Now, we present our main results.
Lemma 2.1
Let \(\alpha \geq 2\), \([\alpha ] =n-1\), \(\lambda _{i} \geq 0\), \(\mu _{i} , \gamma _{i}, \eta \in (0,1)\) for all \(i=1, \ldots, n_{0}\), \(\theta \in (0,1)\) and \(f\in L^{1} [0,1] \). Then the solution of the problem \(D^{\alpha } x(t)=f(t)\) with the boundary conditions \(x^{(j)}(0)= 0\) for \(2 \leq j \leq n -1\), \(x(\theta )=\sum_{i=1}^{n_{0}} \lambda _{i} D^{\mu _{i} }x(\gamma _{i})\) and \(x'(0)= x(\eta )\) is given by
$$\begin{aligned} x(t) =& \frac{1}{\Gamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds \\ &{} + \frac{1 - \eta + t}{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{\alpha - 1} f(s) \,ds \\ &{}- \frac{1 - \eta + t }{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds \\ & {}-\frac{1 - \eta +t}{(\Delta _{\gamma } - \theta - 1 +\eta )} \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds, \end{aligned}$$
where \(\Delta _{\gamma } := \sum_{i=1}^{n_{0}} \frac{\lambda _{i} (\gamma _{i})^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})}\) and \(1- \Delta _{\gamma } \neq \eta - \theta \).
Proof
By using a similar method to [30], we conclude that Lemma 1.2 holds on \(L^{1}[0,1]\). Let x be a solution for the problem. By using Lemma 1.2, we have
$$\begin{aligned} x(t)= \frac{1}{\Gamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds +c_{0} + c_{1} t + \cdots + c_{n-1} t^{n-1}. \end{aligned}$$
Since \(x^{(j)}(0)= 0\) for \(2 \leq j \leq n -1\), we conclude that
$$\begin{aligned} x(t)= \frac{1}{\Gamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds +c_{0} + c_{1} t \end{aligned}$$
(2)
and so \(x(\eta )= \frac{1}{\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{ \alpha - 1} f(s) \,ds +c_{0} + c_{1} \eta \) and \(x'(t)= \frac{1}{\Gamma (\alpha -1)} \int ^{t}_{0} (t-s)^{\alpha - 2} f(s) \,ds +c_{1}\). Thus, \(x'(0) = c_{1}\) and by using the boundary condition \(x'(0) =x(\eta )\) we get
$$\begin{aligned} \frac{1}{\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds +c_{0} + c_{1} \eta = c_{1}. \end{aligned}$$
Hence,
$$\begin{aligned} c_{1} = \frac{1}{(1- \eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{ \alpha - 1} f(s) \,ds + \frac{1}{1- \eta }c_{0}. \end{aligned}$$
(3)
On the other hand by using (2), for each \(i= 1, \ldots, n_{0}\) we have
$$\begin{aligned} D^{\mu _{i}} x(t)= \frac{1}{\Gamma (\alpha -\mu _{i})} \int ^{t}_{0} (t-s)^{ \alpha - \mu _{i} -1} f(s) \,ds +c_{1} \frac{t^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})} \end{aligned}$$
which implies \(\lambda _{i} D^{\mu _{i}} x(\gamma _{i})= \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds +c_{1} \frac{\lambda _{i} (\gamma _{i})^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})}\). Hence,
$$\begin{aligned} \sum_{i=1}^{n_{0}} \lambda _{i} D^{\mu _{i}} x(\gamma _{i})= \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds +c_{1} \sum_{i=1}^{n_{0}} \frac{\lambda _{i} (\gamma _{i})^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})}. \end{aligned}$$
Since \(x(\theta )= \frac{1}{\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{ \alpha - 1} f(s) \,ds +c_{0} + c_{1} \theta \) and \(x(\theta ) = \sum_{i=1}^{n_{0}} \lambda _{i} D^{\mu _{i}} x(\gamma _{i})\), we obtain
$$\begin{aligned}& \frac{1}{\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{ \alpha - 1} f(s) \,ds +c_{0} + c_{1} \theta \\& \quad = \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds +c_{1} \sum_{i=1}^{n_{0}} \frac{\lambda _{i} (\gamma _{i})^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})} \end{aligned}$$
and so by using (3), we have
$$\begin{aligned}& \frac{1}{\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{ \alpha - 1} f(s) \,ds +c_{0} + \frac{\theta }{(1- \eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{ \alpha - 1} f(s) \,ds \\& \quad {} + \frac{\theta }{1- \eta }c_{0} = \sum _{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds \\& \quad {}+ \frac{ \sum_{i=1}^{n_{0}} \frac{\lambda _{i} (\gamma _{i})^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})}}{(1- \eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds + \frac{ \sum_{i=1}^{n_{0}} \frac{\lambda _{i} (\gamma _{i})^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})}}{1- \eta }c_{0}. \end{aligned}$$
If \(\Delta _{\gamma } := \sum_{i=1}^{n_{0}} \frac{\lambda _{i} (\gamma _{i})^{1- \mu _{i}}}{\Gamma (2 -\mu _{i})}\), then
$$\begin{aligned} c_{0} \biggl(\frac{ \Delta _{\gamma } - \theta - 1 +\eta }{1 - \eta } \biggr) =& \frac{1}{\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{ \alpha - 1} f(s) \,ds \\ &{} + \frac{\theta }{(1- \eta )\Gamma (\alpha )} \int ^{\eta }_{0} ( \eta -s)^{\alpha - 1} f(s) \,ds \\ &{} - \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds \\ &{}-\frac{ \Delta _{\gamma }}{(1- \eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds \end{aligned}$$
so
$$\begin{aligned} c_{0} =& \frac{1 - \eta }{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{\alpha - 1} f(s) \,ds \\ &{} -\frac{1 - \eta }{(\Delta _{\gamma } - \theta - 1 +\eta )} \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds \\ &{}+ \frac{\theta - \Delta _{\gamma }}{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds. \end{aligned}$$
Thus, by using (2) and (3) we get
$$\begin{aligned} c_{1} =& \frac{1}{(1 - \eta )\Gamma (\alpha )} \int ^{ \eta }_{0} ( \eta -s)^{\alpha - 1} f(s) \,ds \\ &{} + \frac{1}{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{\alpha - 1} f(s) \,ds \\ &{}+ \frac{\theta - \Delta _{\gamma }}{(\Delta _{\gamma } - \theta - 1 +\eta )(1 - \eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds \\ & {}-\frac{1}{(\Delta _{\gamma } - \theta - 1 +\eta )} \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds \end{aligned}$$
and
$$\begin{aligned} x(t) =& \frac{1}{\Gamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds \\ &{} + \frac{1 - \eta }{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{\alpha - 1} f(s) \,ds \\ &{}+ \frac{\theta - \Delta _{\gamma }}{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds \\ & {}-\frac{1 - \eta }{(\Delta _{\gamma } - \theta - 1 +\eta )} \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds \\ &{}+ \frac{t}{(1 - \eta )\Gamma (\alpha )} \int ^{ \eta }_{0} ( \eta -s)^{ \alpha - 1} f(s) \,ds \\ & {}+ \frac{t}{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{\alpha - 1} f(s) \,ds \\ &{}+ \frac{(\theta - \Delta _{\gamma })t}{(\Delta _{\gamma } - \theta - 1 +\eta )(1 - \eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds \\ & {}-\frac{t}{(\Delta _{\gamma } - \theta - 1 +\eta )} \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds. \end{aligned}$$
Hence,
$$\begin{aligned} x(t) =& \frac{1}{\Gamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds \\ & {}+ \frac{1 - \eta + t}{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{\alpha - 1} f(s) \,ds \\ &{}+ \frac{-(1 - \eta )^{2} +(\theta - \Delta _{\gamma })t + t (\Delta _{\gamma } - \theta - 1 +\eta )}{(1 - \eta )(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds \\ &{} -\frac{1 - \eta +t}{(\Delta _{\gamma } - \theta - 1 +\eta )} \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds \end{aligned}$$
and so
$$\begin{aligned} x(t) =& \frac{1}{\Gamma (\alpha )} \int ^{t}_{0} (t-s)^{\alpha - 1} f(s) \,ds \\ & {}+ \frac{1 - \eta + t}{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\theta }_{0} (\theta -s)^{\alpha - 1} f(s) \,ds \\ &{}- \frac{1 - \eta + t }{(\Delta _{\gamma } - \theta - 1 +\eta )\Gamma (\alpha )} \int ^{\eta }_{0} (\eta -s)^{\alpha - 1} f(s) \,ds \\ &{} -\frac{1 - \eta +t}{(\Delta _{\gamma } - \theta - 1 +\eta )} \sum_{i=1}^{n_{0}} \frac{\lambda _{i}}{\Gamma (\alpha -\mu _{i})} \int ^{\gamma _{i}}_{0} (\gamma _{i} -s)^{\alpha - \mu _{i} -1} f(s) \,ds. \end{aligned}$$
This completes the proof. □
Consider the space \(X= C^{1}[0,1]\) with the norm \(\| \cdot \|_{*}\) and the space \(X^{m}\) with the norm \(\| \cdot \|_{**}\), where \(\|(x_{1},\ldots,x_{m})\|_{**} = \max \{ \|x_{1}\|_{*},\ldots, \|x_{m}\|_{*} \}\), \(\|x\|_{*} = \max \{ \|x\|, \|x'\| \}\) and \(\| . \|\) is the supremum norm on \(C[0,1]\). Let \(f_{k}\) be a map \([0,1]\times X^{4m}\) that is singular at some points of \([0,1]\), for \(k=1, \ldots, m\). Define \(F:X^{m} \to X^{m}\) as
$$ F(x_{1},\ldots,x_{m}) (t) = \begin{pmatrix} \phi _{1}(x_{1},\ldots,x_{m})(t) \\ \vdots \\ \phi _{m}(x_{1},\ldots,x_{m})(t) \end{pmatrix} , $$
where
$$\begin{aligned} &\phi _{k}(x_{1},\ldots,x_{m}) (t) \\ &\quad = \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\ &\qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds \\ &\qquad {} + \frac{1 - \eta _{k} + t}{(\Delta _{\gamma } - \theta _{k} - 1 +\eta )\Gamma (\alpha )} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\ &\qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds \\ &\qquad {}- \frac{1 - \eta _{k} + t }{(\Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k})\Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\ &\qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds \\ &\qquad {} - \frac{1 - \eta _{k} +t}{(\Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k})} \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), \\ &\qquad D^{\beta _{1}}x_{1}(s), I^{p_{1}}x_{1}(s), \ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds, \end{aligned}$$
for \(1 \leq k \leq m\), where \(\Delta _{\gamma _{k}} := \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k} (\gamma _{i,k})^{1- \mu _{i,k}}}{\Gamma (2 -\mu _{i,k})}\). Then we have
$$ F'(x_{1},\ldots,x_{m}) (t) = \begin{pmatrix} \phi '_{1}(x_{1},\ldots,x_{m})(t) \\ \vdots \\ \phi '_{m}(x_{1},\ldots,x_{m})(t) \end{pmatrix} , $$
where for each \(1 \leq k \leq m\) we have
$$\begin{aligned} &\phi '_{k}(x_{1},\ldots,x_{m}) (t) \\ &\quad = \frac{1}{\Gamma (\alpha _{k} - 1)} \int ^{t}_{0} (t-s)^{\alpha _{k} - 2} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\ & \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds \\ & \qquad {}+ \frac{1}{(\Delta _{\gamma } - \theta _{k} - 1 +\eta )\Gamma (\alpha )} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\ & \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds \\ & \qquad {}- \frac{1 }{(\Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k})\Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\ & \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds \\ & \qquad {} -\frac{1 }{(\Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k})} \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} f_{k} \bigl(s, x_{1}(s), x'_{1}(s), \\ & \qquad D^{\beta _{1}}x_{1}(s), I^{p_{1}}x_{1}(s), \ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \,ds. \end{aligned}$$
It is obvious that the singular pointwise defined equation (1) has a solution u if and only if u is a fixed point of the map F.
Theorem 2.2
Let m, n and \(n_{0}\) be natural numbers, \(\alpha _{k} \geq 2\), \([\alpha _{k}] =n_{k}-1\), \(\lambda _{i,k} \geq 0\), \(\gamma _{i,k}, \mu _{i,k}, \theta _{k}, \eta _{k} \in (0,1)\), \(p_{k} >0\) for \(i=1,\ldots,n_{0}\) and \(k=1,2,\ldots,m\), \(f_{k}: [0, 1] \times X^{m} \to \mathbb{R}\) some singular mappings on some points of \([0, 1]\) such that
$$ \bigl\vert f_{k}(t, x_{1}, \ldots, x_{4m}) - f_{k}(t, y_{1}, \ldots, y_{4m}) \bigr\vert \leq \Phi _{k}(t) M_{k} \bigl( \vert x_{1} - y_{1} \vert , \ldots, \vert x_{4m} - y_{4m} \vert \bigr) $$
for all \(x_{1}, \ldots, x_{4m}, y_{1}, \ldots, y_{4m} \in X\) and almost all \(t \in [0,1]\). Assume that
$$ \bigl\vert f_{i}(t, x_{1}, \ldots, x_{4m}) \bigr\vert \leq \sum_{j=1}^{4m} T_{k,j} \bigl(t, \vert x_{k} \vert \bigr), $$
where \(M_{k} :X^{4m} \to \mathbb{R}^{+}\) is non-decreasing mapping respect to all components such that \(\lim_{z \to 0^{+}} \frac{M_{k}(z,\ldots,z)}{z} :=q_{k} \in [0,\infty )\) and \(T_{k,j} : [0, 1] \times X \to \mathbb{R}^{+}\) is a map with \(T_{k,j}(\cdot,z)\) is nondecreasing respect to z and \(\lim_{z \to 0^{+}} \frac{T_{k,j}(t,z)}{z} :=b_{k,j}(t)\) for almost all \(t \in [0,1]\) and for some \(b_{k,j}: \in \mathbb{R}^{+}[0,1] \to \mathbb{R^{+}}\) such that \((1-t)^{\alpha _{k} -2}b_{k,j}(t) \in L^{1}[0,1]\) for \(1 \leq j \leq 4m\) and \(1 \leq {k \leq m}\). Let \(\Delta = \max \{ 1, \frac{1}{\Gamma (2- \beta _{1})}, \ldots, \frac{1}{\Gamma (2- \beta _{m})},\frac{1}{\Gamma (p_{1} +1)}, \ldots, \frac{1}{\Gamma (p_{m} +1)} \} \) and \(\hat{ b}_{i,k}, \hat{\phi }_{k} \in L^{1}[0,1]\), \(\Delta _{\gamma _{k}} := \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k} (\gamma _{i,k})^{1- \mu _{i,k}}}{\Gamma (2 -\mu _{i,k})}\) and \(1- \Delta _{\gamma _{k}} \neq \eta _{k} - \theta _{k}\), where \(\hat{\phi }_{k}(s) = (1-s)^{\alpha _{i} -2} a_{i,j}(s)\). If
$$\begin{aligned}& \max_{1 \leq k \leq m} \Biggl[ \frac{1 }{\Gamma (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \quad {}+ \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \max \Biggl\{ \sum _{j=1}^{m} \Vert \hat{b}_{k,j} \Vert , q_{k} \hat{\Phi }_{k} \Biggr\} \in \biggl[0, \frac{1}{\Delta } \biggr), \end{aligned}$$
then the pointwise defined system
$$\begin{aligned} \textstyle\begin{cases} D^{\alpha _{1}} x_{1}(t) = f_{1}(t, x_{1}(t), x'_{1}(t), D^{\beta _{1}}x_{1}(t), I^{p_{1}}x_{1}(t), \\ \hphantom{D^{\alpha _{1}} x_{1}(t) =}\ldots, x_{m}(t), x'_{m}(t), D^{\beta _{m}}x_{m}(t), I^{p_{m}}x_{m}(t)), \\ D^{\alpha _{m}} x_{m}(t) + f_{m}(t, x_{1}(t), x'_{1}(t), D^{\beta _{1}}x_{1}(t), I^{p_{1}}x_{1}(t), \\ \quad \ldots, x_{m}(t), x'_{m}(t), D^{\beta _{m}}x_{m}(t), I^{p_{m}}x_{m}(t)), \end{cases}\displaystyle \end{aligned}$$
with boundary conditions \(x^{(j)}_{k}(0)= 0\), \(x_{k}(\theta _{k})=\sum_{i=1}^{n_{0}} \lambda _{i,k} D^{\mu _{i,k} }x_{k}( \gamma _{i,k})\) and \(x'_{k}(0)= x_{k}(\eta _{k})\) for \(2 \leq j \leq n_{k}-1\) and \(1\leq k\leq m\), has a solution.
Proof
First, we prove F is continuous on \(X^{m}\). Let \(\epsilon >0\) and \(\|(x_{1},\ldots,x_{m}) - (y_{1},\ldots,y_{m})\|_{**} < \epsilon \). Then \(\max_{1 \leq k \leq m} \|x_{k} - y_{k} \|_{*} < \epsilon \) and so \(\|x_{k} - y_{k} \|_{*} < \epsilon \) for all \(1 \leq k \leq m\). Thus,
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t)-\phi _{k}(y_{1},\ldots, y_{n}) (t) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), \\& \qquad D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{m}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} + \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), \\& \qquad D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{m}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), \\& \qquad D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{m}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} + \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), \\& \qquad I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \\& \qquad \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{m}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y'_{1}(s) \bigr\vert , \\& \qquad \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert , \ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \\& \qquad \bigl\vert x'_{m}(s) - y'_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \\& \qquad {} + \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y'_{1}(s) \bigr\vert , \\& \qquad \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert , \ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \\& \qquad \bigl\vert x'_{m}(s) - y'_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1}\Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y'_{1}(s) \bigr\vert , \\& \qquad \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert , \ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \\& \qquad \bigl\vert x'_{m}(s) - y'_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \\& \qquad {}+ \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y'_{1}(s) \bigr\vert , \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert , \\& \qquad\ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \bigl\vert x'_{m}(s) - y'_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \end{aligned}$$
for all \(1 \leq k \leq m\) and \(t \in [0,1]\). Now for \(\beta \in (0,1)\) and \(t \in [0,1]\), we have
$$ D^{\beta }( x-y) (t)=\frac{1}{\Gamma (1- \beta )} \int _{0}^{t} (t-s)^{ \beta -2} \bigl(x' - y' \bigr) (s) \,ds $$
and so \(|D^{\beta } (x-y)(t)|\leq \frac{\|x' -y'\|}{\Gamma (1- \beta )} \int _{0}^{t} (t-s)^{\beta -2} \,ds = \frac{\|x' -y'\|}{\Gamma (2- \beta )} t^{\beta -1}\). Hence, \(|D^{\beta } (x-y)(t)|\leq \frac{\|x' -y'\|}{\Gamma (2- \beta )}\) and \(|I^{p} (x-y)(t)|\leq \frac{\|x -y\|}{\Gamma (p)} \int _{0}^{t} (t-s)^{p-1} \,ds = \frac{\|x -y\|}{\Gamma (p +1)} t^{p}\). Thus, \(|I^{p} (x-y)(t)|\leq \frac{\|x -y\|}{\Gamma (p +1)}\) and
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t)-\phi _{k}(y_{1},\ldots, y_{n}) (t) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \Phi _{k}(s) M_{k} \biggl( \Vert x_{1} - y_{1} \Vert , \bigl\Vert x'_{1} - y'_{1} \bigr\Vert , \frac{ \Vert x'_{1} -y'_{1} \Vert }{\Gamma (2- \beta _{1})}, \\& \qquad \frac{ \Vert x_{1} -y_{1} \Vert }{\Gamma (p_{1} +1)},\ldots, \Vert x_{m} - y_{m} \Vert , \bigl\Vert x'_{m} - y'_{m} \bigr\Vert , \frac{ \Vert x'_{m} -y'_{m} \Vert }{\Gamma (2- \beta _{m})}, \frac{ \Vert x_{m} -y_{m} \Vert }{\Gamma (p_{m} +1)} \biggr) \,ds \\& \qquad {} + \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \Phi _{k}(s) M_{k} \biggl( \Vert x_{1} - y_{1} \Vert , \bigl\Vert x'_{1} - y'_{1} \bigr\Vert , \\& \qquad \frac{ \Vert x'_{1} -y'_{1} \Vert }{\Gamma (2- \beta _{1})}, \frac{ \Vert x_{1} -y_{1} \Vert }{\Gamma (p_{1} +1)},\ldots, \Vert x_{m} - y_{m} \Vert , \bigl\Vert x'_{m} - y'_{m} \bigr\Vert , \frac{ \Vert x'_{m} -y'_{m} \Vert }{\Gamma (2- \beta _{m})}, \frac{ \Vert x_{m} -y_{m} \Vert }{\Gamma (p_{m} +1)} \biggr) \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1}\Phi _{k}(s) M_{k} \biggl( \Vert x_{1} - y_{1} \Vert , \bigl\Vert x'_{1} - y'_{1} \bigr\Vert , \\& \qquad \frac{ \Vert x'_{1} -y'_{1} \Vert }{\Gamma (2- \beta _{1})}, \frac{ \Vert x_{1} -y_{1} \Vert }{\Gamma (p_{1} +1)},\ldots, \Vert x_{m} - y_{m} \Vert , \bigl\Vert x'_{m} - y'_{m} \bigr\Vert , \frac{ \Vert x'_{m} -y'_{m} \Vert }{\Gamma (2- \beta _{m})}, \frac{ \Vert x_{m} -y_{m} \Vert }{\Gamma (p_{m} +1)} \biggr) \,ds \\& \qquad {}+ \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \Phi _{k}(s)M_{k} \biggl( \Vert x_{1} - y_{1} \Vert , \bigl\Vert x'_{1} - y'_{1} \bigr\Vert , \frac{ \Vert x'_{1} -y'_{1} \Vert }{\Gamma (2- \beta _{1})}, \frac{ \Vert x_{1} -y_{1} \Vert }{\Gamma (p_{1} +1)},\ldots, \\& \qquad \Vert x_{m} - y_{m} \Vert , \bigl\Vert x'_{m} - y'_{m} \bigr\Vert , \frac{ \Vert x'_{m} -y'_{m} \Vert }{\Gamma (2- \beta _{m})}, \frac{ \Vert x_{m} -y_{m} \Vert }{\Gamma (p_{m} +1)} \biggr) \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \Phi _{k}(s) M_{k} \bigl( \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*}, \ldots, \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*}, \\& \qquad \ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*},\ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*}, \bigr) \,ds \\& \qquad {} + \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \Phi _{k}(s)M_{k} \bigl( \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*}, \\& \qquad \ldots, \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*},\ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*},\ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*} \bigr) \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1}\Phi _{k}(s) M_{k} \bigl( \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*}, \\& \qquad \ldots, \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*},\ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*},\ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*} \bigr) \,ds \\& \qquad {} + \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \Phi _{k}(s) M_{k} \bigl( \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*}, \ldots, \Delta _{1} \Vert x_{1} - y_{1} \Vert _{*}, \\& \qquad \ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*},\ldots, \Delta _{m} \Vert x_{m} - y_{m} \Vert _{*} \bigr) \,ds, \end{aligned}$$
where for \(1 \leq j \leq m\) \(\Delta _{j} = \max \{ 1, \frac{1}{\Gamma (2 - \beta _{j})}, \frac{1}{\Gamma (p_{j} + 1)} \}\) and \(\|x_{j} - y_{j}\|_{*} = \max \{ \|x_{j} - y_{j}\|, \|x'_{j} - y'_{j} \| \} \). Let \(\Delta = \max_{1 \leq j \leq m} \Delta _{j}\) and \(\|(x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n})\|_{**} = \max_{1 \leq j \leq m} \{ \|x_{j} - y_{j}\|_{*} \} \). Then, for each \(t \in [0,1]\) and \(1 \leq j \leq m\), we have
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t)-\phi _{k}(y_{1},\ldots, y_{n}) (t) \bigr\vert \\& \quad \leq \frac{ M_{k} ( \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**},\ldots, \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**})}{\Gamma (\alpha _{k})} \\& \qquad {} \times \int ^{1}_{0} (1-s)^{\alpha _{k} - 1} \Phi _{k}(s) \,ds \\& \qquad {} + (1 - \eta _{k} + t ) M_{k} \bigl( \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**},\ldots, \\& \qquad \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}\bigr)/ \bigl( \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})\bigr) \\& \qquad {}\times \int ^{1}_{0} (1 -s)^{\alpha _{k} - 1} \Phi _{k}(s) \,ds \\& \qquad {} + (1 - \eta _{k} + t ) M_{k} \bigl( \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**},\ldots, \\& \qquad \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}\bigr)/ \bigl( \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})\bigr) \\& \qquad {}\times \int ^{1}_{0} (1 -s)^{\alpha - 1}\Phi _{k}(s) \,ds \\& \qquad {}+ (1 - \eta _{k} + t ) M_{k} \bigl( \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**},\ldots, \\& \qquad {}\Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}\bigr)/ \bigl( \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \bigr) \\& \qquad {}\times \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{1}_{0} (1 -s)^{\alpha _{k} - \mu _{i,k} -1} \Phi _{k}(s) \,ds. \end{aligned}$$
(4)
Since \(\lim_{z \to 0^{+}} \frac{M_{k}(\Delta z,\ldots, \Delta z)}{\Delta z}= q_{k}\), for each \(\epsilon >0\) there exists \(\delta (\epsilon )> 0\) such that \(0 < z < \delta (\epsilon )\) implies \(\frac{M_{k}(\Delta z,\ldots, \Delta z)}{\Delta z} < q_{k} + \epsilon \) for all \(1 \leq k \leq m\). Thus,
$$\begin{aligned} M_{k}(\Delta z,\ldots, \Delta z) < (q_{k} + \epsilon ) \Delta z \end{aligned}$$
(5)
for \(0 < z < \delta (\epsilon )\). Put \(\delta _{M}(\epsilon ) = \min \{\delta (\epsilon ), \epsilon \}\). Then, for each \(0 < z < \delta _{M}(\epsilon )\), we have
$$ M_{k}(\Delta z,\ldots, \Delta z) < (q_{k} + \epsilon ) \Delta \epsilon $$
for all \(1 \leq k \leq m\). Let \(\|(x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n})\|_{**} < \delta _{M}( \epsilon )\). Then we have
$$ M_{k} \bigl(\Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**},\ldots, \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \bigr) < (q_{k} + \epsilon ) \Delta \epsilon $$
for all \(1 \leq k \leq m\) and so
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t)-\phi _{k}(y_{1},\ldots, y_{n}) (t) \bigr\vert \\& \quad \leq \frac{(q_{k} + \epsilon ) \Delta \epsilon }{\Gamma (\alpha _{k})} \int ^{1}_{0} (1-s)^{\alpha _{k} - 1} \Phi _{k}(s) \,ds \\& \qquad {} + \frac{(1 - \eta _{k} + t ) (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{1}_{0} (1 -s)^{\alpha _{k} - 1} \Phi _{k}(s) \,ds \\& \qquad {} + \frac{(1 - \eta _{k} + t ) (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{1}_{0} (1 -s)^{\alpha - 1}\Phi _{k}(s) \,ds \\& \qquad {} + \frac{(1 - \eta _{k} + t ) (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{1}_{0} (1 -s)^{\alpha _{k} - \mu _{i,k} -1} \Phi _{k}(s) \,ds \\& \quad \leq \frac{(q_{k} + \epsilon ) \Delta \epsilon }{\Gamma (\alpha _{k})} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} + \frac{(1 - \eta _{k} + t ) (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \\& \qquad {} + \frac{(1 - \eta _{k} + t ) (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \\& \qquad {} + \frac{(1 - \eta _{k} + t ) (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})}. \end{aligned}$$
Hence,
$$\begin{aligned}& \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) - \phi _{k}(y_{1},\ldots, y_{n}) \bigr\Vert \\& \quad \leq \Biggl( \frac{1}{\Gamma (\alpha _{k})} + \frac{2 (2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {} + \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) (q_{k} + \epsilon ) \Delta \Vert \hat{ \Phi }_{k} \Vert _{[0,1]} \epsilon . \end{aligned}$$
If \(\|(x_{1},\ldots,x_{m}) - (y_{1},\ldots,y_{m})\|_{**} < \epsilon \) for all \(t \in [0,1]\) and \(k=1, \ldots, m\), then we get
$$\begin{aligned}& \bigl\vert \phi '_{k}(x_{1},\ldots, x_{n}) (t)-\phi '_{k}(y_{1},\ldots, y_{n}) (t) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha _{k} -1)} \int ^{t}_{0} (t-s)^{ \alpha _{k} - 2} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{1}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), \\& \qquad D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{1}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} + \frac{1}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{1}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), \\& \qquad D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{1}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {}+ \frac{1}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{1}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), \\& \qquad D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{1}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} +\frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \\& \qquad {}\times\int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{1}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) - f_{k} \bigl(s, y_{1}(s), y'_{1}(s), \\& \qquad D^{\beta _{1}}y_{1}(s), I^{p_{1}}y_{1}(s), \ldots, y_{m}(s), y'_{m}(s), D^{\beta _{1}}y_{m}(s), I^{p_{m}}y_{m}(s) \bigr) \bigr\vert \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k} -1)} \int ^{t}_{0} (t-s)^{ \alpha _{k} - 2} \Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y_{1}(s) \bigr\vert , \\& \qquad \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert , \ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \\& \qquad \bigl\vert x'_{m}(s) - y_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \\& \qquad {} + \frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y_{1}(s) \bigr\vert , \\& \qquad {} \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert , \ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \\& \qquad \bigl\vert x'_{m}(s) - y_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \\& \qquad {}+ \frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1}\Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y_{1}(s) \bigr\vert , \\& \qquad \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert , \ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \\& \qquad \bigl\vert x'_{m}(s) - y_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \\& \qquad {}+\frac{1}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \Phi _{k}(s) M_{k} \bigl( \bigl\vert x_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert x'_{1}(s) - y_{1}(s) \bigr\vert , \bigl\vert D^{\beta _{1}}(x_{1}- y_{1}) (s) \bigr\vert , I^{p_{1}}(x_{1} - y_{1}) (s) \vert \\& \qquad ,\ldots, \bigl\vert x_{m}(s) - y_{m}(s) \bigr\vert , \bigl\vert x'_{m}(s) - y_{m}(s) \bigr\vert , \bigl\vert D^{\beta _{m}}(x_{m}- y_{m}) (s) \bigr\vert , I^{p_{m}}(x_{m} - y_{m}) (s) \vert \bigr) \,ds \\& \quad \leq \frac{ M_{k} ( \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**},\ldots, \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**})}{\Gamma (\alpha _{k} -1)} \\& \qquad {}\times \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} \Phi _{k}(s) \,ds \\& \qquad {}+ \frac{ M_{k} ( \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**},\ldots, \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**})}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {}\times \int ^{1}_{0} (1 -s)^{\alpha _{k} - 1} \Phi _{k}(s) \,ds \\& \qquad {} + \frac{ M_{k} ( \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**},\ldots, \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**})}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {} \times \int ^{1}_{0} (1 -s)^{\alpha - 1}\Phi _{k}(s) \,ds \\& \qquad {} + \frac{ M_{k} ( \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**},\ldots, \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**})}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \\& \qquad {}\times \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{1}_{0} (1 -s)^{\alpha _{k} - \mu _{i,k} -1} \Phi _{k}(s) \,ds \\& \quad \leq \frac{(q_{k} + \epsilon ) \Delta \epsilon }{\Gamma (\alpha _{k} -1)} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} + \frac{ (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \\& \qquad {} + \frac{ (q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \\& \qquad {} + \frac{(q_{k} + \epsilon ) \Delta \epsilon }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})}. \end{aligned}$$
This implies that
$$\begin{aligned}& \bigl\Vert \phi '_{k}(x_{1},\ldots, x_{n}) -\phi '_{k}(y_{1},\ldots, y_{n}) \bigr\Vert \\& \quad \leq \Biggl( \frac{1}{\Gamma (\alpha _{k} -1)} + \frac{2}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {} + \frac{1}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) (q_{k} + \epsilon ) \Delta \Vert \hat{ \Phi }_{k} \Vert _{[0,1]} \epsilon \end{aligned}$$
and so
$$\begin{aligned}& \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) - \phi _{k}(y_{1},\ldots, y_{n}) \bigr\Vert _{*} \\& \quad = \max \bigl\{ \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) -\phi _{k}(y_{1},\ldots, y_{n}) \bigr\Vert , \bigl\Vert \phi '_{k}(x_{1}, \ldots, x_{n}) -\phi '_{k}(y_{1}, \ldots, y_{n}) \bigr\Vert \bigr\} \\& \quad \leq \Biggl( \frac{1}{\Gamma (\alpha _{k} -1)} + \frac{2 (2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {} + \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) (q_{k} + \epsilon ) \Delta \Vert \hat{ \Phi }_{k} \Vert _{[0,1]} \epsilon . \end{aligned}$$
Thus, we get
$$\begin{aligned}& \bigl\Vert F(x_{1},\ldots, x_{n}) - F(y_{1}, \ldots, y_{n}) \bigr\Vert _{**} \\& \quad = \max_{1 \leq k \leq m} \bigl\Vert \phi _{k}(x_{1}, \ldots, x_{n}) -\phi _{k}(y_{1},\ldots, y_{n}) \bigr\Vert _{*} \\& \quad \leq \max_{1 \leq k \leq m} \Biggl\{ \Biggl( \frac{1}{\Gamma (\alpha _{k} -1)} + \frac{2 (2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {} + \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) (q_{k} + \epsilon ) \Delta \Vert \hat{ \Phi }_{k} \Vert _{[0,1]} \Biggr\} \epsilon . \end{aligned}$$
This implies \(F(x_{1},\ldots, x_{n}) \to F(y_{1},\ldots, y_{n})\) in \(X^{m}\) when \((x_{1},\ldots, x_{n}) \to (y_{1},\ldots, y_{n})\). Hence, F is continuous on \(X^{m}\). Since \(\lim_{z \to 0^{+}} \frac{T_{k,j}(t,\Delta z)}{\Delta z} =b_{k,j}(t)\) for \(1 \leq k \leq m\) and \(1 \leq j \leq 4m\), for every \(\epsilon >0\) we can choose \(\delta (\epsilon )\) such that \(z \in (0, \delta (\epsilon )]\) implies \(\frac{T_{k,j}(t,\Delta z)}{\Delta z} \leq b_{k,j}(t) + \epsilon \) for almost all \(t \in [0,1]\). Thus,
$$\begin{aligned} T_{k,j}(t,\Delta z) \leq \bigl(b_{k,j}(t) + \epsilon \bigr) \Delta z \end{aligned}$$
(6)
for \(z \in (0, \delta (\epsilon )]\) and almost all \(t \in [0,1]\). On the other hand, by using the assumptions we have
$$\begin{aligned}& \max_{1 \leq k \leq m} \Biggl[ \frac{1 }{\Gamma (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \quad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \Delta < 1. \end{aligned}$$
Choose \(\epsilon _{0} >0\) such that
$$\begin{aligned}& \max_{1 \leq k \leq m} \Biggl( \Biggl[ \frac{1 }{\Gamma (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \quad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \\& \quad {}+ m \epsilon _{0} \Biggl[ \frac{1 }{\Gamma ^{2} (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma ^{2} (\alpha _{k})} \\& \quad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma ^{2} (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \Biggr) \Delta < 1. \end{aligned}$$
Since
$$\begin{aligned}& \max_{1 \leq k \leq m} \Biggl\{ \Biggl( \frac{1}{\Gamma (\alpha _{k} -1)} + \frac{2 (2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \quad {} + \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) q_{k} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \Biggr\} \in \biggl[0, \frac{1}{\Delta }\biggr), \end{aligned}$$
so we can choose \(\epsilon _{1}>0\) such that
$$\begin{aligned}& \max_{1 \leq k \leq m} \Biggl\{ \Biggl( \frac{1}{\Gamma (\alpha _{k} -1)} + \frac{2 (2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \quad {} + \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) (q_{k}+ \epsilon _{1}) \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \Biggr\} \in \biggl[0, \frac{1}{\Delta }\biggr). \end{aligned}$$
Let \(\delta _{0}= \delta (\epsilon _{0})\) and put \(r := \min \{ \delta _{0}, \epsilon _{0}, \frac{\delta _{M}(\epsilon _{1})}{2} \}\). By using (6), \(z \in (0, r]\) implies
$$\begin{aligned} T_{k,j}(t,\Delta z) \leq \bigl(b_{k,j}(t) + \epsilon _{0} \bigr) \Delta r \end{aligned}$$
and specially for \(z=r\) we have
$$\begin{aligned} T_{k,j}(t,\Delta r) \leq \bigl(b_{k,j}(t) + \epsilon _{0} \bigr) \Delta r \end{aligned}$$
(7)
for almost all \(t\in [0,1]\). Let \(C= \{ (x_{1},\ldots,x_{m}) \in X^{m} : \|(x_{1},\ldots,x_{m})\|_{**} \leq r \}\). Define the mapping \(\alpha : X^{2m} \to \mathbb{R}\) by \(\alpha ( (x_{1},\ldots,x_{m}), (y_{1},\ldots,y_{m}) ) =1\) when \((x_{1},\ldots,x_{m})\) and \((y_{1},\ldots,y_{m})\) both are in C and \(\alpha ( (x_{1},\ldots,x_{m}), (y_{1},\ldots,y_{m}) ) =0\) otherwise. If
$$ \alpha \bigl( (x_{1},\ldots,x_{m}), (y_{1}, \ldots,y_{m}) \bigr) \geq 1, $$
then \(\|(x_{1},\ldots,x_{m})\|_{**} \leq r\) and \(\|y_{1},\ldots,y_{m})\|_{**} \leq r\) and so \(\|x_{k}\|_{*} \leq r\) and \(\|y_{k}\|_{*} \leq r\) for all \(1\leq k \leq m\). Thus, for each \(t \in [0,1]\), we have
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} + \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad {}I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} + \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \bigl[ T_{k,1} \bigl(s, \bigl\vert x_{1}(s) \bigr\vert \bigr)+ T_{k,2} \bigl(s, \bigl\vert x'_{1}(s) \bigr\vert \bigr)+ T_{k,3} \bigl(s, \bigl\vert D^{\beta _{1}}x_{1}(s) \bigr\vert \bigr) \\& \qquad {}+ T_{k,4} \bigl(s, \bigl\vert I^{p_{1}}x_{1}(s) \bigr\vert \bigr) +\cdots+T_{k,4m-3} \bigl(s, \bigl\vert x_{m}(s) \bigr\vert \bigr)+ T_{k,4m-2} \bigl(s, \bigl\vert x'_{m}(s) \bigr\vert \bigr) \\& \qquad {} +T_{k,4m-1} \bigl(s, \bigl\vert D^{\beta _{m}}x_{m}(s) \bigr\vert \bigr) +T_{k,4m} \bigl(s, \bigl\vert I^{p_{m}}x_{m}(s) \bigr\vert \bigr) \bigr] \,ds \\& \qquad {} + \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl[ T_{k,1} \bigl(s, \bigl\vert x_{1}(s) \bigr\vert \bigr)+ T_{k,2} \bigl(s, \bigl\vert x'_{1}(s) \bigr\vert \bigr) \\& \qquad {}+ T_{k,3} \bigl(s, \bigl\vert D^{\beta _{1}}x_{1}(s) \bigr\vert \bigr) + T_{k,4} \bigl(s, \bigl\vert I^{p_{1}}x_{1}(s) \bigr\vert \bigr) +\cdots+T_{k,4m-3} \bigl(s, \bigl\vert x_{m}(s) \bigr\vert \bigr) \\& \qquad {}+ T_{k,4m-2} \bigl(s, \bigl\vert x'_{m}(s) \bigr\vert \bigr) +T_{k,4m-1} \bigl(s, \bigl\vert D^{\beta _{m}}x_{m}(s) \bigr\vert \bigr) +T_{k,4m} \bigl(s, \bigl\vert I^{p_{m}}x_{m}(s) \bigr\vert \bigr) \bigr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl[ T_{k,1} \bigl(s, \bigl\vert x_{1}(s) \bigr\vert \bigr)+ T_{k,2} \bigl(s, \bigl\vert x'_{1}(s) \bigr\vert \bigr) \\& \qquad {}+ T_{k,3} \bigl(s, \bigl\vert D^{\beta _{1}}x_{1}(s) \bigr\vert \bigr) +T_{k,4} \bigl(s, \bigl\vert I^{p_{1}}x_{1}(s) \bigr\vert \bigr) +\cdots+T_{k,4m-3} \bigl(s, \bigl\vert x_{m}(s) \bigr\vert \bigr) \\& \qquad {} + T_{k,4m-2} \bigl(s, \bigl\vert x'_{m}(s) \bigr\vert \bigr)+T_{k,4m-1} \bigl(s, \bigl\vert D^{\beta _{m}}x_{m}(s) \bigr\vert \bigr) +T_{k,4m} \bigl(s, \bigl\vert I^{p_{m}}x_{m}(s) \bigr\vert \bigr) \bigr] \,ds \\& \qquad {} + \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {} \times \bigl[ T_{k,1} \bigl(s, \bigl\vert x_{1}(s) \bigr\vert \bigr)+ T_{k,2} \bigl(s, \bigl\vert x'_{1}(s) \bigr\vert \bigr)+ T_{k,3} \bigl(s, \bigl\vert D^{ \beta _{1}}x_{1}(s) \bigr\vert \bigr) \\& \qquad {} +T_{k,4} \bigl(s, \bigl\vert I^{p_{1}}x_{1}(s) \bigr\vert \bigr) +\cdots+T_{k,4m-3} \bigl(s, \bigl\vert x_{m}(s) \bigr\vert \bigr)+ T_{k,4m-2} \bigl(s, \bigl\vert x'_{m}(s) \bigr\vert \bigr) \\& \qquad {} +T_{k,4m-1} \bigl(s, \bigl\vert D^{\beta _{m}}x_{m}(s) \bigr\vert \bigr) +T_{k,4m} \bigl(s, \bigl\vert I^{p_{m}}x_{m}(s) \bigr\vert \bigr) \bigr] \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \biggl[ T_{k,1} \bigl(s, \Vert x_{1} \Vert \bigr)+ T_{k,2} \bigl(s, \bigl\Vert x'_{1} \bigr\Vert \bigr)+ T_{k,3} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (2- \beta _{1})} \biggr) \\& \qquad {}+ T_{k,4} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (p_{1}+1)} \biggr) + \cdots+T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert \bigr)+ T_{k,4m-2} \bigl(s, \bigl\Vert x'_{m} \bigr\Vert \bigr) \\& \qquad {} +T_{k,4m-1} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (2- \beta _{m})} \biggr) +T_{k,4m} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (p_{m}+1)} \biggr) \biggr] \,ds \\& \qquad {} + \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \biggl[ T_{k,1} \bigl(s, \Vert x_{1} \Vert \bigr)+ T_{k,2} \bigl(s, \bigl\Vert x'_{1} \bigr\Vert \bigr) \\& \qquad {}+ T_{k,3} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (2- \beta _{1})} \biggr) + T_{k,4} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (p_{1}+1)} \biggr) +\cdots+T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert \bigr) \\& \qquad {}+T_{k,4m-2} \bigl(s, \bigl\Vert x'_{m} \bigr\Vert \bigr) +T_{k,4m-1} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (2- \beta _{m})} \biggr) +T_{k,4m} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (p_{m}+1)} \biggr) \biggr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \biggl[ T_{k,1} \bigl(s, \Vert x_{1} \Vert \bigr)+ T_{k,2} \bigl(s, \bigl\Vert x'_{1} \bigr\Vert \bigr) \\& \qquad {}+ T_{k,3} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (2- \beta _{1})} \biggr) + T_{k,4} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (p_{1}+1)} \biggr) +\cdots+T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert \bigr) \\& \qquad {}+T_{k,4m-2} \bigl(s, \bigl\Vert x'_{m} \bigr\Vert \bigr) +T_{k,4m-1} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (2- \beta _{m})} \biggr) +T_{k,4m} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (p_{m}+1)} \biggr) \biggr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {} \times \biggl[ T_{k,1} \bigl(s, \Vert x_{1} \Vert \bigr)+ T_{k,2} \bigl(s, \bigl\Vert x'_{1} \bigr\Vert \bigr)+ T_{k,3} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (2- \beta _{1})} \biggr) \\& \qquad {}+T_{k,4} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (p_{1}+1)} \biggr) +\cdots +T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert \bigr) + T_{k,4m-2} \bigl(s, \bigl\Vert x'_{m} \bigr\Vert \bigr) \\& \qquad {}+T_{k,4m-1} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (2- \beta _{m})} \biggr) +T_{k,4m} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (p_{m}+1)} \biggr) \biggr] \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \bigl[ T_{k,1} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr)+ \cdots + T_{k,4} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr) \\& \qquad {}+\cdots +T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert _{*} \bigr)+\cdots+T_{k,4m} \bigl(s, \Vert x_{m} \Vert _{*} \bigr) \bigr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl[ T_{k,1} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr) \\& \qquad {}+ \cdots + T_{k,4} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr)+\cdots +T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert _{*} \bigr)+\cdots +T_{k,4m} \bigl(s, \Vert x_{m} \Vert _{*} \bigr) \bigr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl[ T_{k,1} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr) \\& \qquad {}+ \cdots + T_{k,4} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr)+\cdots +T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert _{*} \bigr)+\cdots +T_{k,4m} \bigl(s, \Vert x_{m} \Vert _{*} \bigr) \bigr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {} \times \bigl[ T_{k,1} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr)+ \cdots + T_{k,4} \bigl(s, \Delta \Vert x_{1} \Vert _{*} \bigr) \\& \qquad {}+\cdots +T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert _{*} \bigr)+\cdots+T_{k,4m} \bigl(s, \Vert x_{m} \Vert _{*} \bigr) \bigr] \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{t}_{0} (t-s)^{\alpha _{k} - 1} \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {} \times \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds. \end{aligned}$$
Now by using (5), we obtain
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha _{k})} \int ^{1}_{0} (1-s)^{\alpha _{k} - 1} \Biggl[ \sum _{j=1}^{m} \bigl(b_{k,j}(s)+ \epsilon _{0} \bigr) \Delta r \Biggr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} + t }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{1}_{0} (1 -s)^{\alpha - 1} \Biggl[ \sum _{j=1}^{m} \bigl(b_{k,j}(s)+ \epsilon _{0} \bigr) \Delta r \Biggr] \,ds \\& \qquad {}+ \frac{1 - \eta _{k} +t}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{1}_{0} (1 -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \Biggl[ \sum_{j=1}^{m} \bigl(b_{k,j}(s)+\epsilon _{0} \bigr) \Delta r \Biggr] \,ds \\& \quad \leq \frac{\Delta r }{\Gamma (\alpha _{k})} \sum_{j=1}^{m} \biggl[ \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} b_{k,j}(s) \,ds+ \epsilon _{0} \int ^{1}_{0} (1-s)^{\alpha _{k} - 1} \,ds \biggr] \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{j=1}^{m} \biggl[ \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} b_{k,j}(s) \,ds+ \epsilon _{0} \int ^{1}_{0} (1-s)^{\alpha _{k} - 1} \,ds \biggr] \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{j=1}^{m} \biggl[ \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} b_{k,j}(s) \,ds+ \epsilon _{0} \int ^{1}_{0} (1-s)^{\alpha _{k} - 1} \,ds \biggr] \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \\& \qquad {}\times \Biggl( \sum_{j=1}^{m} \biggl[ \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} b_{k,j}(s) \,ds+ \epsilon _{0} \int ^{1}_{0} (1-s)^{\alpha _{k} - \mu _{i,k} -1} \,ds \biggr] \Biggr) \\& \quad \leq \frac{\Delta r }{\Gamma (\alpha _{k})} \sum_{j=1}^{m} \biggl( \Vert \hat{b}_{k,j} \Vert + \frac{\epsilon _{0}}{\Gamma (\alpha _{k})} \biggr) \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{j=1}^{m} \biggl( \Vert \hat{b}_{k,j} \Vert + \frac{\epsilon _{0}}{\Gamma (\alpha _{k})} \biggr) \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{j=1}^{m} \biggl( \Vert \hat{b}_{k,j} \Vert + \frac{\epsilon _{0}}{\Gamma (\alpha _{k})} \biggr) \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \Biggl[ \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \sum_{j=1}^{m} \biggl( \Vert \hat{b}_{k,j} \Vert + \frac{\epsilon _{0}}{\Gamma (\alpha _{k} - \mu _{i,k})} \biggr) \Biggr] \\& \quad = \frac{\Delta r }{\Gamma (\alpha _{k})} \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert + \frac{m \epsilon _{0}}{\Gamma ^{2}(\alpha _{k})} \Delta r \\& \qquad {}+ \frac{2(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert + \frac{2 (1 - \eta _{k} + t ) m \epsilon _{0}}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma ^{2}(\alpha _{k})} \Delta r \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggl( \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \Biggr) \\& \qquad {}+ \frac{ (1 - \eta _{k} + t ) m \epsilon _{0}}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma ^{2} (\alpha _{k} -\mu _{i,k})} \Delta r \\& \quad = \Biggl( \Biggl[ \frac{1 }{\Gamma (\alpha _{k})} + \frac{2(1 - \eta _{k} + t ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \\& \qquad {}+ m \epsilon _{0} \Biggl[ \frac{1 }{\Gamma ^{2} (\alpha _{k})} + \frac{2(1 - \eta _{k} + t ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma ^{2} (\alpha _{k})} \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum _{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma ^{2} (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \Biggr) \Delta r \end{aligned}$$
and so
$$\begin{aligned}& \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) \bigr\Vert \\& \quad \leq \Biggl( \Biggl[ \frac{1 }{\Gamma (\alpha _{k})} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})}\\& \qquad {} + \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \\& \qquad {}+ m \epsilon _{0} \Biggl[ \frac{1 }{\Gamma ^{2} (\alpha _{k})} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma ^{2} (\alpha _{k})} \\& \qquad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum _{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma ^{2} (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \Biggr) \Delta r. \end{aligned}$$
Hence,
$$\begin{aligned}& \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) \bigr\Vert \\& \quad \leq \Biggl( \Biggl[ \frac{1 }{\Gamma (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \\& \qquad {}+ m \epsilon _{0} \Biggl[ \frac{1 }{\Gamma ^{2} (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma ^{2} (\alpha _{k})} \\& \qquad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum _{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma ^{2} (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \Biggr) \Delta r \\& \quad \leq r. \end{aligned}$$
Let \(t \in [0,1]\), \(1 \leq k \leq m\) and \((x_{1},\ldots, x_{n}) \in C\). Then we have
$$\begin{aligned}& \bigl\vert \phi '_{k}(x_{1},\ldots, x_{n}) (t) \bigr\vert \\& \quad \leq \frac{1}{\Gamma (\alpha _{k} -1)} \int ^{t}_{0} (t-s)^{ \alpha _{k} - 2} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{1}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} + \frac{1}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{1}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {}+ \frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), \\& \qquad I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \qquad {} +\frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \bigl\vert f_{k} \bigl(s, x_{1}(s), x'_{1}(s), D^{\beta _{1}}x_{1}(s), I^{p_{1}}x_{1}(s),\ldots, x_{m}(s), x'_{m}(s), D^{\beta _{m}}x_{m}(s), I^{p_{m}}x_{m}(s) \bigr) \bigr\vert \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k} -1)} \int ^{t}_{0} (t-s)^{ \alpha _{k} - 2} \biggl[ T_{k,1} \bigl(s, \Vert x_{1} \Vert \bigr)+ T_{k,2} \bigl(s, \bigl\Vert x'_{1} \bigr\Vert \bigr)+ T_{k,3} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (2- \beta _{1})} \biggr) \\& \qquad {}+ T_{k,4} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (p_{1}+1)} \biggr) + \cdots+T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert \bigr)+ T_{k,4m-2} \bigl(s, \bigl\Vert x'_{m} \bigr\Vert \bigr) \\& \qquad {} +T_{k,4m-1} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (2- \beta _{m})} \biggr) +T_{k,4m} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (p_{m}+1)} \biggr) \biggr] \,ds \\& \qquad {} + \frac{1}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \biggl[ T_{k,1} \bigl(s, \Vert x_{1} \Vert \bigr)+ T_{k,2} \bigl(s, \bigl\Vert x'_{1} \bigr\Vert \bigr) \\& \qquad {}+ T_{k,3} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (2- \beta _{1})} \biggr) + T_{k,4} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (p_{1}+1)} \biggr) +\cdots+T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert \bigr) \\& \qquad {}+T_{k,4m-2} \bigl(s, \bigl\Vert x'_{m} \bigr\Vert \bigr) +T_{k,4m-1} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (2- \beta _{m})} \biggr) +T_{k,4m} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (p_{m}+1)} \biggr) \biggr] \,ds \\& \qquad {}+ \frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \biggl[ T_{k,1} \bigl(s, \Vert x_{1} \Vert \bigr)+ T_{k,2} \bigl(s, \bigl\Vert x'_{1} \bigr\Vert \bigr) \\& \qquad {}+ T_{k,3} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (2- \beta _{1})} \biggr) + T_{k,4} \biggl(s, \frac{ \Vert x_{1} \Vert }{\Gamma (p_{1}+1)} \biggr) +\cdots+T_{k,4m-3} \bigl(s, \Vert x_{m} \Vert \bigr) \\& \qquad {}+T_{k,4m-2} \bigl(s, \bigl\Vert x'_{m} \bigr\Vert \bigr) +T_{k,4m-1} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (2- \beta _{m})} \biggr) +T_{k,4m} \biggl(s, \frac{ \Vert x_{m} \Vert }{\Gamma (p_{m}+1)} \biggr) \biggr] \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k} -1)} \int ^{t}_{0} (t-s)^{ \alpha _{k} - 2} \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds \\& \qquad {}+ \frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\theta _{k}}_{0} (\theta _{k} -s)^{\alpha _{k} - 1} \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds \\& \qquad {}+ \frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{\eta _{k}}_{0} (\eta _{k} -s)^{\alpha - 1} \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds \\& \qquad {}+\frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{ \gamma _{i,k}}_{0} (\gamma _{i,k} -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \bigl[ T_{k,1}(s, \Delta r)+ \cdots+T_{k,4m}(s, \Delta r) \bigr] \,ds \\& \quad \leq \frac{1}{\Gamma (\alpha _{k} -1)} \int ^{1}_{0} (1-s)^{ \alpha _{k} - 2} \Biggl[ \sum _{j=1}^{m} \bigl(b_{k,j}(s)+ \epsilon _{0} \bigr) \Delta r \Biggr] \,ds \\& \qquad {}+ \frac{2}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \int ^{1}_{0} (1 -s)^{\alpha - 1} \Biggl[ \sum _{j=1}^{m} \bigl(b_{k,j}(s)+ \epsilon _{0} \bigr) \Delta r \Biggr] \,ds \\& \qquad {}+\frac{1 }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \int ^{1}_{0} (1 -s)^{\alpha _{k} - \mu _{i,k} -1} \\& \qquad {}\times \Biggl[ \sum_{j=1}^{m} \bigl(b_{k,j}(s)+\epsilon _{0} \bigr) \Delta r \Biggr] \,ds \\& \quad \leq \frac{\Delta r }{\Gamma (\alpha _{k} -2)} \sum_{j=1}^{m} \biggl[ \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} b_{k,j}(s) \,ds+ \epsilon _{0} \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} \,ds \biggr] \\& \qquad {}+ \frac{2 \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{j=1}^{m} \biggl[ \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} b_{k,j}(s) \,ds+ \epsilon _{0} \int ^{1}_{0} (1-s)^{\alpha _{k} - 1} \,ds \biggr] \\& \qquad {}+ \frac{ \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \\& \qquad {}\times \Biggl( \sum_{j=1}^{m} \biggl[ \int ^{1}_{0} (1-s)^{\alpha _{k} - 2} b_{k,j}(s) \,ds+ \epsilon _{0} \int ^{1}_{0} (1-s)^{\alpha _{k} - \mu _{i,k} -1} \,ds \biggr] \Biggr) \\& \quad = \frac{\Delta r }{\Gamma (\alpha _{k} -1)} \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert + \frac{m \epsilon _{0}}{\Gamma ^{2}(\alpha _{k} -1)} \Delta r \\& \qquad {}+ \frac{2 \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert + \frac{2 m \epsilon _{0}}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma ^{2}(\alpha _{k})} \Delta r \\& \qquad {}+ \frac{ \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggl( \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \Biggr) \\& \qquad {}+ \frac{ m \epsilon _{0}}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma ^{2} (\alpha _{k} -\mu _{i,k})} \Delta r. \end{aligned}$$
Thus, we get
$$\begin{aligned}& \bigl\Vert \phi '_{k}(x_{1},\ldots, x_{n}) \bigr\Vert \\& \quad \leq \Biggl( \Biggl[ \frac{1 }{\Gamma (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \qquad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \sum_{j=1}^{m} \Vert \hat{b}_{k,j} \Vert \\& \qquad {}+ m \epsilon _{0} \Biggl[ \frac{1 }{\Gamma ^{2} (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma ^{2} (\alpha _{k})} \\& \qquad {}+ \frac{(2 - \eta _{k} ) \Delta r}{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum _{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma ^{2} (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \Biggr) \Delta r \\& \quad \leq r \end{aligned}$$
and so \(\|\phi _{k}(x_{1},\ldots, x_{n}) \|_{*} \leq r\) for all \(1 \leq k \leq m\). Hence,
$$ \bigl\Vert F(x_{1},\ldots, x_{n}) \bigr\Vert _{**} = \max_{1 \leq k \leq m} \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) \bigr\Vert _{*} \leq r $$
and so \(F(x_{1},\ldots, x_{n}) \in C\). For similar reasons, we find \(F(y_{1},\ldots, y_{n}) \in C\) and so
$$ \alpha \bigl(F(x_{1},\ldots, x_{n}), F(y_{1}, \ldots, y_{n}) \bigr) \geq 1. $$
Since \(C \neq \phi \), for each \((x_{1},\ldots, x_{n}) \in C\), \(F(x_{1},\ldots, x_{n}) \in C\) and so
$$ \alpha \bigl((x_{1},\ldots, x_{n}), F(x_{1}, \ldots, x_{n}) \bigr) \geq 1. $$
Let
$$ \begin{aligned} \lambda &:= \max_{1 \leq k \leq m} \Biggl\{ \Biggl( \frac{1}{\Gamma (\alpha _{k} -1)} + \frac{2 (2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\ &\quad {}+ \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) q_{k} \Vert \hat{\Phi }_{k} \Vert _{[0,1]} \Biggr\} \Delta \\ &< 1 \end{aligned} $$
and \((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) \in C\), Then \(\alpha ((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) ) = 1\). On the other hand by using (4), for each \((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) \in X^{m}\), \(t \in [0,1]\) and \(1 \leq k \leq m\) we have
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t)-\phi _{k}(y_{1},\ldots, y_{n}) (t) \bigr\vert \\& \quad \leq \frac{ M_{k} ( \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**},\ldots, \Delta \Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \Vert _{**})}{\Gamma (\alpha _{k})} \Vert \hat{\Phi _{k}} \Vert \\& \qquad {}+ \bigl((1 - \eta _{k} + t ) M_{k} \bigl( \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**},\ldots,\\& \qquad \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}\bigr)/\bigl( \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})\bigr)\bigr) \Vert \hat{\Phi _{k}} \Vert \\& \qquad {}+ \bigl((1 - \eta _{k} + t ) M_{k} \bigl( \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**},\ldots,\\& \qquad \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}\bigr)/\bigl( \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})\bigr)\bigr) \Vert \hat{\Phi _{k}} \Vert \\& \qquad {}+ (1 - \eta _{k} + t ) M_{k} \bigl( \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**},\ldots,\\& \qquad \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}\bigr)/\bigl( \bigl\vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \bigr\vert \bigr) \\& \qquad {}\times \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Vert \hat{\Phi _{k}} \Vert . \end{aligned}$$
Since \((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) \in C\), \(\| (x_{1},\ldots, x_{n}) \|_{**} \leq r\) and \(\| (y_{1},\ldots, y_{n}) \|_{**} \leq r\). Thus,
$$\begin{aligned}& \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1}, \ldots, y_{n}) \bigr\Vert _{**} \\& \quad \leq \bigl\Vert (x_{1},\ldots, x_{n}) \bigr\Vert _{**} + \bigl\Vert (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \leq r + r \leq \frac{\delta _{M}}{2} + \frac{\delta _{M}}{2} = \delta _{M}. \end{aligned}$$
By using (5), for each \(1 \leq k \leq m\) we get
$$\begin{aligned}& M_{k} \bigl(\Delta \bigl\Vert (x_{1},\ldots, x_{n}) \bigr\Vert _{**} - \bigl\Vert (y_{1}, \ldots, y_{n}) \bigr\Vert _{**} ,\ldots, \Delta \bigl\Vert (x_{1},\ldots, x_{n}) \bigr\Vert _{**} - \bigl\Vert (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \bigr) \\& \quad < (q_{k} + \epsilon _{1}) \Delta \bigl\Vert (x_{1},\ldots, x_{n}) \bigr\Vert _{**} - \bigl\Vert (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \end{aligned}$$
and so
$$\begin{aligned}& \bigl\vert \phi _{k}(x_{1},\ldots, x_{n}) (t)-\phi _{k}(y_{1},\ldots, y_{n}) (t) \bigr\vert \\& \quad \leq \frac{\hat{\| \Phi _{k}} \| }{\Gamma (\alpha _{k})} (q_{k} + \epsilon _{1}) \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \Vert \hat{\Phi _{k}} \Vert (q_{k} + \epsilon _{1}) \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \Vert \hat{\Phi _{k}} \Vert (q_{k} + \epsilon _{1}) \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \\& \qquad {}+ \frac{(1 - \eta _{k} + t ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \\& \qquad {}\times \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Vert \hat{\Phi _{k}} \Vert (q_{k} + \epsilon _{1}) \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}. \end{aligned}$$
It implies that
$$\begin{aligned}& \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) - \phi _{k}(y_{1},\ldots, y_{n}) \bigr\Vert \\& \quad \leq \Biggl( \frac{1 }{\Gamma (\alpha _{k})} + \frac{2 (2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} + \frac{(1 - \eta _{k} + t ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr) \\& \qquad {}\times (q_{k} + \epsilon _{1}) \Delta \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \\& \quad \leq \lambda \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**} \end{aligned}$$
and so
$$\begin{aligned}& \bigl\Vert \phi _{k}(x_{1},\ldots, x_{n}) - \phi _{k}(y_{1},\ldots, y_{n}) \bigr\Vert \leq \lambda \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}. \end{aligned}$$
Similarly, we can find
$$ \bigl\Vert \phi '_{k}(x_{1}, \ldots, x_{n}) -\phi '_{k}(y_{1}, \ldots, y_{n}) \bigr\Vert \leq \lambda \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**} $$
and so \(\|\phi _{k}(x_{1},\ldots, x_{n}) -\phi _{k}(y_{1},\ldots, y_{n}) \| _{*} \leq \lambda \|(x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n})\|_{**}\). Hence,
$$\begin{aligned} \bigl\Vert F(x_{1},\ldots, x_{n}) - F(y_{1}, \ldots, y_{n}) \bigr\Vert _{**} =& \max _{1 \leq k \leq m} \bigl\Vert \phi _{k}(x_{1}, \ldots, x_{n}) -\phi _{k}(y_{1},\ldots, y_{n}) \bigr\Vert _{*} \\ \leq& \lambda \bigl\Vert (x_{1},\ldots, x_{n})- (y_{1},\ldots, y_{n}) \bigr\Vert _{**}. \end{aligned}$$
(8)
Now, consider the map \(\psi : [0, \infty ) \to [0, \infty )\) defined by \(\psi (t)= \lambda t\). If \((x_{1},\ldots, x_{n}) \notin C\) or \((y_{1},\ldots, y_{n}) \notin C\), then \(\alpha ((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) ) = 0\) and so
$$\begin{aligned}& \alpha \bigl((x_{1},\ldots, x_{n}), (y_{1}, \ldots, y_{n}) \bigr) \,d \bigl(F(x_{1},\ldots, x_{n}), F(y_{1},\ldots, y_{n}) \bigr) \\& \quad \leq \psi \bigl((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) \bigr). \end{aligned}$$
If \((x_{1},\ldots, x_{n})\), \((y_{1},\ldots, y_{n}) \notin C\), then \(\alpha ((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) ) =1\) and so by using (8), we obtain
$$\begin{aligned}& \alpha \bigl((x_{1},\ldots, x_{n}), (y_{1}, \ldots, y_{n}) \bigr) \,d \bigl(F(x_{1},\ldots, x_{n}), F(y_{1},\ldots, y_{n}) \bigr) \\& \quad \leq \psi \bigl((x_{1},\ldots, x_{n}), (y_{1},\ldots, y_{n}) \bigr). \end{aligned}$$
Now by using Lemma 1.1, we conclude that F has a fixed point in \(X^{m}\) which is a solution for the problem. □
Here, we present an example to illustrate our main result.
Example 2.3
Consider the pointwise defined problem
$$\begin{aligned} \textstyle\begin{cases} D^{\frac{5}{2}} x(t) = f_{1}(t, x(t), x'(t), D^{\frac{1}{2}}x(t), I^{ \frac{1}{3}}x(t), y(t), y'(t), D^{\frac{1}{3}}y(t), I^{\frac{1}{2}}y(t)), \\ D^{\frac{7}{3}} y(t) =f_{2}(t, x(t), x'(t), D^{\frac{1}{2}}x(t), I^{ \frac{1}{3}}x(t), y(t), y'(t), D^{\frac{1}{3}}y(t), I^{\frac{1}{2}}y(t)), \end{cases}\displaystyle \end{aligned}$$
(9)
with boundary conditions \(x^{\prime \prime }(0)= y^{\prime \prime }(0)=0\), \(x(\frac{1}{2})=y(\frac{1}{2})= D^{\frac{1}{2} }x(\frac{1}{3})\), \(x'(0)= x(\frac{1}{4})\) and \(y'(0)= y(\frac{1}{3})\), where \(f_{1}(t, x_{1},\ldots, x_{8})= \sum_{j=1}^{8} \frac{1}{t^{\sigma _{j}}} |x_{j}| \), \(f_{2}(t, x_{1},\ldots, x_{8})=\frac{c(t)}{p(t)} \sum_{j=1}^{8} |x_{j}|\), \(c(t)=1\) and \(p(t)=0\) whenever \(t\in [0, 1]\cap \mathbb{Q}\), \(c(t)=0\) and \(p(t)=1\) whenever \(t\in [0, 1]\cap \mathbb{Q}^{c}\) and \(\sigma _{1},\ldots, \sigma _{8} \in (0,\frac{1}{2})\). Note that
$$\begin{aligned}& \bigl\vert f_{1}(t, x_{1},\ldots. x_{8}) - f_{1}(t, y_{1},\ldots,y_{8}) \bigr\vert \leq \sum_{k=1}^{8} \frac{1}{50 t^{\sigma _{i}}} \vert x_{k} - y_{k} \vert , \\& \bigl\vert f_{2}(t, x_{1},\ldots. x_{8}) - f_{2}(t, y_{1},\ldots,y_{8}) \bigr\vert \leq \frac{c(t)}{40 p(t)} \sum_{k=1}^{8} \vert x_{k} - y_{k} \vert , \\& \bigl\vert f_{1}(t, x_{1}, \ldots, x_{8}) - f_{1}(t, y_{1}, \ldots, y_{8}) \bigr\vert \leq \Phi _{1}(t) M_{1} \bigl( \vert x_{1} - y_{1} \vert , \ldots, \vert x_{8} - y_{8} \vert \bigr) \end{aligned}$$
and \(|f_{2}(t, x_{1}, \ldots, x_{4m}) - f_{1}(t, y_{1}, \ldots, y_{4m})| \leq \Phi _{2}(t) M_{2}(|x_{1} - y_{1}|, \ldots, |x_{8} - y_{8}|)\), where \(\Phi _{1}(t) = \frac{1}{50t^{\sigma }}\), \(\Phi _{2}(t) = \frac{c(t)}{40p(t)}\), \(\sigma := \min \{\sigma _{1}, \ldots, \sigma _{8} \}\) \(M_{1}(x_{1} , \ldots, x_{8} ) = M_{2}(x_{1}, \ldots, x_{8})=\sum_{k=1}^{8} |x_{k} |\). Also,
$$ \bigl\vert f_{k}(t, x_{1}, \ldots, x_{8}) \bigr\vert \leq \sum_{j=1}^{8} T_{k,j} \bigl(t, \vert x_{k} \vert \bigr) $$
for \(k =1,2\), where \(T_{1,j}(t, |x_{k}|) = \frac{1}{50 t^{\sigma _{j}}} |x_{j}|\) and \(T_{2,j}(t, |x_{k}|) = \frac{c(t)}{40 p(t)} |x_{j}|\) for \(j =1, \dots , 8\). Then \(M_{k} :X^{8} \to \mathbb{R}^{+}\) is nondecreasing with respect to all components, \(\lim_{z \to 0^{+}} \frac{M_{k}(z,\ldots,z)}{z}= 8 :=q_{k} \in [0, \infty )\) for \(k =1,2\), \(T_{k,j}(\cdot,z)\) is nondecreasing with respect to z, \(\lim_{z \to 0^{+}} \frac{T_{1,j}(t,z)}{z} = \frac{1}{50 t^{\sigma _{j}}} :=b_{1,j}(s)\),
$$ \lim_{z \to 0^{+}} \frac{T_{2,j}(t,z)}{z} =\frac{c(t)}{40 p(t)} :=b_{1,j}(s) $$
for \(j =1, \ldots, 8\) and almost all \(t \in [0,1]\), \(\| \hat{\phi }_{1}\| \leq \frac{1}{50(1- \sigma )}\), \(\| \hat{\phi }_{2}\| \leq \frac{1}{60}\) \(\| \hat{b}_{1,j}\| \leq \frac{1}{50(1- \sigma _{j})}\), \(\| \hat{b}_{2,j}\| \leq \frac{1}{60}\),
$$\begin{aligned} \Delta _{\gamma _{1}} =& \Delta _{\gamma _{2}} =\sum _{i=1}^{n_{0}} \frac{\lambda _{i,k} (\gamma _{i,k})^{1- \mu _{i,k}}}{\Gamma (2 -\mu _{i,k})} \\ =& \frac{\lambda _{1} (\gamma _{1})^{1- \mu _{1}}}{\Gamma (2 -\mu _{1})} = \frac{ (\frac{1}{3})^{\frac{1}{2}}{\Gamma (\frac{3}{2})}}{\Gamma (2 -\mu _{1})} \frac{2}{\sqrt{3 \pi }} \end{aligned}$$
and \(1- \Delta _{\gamma _{k}} \neq \eta _{k} - \theta _{k}\). Put
$$\begin{aligned} \Delta =& \max \biggl\{ 1, \frac{1}{\Gamma (2- \beta _{1})}, \frac{1}{\Gamma (2- \beta _{2})}, \frac{1}{\Gamma (p_{1} +1)}, \frac{1}{\Gamma (p_{m} +1)} \biggr\} \\ =& \max \biggl\{ 1, \frac{1}{\Gamma (\frac{3}{2})}, \frac{1}{\Gamma (\frac{5}{3})}, \frac{1}{\Gamma (\frac{5}{3})}, \frac{1}{\Gamma (\frac{1}{2})} \biggr\} =\frac{2}{\sqrt{\pi }}. \end{aligned}$$
Then we have
$$\begin{aligned}& \max_{1 \leq k \leq m} \Biggl[ \frac{1 }{\Gamma (\alpha _{k} -1)} + \frac{2(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert \Gamma (\alpha _{k})} \\& \quad {}+ \frac{(2 - \eta _{k} ) }{ \vert \Delta _{\gamma _{k}} - \theta _{k} - 1 +\eta _{k} \vert } \Biggl( \sum_{i=1}^{n_{0}} \frac{\lambda _{i,k}}{\Gamma (\alpha _{k} -\mu _{i,k})} \Biggr) \Biggr] \max \Biggl\{ \sum _{j=1}^{m} \Vert \hat{b}_{k,j} \Vert , q_{k} \hat{\Phi }_{k} \Biggr\} \\& \quad \max \biggl\{ \biggl[ \frac{1 }{\Gamma (\frac{3}{2})} + \frac{2(\frac{7}{4} ) }{ \vert \frac{2}{\sqrt{3 \pi }} - \theta _{k} - 1 + \frac{1}{4} \vert \Gamma (\frac{5}{2})} + \frac{\frac{7}{4}}{ \vert \frac{2}{\sqrt{3 \pi }} - \frac{1}{2} - 1 +\frac{7}{4} \vert } \biggl( \frac{1}{\Gamma (2)} \biggr) \biggr] \times \frac{16}{50}, \\& \quad \biggl[ \frac{1 }{\Gamma (\frac{5}{2})} + \frac{2(\frac{5}{3} ) }{ \vert \frac{2}{\sqrt{3 \pi }} - \frac{1}{2}- 1 + \frac{1}{3} \vert \Gamma (\frac{7}{2})} + \frac{\frac{5}{3}}{ \vert \frac{2}{\sqrt{3 \pi }} - \frac{1}{2} - 1 +\frac{7}{4} \vert } \biggl( \frac{1}{\Gamma (3)} \biggr) \biggr] \times \frac{2}{15}\biggr\} \in \biggl[0, \frac{1}{\Delta }\biggr). \end{aligned}$$
Now by using Theorem 2.2, problem (9) has a solution.
