Analysis on Existence of Positive Solutions for a Class Second Order Semipositone Differential Equations

In this paper, we study the existence of positive solutions of the following second-order semipositone system (see equation 1). By applying a well-known fixed-point theorem, we prove that the problem admits at least one positive solution, if f is bounded below.


Introduction
This paper is focused on the existence of positive solutions of a second-order semipositone system where μ is a positive constant and f satisfies the following assumption: ðF 0 Þf : ½0, 1 × ℝ 2 + ⟶ ℝ is continuous, and where e : ½0, 1 ⟶ ℝ + is continuous and eðtÞ ≢ 0 on [0,1].
The second-order elliptic systems have a strong physical meaning in quantum mechanics models [1,2], in semiconductor theory [3], or in a timeand space-dependent mathematical model of nuclear reactors in a closed container [4]. To the best of our knowledge, existence and multiplicity of nontrivial solutions of BVP (1) have been widely studied by using the variational method [5], bifurcation techniques [6,7], or fixed-point theorems [8][9][10][11]. In general, in order to ensure the positivity of the solutions of Equation (1), one of the crucial assumptions is that the nonlinearity f is nonnegative. Of course, the natural question is whether Equation (1) has a positive solution or not if f satisfies the assumption ðF 0 Þ.
On the other hand, many authors have been interested in finding the relations between the positivity of solutions and the changing sign of the nonlinearity in order to prove the existence of the positive solutions. We refer the readers to [12][13][14][15][16] and the references.
Inspired by these references, the purpose of this paper is to find some new conditions, which are used to study the existence and multiplicity of positive solutions of the semipositone Equation (1). The main tool is the following wellknown fixed-point theorem.
Lemma 1 [17]. Let E be a Banach space and K be a cone in E.
Then, T has a fixed point in Ω R ∩ K\Ω r ∩ K. The same conclusion remains valid if (a) holds on ∂ðΩ R ∩ KÞ and (b) holds on ∂ðΩ r ∩ KÞ.
The paper is organized as follows: in Section 2, we give some preliminaries, which are about the properties of the Green functions, the notations of some sets, etc.; in Section 3, we give the main results and the corresponding proof. In Section 4, some examples are given to illustrate the main results.

Preliminary
Let Gðt, sÞ be the Green function of linear boundary value problem where ρ > −π 2 .
(i) For ρ > 0, we have ð7Þ Let Since J 1 ðtÞ is positive and continuous on [0,1] and J 1 ð1Þ = 0, we have In the similar way, we also have Journal of Function Spaces Therefore, there exists a ξ > 0 such that (ii) For ρ = 0, it is obvious that Using the similar discussion of Case (i), it follows that there exists a ξ > 0 such that For convenience, let Kðt, sÞ denote the Green function for ρ = 0. Then, Equation (1) can be rewritten as Furthermore, let x = u + ω, where ωðtÞ = Ð 1 0 Gðt, sÞeðsÞds is the unique solution of the linear boundary value problem Then, we rewrite (17) as From the above discussion, then we have the following lemma.
Lemma 5. Assume that ðF 0 Þ holds. Then, uðtÞ is a positive solution of (1) if only if xðtÞ is a positive solution of the following problem: with xðtÞ ≥ ωðtÞ. Here, HðtÞ denotes the Heaviside function of a single real variable: Let E denote the Banach space C½0, 1 with the norm kxk = max t∈½0,1 jxðtÞj.
Proof. For any xðtÞ ∈ K, from Lemma 3, it follows that which implies that TðKÞ ⊆ K. Now, we show that T : K ⟶ K is completely continuous.

Journal of Function Spaces
First, we show that T maps the bounded set into itself.
Since e and f are continuous, for any given c > 0, let Then, for x ∈ K c , we have which implies that Tð K c Þ is uniformly bounded. Second, for t 1 , t 2 ∈ ½0, 1, we have which implies that the operator T is equicontinuous. Thus, by applying the Arzela-Ascoli theorem [17], we obtain that TðK c Þ is relatively compact, namely, the operator T is compact.
Finally, we claim that T : K c ⟶ K is continuous. Assume that fx n g ∞ n=1 ⊂ K c which converges to xðtÞ uniformly on [0,1]. By Lebesgue's dominated convergence theorem and letting n ⟶ ∞, we have So, T is continuous on K c . The proof is completed. At the end of this section, let Define the height functions In addition, we need to select some suitable open bounded sets. For any γ > 0, let From [19,20], we can conclude the lemma below.

Main Results
Theorem 8. Assume that (F 0 ) holds. In addition, the function f satisfies the following assumption: (F 1 ) There exists a α > 0 such that Φ * ðαÞ ≥ 0 and Then, we have (i) If σα > ω * , then (1) has at least one partly positive solution ðu, ϕÞ, namely, (ii) If αδξ > Ce * , then (1) has at least one positive solution ðu, ϕÞ, which satisfies Proof. For any x ∈ ∂B α K , it is obvious that Journal of Function Spaces Then, from (F 1 ) it follows that which implies that (a) of Lemma 1 holds. Let From [21], we have that Then, there exists a β > with σβ > α such that and λ 0 > 0 such that x 0 ðtÞ = Tðx 0 ÞðtÞ + λ 0 , then from (iv) of Lemma 7, it follows that Furthermore, for t ∈ ½θ, 1 − θ, we have