We begin by considering the homogeneous BVP

of which we seek an equivalent integral formulation of the form

Due to the nature of these particular BCs, the Green's function

can be constructed (as in [

4]) by means of an auxiliary second-order BVP, namely,

The solutions of the BVP (2.3) can be written in the form

Therefore the function

in (2.2) is given by

In order to use the approach of [

21,

25,

28], we need to use some monotonicity properties of

. Now, since

we obtain the following formulation for the Green's function:

We now look for a suitable interval

, a function

, and a constant

such that

Since

is continuous on

and

for

, a natural choice could be

here we look for a better
, since this enables us to weaken the growth requirements on the nonlinearity
.

An upper bound for
is obtained by finding
for each fixed
. Since
for
,
is a nondecreasing function of
that attains its maximum, for each fixed
, when
.

Therefore, for

, we have

Now, one can see that the derivative of the function

with respect to

is non-positive for all

, that is, the function

is a non-increasing function of

. Therefore, for an arbitrary

, we have

We now turn our attention to the BVP (1.1)–(1.6)

and we show that we can study this problem by means of a perturbation of the Hammerstein integral equation (2.2).

In order to do this, we look for the (unique) solutions of the linear problems

We observe that, for an arbitrary

, we have

where

, and therefore

By a solution of the BVP (1.1)–(1.6) we mean a solution of the perturbed integral equation

and we work in a suitable cone in the Banach space
of continuous functions defined on the interval
endowed with the usual supremum norm.

Our assumptions are the following:

satisfies Carathéodory conditions, that is, for each

,

is measurable, for almost every

,

is continuous, and for every

, there exists an

-function

such that

,
almost everywhere, and
;

are positive continuous functions such that there exist

with

for every
;

are positive bounded linear functionals on

given by

involving Stieltjes integrals with *positive* measures
;

,
and
.

It follows from this last hypothesis that

The above hypotheses enable us to utilize the cone

for an arbitrary

and

and to use the classical fixed point index for compact maps (see e.g., [29] or [30]).

We observe, as in [21], that
leaves
invariant and is compact. We give the proof in the Carathéodory case for completeness.

Lemma 2.1.

If the hypotheses
hold, then
maps
into
. Moreover,
is a compact map.

Proof.

Take

such that

. Then we have, for

,

Hence we have
. Moreover, the map
is compact since it is a sum of two compact maps: the compactness of
is well known, and since
, and
are continuous, the perturbation
maps bounded sets into bounded subsets of a 1-dimensional space.

For

, we use, as in [

23,

31], the following bounded open subsets of

:

Note that
.

We employ the following numbers:

The proofs of the following results can be immediately deduced from the analogous results in [21], where the proofs involve a careful analysis of fixed point index and utilize order-preserving matrices. The only difference here is that we allow nonlinearity
to be Carathéodory instead of continuous. The lines of proof are not effected and therefore the proofs are omitted.

Firstly we give conditions which imply that the fixed point index is
on the set
.

Lemma 2.2.

Suppose that

hold. Assume that there exist

such that

Then the fixed point index,
, is
.

Now, we give conditions which imply that the fixed point index is
on the set
.

Lemma 2.3.

Suppose

hold. Assume that there exists

such that

Then
.

The two lemmas above lead to the following result on existence of one or two positive solutions for the integral equation (2.20). Note that, if the nonlinearity
has a suitable oscillatory behavior, it is possible to state, with the same arguments as in [23], a theorem on the existence of more than two positive solutions.

Theorem 2.4.

Suppose
hold. Let
and
be as in (2.26). Then (2.20) has one positive solution in
if either of the following conditions holds:

there exist
with
such that (2.34) is satisfied for
and (2.33) is satisfied for
;

there exist
with
such that (2.33) is satisfied for
and (2.34) is satisfied for
.

Equation (2.20) has at least two positive solutions in
if one of the following conditions hold.

there exist
, with
, such that (2.34) is satisfied for
, (2.33) is satisfied for
, and (2.34) is satisfied for
;

there exist
, with
and
, such that (2.33) is satisfied for
, (2.34) is satisfied for
, and (2.33) is satisfied for
.