Lemma 2.1 (see [10]).

Assume that (H4) holds. Then for any

, the boundary value problem

has a unique solution

which is given by

Lemma 2.2 (see [10]).

Assume that (H4) holds. Then for any

, the boundary value problem

has a unique solution

which is given by

Lemma 2.3 (see [10]).

Assume that (H4) holds. Then one has

Let

(H5)
be two given constants with
.

Definition 2.4.

*One says that*
*is a generalized eigenvalue of linear problem*
if (2.8) and (2.9) have nontrivial solutions.

Let
with the norm
. Let
with the norm
.

For

, from Lemma 2.1, it follows that

By simple calculations, we have

Combining this with (H4), we conclude that

This together with (2.12) and the fact that

imply that

Since

, we may define the norm of

by

We claim that
is a Banach space.

In fact, let

be a Cauchy sequence, that is,

as

. From the definition of

, it follows that

where

is a normal in

defined by

. Thus,

By the completeness of

, there exists

, such that

From the fact that

, we have that for arbitrary

, there exists

, such that

Fixed

and let

, we get

Therefore,
is a Banach space.

Then the cone
is normal and nonempty interior
and
.

In fact, for any
, it follows from the definition of
that

(1) there exist real number

, such that

(2)
,
.

From

and (H4), we obtain that

for some

. Moreover,

for some

. We may take

satisfying

Then

, and

Thus,
. Obviously,
.

Lemma 2.5.

Assume that

holds. Then for any

, one has

Proof.

In fact, for

, we have that

From
, we have that
, and so
, and accordingly
.

We have from the fact that

,

, that

which implies that
, and consequently
.

For

, define a linear operator

by

Theorem 2.6.

Assume that (H4) and (H5) hold. Let
be the spectral radius of
. Then (2.8) and (2.9) has an algebraically simple eigenvalue,
, with a positive eigenfunction
. Moreover, there is no other eigenvalue with a positive eigenfunction.

Remark 2.7.

If

, then

can be explicitly given by

and the corresponding eigenfunction
.

Proof of Theorem 2.6.

From Lemma 2.2, it is easy to check that (2.8) and (2.9) is equivalent to the integral equation

We claim that
.

In fact, for

, we have that

and for some constant

, it concludes that

where
, it follows that
, and accordingly
.

If

, then

on

, and accordingly

Thus
, and accordingly
.

Now, since
, and
is compactly embedded in
, we have that
is compact.

Next, we show that
is positive.

For

, if

, from Lemma 2.3, we have

Combining this with (2.39), there exist

such that

For

, if

, applying a similar proof process of (2.43), we have

Combining this with (2.39), there exist

such that

This together with (2.9) and (H4) imply
on
.

Therefore, it follows from (2.43) and (2.45) that
.

Now, by the Krein-Rutman theorem ([16, Theorem
C]; [17, Theorem
]),
has an algebraically simple eigenvalue
with an eigenfunction
. Moreover, there is no other eigenvalue with a positive eigenfunction. Correspondingly,
with a positive eigenfunction of
, is a simple eigenvalue of (2.8) and (2.9). Moreover, for (2.8) and (2.9), there is no other eigenvalue with a positive eigenfunction.