Observe that the equation of (1) is from the algebraic point of view a second-order equation in the variable . So, solving it algebraically in order of , we obtain the equivalent form
which leads us to consider the equation
This fact suggests the study of the auxiliary problem
The main argument to prove Theorem 2 relies on the method of upper and lower solutions.
We recall that is a lower solution of (2) if
Similarly, an upper solution of (2) is defined by reversing the inequalities in (3). A solution of (2) is a function u which is simultaneously a lower and an upper solution. A function f is said to satisfy the Nagumo condition on some given subset if there exists a positive continuous function , , such that
The function is a lower solution of the problem (2).
If is small enough, then the function
is a lower solution of the problem (2).
is an upper solution of the problem (2).
Proof 1. If we plug in the first member of the equation in (2), we have that
and by (5)
Consider now the function
Observe first that
On the other hand, we have that
So, if we plug the function in the first member of the equation in (2), we obtain
which, for k small enough, is non-negative.
Consider now the function
Then, in an analogous way, we plug in the first member of the equation in (2). We observe that
So, the three assertions of the lemma are proved. □
Remark 1 The value is a suitable upper bound for the possible values that can take in the above assertion 2. In fact, for and since , easy computations show that
Therefore, if , we have
which is used in the last step of the proof of assertion 2.
Now we state an existence and localisation result for the auxiliary problem (2).
Theorem 1 Suppose that . Then:
The problem (2) has a solution V such that
If is small enough, the problem (2) has a solution V such that
If , the function is a solution of the problem (2).
Proof By the previous lemma, we know already that there are lower and upper solutions for the problem (2). It is also clear that they are well ordered, that is,
So, if the function H satisfies the Nagumo condition, the thesis will follow by  or .
In order to prove assertions 1 and 2, we consider the sets, respectively,
Function H satisfies the Nagumo condition in and in . In fact, we have that
Thus, for some positive constants and , we have that
So, by what was said above, the first two assertions of the thesis hold.
The third assertion follows directly from obvious computations. □
Proposition 1 Consider the problem (2) and the solution V given by Theorem 1. Then V is convex and satisfies in .
Proof The convexity of the solution V follows easily by the equation of the problem (2)
But, using Theorem 1, assertion 1, we have that . In fact, since , it follows that , for . Letting , we obtain that .
Then we have