Existence result for a singular semipositone dynamic system on time scales

Abstract: We concentrate on investigating the existence of positive solutions for the system of second order singular semipositone m-point boundary value problems in this article. We emphasize that the nonlinear term may take a negative value and be singular. By the properties of Green’s function and applying fixed point theorem in cones, existence results for positive solutions are obtained. Also, we provide an example to make our results clear and easy for readers to understand the existence result.


Introduction
M ulti-point boundary value problems for second order and higher order ordinary differential equations and systems arise from many fields in physics, biology and chemistry. These problems play very important role in both theory and applications [1][2][3][4][5].
Problems where the nonlinear terms have some singularities are referred to as singular problems in the literature and this type of differential systems appear in the study of gas dynamics, fluid mechanics, in the theory of boundary layer and so on. Because of its applications in physics, singular problems have extensively study in recent years, for example see [6][7][8][9].
Although much interest has been observed in investigating the existence of positive solutions of dynamic equations on measure chains [9][10][11][12][13], very few research articles has been seen on the existence of positive solutions of dynamic systems on measure chains [14,15].
In [16], Prasad, Rao and Bharathi interested in the existence of positive solutions to the system of dynamic equations: (−1) n v 2n (t) + µq(t) f (u(σ(t))) = 0, t ∈ [a, b], Problems of this type where the nonlinear term may change sign are referred to as semipositone problems in the literature. Semipositone differential systems appear in the study of chemical reactors [17].
The above works motivates us to consider the nonlinear singular semipositone system of m-point boundary value problem (SSS) in this paper.
Compared to previous work in this field, this study presented three new features. Firstly, the nonlinear term is allowed to change sign and tends to negative infinity. Secondly, is allowed to have finitely many singularities in [a, b]. Lastly, the boundary condition taken up generalizes the conditions of many problems in the literature. By using the cone theory technique, we establish some sufficient conditions for the existence of multiple positive solutions to the SSS (1). The rest of the paper is organized as follows: Section 2 gives some inequalities for Green's function and some results which are needed later. Criteria for the existence of positive solutions of the SSS (1) is established in Section 3 and uses fixed point index theorem. In addition, an example is given to illustrate the applications of main result.

The preliminary Lemmas
We shall work in the space E = C([a, b]; R) × C([a, b]; R). The space E is a Banach space if it is endowed with the norm as follows: for any (u 1 , u 2 ) ∈ E. For any u = (u 1 , In the following, let us define a cone P of E by where g is defined by and φ 1 , φ 2 are the solutions of the linear problems respectively. Let G(t, s) be the Green's function for the boundary value problem is given by where Let us define and assume that the following conditions are satisfied: To prove the main results, we will employ following lemmas. Lemma 1. [18] Under the conditions (H 1 ) and (H 2 ), the solutions φ 1 (t) and φ 2 (t) posses the following properties: where g is given in equation (2).
We consider the following boundary value problem where G(t, s) is given in equation (3), , then the solution u of the boundary value problem (4)-(5) satisfies u(t) ≥ 0, for t ∈ [a, b].

Lemma 6.
If b a G(s, s)y(s)∇s < ∞, then the following inequalities are satisfied:

Main result
In this section, we apply the following fixed point index theorem to prove the existence of at least one positive solution for the SSS (1).
) be a Banach space, Ω be a bounded open subset of E with 0 ∈ Ω, P ⊂ E be a cone in E and F : P ∩ Ω → P be a completely continuous operator.
In the remaining part of the paper, we assume that the following conditions are satisfied: In fact, from the properties of φ 1 , φ 2 and Green function, we get Using the expression for Green's function, the definition of the function g, the properties of φ 1 and φ 2 , the assumption (H 6 ) and Lemma 6, we obtain Therefore, we can write where and g is given in equation (2). Therefore, w i (t), i = 1, 2 are well defined in E. By direct computation, we have which implies that w i (t), i = 1, 2 are positive solutions of the following boundary value problems: Now, we consider the following dynamic system and we define the operator F : E → E by It is well known that the existence of the solution to the system (7) is equivalent to the existence of fixed point of the operator F. Therefore, we shall seek a fixed point of F in our cone P.
is a positive solution of the system (7), then (v 1 − w 1 , v 2 − w 2 ) is a positive solution of the SSS (1).

Proof. Suppose that
is a positive solution of system (7), then from (7) Let Notice that h i (t) = h i + (t) − h i − (t), i = 1, 2 and (9). We know that (u 1 , is a positive solution of the SSS (1). This completes the proof. Now, we want to give the main result of this paper. To prove the main theorem, we need the following assumptions for the functions f i , i= 1, 2.
In fact, from the assumption (H 7 ), for c 1 , At the same time, we have Therefore, when c 1 , c 2 ≥ 1, we have Lemma 8. If f i (t, u 1 , u 2 )(i = 1, 2) satisfies (H 7 ), then for (t, u 1 , If v 1 = 0, let a 1 = u 1 /v 1 , then 0 ≤ a 1 ≤ 1. Now, using the assumption (H 7 ), we obtain Thus, we get that f i (t, u 1 , u 2 ) is increasing on u 1 . Similarly, we can prove that f i (t, u 1 , u 2 ) is increasing on u 2 . On the other hand, choose u 1 , u 2 > 1. Considering the Remark 2, we get Therefore, we obtain Lemma 9. Assume that (H 1 ) − (H 7 ) hold. Then F : P → P is a completely continuous operator.
Proof. First, we shall show that the operator F : P → P is well defined. Therefore, for any fixed (u 1 , u 2 ) ∈ P, Thus, using Remark 2 and Lemma 8, we get from which the assumption (H 6 ), the properties of φ 1 , φ 2 and Lemma 6, for any t ∈ [a, b] gives us: Thus F : P → E is well defined. Now we shall prove that F(P) ⊆ P. For any (u 1 , u 2 ) ∈ P, let (v 1 (t), v 2 (t)) = F(u 1 , u 2 )(t).
For t ∈ [a, b], the above relation and Lemma 3 gives: This yields that F(P) ⊆ P. Let D ⊂ P be any bounded set. Then there exists a constant M > 0 such that u i ≤ M, i = 1, 2 for any (u 1 , u 2 ) ∈ D. Furthermore for any (u 1 , u 2 ) ∈ D and t ∈ [a, b], we find Thus, by Remark 2 and Lemma 8, for any s ∈ [a, b], we have Consequently, Similarly, we can easily find F(D) is equicontinuous on [a, b]. Thus from the Ascoli-Arzela Theorem, we know that F(D) is a relatively compact set.
Finally, from the continuity of f i , i = 1, 2, it is not difficult to check that F : P → P is continuous. Hence F : P → P is a completely continuous operator.
On the other hand, let us choose the constant K such that In view of Lemma 8, there exists N > 0 such that f i (t, u 1 , u 2 ) ≥ K(u 1 + u 2 ), u 1 ≥ N, u 2 ≥ N and t ∈ [t 1 , t 2 ], i = 1, 2.
Considering this, for t ∈ [a, b], we get This contradicts the K that we choose. So from Theorem 1, we get i(F, P R , P) = 0.
Therefore, by equations (10) and (13), we have i(F, P R \P r , P) = −1. Then we see that the operator F has a fixed point (ũ 1 ,ũ 2 ) in P such that r < ũ i < R, i = 1, 2.