This paper is devoted to derive some sufficient conditions for the existence and uniqueness of positive solutions to a singular second-order dynamic equation with Dirichlet boundary conditions.
1. Introduction
Hilger [1] introduced the notion of time scale in 1990 in order to unify the theory of continuous and discrete calculus. The field of dynamic equations on time scale contains, links, and extends the classical theory of differential and difference equations, besides many others. There are more time scales than just (corresponding to the continuous case) and (discrete case) and hence many more classes of dynamic equations.
By time scale we mean a closed subset of the real numbers. Let be an arbitrary time scale. We assume that has the topology that it inherits from the standard topology on . Assume that are points in and define the time scale interval . For , define the forward jump operator by and the backward jump operator by . In this definition we put if attains a maximum and if attains a minimum . If is said to be right-scattered and if is said to be right-dense. If is said to be left-scattered and if is said to be left-dense.
A function is said to be rd-continuous provided it is continuous at all right-dense points of , and its left-sided limit exists at left-dense points of . For , the delta derivative of at is defined to be the number (if exists) such that for given , there exists a neighborhood of such that
See [2] for general theory about time scales.
The problem we will consider in this work is of the type
Under this general form it included the Emden-Fowler equation, which arises in several fields, such as the following:
(i)Astrophysics: related to the stellar structure (gaseous dynamics). In this case the fundamental problem is to investigate the equilibrium configuration of the mass of spherical clouds of gas. (ii)Gas dynamics and fluid mechanics. The solutions of physical interest in this context are bounded nonoscillatory and possess a positive zero. (iii)Relativistic mechanics. (iv)Nuclear physics. (v)Chemically reacting systems: in the theory of diffusion and reaction this equation appears as governing the concentration of a substance which disappears by an isothermal reaction at each point of a slab of catalyst. We refer to Wong [3], for a general historical overview about this equation.
Many works on this equation have been written in the continuous case, and we can cite among others, [4, 5] or [6].
On the discrete case we find the book [7] which studies the oscillation properties of the solutions of different difference equations. For the specific problem , where and quotient of odd positive numbers, also oscillation properties were studied in [8].
On time scales some results on existence and uniqueness of solutions in the sense of distribution for this equation can be found in the article [9]. Considering classical solutions, oscillation properties have also been studied, in works such as [10] (with delay) or [11].
In the present paper we present some results on time scales considering classical solutions which generalize the ones from the continuous case.
2. Lower and Upper Solutions Method
Let , such that . Let us put , and if and , and if .
We consider the second-order dynamic equation with Dirichlet boundary conditions,
where , satisfies the following condition.
(i) For every , , (ii) is continuous on uniformly in . For convenience, we denote
We say that satisfies the condition on if there exists a function such that
Definition 2.1. A solution of is a function such that , for all , which satisfies the equalities on for each , where
Definition 2.2. We say that is a lower solution of if , for all and
An upper solution of is defined similarly by reversing the previous inequalities.
We have the following result.
Theorem 2.3. Let and be, respectively, a lower and upper solution for problem , such that on . If satisfies and the conditions on
then problem has at least one solution such that on .
Proof. We consider the following modified problem:
with
Due to the hypothesis it can be easily checked that and are satisfied by the function .
Note that, if is a solution of such that on , then is a solution of , also satisfying on .
To show that any solution of is between and , let , and suppose that there exists such that . As and , then there exists with
and for . The point is not simultaneously left-dense and right-scattered (see [12, Theorem 2.1]) (this implies that ), and we have that (see [12]), so
So , that is a contradiction. And so we have proved that .
Analogously it can be proved that .
We only need to prove that problem has at least one solution.
Consider now the operator , defined by
for each , where (see [2])
is Green's function of the problem
and for
is the solution of such that and .
Clearly, on is rd-continuous on and is continuous on .
The function defined by (2.9) belongs to because checks the conditions and on and , for each .
It is obvious that is a solution of if and only if . So the problem now is ensuring the existence of fixed-points of .
First of all, is well defined, is continuous, and is a bounded set. The existence of a fixed-point of follows from the Schauder fixed-point theorem, once we have checked that is relatively compact, that using the Ascoli-Arzela theorem is equivalent to proving that is an equicontinuous family.
Let be the function related to by condition . We compute the first derivative of using [2, Theorem 1.117]
Finally it is enough to check that , using integration by parts we obtain
due to , and the fact
And so the result is proved.
3. Existence and Uniqueness of Positive Solution
Let in the condition , where , and consider the problem
We will deduce the existence of solution to by supposing that the following hypothesis holds.
There exists a constant such that for any compact set , there is :
Theorem 3.1. Suppose that and hold. If, for any satisfies the condition on , then problem has at least one solution.
Proof. Let's consider as two sequences such that is strictly decreasing to if , and for all if , and is strictly increasing to if for all if . We denote as .
Due to the first hypothesis, we can then ensure the existence of such that , for all . We can suppose, without restriction that is a decreasing sequence and .
Consider the function , such that , and if then for , for all , with . Since is nondecreasing, we obtain that for
is continuous and increasing. Repeating this argument twice, for we define
So is a convex function verifying , .
Analogously, we can consider, such that , for all and , for all , with . Taking , we obtain, for , that for
is continuous and decreasing, and a convex function such that .
We now define
with a convenient function, so that , and for all .
So is a function such that for for , and , for .
In this way, we note that
Let .
Let , for all , with in the conditions . We will prove that if is any solution of
with and for all , then,
Suppose there exists , such that . Then, we can assure, using arguments analogous to the ones in the proof of Theorem 2.3, that there exists verifying
which implies that
which is a contradiction.
We define now, for each
where, for each
For each we have that is a function verifying on , and , for all and , for .
The sequence converges to uniformly in every set of the form , where is a compact set.
Defining, by induction, and for
we have that, for each , the function satisfies the condition on . As well and converges to uniformly, in every set of the form , where is a compact set.
It is also verified that .
Now we define the following problems:
We will prove that for any , the constant function is a (strict) lower solution for .
It is obvious that , and . Now we have to prove that
For , let such that
Suppose now that , for a given , and we will check that . Let such that
Thus the assertion is proved.
Moreover, as on , it can be easily checked that any solution of is an upper solution for .
To show that problem has at least one solution. We fix a constant . From the assumption imposed, there exists a function such that
Note that
with is a suitable constant.
Set . We have that , and
Let be a solution of the boundary value problem:
It is easy to check that such a function exists and that , for all , and thus
which implies that is an upper solution of .
Taking , we have that and are, respectively, a lower and upper solutions of . Moreover , for all . Then, by Theorem 2.3 there exists a solution of such that
Proceeding by induction, using , lower and upper solution of , we obtain (via Theorem 2.3) a sequence of solutions of , such that , for all and .
Since on , and is a solution of with for all , then , for every .
To end, we will prove that
is a solution of problem .
We have that, for any ,
Now let be a compact interval. There exists an index such that for all . Let , if , then
Hence, verifies the first equality of problem for all , and for all . Moreover
Then by the Ascoli-Arzelá theorem, we can conclude that is a solution of in . Since the compact was arbitrary, we have that , and
As , we just have to check the continuity of in and .
Let . Take in such a way that . From the continuity of at , it follows that we can find a constant such that
So
Using the same argument the continuity at is proved.
In order to prove the uniqueness of the solution to problem we have the following result.
Theorem 3.2. Under the hypotheses of Theorem 3.1, if is strictly decreasing in , for each , then problem has a unique solution.
Proof. By Theorem 3.1 the problem has at least one solution.
Suppose that there exist and , solutions of and such that . If this occurs, setting we can find such that