Positive Solution of Singular BVPs for System of Dynamic Equations on Time Scales

This paper is devoted to derive some necessary and suficient 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 the fixed-point theorems and the method of the lower and upper solutions.


Introduction
The main purpose of this paper is to establish existence results for the second order Dirichlet system with f = (f 1 , . . . , f n ), i = 1, 2, . . . , n, where f i : (J κ ) o × A → R, A ⊂ R n and J is a time scale interval. The nonlinearity f i (t, x) may be singular at x i , i = 1, . . . , n and/or t.
Stefan Hilger [1] 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 powerfully, such as a mix of stop-start and continuous behaviors [2], [3]. 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, 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. 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. 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 [4], for a general historical overview about this equation. Many works on this system have been written in the continuous case, we can cite among others, [5], [6] or [7] for n = 1 or [8] for n = 2.
On the discrete case we find the book [9] which studies the oscillation properties of the solutions of different difference equations. For the specific problem u ∆∆ (t) + p(t)u γ (σ(t)) = 0, where p ≥ 0 and γ quotient of odd positive numbers, also oscillation properties were studied in [10].
On time scales some results on existence and uniqueness of classical solutions or solutions in the sense of distribution for n = 1 can be found in the articles [11], [12], [13] and [14]. Considering classical solutions, oscillation properties have also been studied, in works such as [15] (with delay) or [16].
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 Sect. 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 Sect. 3, we shall give a necessary and sufficient condition for the existence of positive solutions of the singular boundary value problem (P ) by constructing a lower solution.
We say that f verifies the hypothesis (H 1 ) if for every i = 1, 2, . . . , n are satisfied the following conditions: For convenience, we denote We say that f satisfies the condition (H 2 ) on B ⊂ (J κ ) o × A if for i = 1, 2, . . . , n there exists a function h i ∈ E such that: An upper solution β = (β 1 , . . . , β n ) 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 α ≤ β on [a, σ 2 (b)] T . If f satisfies (H 1 ) and the conditions for all i = 1, . . . , n.
Then problem (P ) has at least one solution x such that α ≤ x ≤ β on [a, σ 2 (b)] T .

Proof:
We consider the following modified problem Due to the hypothesis it is easy to see that (H 1 ) is satisfied and that there exist h * i ∈ E such that (H 2 ) holds for the function f * . Note that, if u is a solution of (P m ) such that α ≤ u ≤ β on [a, σ 2 (b)] T then u is a solution of (P ).
To show that any solution u of (P m ), is between α and β, suppose that there exists i = 1, . . . , n and t * ∈ [a, and v i (t) < v i (t 0 ) for t ∈ (t 0 , σ 2 (b)] T . The point t 0 is not simultaneously leftdense and right-scattered (see theorem 2.1 in [12]) (this implies that (σ•ρ)(t 0 ) = t 0 ) and we have that v ∆∆ i (ρ(t 0 )) ≤ 0 (see [12]), so given that We only need to prove that problem (P m ) has at least one solution.
Consider now the operator N : where (see [17]) is the Green's function of the problem The function N u defined by (2.1) belongs to C([a, σ 2 (b)] T ) because f * satisfies the conditions (H 1 ) and ( It is obvious that u ∈ C([a, σ 2 (b)] T ) is a solution of (P m ) if and only if u = N u. So the problem now is to ensure the existence of fixed points of N .
First of all, N is well defined, is continuous and N (C([a, σ 2 (b)] T )) is a bounded set. The existence of a fixed point of N follows from the Schauder fixed point theorem, once we have checked that N (C([a, σ 2 (b)] T )) is relatively compact, that using the Ascoli-Arzela theorem, is equivalent to proving that N (C([a, σ 2 (b)] T )) is an equicontinuous family.
Let h * ∈ E be the function related to f * by condition (H 2 ). We compute the first derivative of N u using Theorem 1.117 of [17] Finally it is enough to check that λ ∈ L 1 ((J κ ) o ), using integration by parts we obtain due to h * ∈ E, and the fact And so the result is proved.
⊓ ⊔ Remark 2.1 The above theorem is true also if we change (H 3 ) by:

Remark 2.2
The existence of lower solution and upper solution with 0 < α ≤ β can be obtained through conditions of f i . For instance, if f i ∈ C rd (J) and f i is bounded the existence holds.

Existence of positive solution
Consider the problem We will deduce the existence of solution to (P 0 ) by supposing that the following hypothesis hold ( H 2 ) For every i = 1, . . . , n, and j = 1, . . . , n there exists constants λ ij , µ ij , for each t ∈ (J κ ) o and x ∈ A We consider a solutions to the problem Definition 3.1 A positive solution of type 1 of (P 0 ) is a function x = (x 1 , . . . , x n ), with x i ∈ C 2 rd ((a, b) T ) for all i = 1, . . . , n such that x(t) ∈ A and x i (t) > 0, for all t ∈ [a, σ 2 (b)] T , which satisfies the equalities on (P 0 ) for each t ∈ (J κ ) o and i = 1, . . . , n, and exist and are finite the limits Definition 3.2 We say that α ∈ C 2 rd is a lower solution of (P 0 ) if for each i = 1, 2, . . . , n we have, Similary, β ∈ C 2 rd is called an upper solution of (P 0 ) if for each i = 1, 2, . . . , n, Proof: From (2.2), we have Thus, if we consider α ≤ x ≤ β.

If in addition there exists a function h(t)
then solution x is a positive solution of type 1.

Proof:
Let's consider {a k } k≥1 , {b k } k≥1 ⊂ (J κ ) o two sequences such that {a k } k≥1 ⊂ (a, (a + σ(b))/2) T is strictly decreasing to a if a = σ(a), and a k = a for all k ≥ 1 if a < σ(a), and {b k } k≥1 ⊂ ((a + σ(b))/2, σ(b)) T is strictly We denote as , for all k ∈ N, k ≥ 1 and i = 1, . . . , n, as Consider the problems Due to the hypothesis f i , i = 1, . . . , n, by theorem 2.1 we can ensure that there exists solution (x k1 , . . . , Thus, we can find a sequence {t k } which converges to We note that x ki is the solution of . Hence, due to an adaptation of Theorem 3.2 in [18] 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 {x k (t)} that converges uniformly to x(t) in the compact subintervals of W .
On the other hand, From the conditions on α and β on the boundary it follows that so that x is a solution of the problem (P 0 ).

Sufficiency.
Suppose that there exists a constant C ≥ 1 such that CI i1 ≥ 1 and I i2 ≤ C. We consider α i (t) = k i1 y i (t), where G(t, s) is Green's function 2.2 and k i1 and k i2 be determined below. Note that y i satisfies Given that e σ (t)I i1 ≤ y σ i (t) and CI i1 ≥ 1, then e σ (t) ≤ Cy σ i (t). Since f i (t, α σ 1 (t), . . . , α σ n (t)) = f i t, we have that which implies that In a similar way Thus there is a lower solution α and an upper solution β of the problem
Integrating by parts, we have In a similar way and integrating by parts Then, we concluded that

⊓ ⊔
On the other hand, Thus, if we note that 0 < µ ii < 1 and µ ij < 0 if i = j, we obtained This implies that α is a lower solution of the problem (P 0 ).
Theorem 3.4 Suppose that the conditions of above theorem are satisfied, and consider α the lower solution of the problem (P 0 ) provided. If there exists, β, an upper solution of (P 0 ) with 0 < α ≤ β and ( H 1 ) and (H 3 ) or (H 3 ) hold. Then there exists x a positive solution of (P 0 ).

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 0 < α ≤ β and the implementation of the lemma 3.2. (σ(t) − t) f (t).

Particular cases
2. Let q > 1 fixed, the quantum time scale q Z 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 containing it. We consider a = 0 and σ(b) = 1.
Taking into account the fact that we have with E σ (q −k ) = q −k+1 (1 − q −k ), . . . , q −k+1 (1 − q −k ) . Hence, the convergence of this series is the necessary and sufficient condition in Theorem 3.1.