We now state and prove several lemmas before stating our main results.
Lemma 2.1.
Let
. Then for
, the problem
has a unique solution
Proof.
From (2.1), we have
For
, integration from
to
, gives
For
, integration from
to
yields that
that is,
So,
Integrating (2.7) from
to
, where
, we have
From (2.2), we obtain that
Thus,
Therefore, (2.1)-(2.2) has a unique solution
Lemma 2.2.
Let
. If
and
on
, then the unique solution
of (2.1)-(2.2) satisfies
for
.
Proof.
If
, then, by the concavity of
and the fact that
, we have
for
.
Moreover, we know that the graph of
is concave down on
, we get
where
is the area of triangle under the curve
from
to
for
.
Assume that
. From (2.2), we have
By concavity of
and
, it implies that
.
Hence,
which contradicts the concavity of
.
Lemma 2.3.
Let
. If
and
for
, then (2.1)-(2.2) has no positive solution.
Proof.
Assume (2.1)-(2.2) has a positive solution
.
If
, then
, it implies that
and
which contradicts the concavity of
.
If
, then
, this is
for all
. If there exists
such that
, then
, which contradicts the concavity of
. Therefore, no positive solutions exist.
In the rest of the paper, we assume that
. Moreover, we will work in the Banach space
, and only the sup norm is used.
Lemma 2.4.
Let
. If
and
, then the unique solution
of the problem (2.1)-(2.2) satisfies
where
Proof.
Set
. We divide the proof into three cases.
Case 1.
If
and
, then the concavity of
implies that
Thus,
Case 2.
If
and
, then (2.2), (2.13), and the concavity of
implies
Therefore,
Case 3.
If
, then
. Using the concavity of
and (2.2), (2.13), we have
This implies that
This completes the proof.