For the sake of convenience, we set
$$\begin{aligned}& M_{1}= \biggl(\frac{1}{\varGamma(\theta^{*}_{1}+1)} \biggr)^{q-1} \biggl[\frac{1}{\varGamma (\theta_{1}+1)} +\frac{\gamma\eta^{\theta_{1}+\alpha-1}}{\Delta_{1}\varGamma(\theta _{1}+\alpha)} \biggr], \end{aligned}$$
(3.1)
$$\begin{aligned}& M_{2}= \biggl(\frac{1}{\varGamma(\theta^{*}_{1}+1)} \biggr)^{q-1} \biggl[\frac{1}{\varGamma (\theta_{2}+1)} +\frac{\delta\xi^{\theta_{2}+\beta-1}}{\Delta_{2}\varGamma(\theta_{2}+\beta )} \biggr], \end{aligned}$$
(3.2)
$$\begin{aligned}& M_{0}=\min\bigl\{ 1-(M_{1}k_{1}+M_{2} \lambda_{1}),1-(M_{1}k_{2}+M_{2} \lambda _{2})\bigr\} , \end{aligned}$$
(3.3)
$$\begin{aligned}& \blacktriangle_{f}=\frac{(q-1)J_{1}^{q-2}}{\varGamma(\theta^{*}_{1}+1)} \biggl[ \frac{1}{\varGamma(\theta_{1}+1)} +\frac{\gamma\eta^{\theta_{1}+\alpha-1}}{\Delta_{1}\varGamma(\theta _{1}+\alpha)} \biggr], \end{aligned}$$
(3.4)
$$\begin{aligned}& \blacktriangle_{g}=\frac{(q-1)J_{2}^{q-2}}{\varGamma(\theta^{*}_{1}+1)} \biggl[ \frac{1}{\varGamma(\theta_{2}+1)} +\frac{\delta\xi^{\theta_{2}+\beta-1}}{\Delta_{2}\varGamma(\theta_{2}+\beta )} \biggr]. \end{aligned}$$
(3.5)
We define operators \(T_{1}, T_{2}:\mathcal{X}\times\mathcal{Y}\rightarrow \mathcal{X} \times\mathcal{Y}\) as
$$ \begin{aligned} &T_{1}(\omega_{1},\omega_{2}) (r)=\frac{1}{\varGamma( \theta_{1})} \int _{0}^{r}(r-s)^{\theta_{1}-1} \varphi_{q} \bigl(I^{\theta^{*}_{1}} f \bigl(s,\omega _{1}(s), \omega_{2}(s) \bigr) \bigr) )\,ds \\ &\hphantom{T_{1}(\omega_{1},\omega_{2}) (r)=} {}+\frac{\gamma}{\Delta_{1}\varGamma(\theta_{1}+\alpha-1)} \int_{0}^{\eta }(\eta-s)^{\theta_{1}+\alpha-2} \varphi_{q} \bigl(I^{\theta^{*}_{1}}f \bigl(s,\omega_{1}(s), \omega_{2}(s) \bigr) \bigr)\,ds, \\ &T_{2}(\omega_{1},\omega_{2}) (r)= \frac{1}{\varGamma(\theta_{2})} \int _{0}^{r}(r-s)^{\theta_{2}-1} \varphi_{q} \bigl(I^{\theta^{*}_{2}} g \bigl(s,\omega _{1}(s), \omega_{2}(s) \bigr) \bigr) )\,ds \\ &\hphantom{T_{2}(\omega_{1},\omega_{2}) (r)=}{}+\frac{\delta}{\Delta_{2}\varGamma(\theta_{2}+\beta-1)} \int_{0}^{\xi }(\xi-s)^{\theta_{2}+\beta-2} \varphi_{q} \bigl(I^{\theta^{*}_{2}}g \bigl(s,\omega_{1}(s), \omega_{2}(s) \bigr) \bigr)\,ds. \end{aligned} $$
(3.6)
Lemma 3.1
([9, 26, 30])
Let
\(\mathcal{F}:\mathcal{A} \rightarrow\mathcal{A}\)
be a completely continuous operator (i.e., a map which, restricted to any bounded set in
\(\mathcal{A,}\)
is compact). Let
$$ \varepsilon(\mathcal{F})=\bigl\{ x\in\mathcal{A}:x=\lambda\mathcal{F}(x) , \textit{ for some } 0< \lambda< 1\bigr\} . $$
(3.7)
Then either the set
\(\varepsilon(F)\)
is unbounded, or
\(\mathcal{F}\)
has at least one fixed point.
Theorem 3.2
Suppose that
\(\gamma\neq\frac{\eta^{(\theta_{1}+\alpha-1)}}{\varGamma (\theta_{1}+\alpha)}\)
and
\(\delta\neq\frac{\xi^{(\theta_{2}+\beta -1)}}{\varGamma(\theta_{2}+\beta)}\). Assume that there exist real constants
\(k_{i},\lambda_{i}\geq0\) (\(i=1,2\)) and
\(k_{0}>0\), \(\lambda_{0}>0\)
such that for all
\(x_{i} \in\mathcal{R}\) (\(i=1,2\)), we have
$$ \begin{aligned} &\bigl\vert f(r,x_{1},x_{2}) \bigr\vert \leq\varphi_{p}\bigl(k_{0}+k_{1} \vert x_{1} \vert +k_{2} \vert x_{2} \vert \bigr), \\ &\bigl\vert g(r,x_{1},x_{2}) \bigr\vert \leq \varphi_{p}\bigl(\lambda_{0}+\lambda_{1} \vert x_{1} \vert +\lambda_{2} \vert x_{2} \vert \bigr). \end{aligned} $$
(3.8)
In addition, it is assumed that
$$\begin{aligned} M_{1}k_{1}+M_{2}\lambda_{1}< 1 \quad \textit{and}\quad M_{1}k_{2}+M_{2}\lambda _{2}< 1, \end{aligned}$$
where
\(M_{1}\)
and
\(M_{2}\)
are given by (3.1) and (3.2) respectively. Then the boundary value problem (1.1) has at least one solution.
Proof
First, we show that the operator \(T: \mathcal{X}\times\mathcal {Y}\rightarrow\mathcal{X}\times\mathcal{Y}\) is completely continuous. By the continuity of functions f and g, the operator T is continuous. Let \(\varOmega\subset\mathcal{X}\times\mathcal{Y}\) be bounded. Then there exist positive constants \(L_{1}\) and \(L_{2}\) such that
$$ \bigl\vert f\bigl(r,\omega_{1}(r), \omega_{2}(r)\bigr) \bigr\vert \leq\varphi_{p}(L_{1}), \qquad \bigl\vert g\bigl(r,\omega_{1}(r),\omega_{2}(r) \bigr) \bigr\vert \leq\varphi_{p}(L_{2}),\quad\forall ( \omega_{1},\omega_{2})\in\varOmega. $$
(3.9)
Then for any \((\omega_{1},,\omega_{2})\in\varOmega\), we have
$$ \begin{gathered}[b] \bigl\vert T_{1}(\omega_{1}, \omega_{2}) (r) \bigr\vert \\ \quad = \biggl\vert \frac{1}{\varGamma(\theta _{1})} \int_{0}^{r}(r-s)^{\theta_{1}-1} \varphi_{q} \bigl(I^{\theta^{*}_{1}}f \bigl(s,\omega_{1}(s), \omega_{2}(s) \bigr) \bigr)\,ds \\ \qquad {}+\frac{\gamma}{\Delta_{1}\varGamma(\theta_{1}+\alpha-1)} \int_{0}^{\eta }(\eta-s)^{\theta_{1}+\alpha-2} \varphi_{q} \bigl(I^{\theta^{*}_{1}}f \bigl(s,\omega_{1}(s), \omega_{2}(s) \bigr) \bigr)\,ds \biggr\vert \\ \quad \leq \frac{1}{\varGamma(\theta_{1})} \int_{0}^{r}(r-s)^{\theta_{1}-1}\varphi _{q} \biggl( \frac{1}{\varGamma( \theta^{*}_{1})} \int_{0}^{r}(r-s)^{\theta ^{*}_{1}-1} \bigl\vert f \bigl(s,\omega_{1}(s),\omega_{2}(s) \bigr) \bigr\vert \,ds \biggr) \\ \qquad {}+\frac{\gamma}{\Delta_{1}\varGamma(\theta_{1}+\alpha-1)}\\ \qquad {}\times \int_{0}^{\eta }(\eta-s)^{\theta_{1}+\alpha-2} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{1})} \int_{0}^{\eta}(\eta-s)^{\theta^{*}_{1}-1} \bigl\vert f \bigl(s,\omega_{1}(s),\omega _{2}(s) \bigr) \bigr\vert \biggr)\,ds \\ \quad \leq \frac{1}{\varGamma(\theta_{1})} \int_{0}^{r}(r-s)^{\theta_{1}-1}\varphi _{q} \biggl( \frac{1}{\varGamma( \theta^{*}_{1})} \int_{0}^{r}(r-s)^{\theta ^{*}_{1}-1} \varphi_{p}(L_{1})\,ds \biggr) \\ \qquad {}+\frac{\gamma}{\Delta_{1}\varGamma(\theta_{1}+\alpha-1)} \int_{0}^{\eta }(\eta-s)^{\theta_{1}+\alpha-2} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{1})} \int_{0}^{\eta}(\eta-s)^{\theta^{*}_{1}-1} \varphi_{p}(L_{1}) \biggr)\,ds \\ \quad \leq L_{1} \biggl(\frac{1}{\varGamma(\theta^{*}_{1}+1)} \biggr)^{q-1} \biggl[ \frac {1}{\varGamma(\theta_{1}+1)} +\frac{\gamma\eta^{\theta_{1}+\alpha-1}}{\Delta_{1}\varGamma(\theta _{1}+\alpha)} \biggr]=L_{1}M_{1}. \end{gathered} $$
(3.10)
And also,
$$ \begin{gathered}[b] \bigl\vert T_{2}(\omega_{1}, \omega_{2}) (r) \bigr\vert \\ \quad = \biggl\vert \frac{1}{\varGamma(\theta _{2})} \int_{0}^{r}(r-s)^{\theta_{2}-1} \varphi_{q} \bigl(I^{\theta^{*}_{2}}g \bigl(s,\omega_{1}(s), \omega_{2}(s) \bigr) \bigr)\,ds \\ \qquad {}+\frac{\delta}{\Delta_{2}\varGamma(\theta_{2}+\beta-1)} \int_{0}^{\xi }(\xi-s)^{\theta_{2}+\beta-2} \varphi_{q} \bigl(I^{\theta^{*}_{2}}g \bigl(s,\omega_{1}(s), \omega_{2}(s) \bigr) \bigr)\,ds \biggr\vert \\ \quad \leq \frac{1}{\varGamma(\theta_{2})} \int_{0}^{r}(r-s)^{\theta_{2}-1}\varphi _{q} \biggl( \frac{1}{\varGamma( \theta_{2}{2})} \int_{0}^{r}(r-s)^{\theta ^{*}_{2}-1} \bigl\vert g \bigl(s,\omega_{1}(s),\omega_{2}(s) \bigr) \bigr\vert \,ds \biggr) \\ \qquad {}+\frac{\delta}{\Delta_{2}\varGamma(\theta_{2}+\beta-1)}\\ \qquad {}\times \int_{0}^{\xi }(\xi-s)^{\theta_{2}+\beta-2} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int_{0}^{\xi}(\xi-s)^{\theta^{*}_{2}-1} \bigl\vert g \bigl(s,\omega_{1}(s),\omega _{2}(s) \bigr) \bigr\vert \biggr)\,ds \\ \quad \leq \frac{1}{\varGamma(\theta_{2})} \int_{0}^{r}(r-s)^{\theta_{2}-1}\varphi _{q} \biggl( \frac{1}{\varGamma( \theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta ^{*}_{2}-1} \varphi_{p}(L_{2})\,ds \biggr) \\ \qquad {}+\frac{\delta}{\Delta_{2}\varGamma(\theta_{2}+\beta-1)} \int_{0}^{\xi }(\xi-s)^{\theta_{2}+\beta-2} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int_{0}^{\xi}(\xi-s)^{\theta^{*}_{2}-1} \varphi_{p}(L_{2}) \biggr)\,ds \\ \quad \leq L_{2} \biggl(\frac{1}{\varGamma(\theta^{*}_{2}+1)} \biggr)^{q-1} \biggl[ \frac {1}{\varGamma(\theta_{2}+1)} +\frac{\delta\xi^{\theta_{2}+\beta-1}}{\Delta_{2}\varGamma(\theta_{2}+\beta )} \biggr]=L_{2}M_{2}. \end{gathered} $$
(3.11)
Thus, it follows from the above inequalities that the operator T is uniformly bounded. Next we show that T is equicontinuous. Let \(0\leq r_{1}\leq r_{2}\leq1\). Then we have
$$\begin{aligned}& \begin{gathered}[b] \bigl\vert T_{1} \bigl(\omega_{1}(r_{2}), \omega_{2}(r_{2}) \bigr)-T_{1} \bigl(\omega _{1}(r_{1}),\omega_{2}(r_{1}) \bigr) \bigr\vert \\ \quad= \biggl\vert \frac{1}{\varGamma( \theta_{1})} \int _{0}^{r_{2}}(r_{2}-s)^{\theta_{1}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{1})} \int_{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ \qquad{}-\frac{1}{\varGamma( \theta_{1})} \int_{0}^{r_{1}}(r_{1}-s)^{\theta _{1}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{1})} \int_{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \biggr\vert \\ \quad \leq \biggl\vert \frac{1}{\varGamma( \theta_{1})} \int _{0}^{r_{2}}\bigl[(r_{2}-s)^{\theta^{*}_{1}-1}-(r_{1}-s)^{\theta^{*}_{1}-1} \bigr]\\ \qquad {}\times \varphi_{q} \biggl(\frac{1}{\varGamma (\theta^{*}_{1})} \int_{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ \qquad{} -\frac{1}{\varGamma( \theta_{1})} \int _{r_{1}}^{r_{2}}(r_{2}-s)^{\theta_{1}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{1})} \int_{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \biggr\vert \\ \quad \leq\frac{L_{1}}{\varGamma(\theta_{1}+1)(\varGamma\theta^{*}_{1}+1)^{q-1}} \bigl(r_{1}^{\theta_{1}}-r_{2}^{\theta_{1}} \bigr), \end{gathered} \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{gathered}[b] \bigl\vert T_{2} \bigl(\omega_{1}(r_{2}), \omega_{2}(r_{2}) \bigr)-T_{2} \bigl(\omega _{1}(r_{1}),\omega_{2}(r_{1}) \bigr) \bigr\vert \\ \quad= \biggl\vert \frac{1}{\varGamma( \theta_{2})} \int _{0}^{r_{2}}(r_{2}-s)^{\theta_{2}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta^{*}_{2}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ \qquad{} -\frac{1}{\varGamma( \theta_{2})} \int_{0}^{r_{1}}(r_{1}-s)^{\theta _{2}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta^{*}_{2}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \biggr\vert \\ \quad \leq \biggl\vert \frac{1}{\varGamma( \theta_{2})} \int _{0}^{r_{2}}\bigl[(r_{2}-s)^{\theta^{*}_{2}-1}-(r_{1}-s)^{\theta^{*}_{2}-1} \bigr]\\ \qquad {}\times \varphi_{q} \biggl(\frac{1}{\varGamma (\theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta^{*}_{2}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ \qquad{} -\frac{1}{\varGamma( \theta_{2})} \int _{r_{1}}^{r_{2}}(r_{2}-s)^{\theta_{2}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta^{*}_{2}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \biggr\vert \\ \quad \leq\frac{L_{1}}{\varGamma(\theta_{2}+1)(\varGamma\theta^{*}_{2}+1)^{q-1}} \bigl(r_{1}^{\theta_{2}}-r_{2}^{\theta_{2}} \bigr). \end{gathered} \end{aligned}$$
(3.13)
Therefore, the operator \(T(\omega_{1},\omega_{2})\) is equicontinuous, and thus the operator \(T(\omega_{1},\omega_{2})\) is completely continuous. Finally, it will be verified that the set \(\varepsilon=\{(\omega _{1},\omega_{2})\in\mathcal{X}\times\mathcal{Y}|(\omega_{1},\omega _{2})=\lambda T(\omega_{1},\omega_{2}), 0\leq\lambda\leq1\}\) is bounded. Let \((\omega_{1},\omega_{2})\in\varepsilon\), then \((\omega _{1},\omega_{2})=\lambda T(\omega_{1},\omega_{2})\). For any \(r\in [0,1]\), we have
$$ \omega_{1}(r)=\lambda r_{1}(\omega_{1}, \omega_{2}),\qquad\omega _{2}(r)=\lambda T_{2}( \omega_{1},\omega_{2}). $$
Then
$$ \begin{aligned}[b] \bigl\vert \omega_{1}(r) \bigr\vert &= \biggl( \frac{1}{\varGamma(\theta^{*}_{1}+1)} \biggr)^{q-1} \biggl[\frac{1}{\varGamma(\theta_{1}+1)} + \frac{\gamma\eta^{\theta_{1}+\alpha-1}}{\Delta_{1}\varGamma(\theta _{1}+\alpha)} \biggr] \\ &\quad {}\times\bigl(k_{0}+k_{1} \bigl\vert \omega_{1}(r) \bigr\vert +k_{2} \bigl\vert \omega_{2}(r) \bigr\vert \bigr) \end{aligned} $$
(3.14)
and
$$\begin{aligned} \bigl\vert \omega_{2}(r) \bigr\vert =& \biggl( \frac{1}{\varGamma(\theta^{*}_{2}+1)} \biggr)^{q-1} \biggl[\frac{1}{\varGamma(\theta_{2}+1)} + \frac{\delta\xi^{\theta_{2}+\beta-1}}{\Delta_{2}\varGamma(\theta_{2}+\beta )} \biggr] \\ &{}\times\bigl(\lambda_{0}+\lambda_{1} \bigl\vert \omega_{1}(r) \bigr\vert +\lambda_{2} \bigl\vert \omega _{2}(r) \bigr\vert \bigr). \end{aligned}$$
(3.15)
Hence we have
$$ \begin{aligned} &\Vert \omega_{1} \Vert =M_{1} \bigl(k_{0}+k_{1} \bigl\Vert \omega_{1}(r) \bigr\Vert +k_{2} \bigl\Vert \omega _{2}(r) \bigr\Vert \bigr)\quad \text{and} \\ & \Vert \omega_{2} \Vert =M_{2} \bigl(\lambda_{0}+\lambda_{1} \bigl\Vert \omega_{1}(r) \bigr\Vert +\lambda_{2} \bigl\Vert \omega_{2}(r) \bigr\Vert \bigr). \end{aligned} $$
(3.16)
From (3.16) we have
$$ \Vert \omega_{1} \Vert + \Vert \omega_{2} \Vert = (M_{1}k_{0}+M_{2} \lambda_{0} )+ (M_{1}k_{1}+M_{2} \lambda_{1} ) \Vert \omega_{1} \Vert + (M_{1}k_{2}+M_{2}\lambda_{2} ) \Vert \omega_{2} \Vert . $$
(3.17)
Consequently,
$$ \bigl\Vert (\omega_{1}, \omega_{2}) \bigr\Vert \leq\frac{M_{1}k_{0}+M_{2}\lambda_{0}}{M_{0}}, $$
(3.18)
for any \(r\in[0,1]\), where \(M_{0}\) is defined in (3.3), which proves that ε is bounded. Thus, by Lemma 3.2, operator T has at leat one fixed point. Hence, the boundary value problem (1.1) has at least one solution. □
Theorem 3.3
Assume that
\(f,g:[0,1]\times\mathcal{R}^{2}\rightarrow\mathcal{R}\)
are continuous functions and there exist constants
\(m_{i},n_{i}\), \(i=1,2\)
such that for all
\(r\in[0,1]\)
and
\(\omega_{1}, \omega_{2}, \hbar_{1}, \hbar_{2}\in\mathcal{R}\),
$$ \begin{aligned}[b] \bigl\vert f(r, \omega_{1},\hbar_{1})-f(r,\omega_{2}, \hbar_{2}) \bigr\vert \leq m_{1} \vert \omega _{1}-\omega_{2} \vert +n_{1} \vert \hbar_{1}-\hbar_{2} \vert , \\ \bigl\vert g(r,\omega_{1},\hbar_{1})-g(r, \omega_{2},\hbar_{2}) \bigr\vert \leq m_{2} \vert \omega _{1}-\omega_{2} \vert +n_{2} \vert \hbar_{1}-\hbar_{2} \vert . \end{aligned} $$
(3.19)
In addition, assume that
$$ \blacktriangle_{f}(m_{1}+m_{2})+\blacktriangle _{g}(n_{1}+n_{2})< 1, $$
where
\(\blacktriangle_{f}\)
and
\(\blacktriangle_{g}\)
are given by (3.4) and (3.5), respectively. Then the boundary value problem (1.1) has a unique solution.
Proof
Consider a bounded set \(\|T(\omega_{1},\omega_{2})(r)\|\leq r\). For \((\hbar_{1},\hbar_{2}),(\omega_{1},\omega_{2})\in\mathcal{X}\times \mathcal{Y}\), and for any \(r\in[0,1]\), we get
$$\begin{aligned} & \bigl\vert r_{1}( \hbar_{1},\hbar_{2}) (r)-r_{1}( \omega_{1},\omega_{2}) (r) \bigr\vert \\ &\quad = \biggl\vert \frac{1}{\varGamma( \theta_{1})} \int_{0}^{r}(r-s)^{\theta _{1}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{1})} \int _{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \hbar_{1}(s),\hbar_{2}(s) \bigr) \biggr)\,ds \\ &\qquad {} +\frac{\gamma}{\Delta_{1}\varGamma(\theta_{1}+\alpha-1)} \int_{0}^{\eta }(\eta-s)^{\theta_{1}+\alpha-2} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta ^{*}_{1})} \int_{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \hbar_{1}(s),\hbar_{2}(s) \bigr) \biggr)\,ds \\ &\qquad{} -\frac{1}{\varGamma( \theta_{1})} \int_{0}^{r}(r-s)^{\theta _{1}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{1})} \int _{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ &\qquad {} -\frac{\gamma}{\Delta_{1}\varGamma(\theta_{1}+\alpha-1)} \int_{0}^{\eta }(\eta-s)^{\theta_{1}+\alpha-2} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta ^{*}_{1})} \int_{0}^{r}(r-s)^{\theta^{*}_{1}-1} f \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \biggr\vert \\ &\quad\leq\frac{(q-1)J_{1}^{q-2}}{\varGamma( \theta_{1})} \int _{0}^{r} \bigl\vert (r-s)^{\theta_{1}-1} \bigr\vert \frac{1}{\varGamma(\theta^{*}_{1})} \int _{0}^{r} \bigl\vert (r-s)^{\theta^{*}_{1}-1} \bigr\vert \bigl\vert f \bigl(s,\hbar_{1}(s),\hbar_{2}(s) \bigr) \\ &\qquad {} -f \bigl(s,\omega_{1}(s),\omega_{2}(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{(q-1)J_{1}^{q-2}\gamma}{\Delta_{1}\varGamma(\theta _{1}+\alpha-1)} \int_{0}^{\eta} \bigl\vert (\eta-s)^{\theta_{1}+\alpha-2} \bigr\vert \frac {1}{\varGamma(\theta^{*}_{1})} \int_{0}^{r} \bigl\vert (r-s)^{\theta^{*}_{1}-1} \bigr\vert \bigl\vert f \bigl(s,\hbar_{1}(s),\hbar_{2}(s) \bigr) \\ &\qquad {} -f \bigl(s,\omega_{1}(s),\omega_{2}(s) \bigr) \bigr\vert \,ds \\ &\quad\leq\frac{(q-1)J_{1}^{q-2}}{\varGamma(\theta^{*}_{1}+1)} \biggl[\frac {1}{\varGamma(\theta_{1}+1)} +\frac{\gamma\eta^{\theta_{1}+\alpha-1}}{\Delta_{1}\varGamma(\theta _{1}+\alpha)} \biggr] \bigl(m_{1} \vert \hbar_{1}-\omega_{1} \vert +m_{2} \vert \hbar _{2}-\omega_{2} \vert \bigr) \\ &\quad\leq\blacktriangle_{f}(m_{1}+m_{2}) \bigl( \vert \hbar_{1}-\omega _{1} \vert + \vert \hbar_{2}-\omega_{2} \vert \bigr). \end{aligned}$$
(3.20)
Similarly, we have
$$\begin{aligned} & \bigl\vert T_{2} ( \hbar_{1},\hbar_{2} ) (r)-T_{2} ( \omega_{1},\omega _{2} ) (r) \bigr\vert \\ &\quad= \biggl\vert \frac{1}{\varGamma(\theta_{2})} \int_{0}^{r}(r-s)^{\theta _{2}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int _{0}^{r}(r-s)^{\theta^{*}_{2}-1} g \bigl(s, \hbar_{1}(s),\hbar_{2}(s) \bigr) \biggr)\,ds \\ &\qquad{} +\frac{\delta}{\Delta_{2}\varGamma(\theta_{2}+\beta-1)} \int _{0}^{\xi}(\xi-s)^{\theta_{2}+\beta-2} \varphi_{q} \biggl(\frac{1}{\varGamma (\theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta^{*}_{2}-1} g \bigl(s, \hbar_{1}(s),\hbar_{2}(s) \bigr) \biggr)\,ds \\ &\qquad{}-\frac{1}{\varGamma( \theta_{2})} \int_{0}^{r}(r-s)^{\theta _{2}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int _{0}^{r}(r-s)^{\theta^{*}_{2}-1} g \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ &\qquad{}-\frac{\delta}{\Delta_{2}\varGamma(\theta_{2}+\beta-1)} \int _{0}^{\xi}(\xi-s)^{\theta_{2}+\beta-2} \varphi_{q} \biggl(\frac{1}{\varGamma (\theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta^{*}_{2}-1} g \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ &\qquad{}-\frac{1}{\varGamma( \theta_{2})} \int_{0}^{r}(r-s)^{\theta _{2}-1} \varphi_{q} \biggl(\frac{1}{\varGamma(\theta^{*}_{2})} \int _{0}^{r}(r-s)^{\theta^{*}_{2}-1} g \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds \\ &\qquad{}-\frac{\delta}{\Delta_{2}\varGamma(\theta_{2}+\beta-1)} \int _{0}^{\xi}(\xi-s)^{\theta_{2}+\beta-2} \varphi_{q} \biggl(\frac{1}{\varGamma (\theta^{*}_{2})} \int_{0}^{r}(r-s)^{\theta^{*}_{2}-1} g \bigl(s, \omega_{1}(s),\omega_{2}(s) \bigr) \biggr)\,ds\biggr\vert \\ &\quad\leq\frac{(q-1)J_{2}^{q-2}}{\varGamma( \theta_{2})} \int _{0}^{r} \bigl\vert (r-s)^{\theta_{2}-1} \bigr\vert \frac{1}{\varGamma(\theta^{*}_{2})} \int _{0}^{r} \bigl\vert (r-s)^{\theta^{*}_{2}-1} \bigr\vert \bigl\vert g \bigl(s,\hbar_{1}(s),\hbar_{2}(s) \bigr) \\ &\qquad{}-g \bigl(s,\omega_{1}(s),\omega_{2}(s) \bigr) \bigr\vert \,ds \\ &\qquad{}+\frac{(q-1)J_{2}^{q-2}\delta}{\Delta_{2}\varGamma(\theta _{2}+\beta-1)} \int_{0}^{\xi} \bigl\vert (\xi-s)^{\theta_{2}+\beta-2} \bigr\vert \frac {1}{\varGamma(\theta^{*}_{2})} \int_{0}^{r} \bigl\vert (r-s)^{\theta^{*}_{2}-1} \bigr\vert \bigl\vert g \bigl(s,\hbar_{1}(s),\hbar_{2}(s) \bigr) \\ &\qquad{}-g \bigl(s,\omega_{1}(s),\omega_{2}(s) \bigr) \bigr\vert \,ds \\ &\quad\leq\frac{(q-1)J_{2}^{(q-2)}}{\varGamma(\theta^{*}_{2}+1)} \biggl[\frac {1}{\varGamma(\theta_{2}+1)} +\frac{\delta\xi^{\theta_{2}+\beta-1}}{\Delta_{2}\varGamma(\theta_{2}+\beta )} \biggr] \bigl(n_{1} \vert \hbar_{1}-\omega_{1} \vert +n_{2} \vert \hbar_{2}-\omega_{2} \vert \bigr) \\ &\quad\leq\blacktriangle_{g}(n_{1}+n_{2}) \bigl( \Vert \hbar_{1}-\omega_{1} \Vert + \Vert \hbar_{2}-\omega_{2} \Vert \bigr). \end{aligned}$$
(3.21)
Therefore, by (3.20) and (3.21), we have
$$ \begin{aligned}[b] &\bigl\Vert T ( \hbar_{1},\hbar_{2} ) (r)-T (\omega_{1}, \omega_{2} ) (r) \bigr\Vert \\ &\quad \leq \bigl[\blacktriangle_{f}(m_{1}+m_{2})+ \blacktriangle _{g}(n_{1}+n_{2})\bigr] \bigl( \Vert \hbar_{1}-\omega_{1} \Vert + \Vert \hbar_{2}-\omega_{2} \Vert \bigr). \end{aligned} $$
(3.22)
Hence \(\blacktriangle_{f}(m_{1}+m_{2})+\blacktriangle _{g}(n_{1}+n_{2})<1\), and therefore T is a contraction operator. So by Banach’s fixed point theorem, the operator T has a unique fixed point, which is the unique solution of problem (1.1). □