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Positive solution of singular BVPs for a system of dynamic equations on time scales
Advances in Difference Equations volume 2012, Article number: 185 (2012)
This paper is devoted to derivation of some necessary and sufficient conditions for the existence of positive solutions to a singular second-order system of dynamic equations with Dirichlet boundary conditions. The results are obtained by employing fixed-point theorems and the method of lower and upper solutions.
MSC:34B16, 34K10, 39A13.
The main purpose of this paper is to establish existence results for the second-order Dirichlet system
with , , where , and J is a time scale interval. The nonlinearity may be singular at , and/or t.
Stefan Hilger  introduced the notion of time scale in 1988 in order to unify the theory of continuous and discrete calculus. The time scales approach not only unifies differential and difference equations, but also solves some other problems such as a mix of stop-start and continuous behaviors [2, 3] powerfully. Nowadays the theory on time scales has been widely applied to several scientific fields such as biology, heat transfer, stock market, wound healing and epidemic models.
Under the general form of problem (P), it included the Emden-Fowler equation which arises, for example, in astrophysics in relation 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. It also arises in gas dynamics and fluid mechanics. The solutions of physical interest in this context are bounded non-oscillatory and possess a positive zero. It is also encountered in the relativistic mechanics and nuclear physics; and in chemically reacting systems: in the theory of diffusion and reaction this equation appears as governing the concentration u of a substance which disappears by an isothermal reaction at each point of a slab of catalyst. We refer to Wong  for a general historical overview of this equation.
Many works on this system have been written in the continuous case. We can cite, among others, [5, 6] or  for or  for . On the discrete case, we find the book  which studies the oscillation properties of the solutions of different difference equations. For the specific problem , where and γ is the quotient of odd positive numbers, oscillation properties were also studied in .
Regarding time scales, some results on the existence and uniqueness of classical solutions or solutions in the sense of distribution for can be found in the articles [11–13] and . Considering classical solutions, oscillation properties have also been studied in works such as  (with delay) or .
In the present paper, we present some results on time scales considering classical solutions which generalize the ones from the continuous case. The remainder of the paper is organized as follows. In Section 2, we state some existence results supposing the existence of a pair of lower and upper solutions and employing the Schauder fixed-point theorem. In Section 3, we shall give a necessary and sufficient condition for the existence of positive solutions of singular boundary value problem (P) by constructing a lower solution.
2 Lower and upper solutions method
Let be an arbitrary time scale. We assume that has the topology that it inherits from the standard topology on ℝ. See  for general theory on time scales.
Let be such that . If a is a right-dense point, we consider , and . In the other case, , and .
The problem we will consider in this section is
with , , where , .
We say that f verifies the hypothesis (H1) if for every , the following conditions are satisfied:
For every , ,
is continuous on uniformly in .
For convenience, we denote
We say that f satisfies the condition (H2) on if for there exists a function such that
(H2) , .
Definition 2.1 A solution of (P) is a function , with for all such that for all , which satisfies (P) for each and , where
Definition 2.2 We say that , with , is a lower solution of (P) if for all and
An upper solution of (P) is defined similarly by reversing the previous inequalities.
We have the following result.
Theorem 2.1 Let α and β be, respectively, a lower and upper solution for problem (P) such that on . If f satisfies (H1) and the condition (H2) on
(H3) For and
for all .
Then problem (P) has at least one solution x such that on .
Proof We consider the following modified problem:
where and is defined
We can prove that is continuous on uniformly in t and for every . Hence, the function , verified for each , .
Due to the hypothesis, it is easy to see that (H1) is satisfied and that there exist such that (H2) holds for the function .
Note that if u is a solution of () such that on , then u is a solution of (P).
To show that any solution u of () is between α and β, suppose that there exists and such that . As and , then there exists with
So, , that is a contradiction. And so we have proved that for each .
Analogously, it can be proved that for all .
We only need to prove that problem () has at least one solution.
Consider now the operator defined by
for each , where (see )
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 Nu defined by (2.1) belongs to because satisfies the conditions (H1) and (H2) on and for each .
It is obvious that is a solution of () if and only if . So, the problem now is to ensure the existence of fixed points of N.
First of all, N is well defined, continuous and is a bounded set. The existence of a fixed point of N 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 the condition (H2). We compute the first derivative of Nu using Theorem 1.117 of :
Finally, it is enough to check that . Using integration by parts, we obtain
due to and the fact
And so the result is proved. □
Remark 2.1 The above theorem is also true if we change (H3) by
() For and , there exists such that
Remark 2.2 The existence of a lower solution and an upper solution with can be obtained through conditions of . For instance, if and is bounded, the existence holds.
3 Existence of a positive solution
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 with such that
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
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The authors are grateful to the referees for their valuable suggestions that led to the improvement of the original manuscript.The research of VOE has been partially supported by Ministerio de Educación y Ciencia (Spain) and FEDER, Project MTM2010-15314. The research of AL has been supported by FSE y de la DX de Ordenación y Calidad del Sistema Universitario de Galicia, de la Conselleria de Educación y Ordenación Universitaria-Xunta de Galicia.
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.