Condition
in Theorem 2.7 is rather restrictive and can be relaxed by assuming boundedness of
between a lower and an upper solution.

In this section we will prove the following result.

Theorem 3.1.

Suppose that (1.1) has a lower solution
and an upper solution
such that
for all
and let
.

Suppose that there exists a null-measure set
such that the following conditions hold:

() for every
, the mapping
with domain
is measurable;

() for every
,
, one has either (2.1) or (2.13), and (2.1) can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained in
;

() there exists an integrable function

,

, such that

Then (1.1) has extremal solutions in the set

Moreover the least solution of (1.1) in

is given by

and the greatest solution of (1.1) in

is given by

Proof.

Without loss of generality we suppose that
and
exist and satisfy
,
,
, and
on
. We also may (and we do) assume that every admissible nqsc curve in condition
, say
, satisfies for all
either
or (2.4)-(2.5).

For each

we define

Claim 1.

satisfies conditions
and
in Theorem 2.4 with
replaced by
. First we note that
is an immediate consequence of
and the definition of
.

To show that condition

in Theorem 2.4 is satisfied with

replaced by

, let

be fixed. The verification of (2.1) for

at

is trivial in the following cases:

and

satisfies (2.1) at

,

,

and

. Let us consider the remaining situations: we start with the case

and

satisfies (2.1) at

, for which we have

and

and an analogous argument is valid when
and
satisfies (2.1).

The previous argument shows that

satisfies (2.1) at every

except, at most, over the graphs of the countable family of admissible nonquasisemicontinuity curves in condition

for

. Therefore it remains to show that if

is one of those admissible nqsc curves for

then it is also an admissible nqsc curve for

. As long as the graph of

remains in the interior of

we have nothing to prove because

and

are the same, so let us assume that

on a positive measure set

,

. Since

and

are absolutely continuous there is a null measure set

such that

for all

, thus for

we have

so condition (2.5) with

replaced by

is satisfied on

. On the other hand, we have to check whether

for those

at which we have

We distinguish two cases:

and

. In the first case (3.9) is equivalent to

and therefore either
or condition (2.4) holds, yielding
. If
then we have
.

Analogous arguments show that either
or (2.4)-(2.5) hold for
at almost every point where
coincides with
, so we conclude that
is an admissible nqsc curve for
.

By virtue of Claim 1 and Theorem 2.4 we can ensure that the functions

and

defined as

are absolutely continuous on

and satisfy

and

for a.a.

, where

and for all

the set

contains no positive measure set, and

for a.a.

, where

and for all

the set

contains no positive measure set.

Claim 2.

For all

we have

Let

be an upper solution of (3.6) and let us show that

for all

. Reasoning by contradiction, assume that there exist

such that

,

and

For a.a.

we have

which together with
imply
on
, a contradiction with (3.16). Therefore every upper solution of (3.6) is greater than or equal to
, and, on the other hand,
is an upper solution of (3.6) with
a.e., thus
satisfies (3.14).

One can prove by means of analogous arguments that
satisfies (3.15).

Claim 3.

is the least solution of (1.1) in
and
is the greatest one. From (3.14) and (3.15) it suffices to show that
and
are actually solutions of (3.6). Therefore we only have to prove that
and
are null measure sets.

Let us show that the set

is a null measure set. First, note that

and we can split
, where
and

Let us show that

is a null measure set. Since

and

are absolutely continuous the set

is null. If

then there is some

such that

and

, but then the definitions of

and

yield

Therefore
and thus
is a null measure set.

The set

can be expressed as

, where for each

For

,

, we have

, so the definition of

implies that

which is a measurable set by virtue of Lemma 2.5 and Remark 2.6.

Since
contains no positive measure subset we can ensure that
is a null measure set for all
,
, and since
increases with
and
, we conclude that
is a null measure set. Finally
is null because it is the union of countably many null measure sets.

Analogous arguments show that
is a null measure set, thus the proof of Claim 3 is complete.

Claim 4.

satisfies (3.3) and

satisfies (3.4). Let

be an upper solution of (1.1), let

, and for all

let

Repeating the previous arguments we can prove that also
is the least Carathéodory solution of (1.1) in
, thus
on
. Hence
satisfies (3.3).

Analogous arguments show that
satisfies (3.4).

Remark 3.2.

Problem (3.6) may not satisfy condition
in Theorem 2.7 as the compositions
and
need not be measurable. That is why we used Theorem 2.4, instead of Theorem 2.7, to establish Theorem 3.1.

Next we show that even singular problems may fall inside the scope of Theorem 3.1 if we have adequate pairs of lower and upper solutions.

Example 3.3.

Let us denote by

the integer part of a real number

. We are going to show that the problem

has positive solutions. Note that the limit of the right hand side as
tends to the origin does not exist, so the equation is singular at the initial condition.

In order to apply Theorem 3.1 we consider (1.1) with

,

, and

It is elementary matter to check that

and

,

, are lower and upper solutions for the problem. Condition (2.1) only fails over the graphs of the functions

which are a countable family of admissible nqsc curves at which condition (2.13) holds.

so condition
is satisfied.

Theorem 3.1 ensures that our problem has extremal solutions between
and
which, obviously, are different from zero almost everywhere. Therefore (3.24) has positive solutions.

The result of Theorem 3.1 may fail if we assume that condition
is satisfied only in the interior of the set
. This is shown in the following example.

Example 3.4.

Let us consider problem (1.1) with

,

and

defined as

It is easy to check that
and
for all
are lower and upper solutions for this problem and that all the assumptions of Theorem 3.1 are satisfied in the interior of
. However this problem has no solution at all.

In order to complete the previous information we can say that condition
in the interior of
is enough if we modify the definitions of lower and upper solutions in the following sense.

Theorem 3.5.

Suppose that

and

are absolutely continuous functions on

such that

for all

,

,

and let
.

Suppose that there exists a null-measure set
such that conditions
and
hold and, moreover,

() for every
,
, one has either (2.1) or (2.13), and (2.1) can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained in
.

Then the conclusions of Theorem 3.1 hold true.

Proof (Outline)

It follows the same steps as the proof of Theorem 3.1 but replacing

by

Note that condition (2.1) with
replaced by
is immediately satisfied over the graphs of
and
thanks to the definition of
.

Remarks

(i)The function

in Example 3.4 does not satisfy the conditions in Theorem 3.5.

- (ii)
When
satisfies (2.1) everywhere or almost all
then every couple of lower and upper solutions satisfies the conditions in Theorem 3.5, so this result is not really new in that case (which includes the Carathéodory and continuous cases).