Consider the problem
We will deduce the existence of a solution to () by supposing that the following hypotheses hold:
() For every , , where , verifies
For every , ,
is continuous on uniformly in .
() For every , and , there exist constants , , with , , if , such that if , then
for each and
Remark 3.1 If for every ,
for each and .
We consider solutions to the problem.
Definition 3.1 A positive solution of type 1 of () is a function , with for all such that and for all , which satisfies the equalities on () for each and , and the following limits exist and are finite:
Definition 3.2 We say that is a lower solution of () if for each we have
Similarly, is called an upper solution of () if for each ,
Lemma 3.1 Suppose that () and () hold. If x is a positive solution of type 1 of (), then for each , there are constants , , such that
Proof Integrate the equations of () in for
From (2.2), we have
Thus, if we consider
Lemma 3.2 If α and β are lower and upper solutions of problem () such that for , and () and (H3) or () hold, then problem () has a solution x such that
If in addition there exists a function
then the solution x is a positive solution of type 1.
Proof Let us consider to be two sequences such that is strictly decreasing to a if , and for all if , and is strictly increasing to if , for all if .
We denote , , and let , be sequences so that
For each , define
for all , and , as
Consider the problems
Due to the hypothesis , , by Theorem 2.1, we can ensure that there exists a solution with such that
with . Since for , there exists such that
Thus, we can find a sequence which converges to for , satisfying
for when .
We note that is the solution of
with and .
Hence, due to an adaptation of Theorem 3.2 in  and by existence theorems, we can find a solution of the problem
This solution is defined in a maximal interval W, and we can find at least one sequence that converges uniformly to in the compact subintervals of W.
On the other hand, and for , then x is defined in and for all . From the conditions on α and β on the boundary, it follows that
so that x is a solution of problem ().
Suppose there exists a function with such that
then we can assume that , , which implies that is absolutely integrable on and , , so x is a positive solution of type 1. □
Theorem 3.1 Suppose that (), () and (H3) or () hold. There exists a positive solution of type 1 if and only if the following conditions hold:
for all , where .
Proof Necessity. Suppose that there exists positive solutions of type 1 of (). By Lemma 3.1, there are constants , , for each such that
Let such that , , . By () and the above inequality, it follows that
Sufficiency. Suppose that there exists a constant such that and .
where is Green’s function (2.2) and and are determined below.
Note that satisfies
Given that and , then . Since
we have that
which implies that
In a similar way,
Thus, there is a lower solution α and an upper solution β of problem () that satisfy for , , . Applying Lemma 3.2, problem () has a solution x such that . Note that for and ,
Due to the hypothesis, we can then ensure that
for , which implies the existence of a positive solution of type 1 of problem () such that . □
Theorem 3.2 If there exists a positive solution of the problem and , then the following conditions hold:
for all , with .
Proof Fix , let us consider , to be two constants such that if and , , and .
Integrating by parts, we have
In a similar way,
and integrating by parts,
Then we conclude that
Theorem 3.3 If the following conditions hold:
for all and , then there exists a lower solution to problem (P).
Proof Consider the function
Let us see that
If we consider
On the other hand,
Let , where
with being a constant such that and .
Thus, if we note that and if , we obtain
This implies that α is a lower solution of problem (). □
Theorem 3.4 Suppose that the conditions of the above theorem are satisfied and consider α the lower solution of problem () provided. If there exists β, an upper solution of (), with and () and (H3) or () hold, then there exists x a positive solution of ().
Proof The demonstration of this fact is immediate taking into account the construction of the lower solution α obtained in the previous theorem, the existence of the upper solution β with and the implementation of Lemma 3.2. □
3.1 Particular cases
Let us briefly consider the following examples.
If is bounded and consists of only isolated points such as in the case , then the conditions of Theorems 3.1 and 3.2 are fulfilled. This follows from the fact
Let be fixed, the quantum time scale is defined as
which appears throughout the mathematical physics literature, where the dynamical systems of interest are the q-difference equations.
Since the only non-isolated point is 0, the interesting case is the one in which the interval contains this point. We consider and .
Taking into account the fact that
with . Hence, the convergence of this series is the necessary and sufficient condition in Theorem 3.1.
Analogously, the condition in Theorem 3.2 can be rewritten as