Let

such that

Then by using (2.22) and Lemma 2.4, we have

By Lebesgue dominated convergence theorem, it follows that

Similarly, by using (2.22) and Lemma 2.4, we have

From (2.32) and (2.33), it follows that

that is,
is continuous. Hence,
is completely continuous.