Positive solution of singular BVPs for a system of dynamic equations on time scales
© Lago et al.; licensee Springer 2012
Received: 6 June 2012
Accepted: 15 October 2012
Published: 29 October 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.
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 .
with , , where , .
For every , ,
is continuous on uniformly in .
We say that f satisfies the condition (H2) on if for there exists a function such that
(H2) , .
An upper solution of (P) is defined similarly by reversing the previous inequalities.
We have the following result.
for all .
Then problem (P) has at least one solution x such that on .
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).
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.
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.
And so the result is proved. □
Remark 2.1 The above theorem is also true if we change (H3) by
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
We will deduce the existence of a solution to () by supposing that the following hypotheses hold:
For every , ,
is continuous on uniformly in .
for each and
for each and .
We consider solutions to the problem.
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 .
for when .
with and .
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.
so that x is a solution of problem ().
then we can assume that , , which implies that is absolutely integrable on and , , so x is a positive solution of type 1. □
for all , where .
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
for , which implies the existence of a positive solution of type 1 of problem () such that . □
for all , with .
Proof Fix , let us consider , to be two constants such that if and , , and .
for all and , then there exists a lower solution to problem (P).
with being a constant such that and .
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
- 1If 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
- 2Let 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 .
with . Hence, the convergence of this series is the necessary and sufficient condition in Theorem 3.1.
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.
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