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.
Multi-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].
For example, Asif and Khan [6] studied the existence of positive solution to a nonlinear singular system with four-point boundary conditions of the type \begin{eqnarray*} -x(t)” &=& f(t, x(t), y(t)), t \in (0, 1),\\ -y(t)” &=& g(t, x(t), y(t)), t \in (0, 1),\\ x(0)&=&0,\ \ x(1)=\alpha y(\xi),\ \ y(0)=0,\ \ y(1)=\beta x(\eta). \end{eqnarray*} In [7], Liu and Yan considered the following singular boundary value problem of Sturm Liouville differential system: \begin{eqnarray*} (p(t)x(t)’)’ + \lambda f(t, x(t), y(t))= 0, t \in (0, 1),\\ (p(t)y(t)’)’ + \lambda g(t, x(t), y(t))= 0, t \in (0, 1), \\ \alpha x(0)- \beta p(0) x'(0)= \gamma x(1)- \delta p(1) x'(1) = 0, \\ \alpha y(0)- \beta p(0) y'(0)= \gamma y(1)- \delta p(1) y'(1) = 0. \end{eqnarray*} 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: \begin{eqnarray*} (-1)^{n} u^{\triangle^{2n}}(t) + \lambda p(t)f(v(\sigma(t)))= 0, t \in [a, b], \end{eqnarray*} \begin{eqnarray*} (-1)^{n} v^{\triangle^{2n}}(t) + \mu q(t)f(u(\sigma(t)))= 0, t \in [a, b],\\ \alpha_{i+1} u^{\triangle^{2i}}(a)- \beta_{i+1} u^{\triangle^{2i+1}}(a)= 0, \gamma_{i+1} u^{\triangle^{2i}}(\sigma(b))+\delta_{i+1} u^{\triangle^{2i+1}}(\sigma(b))= 0,\\ \alpha_{i+1} v^{\triangle^{2i}}(a)- \beta_{i+1} v^{\triangle^{2i+1}}(a)= 0, \gamma_{i+1} v^{\triangle^{2i}}(\sigma(b))+\delta_{i+1} v^{\triangle^{2i+1}}(\sigma(b))= 0. \end{eqnarray*} 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.Lemma 1. [18] Under the conditions \((H_{1})\) and \((H_{2})\), the solutions \(\phi_{1}(t)\) and \(\phi_{2}(t)\) posses the following properties: \begin{equation*} \phi_{1}(t), \phi_{2}(t) \geq 0, \quad\phi_{1}^{[\triangle]}(t)\geq 0,\quad \phi_{2}^{[\triangle]}(t)\leq 0, \quad t \in [a, b]. \end{equation*}
Lemma 2. [18] If the conditions \((H_{1})-(H_{3})\) are hold, then \(G(t, s)> 0\) for \(t, s \in [a, b]\).
Lemma 3. [20] Assume that \((H_{1})-(H_{3})\) hold. Then \begin{equation*} g(t)G(s, s)\leq G(t, s) \leq G(s, s), \quad t, s \in [a, b], \end{equation*} where \(g\) is given in equation (2).
We consider the following boundary value problemLemma 4. [12] Let the conditions \((H_{1})-(H_{3})\) be hold. Assume that \(\Omega: \neq 0\). Then for \(y \in C([a, b])\), the boundary value problem given in equations (4)-(5) has a unique solution \begin{equation*} u(t)= \int_{a}^{b} G(t, s) y(s) \nabla s + A(y) \phi_{1}(t) + B(y) \phi_{2}(t), \end{equation*} where \(G(t, s)\) is given in equation (3), \begin{equation*} A(y):=\frac{1}{\Omega}\left|{\begin{array}{cc} \sum_{k=1}^{m-2} \alpha_{k} \int_{a}^{b}G(\xi_{k}, s)y(s)\nabla s & d-\sum_{k=1}^{m-2} \alpha_{k}\phi_{2}(\xi_{k})\\ \sum_{k=1}^{m-2} \beta_{k} \int_{a}^{b}G(\xi_{k}, s)y(s) \nabla s & -\sum_{k=1}^{m-2} \beta_{k}\phi_{2}(\xi_{k}) \end{array} } \right|, \end{equation*} \begin{equation*} B(y):=\frac{1}{\Omega}\left|{\begin{array}{cc} -\sum_{k=1}^{m-2} \alpha_{k}\phi_{1}(\xi_{k}) & \sum_{k=1}^{m-2} \alpha_{k} \int_{a}^{b}G(\xi_{k}, s)y(s)\nabla s \\ d-\sum_{k=1}^{m-2} \beta_{k}\phi_{1}(\xi_{k}) & \sum_{k=1}^{m-2} \beta_{k}\int_{a}^{b}G(\xi_{k}, s)y(s)\nabla s \end{array} } \right|. \end{equation*}
Lemma 5. [5] Let \((H_{1})-(H_{4})\) hold. If \(y \in C([a, b], [0, \infty))\), then the solution \(u\) of the boundary value problem (4)-(5) satisfies \(u(t)\geq 0\), for \(t \in [a, b]\).
Lemma 6. If \(\int_{a}^{b}G(s, s)y(s)\nabla s< \infty\), then the following inequalities are satisfied: \begin{equation*} A(y)\leq A\int_{a}^{b}G(s, s)y(s)\nabla s, \quad B(y)\leq B\int_{a}^{b}G(s,s)y(s)\nabla s, \end{equation*} where \begin{equation*} A=\frac{1}{\Omega}\left|{\begin{array}{cc} \sum_{k=1}^{m-2} \alpha_{k} & d-\sum_{k=1}^{m-2} \alpha_{k}\phi_{2}(\xi_{k})\\ \sum_{k=1}^{m-2} \beta_{k} & -\sum_{k=1}^{m-2} \beta_{k}\phi_{2}(\xi_{k}) \end{array} } \right|, \end{equation*} \begin{equation*} B=\frac{1}{\Omega}\left|{\begin{array}{cc} -\sum_{k=1}^{m-2} \alpha_{k}\phi_{1}(\xi_{k}) & \sum_{k=1}^{m-2} \alpha_{k} \\ d-\sum_{k=1}^{m-2} \beta_{k}\phi_{1}(\xi_{k}) & \sum_{k=1}^{m-2} \beta_{k} \end{array} } \right|. \end{equation*}
Theorem 1. Let \(E=(E, \| . \|)\) be a Banach space, \(\Omega\) be a bounded open subset of \(E\) with \(0 \in \Omega\), \(P\subset E\) be a cone in \(E\) and \(F : P \cap \overline{\Omega}\rightarrow P\) be a completely continuous operator.
where \(h_{i_{+}}(t)= \max \{h_{i}(t), 0\}\) and \(h_{i_{-}}(t)= \max \{-h_{i}(t), 0\}\).
Remark 1. By the assumption \((H_{6})\), we have \(\int_{a}^{b} G(t, s)h_{i_{-}}(s)\nabla s < \infty \), \(i=1, 2\).
In fact, from the properties of \(\phi_{1}\), \(\phi_{2} \) and Green function, we get \begin{equation*} \int_{a}^{b} G(t, s)h_{i_{-}}(s)\nabla s \leq \int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s \leq \phi_{1}(b)\phi_{2}(a) \int_{a}^{b} h_{i_{-}}(s)\nabla s < \infty , \quad i=1, 2. \end{equation*} Let \(w_{i}(t)= \int_{a}^{b} G(t, s)h_{i_{-}}(s) \nabla s + A(h_{i_{-}}) \phi_{1}(t) + B(h_{i_{-}}) \phi_{2}(t)\), \(t \in [a, b], i=1, 2\). Using the expression for Green's function, the definition of the function \(g\), the properties of \(\phi_{1}\) and \(\phi_{2}\), the assumption \((H_{6})\) and Lemma 6, we obtain \begin{align*} w_{i}(t)&= \int_{a}^{b} G(t, s)h_{i_{-}}(s) \nabla s +A(h_{i_{-}}) \phi_{1}(t) + B(h_{i_{-}}) \phi_{2}(t) \nonumber \\ &= \frac{1}{d} \int_{a}^{t} \phi_{1}(s)\phi_{2}(t)h_{i_{-}}(s) \nabla s + \frac{1}{d} \int_{t}^{b} \phi_{1}(t)\phi_{2}(s) h_{i_{-}}(s)\nabla s + A(h_{i_{-}}) \phi_{1}(t) + B(h_{i_{-}}) \phi_{2}(t) \nonumber\\ &\leq \frac{1}{d} \int_{a}^{t} \phi_{1}(t)\phi_{2}(t)h_{i_{-}}(s) \nabla s + \frac{1}{d} \int_{t}^{b} \phi_{1}(t)\phi_{2}(t) h_{i_{-}}(s)\nabla s + A\phi_{1}(t)\int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s\\ &\quad + B\phi_{2}(t) \int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s \nonumber \\ &= \frac{1}{d} \int_{a}^{b} \phi_{1}(t)\phi_{2}(t)h_{i_{-}}(s) \nabla s + A\phi_{1}(t)\int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s + B\phi_{2}(t)\int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s \nonumber \end{align*} \begin{align*}\\ &\leq \frac{1}{d} \phi_{1}(b)\phi_{2}(a)g(t) \int_{a}^{b}h_{i_{-}}(s) \nabla s + \frac{1}{\phi_{2}(t)} A\phi_{1}(b)\phi_{2}(a)g(t)\int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s \nonumber \\ &\quad + \frac{1}{\phi_{1}(t)} B\phi_{1}(b)\phi_{2}(a)g(t)\int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s \nonumber \\ &\leq \frac{1}{d} \phi_{1}(b)\phi_{2}(a) g(t) \int_{a}^{b}h_{i_{-}}(s) \nabla s + \frac{1}{\phi_{2}(b)} A\phi_{1}(b)\phi_{2}(a)g(t)\int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s \nonumber \\ &\quad + \frac{1}{\phi_{1}(a)} B\phi_{1}(b)\phi_{2}(a)g(t)\int_{a}^{b} G(s, s)h_{i_{-}}(s)\nabla s \nonumber \\ &= \Big[ \frac{1}{d}\int_{a}^{b}h_{i_{-}}(s) \nabla s + \Big(\frac{A}{\phi_{2}(b)}+ \frac {B}{\phi_{1}(a)} \Big)\int_{a}^{b}G(s, s)h_{i_{-}}(s) \nabla s\Big] \phi_{1}(b)\phi_{2}(a)g(t)< +\infty, \quad i=1, 2. \nonumber \end{align*} Therefore, we can writeLemma 7. If \((v_{1}, v_{2})\) with \((w_{1}, w_{2})\leq (v_{1},v_{2})\) 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 \((v_{1}, v_{2})\) with \((w_{1}, w_{2})\leq (v_{1},v_{2})\) is a positive solution of system (7), then from (7) and the definition of \([ . ]^{*}\), we have
Remark 2. For \(c_{1}, c_{2} \geq 1\), \((t, u_{1}, u_{2}) \in (a, b)\times[0,\infty) \times[0,\infty)\), we have \begin{equation*} (c_{1} c_{2})^{\lambda_{2}} f_{i}(t, u_{1}, u_{2}) \leq f_{i}(t, c_{1} u_{1}, c_{2} u_{2} ) \leq (c_{1} c_{2})^{\lambda_{1}} f_{i}(t, u_{1}, u_{2}), \quad i=1, 2. \end{equation*}
In fact, from the assumption \((H_{7})\), for \(c_{1}, c_{2} \geq 1\), \((t, u_{1}, u_{2}) \in (a, b)\times[0,\infty) \times[0,\infty)\), we get \begin{equation*} f_{i}(t, u_{1}, u_{2})=f_{i}(t, \frac{1}{c_{1}} c_{1} u_{1}, \frac{1}{c_{2}} c_{2} u_{2}) \leq (\frac{1}{c_{1}c_{2}})^{\lambda_{2}} f_{i}(t, c_{1} u_{1}, c_{2} u_{2} ), \end{equation*} This implies \begin{equation*} (c_{1} c_{2})^{\lambda_{2}} f_{i}(t, u_{1}, u_{2}) \leq f_{i}(t, c_{1} u_{1}, c_{2} u_{2} ), \quad i=1, 2. \end{equation*} At the same time, we have \begin{equation*} f_{i}(t, c_{1} u_{1}, c_{2} u_{2} ) \leq (c_{1} c_{2})^{\lambda_{1}} f_{i}(t, u_{1}, u_{2}), \quad i=1, 2. \end{equation*} Therefore, when \(c_{1}, c_{2} \geq 1\), we have \begin{equation*} (c_{1} c_{2})^{\lambda_{2}} f_{i}(t, u_{1}, u_{2}) \leq f_{i}(t, c_{1} u_{1}, c_{2} u_{2} ) \leq (c_{1} c_{2})^{\lambda_{1}} f_{i}(t, u_{1}, u_{2}), \quad i=1, 2. \end{equation*}Lemma 8. If \(f_{i}(t, u_{1}, u_{2}) (i=1, 2)\) satisfies \((H_{7})\), then for \((t, u_{1}, u_{2}) \in (a, b)\times[0,\infty) \times[0,\infty)\), \(f_{i}(t, u_{1}, u_{2})\) is increasing on \(u_{1}\), \(u_{2}\) and for \([t_{1}, t_{2}] \subset (a, b),\) \begin{equation*} \lim_{ u_{1}, u_{2} \rightarrow + \infty} \min_{t \in [t_{1}, t_{2}]} \frac{f_{i}(t, u_{1}, u_{2})}{|u_{1}|+ |u_{2}|} =+ \infty, \quad i=1, 2. \end{equation*}
Proof. Let \(t\in (a, b)\), \(u_{1}, v_{1}, u_{2} \in [0, \infty)\) such that \(u_{1} \leq v_{1}\). We will show that \(f_{i}(t, u_{1}, u_{2}) \leq f_{i}(t, v_{1}, u_{2})(i=1, 2) \). Clearly, if \(v_{1}=0\), then \(f_{i}(t, u_{1}, u_{2}) \leq f_{i}(t, v_{1}, u_{2}) \). If \(v_{1} \neq 0\), let \(a_{1}=u_{1} / v_{1}\), then \(0 \leq a_{1} \leq 1\). Now, using the assumption \((H_{7})\), we obtain \begin{equation*} f_{i}(t, u_{1}, u_{2})= f_{i}(t, a_{1} v_{1}, u_{2}) \leq a_{1} ^{\lambda_{2}} f_{i}(t, v_{1}, u_{2}) \leq f_{i}(t, v_{1}, u_{2}), \quad i=1, 2. \end{equation*} 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 1, we get \begin{equation*} f_{i}(t, u_{1}, u_{2}) \geq (u_{1} u_{2})^{\lambda_{2}} f_{i}(t, 1, 1), \quad i=1, 2, \end{equation*} and thus, for \([t_{1}, t_{2}] \subset (a, b),\) \( \forall t \in [t_{1}, t_{2}]\), we have \begin{equation*} \min_{t \in [t_{1}, t_{2}]} \frac{f_{i}(t, u_{1}, u_{2})}{|u_{1}|+ |u_{2}|} \geq \min_{t \in [t_{1}, t_{2}]} \frac{(u_{1} u_{2})^{\lambda_{2}}}{|u_{1}|+ |u_{2}|} f_{i}(t, 1, 1) > 0, \end{equation*} Therefore, we obtain \begin{equation*} \lim_{ u_{1}, u_{2} \rightarrow + \infty}\min_{t \in [t_{1}, t_{2}]} \frac{f_{i}(t, u_{1}, u_{2})}{|u_{1}|+ |u_{2}|}=+ \infty, \quad i=1, 2. \end{equation*}
Lemma 9. Assume that \((H_{1})-(H_{7})\) hold. Then \(F:P \rightarrow P\) is a completely continuous operator.
Proof.
First, we shall show that the operator \(F:P
\rightarrow P\) is well defined. Therefore, for any fixed \((u_{1}, u_{2}) \in P\), choose
\(0< d_{1}, d_{2}< 1\) such that \(d_{1}\|u_{1} \|< 1\) and
\(d_{2}\|u_{2}\|< 1\). Then for \(t\in [a, b]\), we get
\begin{equation*}
d_{i}[u_{i}(t)-w_{i}(t)]^{*}\leq d_{i}u_{i}(t)\leq d_{i} \|u_{i}\| < 1, \quad i=1, 2.
\end{equation*}
Thus, using Remark 1 and Lemma 8, we get
\begin{align*}
f_{i}(t, [u_{1}(t)-w_{1}(t)]^{*} , [u_{2}(t)-w_{2}(t)]^{*}) &\leq ( \frac{1}{d_{1} d_{2}})^{\lambda_{1}} f_{i}(t, d_{1} \|u_{1}\|, d_{2}\|u_{2}\|) \nonumber \\
&\leq (d_{1}d_{2})^{\lambda_{2}-\lambda_{1}} \|u_{1}\|^{\lambda_{2}}\|u_{2}\|^{\lambda_{2}} f_{i}(t, 1, 1), \quad i=1, 2,
\end{align*}
from which the assumption \((H_{6})\), the properties of \(\phi_{1}, \phi_{2}\) and Lemma 6, for any \(t \in [a, b]\) gives us:
\[\begin{align}
F_{i}(u_{1},u_{2})(t) &=\int_{a}^{b} G(t, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)]\nabla s \\
&\quad + A(f_{i} + h_{i_{+}}) \phi_{1}(t) + B(f_{i} + h_{i_{+}}) \phi_{2}(t) \\
&\leq \int_{a}^{b} G(s, s) [f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)]\nabla s \\
&\quad +A(f_{i} + h_{i_{+}}) \phi_{1}(t) + B(f_{i} + h_{i_{+}}) \phi_{2}(t) \\
&\leq \int_{a}^{b} G(s, s)[(d_{1}d_{2})^{\lambda_{2}-\lambda_{1}} \|u_{1}\|^{\lambda_{2}}\|u_{2}\|^{\lambda_{2}} f_{i}(s, 1, 1)+h_{i_{+}}(s)] \nabla s \\
&\quad +A(f_{i} + h_{i_{+}}) \phi_{1}(b) + B(f_{i} + h_{i_{+}}) \phi_{2}(a) \\
&\leq ((d_{1}d_{2})^{\lambda_{2}-\lambda_{1}} \|u_{1}\|^{\lambda_{2}}\|u_{2}\|^{\lambda_{2}}+1) \int_{a}^{b} G(s,s)[f_{i}(s, 1, 1)+h_{i_{+}}(s)] \nabla s \\
&\quad + A((d_{1}d_{2})^{\lambda_{2}-\lambda_{1}} \|u_{1}\|^{\lambda_{2}}\|u_{2}\|^{\lambda_{2}}+1)\phi_{1}(b) \int_{a}^{b} G(s,s)[f_{i}(s, 1,1)+h_{i_{+}}(s)] \nabla s\\
&\quad +B((d_{1}d_{2})^{\lambda_{2}-\lambda_{1}} \|u_{1}\|^{\lambda_{2}}\|u_{2}\|^{\lambda_{2}}+1) \phi_{2}(a) \int_{a}^{b} G(s, s)[f_{i}(s, 1, 1)+h_{i_{+}}(s)] \nabla s \\
&=((d_{1}d_{2})^{\lambda_{2}-\lambda_{1}} \|u_{1}\|^{\lambda_{2}}\|u_{2}\|^{\lambda_{2}}+1)(1+A\phi_{1}(b)+B\phi_{2}(a)) \int_{a}^{b}
G(s, s)[f_{i}(s, 1, 1)+h_{i_{+}}(s)] \nabla s\\
&<\infty, \quad i=1, 2.
\end{align}\]
Thus \(F:P \rightarrow E\) is well defined.
Now we shall prove that \(F(P)\subseteq P\). For any \((u_{1}, u_{2})\in P\), let \((v_{1}(t), v_{2}(t)) =F(u_{1}, u_{2})(t)
\). Then for \(t \in [a, b]\), we get
\[\begin{align}
v_{i}(t)&= \int_{a}^{b} G(t, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*}, [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)] \nabla s +A(f_{i} + h_{i_{+}}) \phi_{1}(t)+ B(f_{i} + h_{i_{+}}) \phi_{2}(t) \\
&\leq \int_{a}^{b} G(s, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)] \nabla s +A(f_{i} + h_{i_{+}}) \phi_{1}(b) + B(f_{i} + h_{i_{+}}) \phi_{2}(a)
\end{align}\]
and so
\[\begin{align}
\|v_{i}\| &\leq \int_{a}^{b} G(s, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)]
\nabla s +A(f_{i} + h_{i_{+}}) \phi_{1}(b) + B(f_{i} + h_{i_{+}}) \phi_{2}(a),
\end{align}\]
For \(t\in [a, b]\), the above relation and Lemma 3 gives:
\[\begin{align}
v_{i}(t)&= \int_{a}^{b} G(t, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*}, [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)] \nabla s +A(f_{i} + h_{i_{+}}) \phi_{1}(t) +B(f_{i} + h_{i_{+}}) \phi_{2}(t) \\
&\geq g(t) \int_{a}^{b} G(s, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)]
\nabla s \\
&\quad +A(f_{i} + h_{i_{+}}) \frac{\phi_{1}(t)}{\phi_{1}(b)}\phi_{1}(b) + B(f_{i} + h_{i_{+}}) \frac{\phi_{2}(t)}{\phi_{2}(a)}\phi_{2}(a) \\
&\geq g(t) \int_{a}^{b} G(s, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)]\nabla s \nonumber \\
&\quad +A(f_{i} + h_{i_{+}}) g(t)\phi_{1}(b) + B(f_{i} + h_{i_{+}}) g(t) \phi_{2}(a) \nonumber \\
&=g(t)\Big[ \int_{a}^{b} G(s, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)] \nabla s \nonumber \\
&\quad +A(f_{i} + h_{i_{+}}) \phi_{1}(b) + B(f_{i} + h_{i_{+}}) \phi_{2}(a) \Big] \nonumber \\
&\geq g(t)\|v_{i}\|, \quad i=1, 2.
\end{align}\]
This yields that \(F(P)\subseteq P\).
Let \(D\subset P\) be any bounded set. Then there exists a constant \(M>0\) such that \(\|u_{i}\|\leq M\), \(i=1, 2\) for any
\((u_{1}, u_{2}) \in D\). Furthermore for any \((u_{1}, u_{2}) \in D\) and \(t\in [a, b]\), we find
\[\begin{equation}
0\leq [u_{i}(t)-w_{i}(t)]^{*}\leq u_{i}(t)\leq \|u_{i}\| \leq
M < M+1 , \quad i=1, 2.
\end{equation}\]
Thus, by Remark 1 and Lemma 8, for any \(s \in [a, b]\), we have
\[\begin{align}
&f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*})+ h_{i_{+}}(s) \leq f_{i}(s, M+1 , M+1)+ h_{i_{+}}(s) \\
&\leq (M+1)^{2 \lambda_{1}} f_{i}(s, 1 , 1) + h_{i_{+}}(s)\leq ((M+1)^{2 \lambda_{1}}+1)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)], \quad i=1, 2.
\end{align}\]
Consequently,
\[\begin{align}
F_{i}(u_{1},u_{2})(t)&= \int_{a}^{b} G(t, s)[f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)]
\nabla s +A(f_{i} + h_{i_{+}}) \phi_{1}(t) + B(f_{i} + h_{i_{+}}) \phi_{2}(t) \\
&\leq \int_{a}^{b} G(s, s)((M+1)^{2 \lambda_{1}}+1)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)]\nabla s\\
&\quad +A\phi_{1}(b) \int_{a}^{b} G(s, s)((M+1)^{2 \lambda_{1}}+1)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)]\nabla s \\
&\quad +B\phi_{2}(a) \int_{a}^{b} G(s, s)((M+1)^{2 \lambda_{1}}+1)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)]\nabla s \\
&=((M+1)^{2 \lambda_{1}}+1)(1+A\phi_{1}(b)+B\phi_{2}(a))\int_{a}^{b} G(s, s)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)] \nabla s \\
&<\infty, \quad i=1, 2.
\end{align}\]
Therefore \(F(D)\) is uniformly bounded.
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 \(F:P \rightarrow P\) is continuous. Hence \(F:P
\rightarrow P\) is a completely continuous operator.
Theorem 2. Let \((H_{1})-(H_{7})\) hold. For each \(r\) satisfying \begin{equation*} r> \max \Big \{2C_{1}, 2C_{2}, ((r+1)^{2 \lambda_{1}}+1)(1+A\phi_{1}(b)+B\phi_{2}(a)) \int_{a}^{b}G(s, s)[f_{i}(s, 1 , 1)+ h_{i_{+}}(s)]\nabla s \Big \}, \end{equation*} where \(C_{i}(i=1, 2)\) are given in (6). The SSS (1) has at least one positive solution \((\tilde{u_{1}}, \tilde{u_{2}})\) such that \(\|\tilde{u_{i}}\| > r\), \(i=1, 2\).
Proof. Assume that there exist \(\lambda_{0}\geq 1\) and \((\tilde{u_{1}}, \tilde{u_{2}}) \in \partial P_{r}\) such that \(F(\tilde{u_{1}}, \tilde{u_{2}}) =\lambda_{0}(\tilde{u_{1}}, \tilde{u_{2}})\) where\\ \(P_{r}=\{(u_{1}, u_{2}) \in P: \|u_{1}\|< r, \|u_{2}\|< r \}\). Then \( \frac{1}{\lambda_{0}}(F_{1}(\tilde{u_{1}}, \tilde{u_{2}}), F_{2}(\tilde{u_{1}}, \tilde{u_{2}}))= (\tilde{u_{1}}, \tilde{u_{2}})\) and \(0< \frac{1}{\lambda_{0}}\leq 1\). Moreover for \(t\in [a, b]\), we obtain \begin{equation*} 0\leq [\tilde{u_{i}}(t)-w_{i}(t)]^{*}\leq \tilde{u_{i}}(t) \leq \|\tilde{u_{i}}\| =r < r+1 , \quad i=1, 2, \end{equation*} from which, using Remark 1 and Lemma 8, for \(t \in [a, b]\), we get \begin{equation*} f_{i}(s, [u_{1}(s)-w_{1}(s)]^{*} , [u_{2}(s)-w_{2}(s)]^{*}) \leq f_{i}(s, r+1 , r+1) \leq (r+1)^{2 \lambda_{1}} f_{i}(s, 1 , 1), \quad i=1, 2. \end{equation*} Now, using Lemma 1 and Lemma 6 and the properties of the operators \(A, B\), for \(t \in [a, b]\), we get \begin{align*} \tilde{u_{i}}(t)&=\frac{1}{\lambda_{0}} \Big \{\int_{a}^{b} G(t, s)[f_{i}(s, [\tilde{u_{1}}(s)-w_{1}(s)]^{*} , [\tilde{u_{2}}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)] \nabla s \\&\quad +A(f_{i} +h_{i_{+}}) \phi_{1}(t)+ B(f_{i} + h_{i_{+}}) \phi_{2}(t) \Big \} \\ &\leq \int_{a}^{b} G(s, s)[f_{i}(s, [\tilde{u_{1}}(s)-w_{1}(s)]^{*} , [\tilde{u_{2}}(s)-w_{2}(s)]^{*}) + h_{i_{+}}(s)] \nabla s +A(f_{i} + h_{i_{+}}) \phi_{1}(t)+ B(f_{i} + h_{i_{+}}) \phi_{2}(t)\\ &\leq \int_{a}^{b} G(s, s) ((r+1)^{2 \lambda_{1}}+1)[ f_{i}(s, 1 , 1)+ h_{i_{+}}(s)] \nabla s \\ &\quad +A\phi_{1}(b) \int_{a}^{b} G(s, s)((r+1)^{2 \lambda_{1}}+1)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)]\nabla s \\ &\quad +B\phi_{2}(a)\int_{a}^{b} G(s, s)((r+1)^{2 \lambda_{1}}+1)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)]\nabla s \\ &=((r+1)^{2 \lambda_{1}}+1)(1+A\phi_{1}(b)+B\phi_{2}(a)) \int_{a}^{b}G(s, s)[f_{i}(s, 1 , 1) + h_{i_{+}}(s)] \nabla s. \end{align*} Thus, we get \begin{equation*} r\leq ((r+1)^{2 \lambda_{1}}+1)(1+A\phi_{1}(b)+B\phi_{2}(a)) \int_{a}^{b}G(s, s)[f_{i}(s, 1 , 1)+ h_{i_{+}}(s)]\nabla s, \quad i=1, 2. \end{equation*} This is a contradiction. Then by Theorem 1, we have
Example 1. Let \(\mathrm{T}=\{2^{k}: k \in Z\}\cup \{0\}\). Consider the following SSS, \begin{equation*} \begin{cases} -u_{i}^{\triangle \nabla }(t) =f_{i}(t, u_{1}(t), u_{2}(t))+h_{i}(t), \quad t \in (0, 1), \quad i=1, 2, \\ u_{i}(0)- u_{i}^{\triangle}(0)= u_{i}(1)+ u_{i}^{\triangle}(1) = 0,\quad i=1, 2, \end{cases} \end{equation*} where \begin{align*} &f_{1}(t, u_{1}, u_{2})=t^{2}(1-t)u_{1}^{3/2}u_{2}^{2}+\sqrt{u_{1}},\quad h_{1}(t)=-t,\\ &f_{2}(t, u_{1}, u_{2})=\frac{1}{10^{3}t(1-t)}u_{1}^{3/2}+\frac{1}{10^{2}}\sqrt{u_{1}+u_{2}}, \quad h_{2}(t)=-t^{2}. \end{align*} Clearly \(f_{1}\) and \(f_{2}\) satisfy the condition \((H_{7})\) . We can easily calculate the followings; \begin{align*} &\int_{0}^{1}G(s, s) h_{1_{-}}(s) \nabla s = \int_{0}^{1}G(s, s) s \nabla s =\frac{1}{3}\int_{0}^{1} (1+s)(2-s)s \nabla s = \frac{16}{35}, \\ &\int_{0}^{1}G(s, s) h_{2_{-}}(s) \nabla s = \int_{0}^{1}G(s, s) s^{2} \nabla s =\frac{1}{3}\int_{0}^{1} (1+s)(2-s)s^{2} \nabla s = \frac{3776}{9765}, \\ &\int_{0}^{1} h_{1_{-}}(s) \nabla s = \int_{0}^{1} s \nabla s = \frac{2}{3}, \quad \int_{0}^{1} h_{2_{-}}(s) \nabla s = \int_{0}^{1} s^{2} \nabla s = \frac{4}{7}, \\ &\int_{0}^{1}G(s, s) (1+h_{i_{+}}(s)) \nabla s = \frac{1}{3}\int_{0}^{1} (1+s)(2-s)\nabla s = \frac{44}{63} \quad for \quad i=1,2, \\ &C_{1}= \frac{1}{3}\int_{0}^{1}h_{1_{-}}(s) \nabla s \phi_{1}(1)\phi_{2}(0) = \frac{8}{9}, \quad C_{2}= \frac{1}{3}\int_{0}^{1}h_{2_{-}}(s) \nabla s \phi_{1}(1)\phi_{2}(0) = \frac{16}{21}, \end{align*} And for \((t, u_{1}, u_{2})\in [0,1] \times [0, r] \times [0, r]\) \begin{align*} &K_{1}= \max \Big \{t^{2}(1-t)u_{1}^{3/2}u_{2}^{2}+\sqrt{u_{1}}+ 1 ) \Big\}= \frac{1}{12}r^{7/2}+\sqrt{r}+1, \\ &K_{2}= \max \Big \{\frac{1}{10^{3}t(1-t)}u_{1}^{3/2}+\frac{1}{10^{2}}\sqrt{u_{1}+u_{2}}+ 1 \Big \}= \frac{1}{250}r^{3/2}+\frac{1}{10^{2}}\sqrt{2r}+1. \end{align*} If we choose \(r= \displaystyle \frac{17}{9}\), we have \begin{equation*} r > \max \Big\{ \frac{16}{9}, \frac{32}{21}, K_{1}\frac{44}{63},K_{2}\frac{44}{63} \Big\}. \end{equation*} Then, by Theorem 2, the dynamic system has two positive solutions \((\tilde{u_{1}}, \tilde{u_{2}})\) and \( (\hat{u_{1}}, \hat{u_{2}})\) such that \begin{equation*} 0< \|\hat{u_{i}}\|< \frac{17}{9}< \|\tilde{u_{i}}\|, \quad i=1, 2. \end{equation*}
