In what follows, we denote . The space of continuous functions , and the space of continuously derivable functions are equipped with the usual norms and , respectively. We denote the usual norm in by , where . We set
Definition 2.1 A map is a Carathéodory function if
-
(i)
is continuous for almost every ;
-
(ii)
is measurable for all ;
-
(iii)
for all , there exists such that for all such that , , and for almost every .
Definition 2.2 We say that is a lower solution of (1.1) if
(2.1)
and, in addition,
-
(i)
if ℬ denotes Dirichlet boundary condition (1.2), it satisfies
-
(ii)
if ℬ denotes periodic boundary condition (1.3), it satisfies
-
(iii)
if ℬ denotes Neumann boundary condition (1.4), it satisfies
Similarly, we define an upper solution of (1.1) if the previous conditions are satisfied with the reversed inequalities.
Definition 2.3 We say that is a strict lower solution of (1.1) if for any , there exist and , a neighborhood of in I, such that for almost every and all ,
(2.2)
and, in addition,
-
(i)
if ℬ denotes Dirichlet boundary condition (1.2), it satisfies
-
(ii)
if ℬ denotes periodic boundary condition (1.3), it satisfies
-
(iii)
if ℬ denotes Neumann boundary condition (1.4), it satisfies
Similarly, we define a strict upper solution of (1.1) if the previous conditions are satisfied with the reversed inequalities.
We will use the following general assumptions.
(H
ϕ
) The map is a bijective increasing homeomorphism.
(H
f
) The map is Carathéodory.
() There exist , respectively lower and upper solutions of (1.1), such that for all .
(WN) There exist , , , and such that
and
with if .
In what follows, () will be replaced by (H
D
), (H
N
) or (H
P
) if ℬ denotes (1.2), (1.3) or (1.4), the Dirichlet, periodic or Neumann boundary conditions, respectively.
We present some properties of operators that will be used later. Here is a particular case of Lemmas 2.3 and 2.6 in [10].
Lemma 2.4
Let
be a Carathéodory function and
such that
-
(i)
for any , the map is measurable;
-
(ii)
for any sequence converging to in , there exists such that
and
Then the operator
defined by
with
is continuous and completely continuous. Moreover, for every , is absolutely continuous and
Observe that the previous lemma holds with . In this case, .
Now, we present some results related to the homeomorphism ϕ. The first one is a lemma due to Manásevich and Mawhin [17].
Lemma 2.5 Let satisfy (H
ϕ
), and let be defined by
Then the following statements hold:
-
(1)
For any and any , the equation
has a unique solution .
-
(2)
For any , the function defined in (1) is continuous, and it sends bounded sets into bounded sets.
It is easy to show the following result.
Lemma 2.6 Assume that satisfies (H
ϕ
). Let be defined by
Then Φ is continuous.
We will use the following maximum principle type result.
Lemma 2.7 Assume that (H
ϕ
) is satisfied. Let be such that
Assume that one of the following conditions holds:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, , .
Then for all , or there exists such that for all .
Proof Assume that
Condition (i) (resp. (ii) or (iii)) implies that one of the following statements holds:
(2.3)
(2.4)
or
(2.5)
Indeed, if (2.5) does not hold, let be such that
and
If , then and (2.3) or (2.4) hold.
If , then (i) does not hold. If (ii) holds, then
Similarly, if (iii) holds, since
one has and . Using (iii) and the fact that ϕ is increasing, we get
Hence, (2.3) or (2.4) are satisfied.
Therefore, by assumption,
(2.6)
If (2.3) holds, for every ,
Since ϕ is increasing, we deduce that is nonincreasing in . This is a contradiction. Similarly, we obtain a contradiction if (2.4) holds.
Therefore, for all , or (2.5) is satisfied. □
Lemma 2.8 Assume (H
ϕ
) and (H
f
). Let be respectively strict lower and upper solutions of (1.1) such that for all . If is a solution of (1.1) such that for all , then for all .
Proof Let be a solution of (1.1) such that for all . Assume that
First, we claim that . This is obviously the case if ℬ denotes Dirichlet boundary condition (1.2). If ℬ denotes periodic boundary condition (1.3), then attains a minimum at 0 and T. So, by (H
ϕ
) and Definition 2.3(ii),
a contradiction. If ℬ denotes Neumann boundary condition (1.4), then attains a minimum at 0 or at T. So, by (H
ϕ
) and Definition 2.3(iii),
a contradiction.
Let . So, . By Definition 2.3, there exist and , a neighborhood of , such that a.e. and all x, y such that and . Since , there exists such that and for all . Since , there exists such that . Using the fact that ϕ is increasing, we deduce that
This is a contradiction. Therefore, for all .
Similarly, we show that for all . □
The following result establishes the existence of an a priori bound on the derivative of functions satisfying a suitable inequality.
Lemma 2.9 Assume that (H
ϕ
) is satisfied. Let and be such that
(2.7)
Then, for every , , , , and bounded, there exists such that for every
satisfying
one has .
Proof Let
Assumptions (H
ϕ
) and (2.7) imply that there exists such that
and
(2.8)
Assume that there exists
such that . If , there exist such that , , and for all t between and . Without loss of generality, we assume that . Then, by assumption,
Integrating from to and using the Hölder inequality and the change of variable formula in an integral give us
This contradicts (2.8).
Similarly, if , there exist such that , , and for all t between and . Arguing as above leads to a contradiction. □