## 2 An Auxiliary Lemma  
In this section, we prove a lemma for a linear variant of the problem (1) and (2), which plays a key role in the forthcoming analysis. **Lemma 1**. : _For \(f_{1},g_{1}\in C([a,b],\mathbb{R})\) and \(B\
eq 0\), the solution of the following linear system of differential equations_:__  
\[\left(p(t)u^{\prime}(t)\right)^{\prime}=f_{1}(t),\;\;\;\left(q(t)v^{\prime}(t )\right)^{\prime}=g_{1}(t),\;t\in[a,b], \tag{3}\]  
_subject to the boundary conditions (2), can be expressed in the following formulas:_\[u(t) =\int_{a}^{t}\Bigg{(}\frac{1}{p(s)}\int_{a}^{s}f_{1}(\tau)d\tau\Bigg{)}ds\] \[\quad+\frac{1}{B}\Bigg{[}-\int_{a}^{b}\Bigg{(}\frac{1}{p(s)}\int_{a }^{s}f_{1}(\tau)d\tau\Bigg{)}ds+\sum_{j=1}^{m}\alpha_{j}\int_{a}^{\eta_{j}} \frac{1}{q(s)}\Bigg{(}\int_{a}^{s}g_{1}(\tau)d\tau\Bigg{)}ds \tag{4}\] \[\quad-\int_{a}^{b}\Bigg{(}\frac{\sum_{j=1}^{m}\alpha_{j}}{q(s)} \int_{a}^{s}g_{1}(\tau)d\tau\Bigg{)}ds+\Bigg{(}\sum_{j=1}^{m}\alpha_{j}\Bigg{)} \Bigg{(}\sum_{k=1}^{n}\beta_{k}\int_{a}^{5_{k}}\frac{1}{p(s)}\Bigg{(}\int_{a} ^{s}f_{1}(\tau)d\tau\Bigg{)}ds\Bigg{)}\Bigg{]}\]  
_and_  
\[v(t) =\int_{a}^{t}\Bigg{(}\frac{1}{q(s)}\int_{a}^{s}g_{1}(\tau)d\tau \Bigg{)}ds \tag{5}\] \[\quad+\frac{1}{B}\Bigg{[}-\int_{a}^{b}\Bigg{(}\frac{1}{q(s)}\int_ {a}^{s}g_{1}(\tau)d\tau\Bigg{)}ds+\sum_{k=1}^{n}\beta_{k}\int_{a}^{5_{k}}\frac {1}{p(s)}\Bigg{(}\int_{a}^{s}f_{1}(\tau)d\tau\Bigg{)}ds\] \[\quad-\int_{a}^{b}\Bigg{(}\frac{\sum_{k=1}^{m}\beta_{k}}{p(s)} \int_{a}^{s}f_{1}(\tau)d\tau\Bigg{)}ds+\Bigg{(}\sum_{k=1}^{n}\beta_{k}\Bigg{)} \Bigg{(}\sum_{j=1}^{m}\alpha_{j}\int_{a}^{\eta_{j}}\frac{1}{q(s)}\Bigg{(}\int _{a}^{s}g_{1}(\tau)d\tau\Bigg{)}ds\Bigg{)}\Bigg{]},\] _where_ \[B=\Bigg{[}1-\Bigg{(}\sum_{j=1}^{m}\alpha_{j}\Bigg{)}\Bigg{(}\sum_{k=1}^{n} \beta_{k}\Bigg{)}\Bigg{]}. \tag{6}\]  
**Proof. ** Integrating the linear differential Equation (3) twice from \(a\) to \(t\), and using the conditions \(u^{\prime}(a)=0\), \(v^{\prime}(a)=0\), we get  
\[u(t) =u(a)+\int_{a}^{t}\Bigg{(}\frac{1}{p(s)}\int_{a}^{s}f_{1}(\tau)d \tau\Bigg{)}ds \tag{7}\]  
and  
\[v(t) =v(a)+\int_{a}^{t}\Bigg{(}\frac{1}{q(s)}\int_{a}^{s}g_{1}(\tau)d \tau\Bigg{)}ds. \tag{8}\]  
Using the coupled boundary conditions given by (2) in (7) and (8), we obtain a system of equations:  
\[u(t) =\int_{a}^{t}\Bigg{(}\frac{1}{p(s)}\int_{a}^{s}f_{1}(\tau)d\tau\Bigg{)}ds\]0  
and  
\[u(t) =\int_{a}^{t}\Bigg{(}\frac{1}{p(s)}\int_{a}^{s}f_{1}(\tau)d\tau\Bigg{)}ds\]1  
Solving (9) and (10) for \(u(a)\) and \(v(a)\), together with the notation (6), we find that \[ \begin{array}{rl}u(a)&=\frac{1}{B}\Bigg{[}-\int_{a}^{b}\Bigg{(}\frac{1}{p(s)} \int_{a}^{s}f_{1}(\tau)d\tau\Bigg{)}ds+\sum_{j=1}^{m}\alpha_{j}\int_{a}^{\eta_{ j}}\frac{1}{q(s)}\Bigg{(}\int_{a}^{s}g_{1}(\tau)d\tau\Bigg{)}ds\\ &-\int_{a}^{b}\Bigg{(}\frac{\sum_{j=1}^{m}\alpha_{j}}{q(s)}\int_{a}^{s}g_{1}( \tau)d\tau\Bigg{)}ds+\Bigg{(}\sum_{j=1}^{m}\alpha_{j}\Bigg{)}\Bigg{(}\sum_{k=1 }^{n}\beta_{k}\int_{a}^{\xi_{k}}\frac{1}{p(s)}\Bigg{(}\int_{a}^{s}f_{1}(\tau) d\tau\Bigg{)}ds\Bigg{)}\Bigg{]}\\ \end{array}  \tag{11}\]  
\[ \begin{array}{rl}v(a)&=\frac{1}{B}\Bigg{[}-\int_{a}^{b}\Bigg{(}\frac{1}{q(s )}\int_{a}^{s}g_{1}(\tau)d\tau\Bigg{)}ds+\sum_{k=1}^{n}\beta_{k}\int_{a}^{\xi _{k}}\frac{1}{p(s)}\Bigg{(}\int_{a}^{s}f_{1}(\tau)d\tau\Bigg{)}ds\\ &-\int_{a}^{b}\Bigg{(}\frac{\sum_{k=1}^{m}\beta_{k}}{p(s)}\int_{a}^{s}f_{1}( \tau)d\tau\Bigg{)}ds+\Bigg{(}\sum_{k=1}^{m}\beta_{k}\Bigg{)}\Bigg{(}\sum_{j=1} ^{m}\alpha_{j}\int_{a}^{\eta_{j}}\frac{1}{q(s)}\Bigg{(}\int_{a}^{s}g_{1}(\tau) d\tau\Bigg{)}ds\Bigg{)}\Bigg{]}.\end{array}  \tag{12}\]  
Inserting the values of \(u(a)\) and \(v(a)\) in (7) and (8) respectively, we obtain the solutions (4) and (5). By direct computation, one can obtain the converse of the lemma. This completes the proof. Now we state the fixed point theorems used in establishing the existence theory for the system (1). **Lemma 2** (Banach fixed point theorem [29]).