## 2 An Auxiliary Lemma
: _Let \(X\) be a Banach space, \(D\subset X\) closed and \(F:D\to D\) a strict contraction, i.e., \(|Fx-Fy|\leq k|x-y|\) for some \(k\in(0,1)\) and all \(x,y\in D\). Then \(F\) has a fixed point in \(D\)._  
**Lemma 3** (Leray-Schauder alternative [30]). : _Let \(\Psi\) be a Banach space, and \(Y:\Psi\rightarrow\Psi\) be a completely continuous operator (i.e., a map restricted to any bounded set in \(\Psi\) is compact). Let \(\Theta(Y)=\{y\in\mathbb{V}:y=\epsilon Y(y)\) for some \(0<\epsilon<1\}\). Then either the set \(\Theta(Y)\) is unbounded or \(Y\) has at least one fixed point._  
**Lemma 4** ([PERSON] fixed point theorem [30]). : _Let \(C\) be a convex (not necessary closed) subset of a normed linear space \(E\). Then each compact map \(F:C\to C\) has at least one fixed point._  
**3. Main Results**  
Let \((\mathcal{P},\|\cdot\|)\) be a Banach space, where  
\[\mathcal{P}=\{u(t)|u(t)\in C([a,b],\quad\mathbb{R})\}\qquad\text{and}\qquad\|u \|=\sup\{|u(t)|,\;\;t\in[a,b]\}.\]  
Obviously, the product space \((\mathcal{P}\times\mathcal{P},\|(u,v)\|)\) is a Banach space with the norm given by  
\[\|(u,v)\|=\|u\|+\|v\|\]  
for \((u,v)\in\mathcal{P}\times\mathcal{P}\). Using Lemma 1, we transform the problem (1) and (2) into an equivalent fixed point problem as follows:  
\[(u,v)=\mathcal{T}(u,v), \tag{13}\]  
where \(\mathcal{T}:\mathcal{P}\times\mathcal{P}\rightarrow\mathcal{P}\times\mathcal{P}\) is defined as  
\[\mathcal{T}(u,v)(t):=(\mathcal{T}_{1}(u,v)(t),\mathcal{T}_{2}(u,v)(t)), \tag{14}\]  
\[\mathcal{T}_{1}(u,v)(t)=\int_{a}^{t}\Bigg{(}\frac{1}{p(s)}\int_{a}^{s}f(\tau,u (\tau),v(\tau))d\tau\Bigg{)}ds\]\[+\frac{1}{B}\Bigg{[}-\int_{a}^{b}\Bigg{(}\frac{1}{p(s)}\int_{a}^{s}f(\tau,u(\tau),v( \tau))d\tau\Bigg{)}ds\]  
\[+\sum_{j=1}^{m}\alpha_{j}\int_{a}^{\eta_{j}}\frac{1}{q(s)}\Bigg{(}\int_{a}^{s}g( \tau,u(\tau),v(\tau))d\tau\Bigg{)}ds \tag{15}\]  
\[-\int_{a}^{b}\Bigg{(}\frac{\sum_{j=1}^{m}\alpha_{j}}{q(s)}\int_{a}^{s}g(\tau,u( \tau),v(\tau))d\tau\Bigg{)}ds\]  
\[+\Bigg{(}\sum_{j=1}^{m}\alpha_{j}\Bigg{)}\Bigg{(}\sum_{k=1}^{n}\beta_{k}\int_{ a}^{\beta_{k}}\frac{1}{p(s)}\Bigg{(}\int_{a}^{s}f(\tau,u(\tau),v(\tau))d\tau \Bigg{)}ds\Bigg{)}\Bigg{]}\]  
and  
\[\mathcal{T}_{2}(u,v)(t) =\int_{a}^{t}\Bigg{(}\frac{1}{q(s)}\int_{a}^{s}g(\tau,u(\tau),v( \tau))d\tau\Bigg{)}ds\] \[\|(u,v)\|=\|u\|+\|v\|\]0 \[\|(u,v)\|=\|u\|+\|v\|\]1 \[\|(u,v)\|=\|u\|+\|v\|\]2 \[\|(u,v)\|=\|u\|+\|v\|\]3  
We need the following assumptions in the forthcoming analysis:  
1. (_Growth conditions_) There exist real constants \(\sigma_{i},\theta_{i}\geq 0\)\((i=1,2)\), and \(\sigma_{0}>0\), \(\theta_{0}>0\), such that \(\forall\;u,v\in\mathbb{R}\), we have \[\|(u,v)\|=\|u\|+\|v\|\]4
2. (_Sub-growth conditions_) There exist nonnegative functions \(\omega(t),\;\lambda(t)\in L(a,b)\) such that \[\|(u,v)\|=\|u\|+\|v\|\]5 \[\|(u,v)\|=\|u\|+\|v\|\]6
3. (_Lipschitz conditions_) For all \(t\in[a,b]\) and \(u_{i},v_{i}\in\mathbb{R}\)\((i=1,2)\), there exist \(\ell_{i}>0\)\((i=1,2)\) such that \[\|(u,v)\|=\|u\|+\|v\|\]7 and \[\|(u,v)\|=\|u\|+\|v\|\]8 For the sake of computational convenience, we set \[\|(u,v)\|=\|u\|+\|v\|\]9 (17)where  
\[\left\{ \begin{array}{l} L_{1}=\frac{1}{|B\bar{p}|}\left[\frac{(b-a)^{2}}{2}(|B|+1)+ \Bigg{(}\sum_{j=1}^{m}\alpha_{j}\Bigg{)}\Bigg{(}\sum_{k=1}^{n}\beta_{k}\frac{( \xi_{k}-a)^{2}}{2}\Bigg{)}\right],\\ L_{2}=\frac{\sum_{j=1}^{m}\alpha_{j}}{|B\bar{q}|}\Bigg{[}\frac{(\eta_{j}-a)^{2 }}{2}+\frac{(b-a)^{2}}{2}\Bigg{]},\;\;L_{3}=\frac{\sum_{k=1}^{m}\beta_{k}}{|B \bar{p}|}\Bigg{[}\frac{(b-a)^{2}}{2}+\frac{(\xi_{k}-a)^{2}}{2}\Bigg{]},\\ L_{4}=\frac{1}{|B\bar{q}|}\Bigg{[}\frac{(b-a)^{2}}{2}(|B|+1)+\Bigg{(}\sum_{k=1 }^{n}\beta_{k}\Bigg{)}\Bigg{(}\sum_{j=1}^{m}\alpha_{j}\frac{(\eta_{j}-a)^{2}}{ 2}\Bigg{)}\Bigg{]},\\ \bar{p}=\inf_{s\in[a,b]}|p(s)|,\;\;\bar{q}=\inf_{s\in[a,b]}|q(s)|,\\ \mathcal{O}=\min\{1-(\mathcal{O}_{1}\sigma_{1}+\mathcal{O}_{2}\theta_{1}),1- (\mathcal{O}_{1}\sigma_{2}+\mathcal{O}_{2}\theta_{2})\},\;\sigma_{i},\;\theta _{i}\;\;\text{are given in $(A_{1})$}.\end{array} \right. \tag{18}\]