We define the Banach space \(\mathcal{C}_{\ell }=\{w_{\ell }:w_{\ell },\mathfrak{D}^{k} w_{\ell },w_{ \ell }^{\prime } \in C[0,1]\}\) with the norm
$$ \Vert w_{\ell } \Vert _{\mathcal{C}_{\ell }}= \sup_{s \in {}[ 0,1]} \bigl\vert w_{\ell }(s ) \bigr\vert + \sup_{s \in {}[ 0,1]} \bigl\vert \mathfrak{D}^{k} w_{\ell }(s ) \bigr\vert + \sup _{s \in {}[ 0,1]} \bigl\vert w_{\ell }^{\prime }(s ) \bigr\vert $$
for \(\ell =1,2,\dots ,18\). It is obvious that the product space \(\mathcal{C}=\mathcal{C}_{1}\times \mathcal{C}_{2}\times \cdots \times \mathcal{C}_{18}\) is a Banach space, where the norm is defined by
$$ \big\Vert w=(w_{1},w_{2},\dots ,w_{18})\big\Vert _{\mathcal{C}}=\sum_{\ell =1}^{18} \Vert w_{\ell } \Vert _{ \mathcal{C}_{\ell }}. $$
In order to apply Lemma 2.3, we introduce the operator \(\mathcal{T} \): \(\mathcal{C}\rightarrow \mathcal{C}\) by
$$ \mathcal{T} (w_{1},w_{2},\dots ,w_{18}) (s ) := \bigl( \mathcal{T}_{1}(w_{1},w_{2}, \dots ,w_{18}), \dots , \mathcal{T}_{18}(w_{1},w_{2}, \dots , w_{18}) (s ) \bigr) , $$
(3.1)
where
$$ \begin{aligned} &\mathcal{T} _{\ell }(w_{1},w_{2}, \dots ,w_{18}) (s ) \\ &\quad = \int _{0}^{s }\frac{(s -\omega )^{j -1}}{\Gamma (j )} \mathcal{H} _{ \ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{\ell }( \omega ),w_{\ell }^{\prime }(\omega ) \bigr)\,d\omega \\ & \qquad {}+ \eta _{3} \biggl( \frac{1}{\eta _{1}+\eta _{2} } + \frac{A_{0}+ s }{A_{1} } \biggr) \int _{0}^{1} \int _{0}^{\omega } \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }(\zeta ),\mathfrak{D}^{k}w_{ \ell }(\zeta ),w_{\ell }^{\prime }(\zeta ) \bigr)\,d\zeta \,d\omega \\ &\qquad {}- \frac{\eta _{2}}{\eta _{1}+\eta _{2} } \int _{0}^{1} \frac{(1-\omega )^{j-1}}{\Gamma (j)} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{\ell }( \omega ),w_{\ell }^{ \prime }(\omega ) \bigr)\,d\omega \\ & \qquad {}- \frac{A_{0} + s}{A_{1} } \biggl[ \frac{\eta _{2}}{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{ \ell }( \omega ),w_{\ell }^{\prime }(\omega ) \bigr)\,d\omega \\ & \qquad {}+ \frac{\eta _{1}}{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k -1} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }( \omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr)\,d\omega \biggr] \end{aligned} $$
(3.2)
for all \(s \in {}[ 0,1]\) and \(w_{\ell }\in \mathcal{C}_{\ell }\).
To simplify calculations, we shall use the following notation:
$$\begin{aligned}& A_{0} = \frac{\eta _{3} - \eta _{2} \Gamma (4-j)}{ ( \eta _{1} + \eta _{2} ) \Gamma (4-j)} , \end{aligned}$$
(3.3)
$$\begin{aligned}& A_{1} = \biggl[\frac{\eta _{1}}{\Gamma (2-k)} + \frac{\eta _{2}}{\Gamma (2-2k)} - \frac{\eta _{3}}{\Gamma (4-j)} \biggr] \neq 0 , \end{aligned}$$
(3.4)
$$\begin{aligned}& \mathcal{F}_{0}^{\ast } = \frac{1 }{\Gamma (j +1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } \biggr) \frac{ \vert \eta _{1} \vert }{\Gamma (j-k+1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert } \biggr) \frac{ \vert \eta _{3} \vert }{2} \\& \hphantom{\mathcal{F}_{0}^{\ast } =} {}+ \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert \Gamma (j-2k+1)} + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert \Gamma (j+1)} \biggr) \vert \eta _{2} \vert , \end{aligned}$$
(3.5)
$$\begin{aligned}& \mathcal{F}_{1}^{\ast } = \frac{1}{\Gamma (j -k +1 )}+ \frac{1 }{\Gamma (2-k) \vert A_{1} \vert } \\& \hphantom{\mathcal{F}_{1}^{\ast } =}{}\times \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr), \end{aligned}$$
(3.6)
$$\begin{aligned}& \mathcal{F}_{2}^{\ast } = \frac{1}{\Gamma (j)}+ \frac{1 }{ \vert A_{1} \vert } \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr) , \end{aligned}$$
(3.7)
$$\begin{aligned}& \mathcal{V}_{0}^{\ast } = \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } \biggr) \frac{ \vert \eta _{1} \vert }{\Gamma (j-k+1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert } \biggr) \frac{ \vert \eta _{3} \vert }{2} \\& \hphantom{\mathcal{V}_{0}^{\ast } =} {}+ \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert \Gamma (j-2k+1)} + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert \Gamma (j+1)} \biggr) \vert \eta _{2} \vert , \end{aligned}$$
(3.8)
$$\begin{aligned}& \mathcal{V} _{1}^{\ast } = \frac{1 }{\Gamma (2-k) \vert A_{1} \vert } \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr) , \end{aligned}$$
(3.9)
$$\begin{aligned}& \mathcal{V} _{2}^{\ast } = \frac{1 }{ \vert A_{1} \vert } \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr) . \end{aligned}$$
(3.10)
Theorem 3.1
Consider the fractional boundary value problem (1.6). Suppose that are continuous functions and there exist constants \(\Xi _{\ell }>0\) for all \(\ell =1,2,\dots ,18\) such that \(\vert \mathcal{H} _{\ell }(s ,w, \tilde{w}, \tilde{\tilde{w}}) \vert \leq \Xi _{\ell } \) for all , \(s \in {}[ 0,1]\). Then problem (1.6) has a solution.
Proof
It is obvious from the implication of (3.2) that the fixed points of \(\mathcal{T} \) defined by (3.1) exist if and only if (1.6) has a solution. To prove this, we first show that \(\mathcal{T} \) is completely continuous.
As \(\mathcal{H} _{1},\mathcal{H} _{2},\dots ,\mathcal{H} _{18}\) are continuous, \(\mathcal{T} :\mathcal{C}\rightarrow \mathcal{C}\) is continuous, too. Let \(\mathcal{O} \in \mathcal{C}\) be a bounded set and \(w=(w_{1},w_{2},\dots ,w_{18})\in \mathcal{C}\), so that for each \(s \in {}[ 0,1]\) we have
$$ \begin{aligned} &\bigl\vert (\mathcal{T} _{\ell }w) (s ) \bigr\vert \\ &\quad \leq \int _{0}^{s }\frac{(s -\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ & \qquad {}+ \vert \eta _{3} \vert \biggl( \frac{1}{ \vert \eta _{1}+\eta _{2} \vert } + \frac{ \vert A_{0} \vert +s }{ \vert A_{1} \vert } \biggr) \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }( \zeta ),\mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }(\zeta ) \bigr) \bigr\vert \,d\zeta \,d\omega \\ & \qquad {}+ \frac{ \vert \eta _{2} \vert }{ \vert \eta _{1}+\eta _{2} \vert } \int _{0}^{1} \frac{(1-\omega )^{j-1}}{\Gamma (j)} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{ \prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ & \qquad {}+ \frac{ \vert A_{0} \vert + s}{ \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ &\qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k -1} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }( \omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \biggr] \\ &\quad \leq \Xi _{\ell } \biggl[ \frac{1 }{\Gamma (j +1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } \biggr) \frac{ \vert \eta _{1} \vert }{\Gamma (j-k+1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert } \biggr) \frac{ \vert \eta _{3} \vert }{2} \\ &\qquad {}+ \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert \Gamma (j-2k+1)} + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert \Gamma (j+1)} \biggr) \vert \eta _{2} \vert \biggr] \\ &\quad =\Xi _{\ell }\mathcal{F}_{0}^{\ast }, \end{aligned} $$
where \(\mathcal{F}_{0}^{\ast }\) is given in (3.5). Also,
$$\begin{aligned}& \bigl\vert \bigl(\mathfrak{D}^{k} \mathcal{T} _{\ell }w \bigr) (s ) \bigr\vert \\& \quad \leq \int _{0}^{s }\frac{(s -\omega )^{j -k -1}}{\Gamma (j-k)} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{3} \vert s^{1-k} }{\Gamma (2-k) \vert A_{1} \vert } \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{ \ell }(\zeta ),\mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }( \zeta ) \bigr) \bigr\vert \,d\zeta \, d \omega \\& \qquad {}+ \frac{s^{1-k} }{\Gamma (2-k) \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k-1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d \omega \biggr] \\& \quad \leq \Xi _{\ell } \biggl[ \frac{1}{\Gamma (j -k +1 )}+ \frac{1 }{\Gamma (2-k) \vert A_{1} \vert } \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr) \biggr] \\& \quad =\Xi _{\ell }\mathcal{F}_{1}^{\ast }, \end{aligned}$$
and
$$ \begin{aligned} \bigl\vert \bigl(\mathcal{T} _{\ell }^{\prime }w \bigr) (s ) \bigr\vert \leq{}& \int _{0}^{s }\frac{(s -\omega )^{j -2}}{\Gamma (j-1)} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ & {}+ \frac{ \vert \eta _{3} \vert }{ \vert A_{1} \vert } \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }( \zeta ),\mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }(\zeta ) \bigr) \bigr\vert \,d\zeta \,d\omega \\ & {}+ \frac{1 }{ \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ & {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k-1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \, d \omega \biggr] \\ \leq{}& \Xi _{\ell } \biggl[ \frac{1}{\Gamma (j)}+ \frac{1 }{ \vert A_{1} \vert } \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr) \biggr] \\ &=\Xi _{\ell }\mathcal{F}_{2}^{\ast } \end{aligned} $$
for all \(s \in {}[ 0,1]\), where \(\mathcal{F}_{1}^{\ast }\), \(\mathcal{F}_{2}^{\ast }\) are defined in (3.6) and (3.7), respectively. Therefore
$$ \bigl\Vert (\mathcal{T} _{\ell }w) (s ) \bigr\Vert _{\mathcal{C}_{ \ell }} \leq \Xi _{\ell } \bigl( \mathcal{F}_{0}^{\ast }+ \mathcal{F}_{1}^{ \ast } + \mathcal{F}_{2}^{\ast } \bigr). $$
Hence,
$$ \bigl\Vert (\mathcal{T}w) (s ) \bigr\Vert _{\mathcal{C}} =\sum _{\ell =1}^{18} \bigl\Vert (\mathcal{T} _{\ell }w) (s ) \bigr\Vert _{\mathcal{C}_{\ell }} \leq \sum _{\ell =1}^{18}\Xi _{\ell } \bigl( \mathcal{F}_{0}^{\ast }+\mathcal{F}_{1}^{\ast } + \mathcal{F}_{2}^{\ast } \bigr) < \infty , $$
which shows that \(\mathcal{T} \) is uniformly bounded.
Now, we have to prove that \(\mathcal{T} \) is equicontinuous. As for this purpose, let \(w=(w_{1},w_{2},\dots , w_{18})\in \mathcal{O}\) and \(s _{1},s _{2}\in {}[ 0,1]\) with \(s _{1}< s _{2}\). Then, we have
$$\begin{aligned}& \bigl\vert (\mathcal{T} _{\ell }w) (s _{2})-(\mathcal{T} _{ \ell }w) (s _{1}) \bigr\vert \\& \quad = \int _{0}^{s _{1}} \frac{(s _{2}-\omega )^{j -1}-(s _{1}-\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \int _{s _{1}}^{s _{2}} \frac{(s _{2}-\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{ \prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{3} \vert (s_{2} -s_{1}) }{ \vert A_{1} \vert } \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }(\zeta ), \mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }(\zeta ) \bigr) \bigr\vert \, d \zeta\, d \omega \\& \qquad {}+ \frac{(s_{2} -s_{1}) }{ \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k-1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \, d \omega \biggr] . \end{aligned}$$
It is clear that if \(s _{1}\rightarrow s _{2}\) then, independently, the right-hand side of the above expression converges to zero. Also,
$$ \lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl(\mathfrak{D}^{k} \mathcal{T} _{\ell }w \bigr) (s_{2} ) - \bigl( \mathfrak{D}^{k} \mathcal{T} _{\ell }w \bigr) (s_{1} ) \bigr\vert =0, \qquad \lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathcal{T} _{\ell }^{\prime }w \bigr) (s_{2} ) - \bigl(\mathcal{T} _{\ell }^{\prime } w \bigr) (s_{1} ) \bigr\vert =0. $$
As a result \(\Vert (\mathcal{T} w)(s _{2})-(\mathcal{T} w)(s _{1}) \Vert _{\mathcal{C}}\rightarrow 0\) as \(s _{1}\rightarrow s _{2}\). This proves that \(\mathcal{T} \) is equicontinuous on \(\mathcal{C}=\mathcal{C}_{1}\times \mathcal{C}_{2}\times \cdots \times \mathcal{C}_{18}\). Now, the Arzela–Ascoli theorem implies the complete continuity of the operator.
Now, we define a subset Θ of \(\mathcal{C}\) by
$$ \Theta := \bigl\{ (w_{1},w_{2},\dots ,w_{18})\in \mathcal{C}:(w_{1},w_{2},\dots ,w_{18})=b \mathcal{T} (w_{1},w_{2},\dots ,w_{18}), b \in (0,1) \bigr\} . $$
Here, we shall show that Θ is bounded. For this, let \((w_{1},w_{2},\dots ,w_{18})\in \Theta \). Then, we can write \((w_{1},w_{2},\dots ,w_{18})=b \mathcal{T} (w_{1},w_{2},\dots ,w_{18}) \), and so \(w_{\ell }(s )=b \mathcal{T} _{\ell }(w_{1},w_{2},\dots ,w_{18}) \), for all \(s \in {}[ 0,1]\) and \(\ell =1,2,\dots ,18\). Thus,
$$ \begin{aligned} \bigl\vert w_{\ell }(s ) \bigr\vert \leq{}& b \bigg[ \int _{0}^{s }\frac{(s -\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ & {}+ \vert \eta _{3} \vert \biggl( \frac{1}{ \vert \eta _{1}+\eta _{2} \vert } + \frac{ \vert A_{0} \vert +s }{ \vert A_{1} \vert } \biggr) \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }( \zeta ),\mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }(\zeta ) \bigr) \bigr\vert \,d\zeta \,d\omega \\ & {}+ \frac{ \vert \eta _{2} \vert }{ \vert \eta _{1}+\eta _{2} \vert } \int _{0}^{1} \frac{(1-\omega )^{j-1}}{\Gamma (j)} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{ \prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ & {}+ \frac{ \vert A_{0} \vert + s}{ \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ & {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k -1} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }( \omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \biggr] \\ \leq{}& b \Xi _{\ell }\mathcal{F}_{0}^{\ast }. \end{aligned} $$
By similar computations, we have \(\vert \mathfrak{D}^{k} w_{\ell }(s ) \vert \leq b \Xi _{ \ell }\mathcal{F}_{1}^{\ast }\), \(\vert w_{\ell }^{\prime }(s ) \vert \leq b \Xi _{\ell } \mathcal{F}_{2}^{\ast } \), where \(\mathcal{F}_{0}^{\ast },\dots ,\mathcal{F}_{2}^{\ast }\) are given in (3.5)–(3.7). Hence
$$ \Vert w \Vert _{\mathcal{C}} =\sum_{\ell =1}^{18} \Vert w_{\ell } \Vert _{\mathcal{C}_{\ell }}\leq b \sum _{\ell =1}^{18}\Xi _{\ell } \bigl( \mathcal{F}_{0}^{\ast }+\mathcal{F}_{1}^{\ast } + \mathcal{F}_{2}^{\ast } \bigr) < \infty , $$
which shows the boundedness of Θ. Now, using Theorem 2.4 and Lemma 2.3, we can claim that \(\mathcal{T} \) has a fixed point in \(\mathcal{C}\). This demonstrates that the fractional boundary value problem (1.6) does indeed have a solution. □
We shall now examine the solution of the fractional boundary value problem (1.6) by applying various conditions.
Theorem 3.2
Consider the fractional boundary value problem (1.6). Assume that are continuous functions and there exist bounded continuous functions , \(\mathcal{W} _{1},\mathcal{W} _{2},\dots , \mathcal{W} _{18}:[0,1]\rightarrow [0,\infty ) \) and nondecreasing continuous functions \(\mathcal{P} _{1},\mathcal{P} _{2},\dots ,\mathcal{P}_{18}:[0,1] \rightarrow [0,\infty ) \) such that
$$ \bigl\vert \mathcal{H} _{\ell }(s ,w, \tilde{w}, \tilde{\tilde{w}}) \bigr\vert \leq \mathcal{W}_{\ell }(s )\mathcal{P} _{\ell } \bigl( \vert w \vert + \vert \tilde{w} \vert + \vert \tilde{\tilde{w}} \vert \bigr) $$
and
$$ \bigl\vert \mathcal{H} _{\ell }(s ,w_{1},w_{2},w_{3})- \mathcal{H} _{ \ell }(s ,\tilde{w}_{1},\tilde{w}_{2}, \tilde{w}_{3}) \bigr\vert \leq \mathcal{S}_{\ell }(s ) \bigl( \vert w_{1}-\tilde{w}_{1} \vert + \vert w_{2}-\tilde{w}_{2} \vert + \vert w_{3}- \tilde{w}_{3} \vert \bigr) $$
for all \(s \in {}[ 0,1]\), and \(\ell =1,2,\dots ,18\). If
$$ \mathcal{F}:= \bigl(\mathcal{V} _{0}^{\ast }+\mathcal{V} _{1}^{\ast } + \mathcal{V}_{2}^{\ast } \bigr) \sum_{\ell =1}^{18} \Vert \mathcal{S} _{\ell } \Vert < 1, $$
then (1.6) has a solution, where \(\Vert \mathcal{S} _{ \ell } \Vert =\sup_{s \in {}[ 0,1]} \vert \mathcal{S} _{ \ell }(s ) \vert \) and the constants \(\mathcal{V} _{0}^{\ast },\dots ,\mathcal{V} _{2}^{\ast }\) are given in (3.8)–(3.10), respectively.
Proof
First, we put \(\Vert \mathcal{W} _{\ell } \Vert =\sup_{s \in {}[ 0,1]} \vert \mathcal{W} _{\ell }(s ) \vert \) and choose a suitable real constant \(\kappa _{\ell }\) such that
$$ \kappa _{\ell }\geq \sum_{\ell =1}^{18} \mathcal{P} _{\ell } \bigl( \Vert w_{\ell } \Vert _{\mathcal{C}_{\ell }} \bigr) \Vert \mathcal{W} _{\ell } \Vert \bigl\{ \mathcal{F}_{0}^{ \ast }+\mathcal{F}_{1}^{\ast } +\mathcal{F}_{2}^{\ast } \bigr\} , $$
(3.11)
where \(\mathcal{F}_{0}^{\ast },\dots ,\mathcal{F}_{2}^{\ast }\) are given in (3.5)–(3.7). We define a set
$$ \mathcal{O}_{\kappa _{\ell }}:= \bigl\{ w=(w_{1},w_{2},\dots ,w_{18}) \in \mathcal{C}: \Vert w \Vert _{\mathcal{C}}\leq \kappa _{ \ell } \bigr\} , $$
where \(\kappa _{\ell }\) is defined in (3.11). It is obvious that \(\mathcal{O}_{\kappa _{\ell }}\) is a nonempty closed bounded and convex subset of \(\mathcal{C}=\mathcal{C} _{1}\times \mathcal{C}_{2}\times \cdots \times \mathcal{C}_{18}\). Now, we define \(\mathcal{T} _{1}\) and \(\mathcal{T} _{2}\) on \(\mathcal{O}_{\kappa _{\ell }}\) by
$$\begin{aligned}& \mathcal{T} _{1}(w_{1},w_{2}, \dots , w_{18}) (s ) := \bigl( \mathcal{T} _{1}^{(1)}(w_{1},w_{2}, \dots , w_{18}) (s ),\dots , \mathcal{T} _{1}^{(18)}(w_{1},w_{2}, \dots , w_{18}) (s ) \bigr) , \\& \mathcal{T} _{2}(w_{1},w_{2},\dots , w_{18}) (s ) := \bigl( \mathcal{T} _{2}^{(1)}(w_{1},w_{2}, \dots , w_{18}) (s ),\dots , \mathcal{T} _{2}^{(18)}(w_{1},w_{2}, \dots , w_{18}) (s ) \bigr) , \end{aligned}$$
where
$$ \bigl( \mathcal{T} _{1}^{(\ell )}w \bigr) (s )= \int _{0}^{s } \frac{(s -\omega )^{j -1}}{\Gamma (j )} \mathcal{H} _{\ell } \bigl(\omega , w_{ \ell }(\omega ),\mathfrak{D}^{k}w_{\ell }( \omega ),w_{\ell }^{\prime }( \omega ) \bigr)\,d\omega $$
(3.12)
and
$$\begin{aligned}& \bigl( \mathcal{T} _{2}^{(\ell )}w \bigr) (s ) \\& \quad =\eta _{3} \biggl( \frac{1}{\eta _{1}+\eta _{2} } + \frac{A_{0}+ s }{A_{1} } \biggr) \int _{0}^{1} \int _{0}^{\omega } \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }(\zeta ),\mathfrak{D}^{k}w_{ \ell }(\zeta ),w_{\ell }^{\prime }(\zeta ) \bigr)\,d\zeta \,d\omega \\& \qquad {}- \frac{\eta _{2}}{\eta _{1}+\eta _{2} } \int _{0}^{1} \frac{(1-\omega )^{j-1}}{\Gamma (j)} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{\ell }( \omega ),w_{\ell }^{ \prime }(\omega ) \bigr)\,d\omega \\& \qquad {}- \frac{A_{0} + s}{A_{1} } \biggl[ \frac{\eta _{2}}{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{ \ell }( \omega ),w_{\ell }^{\prime }(\omega ) \bigr)\,d\omega \\& \qquad {}+ \frac{\eta _{1}}{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k -1} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }( \omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr)\,d\omega \biggr] \end{aligned}$$
for all \(s \in {}[ 0,1]\) and \(w= (w_{1},w_{2},\dots ,w_{18}) \in \mathcal{O}_{\kappa _{\ell }}\). Let
$$ \tilde{\mathcal{P}}_{\ell }=\sup_{w_{\ell }\in \mathcal{C}_{\ell }} \mathcal{P} _{\ell } \bigl( \Vert w_{\ell } \Vert _{ \mathcal{C}_{\ell }} \bigr). $$
Now, for every \(\tilde{w}=(\tilde{w}_{1},\tilde{w}_{2},\dots ,\tilde{w}_{18})\), \(w=(w_{1},w_{2}, \dots ,w_{18})\in \mathcal{O}_{\kappa _{\ell }}\), we have
$$\begin{aligned}& \bigl\vert \bigl( \mathcal{T} _{1}^{(\ell )} \tilde{w} + \mathcal{T} _{2}^{(\ell )}w \bigr) (s) \bigr\vert \\& \quad \leq \int _{0}^{s }\frac{(s -\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , \tilde{w}_{\ell }(\omega ), \mathfrak{D}^{k} \tilde{w}_{\ell }(\omega ),\tilde{w}_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \, d \omega \\& \qquad {}+ \vert \eta _{3} \vert \biggl( \frac{1}{ \vert \eta _{1}+\eta _{2} \vert } + \frac{ \vert A_{0} \vert +s }{ \vert A_{1} \vert } \biggr) \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }( \zeta ),\mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }(\zeta ) \bigr) \bigr\vert \,d\zeta \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{2} \vert }{ \vert \eta _{1}+\eta _{2} \vert } \int _{0}^{1} \frac{(1-\omega )^{j-1}}{\Gamma (j)} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{ \prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \frac{ \vert A_{0} \vert + s}{ \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k -1} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }( \omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \biggr] \\& \quad \leq \int _{0}^{s }\frac{(s -\omega )^{j -1}}{\Gamma (j )} \mathcal{W} _{\ell }(\omega )\mathcal{P} _{\ell } \bigl( \bigl\vert \tilde{w}_{\ell }(\omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k} \tilde{w}_{\ell }(\omega ) \bigr\vert + \bigl\vert \tilde{w}_{\ell }^{ \prime }(\omega ) \bigr\vert \bigr)\,d\omega \\& \qquad {}+ \vert \eta _{3} \vert \biggl( \frac{1}{ \vert \eta _{1}+\eta _{2} \vert } + \frac{ \vert A_{0} \vert +s }{ \vert A_{1} \vert } \biggr) \\& \qquad {} \times \int _{0}^{1} \int _{0}^{\omega } \mathcal{W} _{\ell }( \zeta ) \mathcal{P} _{\ell } \bigl( \bigl\vert w_{\ell }(\zeta ) \bigr\vert + \bigl\vert \mathfrak{D}^{k} w_{\ell }(\zeta ) \bigr\vert + \bigl\vert w_{\ell }^{\prime }(\zeta ) \bigr\vert \bigr)\,d\zeta\, d \omega \\& \qquad {}+ \frac{ \vert \eta _{2} \vert }{ \vert \eta _{1}+\eta _{2} \vert } \int _{0}^{1} \frac{(1-\omega )^{j-1}}{\Gamma (j)} \mathcal{W} _{\ell }(\zeta ) \mathcal{P} _{\ell } \bigl( \bigl\vert w_{\ell }(\omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k} w_{\ell }(\omega ) \bigr\vert + \bigl\vert w_{\ell }^{\prime }( \omega ) \bigr\vert \bigr)\,d\omega \\& \qquad {}+ \frac{ \vert A_{0} \vert + s}{ \vert A_{1} \vert } \\& \qquad {} \times \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \mathcal{W} _{\ell }(\zeta ) \mathcal{P} _{\ell } \bigl( \bigl\vert w_{\ell }(\omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k} w_{\ell }(\omega ) \bigr\vert + \bigl\vert w_{\ell }^{\prime }( \omega ) \bigr\vert \bigr)\,d\omega \\& \qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k -1} \mathcal{W} _{\ell }(\zeta )\mathcal{P} _{\ell } \bigl( \bigl\vert w_{\ell }(\omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k} w_{\ell }(\omega ) \bigr\vert + \bigl\vert w_{\ell }^{ \prime }( \omega ) \bigr\vert \bigr)\,d\omega \biggr] \\& \quad \leq \Vert \mathcal{W} _{\ell } \Vert \tilde{\mathcal{P}}_{ \ell } \biggl[ \frac{1 }{\Gamma (j +1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } \biggr) \frac{ \vert \eta _{1} \vert }{\Gamma (j-k+1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert } \biggr) \frac{ \vert \eta _{3} \vert }{2} \\& \qquad {}+ \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert \Gamma (j-2k+1)} + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert \Gamma (j+1)} \biggr) \vert \eta _{2} \vert \biggr] \\& \quad = \Vert \mathcal{W} _{\ell } \Vert \tilde{\mathcal{P}}_{ \ell } \mathcal{F} _{0}^{\ast }, \end{aligned}$$
and
$$\begin{aligned}& \bigl\vert \mathfrak{D}^{k} \mathcal{T} _{1}^{(\ell )} \tilde{w}(s ) + \mathfrak{D}^{k} \mathcal{T} _{2}^{(\ell )} w(s ) \bigr\vert \\& \quad \leq \int _{0}^{s }\frac{(s -\omega )^{j -k-1}}{\Gamma (j-k)} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , \tilde{w}_{\ell }(\omega ), \mathfrak{D}^{k} \tilde{w}_{\ell }(\omega ),\tilde{w}_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \, d \omega \\& \qquad {}+ \frac{ \vert \eta _{3} \vert s^{1-k} }{\Gamma (2-k) \vert A_{1} \vert } \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }(\zeta ),\mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }( \zeta ) \bigr) \bigr\vert \,d\zeta \,d \omega \\& \qquad {}+ \frac{s^{1-k} }{\Gamma (2-k) \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k-1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d \omega \biggr] \\& \quad \leq \Vert \mathcal{W} _{\ell } \Vert \tilde{\mathcal{P}}_{ \ell } \biggl[ \frac{1}{\Gamma (j -k +1 )}+ \frac{1 }{\Gamma (2-k) \vert A_{1} \vert } \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr) \biggr] \\& \quad = \Vert \mathcal{W} _{\ell } \Vert \tilde{\mathcal{P}}_{ \ell } \mathcal{F}_{1}^{\ast }. \end{aligned}$$
By using similar computations, we have
$$ \begin{aligned} & \bigl\vert \bigl( \mathcal{T} _{1}^{(\ell )} \tilde{w} \bigr)^{\prime } (s ) + \bigl( \mathcal{T} _{2}^{(\ell )} w \bigr)^{\prime } (s ) \bigr\vert \\ &\quad \leq \int _{0}^{s }\frac{(s -\omega )^{j -2}}{\Gamma (j-1)} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , \tilde{w}_{\ell }(\omega ), \mathfrak{D}^{k} \tilde{w}_{\ell }(\omega ),\tilde{w}_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \, d \omega \\ &\qquad {}+ \frac{ \vert \eta _{3} \vert }{ \vert A_{1} \vert } \int _{0}^{1} \int _{0}^{\omega } \bigl\vert \mathcal{H} _{\ell } \bigl( \zeta , w_{\ell }(\zeta ), \mathfrak{D}^{k}w_{\ell }( \zeta ),w_{\ell }^{\prime }(\zeta ) \bigr) \bigr\vert \,d \zeta \,d \omega \\ &\qquad {}+ \frac{1 }{ \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ &\qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k-1} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d \omega \biggr] \\ &\quad \leq \Vert \mathcal{W} _{\ell } \Vert \tilde{\mathcal{P}}_{ \ell } \biggl[ \frac{1}{\Gamma (j)}+ \frac{1 }{ \vert A_{1} \vert } \biggl( \frac{ \vert \eta _{1} \vert }{\Gamma (j-k +1)} + \frac{ \vert \eta _{2} \vert }{\Gamma (j - 2k +1)} + \frac{ \vert \eta _{3} \vert }{2} \biggr) \biggr] \\ &\quad = \Vert \mathcal{W} _{\ell } \Vert \tilde{\mathcal{P}}_{ \ell } \mathcal{F}_{2}^{\ast }. \end{aligned} $$
This yields
$$ \Vert \mathcal{T} _{1} \tilde{w} +\mathcal{T} _{2} w \Vert _{\mathcal{C}} =\sum_{\ell =1}^{18} \bigl\Vert \mathcal{T} _{1}^{(\ell )} \tilde{w} + \mathcal{T}_{2}^{(k)}w \bigr\Vert _{\mathcal{C}_{\ell }} \leq \Vert \mathcal{W} _{\ell } \Vert \tilde{\mathcal{P}}_{\ell } \bigl( \mathcal{F}_{0}^{\ast }+\mathcal{F}_{1}^{\ast } +\mathcal{F}_{2}^{\ast } \bigr) \leq \kappa _{\ell }, $$
and so \(\mathcal{T} _{1} \tilde{w} +\mathcal{T} _{2} w\in \mathcal{O}_{ \kappa _{\ell }}\). Also, the continuity of \(\mathcal{H} _{\ell }\) follows from the continuity of the operator \(\mathcal{T} _{1}\).
Now, we shall show that \(\mathcal{T} _{1}\) is uniformly bounded. As for this purpose, we have
$$ \begin{aligned} \bigl\vert \bigl( \mathcal{T} _{1}^{(\ell )} w \bigr) (s ) \bigr\vert &\leq \int _{0}^{s } \frac{(s -\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{\ell } \bigl(\omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{ \ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ &\leq \frac{1}{\Gamma (j +1)} \Vert \mathcal{W} _{\ell } \Vert \mathcal{P} _{\ell } \bigl( \bigl\vert w_{\ell }(\omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k}w_{\ell }(\omega ) \bigr\vert + \bigl\vert w_{\ell }^{\prime }(\omega ) \bigr\vert \bigr) \end{aligned} $$
for all \(w\in \mathcal{O}_{\kappa _{\ell }}\). Also,
$$ \begin{aligned} \bigl\vert \bigl( \mathfrak{D}^{k} \mathcal{T} _{1}^{( \ell )} w \bigr) (s ) \bigr\vert &\leq \int _{0}^{s } \frac{(s -\omega )^{j -k-1}}{\Gamma (j-k)} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{ \prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ &\leq \frac{1}{\Gamma (j -k+1 )} \Vert \mathcal{W} _{\ell } \Vert \mathcal{P} _{\ell } \bigl( \bigl\vert w_{\ell }(\omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k}w_{\ell }(\omega ) \bigr\vert + \bigl\vert w_{\ell }^{\prime }(\omega ) \bigr\vert \bigr) , \end{aligned} $$
and
$$ \bigl\vert \bigl( \mathcal{T} _{1}^{(\ell )}w \bigr)^{\prime } (s ) \bigr\vert \leq \frac{1}{\Gamma (j )} \Vert \mathcal{W} _{ \ell } \Vert \mathcal{P} _{\ell } \bigl( \bigl\vert w_{\ell }( \omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k}w_{\ell }( \omega ) \bigr\vert + \bigl\vert w_{\ell }^{\prime }(\omega ) \bigr\vert \bigr) , $$
for all \(w \in \mathcal{O}_{\kappa _{\ell }}\). Thus,
$$ \Vert \mathcal{T} _{1} w \Vert _{\mathcal{C}} =\sum _{\ell =1}^{18} \bigl\Vert \mathcal{T} _{1}^{(\ell )} w \bigr\Vert _{\mathcal{C}_{\ell }} \leq \biggl\{ \frac{j +1}{\Gamma (j +1)}+\frac{1}{\Gamma (j -k + 1)} \biggr\} \sum_{\ell =1}^{18} \Vert \mathcal{W} _{\ell } \Vert \mathcal{P} _{\ell } \bigl( \Vert w_{\ell } \Vert _{ \mathcal{C}_{\ell }} \bigr) , $$
which shows that \(\mathcal{T} _{1}\) is uniformly bounded on \(\mathcal{O}_{\kappa _{\ell }}\).
Now, we shall prove that \(\mathcal{T} _{1}\) is compact on \(\mathcal{O}_{\kappa _{\ell }}\). For this, let \(s _{1},s _{2}\in {}[ 0,1]\) with \(s _{1}< s _{2}\). Then, we have
$$ \begin{aligned} & \bigl\vert \bigl( \mathcal{T} _{1}^{(\ell )}w \bigr) (s _{2})- \bigl( \mathcal{T} _{1}^{(k)}w \bigr) (s _{1}) \bigr\vert \\ &\quad \leq \biggl\vert \int _{0}^{s _{2}} \frac{(s _{2}-\omega )^{j -1}}{\Gamma (j )} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{\ell }( \omega ),w_{ \ell }^{\prime }(\omega ) \bigr)\,d\omega \\ &\qquad {}- \int _{0}^{s _{1}} \frac{(s _{1}-\omega )^{j-1}}{\Gamma (j )} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{\ell }( \omega ),w_{\ell }^{ \prime }(\omega ) \bigr)\,d\omega \biggr\vert \\ &\quad \leq \biggl\vert \int _{0}^{s _{1}} \frac{(s _{2}-\omega )^{j -1}-(s _{1}-\omega )^{j -1}}{\Gamma (j )} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{ \ell }( \omega ),w_{\ell }^{\prime }(\omega ) \bigr)\,d\omega \biggr\vert \\ &\qquad {}+ \biggl\vert \int _{s _{1}}^{s _{2}} \frac{(s_{2}-\omega )^{j -1}}{\Gamma (j )} \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ),\mathfrak{D}^{k}w_{\ell }( \omega ),w_{\ell }^{ \prime }(\omega ) \bigr)\,d\omega \biggr\vert \\ &\quad \leq \int _{0}^{s _{1}} \frac{(s _{2}-\omega )^{j-1}-(s _{1}-\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{\ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }(\omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ &\qquad {}+ \int _{s _{1}}^{s _{2}} \frac{(s _{2}-\omega )^{j -1}}{\Gamma (j )} \bigl\vert \mathcal{H} _{ \ell } \bigl( \omega , w_{\ell }(\omega ), \mathfrak{D}^{k}w_{\ell }( \omega ),w_{\ell }^{\prime }( \omega ) \bigr) \bigr\vert \,d\omega \\ &\quad \leq \biggl\{ \frac{s _{2}^{j }-s _{1}^{j }- ( s _{2}-s _{1} ) ^{j }}{\Gamma (j +1)}+ \frac{ ( s _{2}-s _{1} ) ^{j }}{\Gamma (j +1)} \biggr\} \Vert \mathcal{W} _{\ell } \Vert \mathcal{P} _{ \ell } \bigl( \Vert w_{\ell } \Vert _{\mathcal{C}_{\ell }} \bigr) . \end{aligned} $$
Moreover, \(\vert ( \mathcal{T} _{1}^{(\ell )}w ) (s _{2})- ( \mathcal{T} _{1}^{(\ell )} w ) (s _{1}) \vert \rightarrow 0\) as \(s _{1}\rightarrow s _{2}\). Also, we have
$$\begin{aligned}& \lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathfrak{D}^{k} \mathcal{T} _{1}^{(\ell )} w \bigr) (s_{2} ) - \bigl( \mathfrak{D}^{k} \mathcal{T} _{1}^{(\ell )} w \bigr) (s_{1} ) \bigr\vert =0, \\& \lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathcal{T} _{1}^{( \ell )}w \bigr)^{\prime } (s_{2} ) - \bigl( \mathcal{T} _{1}^{(\ell )} w \bigr) ^{\prime } (s_{1} ) \bigr\vert =0. \end{aligned}$$
Hence, \(\Vert (\mathcal{T} _{1}w)(s_{2})- (\mathcal{T} _{1}w)(s _{1}) \Vert _{\mathcal{C}}\) converges to zero as \(s _{1}\rightarrow s _{2}\). Thus, \(\mathcal{T} _{1}\) is equicontinuous and so \(\mathcal{T} _{1}\) is relatively compact operator on \(\mathcal{O}_{\kappa _{\ell }}\). Now, by Arzela–Ascoli theorem, we obtain that \(\mathcal{T} _{1}\) is compact on \(\mathcal{O}_{\kappa _{\ell }}\).
Lastly, we need to prove that \(\mathcal{T} _{2}\) is a contraction. For this, let \(\tilde{w},w\in \mathcal{O}_{\kappa _{\ell }}\). Thus, we have
$$\begin{aligned}& \bigl\vert \bigl( \mathcal{T} _{2}^{(\ell )} \tilde{w} \bigr) (s )- \bigl( \mathcal{T} _{2}^{(k)} w \bigr) (s ) \bigr\vert \\& \quad \leq \vert \eta _{3} \vert \biggl( \frac{1}{ \vert \eta _{1}+\eta _{2} \vert } + \frac{ \vert A_{0} \vert +s }{ \vert A_{1} \vert } \biggr) \int _{0}^{1} \int _{0}^{\omega } \mathcal{S}_{\ell }(\omega ) \bigl( \bigl\vert \tilde{w}_{\ell }(\zeta )- w_{\ell }(\zeta ) \bigr\vert \\& \qquad {}+ \bigl\vert \mathfrak{D}^{k} \tilde{w}_{\ell }(\zeta )- \mathfrak{D}^{k} w_{\ell }( \zeta ) \bigr\vert + \bigl\vert \tilde{w}_{\ell }^{\prime }(\zeta )-w_{ \ell }^{\prime }( \zeta ) \bigr\vert \bigr)\,d\zeta \,d\omega \\& \qquad {}+ \frac{ \vert \eta _{2} \vert }{ \vert \eta _{1}+\eta _{2} \vert } \int _{0}^{1} \frac{(1-\omega )^{j-1}}{\Gamma (j)} \mathcal{S}_{\ell }(\omega ) \bigl( \bigl\vert \tilde{w}_{\ell }( \omega )- w_{\ell }(\omega ) \bigr\vert + \bigl\vert \mathfrak{D}^{k} \tilde{w}_{\ell }(\omega )- \mathfrak{D}^{k} w_{\ell }(\omega ) \bigr\vert \\& \qquad {}+ \bigl\vert \tilde{w}_{\ell }^{\prime }(\omega )-w_{\ell }^{ \prime }(\omega ) \bigr\vert \bigr)\,d\omega \\& \qquad {}+ \frac{ \vert A_{0} \vert + s}{ \vert A_{1} \vert } \biggl[ \frac{ \vert \eta _{2} \vert }{\Gamma (j-2k)} \int _{0}^{1}(1-\omega )^{j-2k -1} \mathcal{S}_{\ell }(\omega ) \bigl( \bigl\vert \tilde{w}_{\ell }( \omega )- w_{\ell }(\omega ) \bigr\vert \\& \qquad {}+ \bigl\vert \mathfrak{D}^{k} \tilde{w}_{\ell }(\omega )- \mathfrak{D}^{k} w_{\ell }( \omega ) \bigr\vert + \bigl\vert \tilde{w}_{\ell }^{\prime }(\omega )-w_{ \ell }^{\prime }( \omega ) \bigr\vert \bigr)\,d\omega \\& \qquad {}+ \frac{ \vert \eta _{1} \vert }{\Gamma (j - k)} \int _{0}^{1}(1- \omega )^{j -k -1} \mathcal{S}_{\ell }(\omega ) \bigl( \bigl\vert \tilde{w}_{\ell }( \omega )- w_{\ell }(\omega ) \bigr\vert \\& \qquad {}+ \bigl\vert \mathfrak{D}^{k} \tilde{w}_{\ell }( \omega )- \mathfrak{D}^{k} w_{ \ell }(\omega ) \bigr\vert + \bigl\vert \tilde{w}_{\ell }^{\prime }( \omega )-w_{\ell }^{\prime }( \omega ) \bigr\vert \bigr)\,d\omega \biggr] \\& \quad \leq \Vert \mathcal{S}_{\ell } \Vert \biggl[ \frac{1 }{\Gamma (j +1)} + \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } \biggr) \frac{ \vert \eta _{1} \vert }{\Gamma (j-k+1)} \\& \qquad {}+ \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert \Gamma (j-2k+1)} + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert \Gamma (j+1)} \biggr) \vert \eta _{2} \vert \\& \qquad {}+ \biggl( \frac{1 + \vert A_{0} \vert }{ \vert A_{1} \vert } + \frac{1}{ \vert \eta _{1} + \eta _{2} \vert } \biggr) \frac{ \vert \eta _{3} \vert }{2} \biggr] \Vert \tilde{w}_{\ell }-w_{ \ell } \Vert _{\mathcal{C}_{\ell }} \\& \quad = \Vert \mathcal{S}_{\ell } \Vert \mathcal{V} _{0}^{\ast } \Vert \tilde{w}_{\ell }-w_{\ell } \Vert _{\mathcal{C}_{\ell }} \end{aligned}$$
for each \(\ell = 1, 2, \dots , 18 \), where \(\mathcal{V} _{0}^{\ast }\) is given in (3.8). Also, by similar computations, we have
$$\begin{aligned}& \sup_{s \in {}[ 0,1]} \bigl\vert \bigl( \mathfrak{D}^{k} \mathcal{T} _{2}^{(\ell )}\tilde{w} \bigr) (s )- \bigl( \mathfrak{D}^{k}\mathcal{T} _{2}^{(\ell )} w \bigr) (s) \bigr\vert \leq \Vert \mathcal{S}_{\ell } \Vert \mathcal{V} _{1}^{ \ast } \Vert \tilde{w}_{\ell }-w_{\ell } \Vert _{\mathcal{C}_{ \ell }}, \\& \sup_{s \in {}[ 0,1]} \bigl\vert \bigl( \mathcal{T} _{2}^{( \ell )}z \bigr) ^{\prime }(s )- \bigl( \mathcal{T} _{2}^{(\ell )}y \bigr) ^{\prime }(s ) \bigr\vert \leq \Vert \mathcal{S}_{ \ell } \Vert \mathcal{V} _{2}^{\ast } \Vert z_{k}-w_{\ell } \Vert _{\mathcal{C}_{\ell }}, \end{aligned}$$
where \(\mathcal{V} _{1}^{\ast }\) and \(\mathcal{V} _{2}^{\ast }\) are given in (3.9) and (3.10), respectively. Thus, we have
$$ \Vert \mathcal{T} _{2} \tilde{w} -\mathcal{T} _{2}w \Vert _{\mathcal{C}} =\sum_{\ell =1}^{18} \bigl\Vert \mathcal{T} _{2}^{(\ell )}\tilde{w}- \mathcal{T}_{2}^{(k)}w \bigr\Vert _{\mathcal{C}_{\ell }} \leq \bigl( \mathcal{V}_{0}^{\ast }+ \mathcal{V}_{1}^{\ast } + \mathcal{V}_{2}^{\ast } \bigr) \sum _{\ell =1}^{18} \Vert \mathcal{S}_{\ell } \Vert \Vert \tilde{w}_{k}-w_{\ell } \Vert _{\mathcal{C}_{\ell }}, $$
and so \(\Vert \mathcal{T} _{2} \tilde{w}-\mathcal{T} _{2}w \Vert _{\mathcal{C}}\leq \mathcal{F} \Vert \tilde{w}-w \Vert _{\mathcal{C}} \). As \(\mathcal{F}<1 \), which means that \(\mathcal{T} _{2}\) is a contraction on \(\mathcal{O}_{\kappa _{\ell }}\). As a result of Theorem 2.5, we infer that \(\mathcal{T} \) contains a fixed point that is a solution to the fractional boundary value problem (1.6). □
To show the significance of our results, we present the following example.
Example 3.3
Consider the following system of fractional differential equations:
$$\begin{aligned}& \textstyle\begin{cases} \mathfrak{D}^{1.7}w_{1}(s ) = \frac{8e^{s} \vert w_{1}(s ) \vert }{20{,}000(1 + \vert w_{1}(s) \vert )} +0.0004 e^{s} \vert \mathfrak{D}^{0.04} w_{1}(s) \vert + \frac{2 e^{s} \vert \arcsin w_{1}^{\prime }(s ) \vert }{5000 }, \\ \mathfrak{D}^{1.7}w_{2}(s ) = \frac{e^{s} \vert \arctan w_{2}(s ) \vert }{5000}+0.0002 e^{s} \vert \sin ( \mathfrak{D}^{0.04} w_{2}(s) ) \vert + \frac{30 e^{s} \vert w_{2}^{\prime }(s ) \vert }{150{,}000 ( 1+ \vert w_{2}^{\prime }(s) \vert ) }, \\ \mathfrak{D}^{1.7}w_{3}(s ) = \frac{s \vert w_{3}(s ) \vert }{6000(1 + \vert w_{3}(s) \vert )} + \frac{4 s \vert \mathfrak{D}^{0.04} w_{3}(s ) \vert }{24{,}000 } + \frac{12 s \vert \arcsin w_{3}^{\prime }(s ) \vert }{72{,}000 }, \\ \mathfrak{D}^{1.7}w_{4}(s ) = 0.0009s \vert \sin w_{4}(s ) \vert + \frac{180s \vert \mathfrak{D}^{0.04} w_{4}(s) \vert }{200{,}000 + 200{,}000 \vert \mathfrak{D}^{0.04} w_{4}(s) \vert } + \frac{36s \vert \arcsin w_{4}^{\prime }(s ) \vert }{40{,}000 }, \end{cases}\displaystyle \end{aligned}$$
(3.13)
with boundary conditions
$$\begin{aligned}& \textstyle\begin{cases} \frac{3}{2} w_{1} (0) + \frac{7}{4} w_{1}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{1} (\omega )\,d\omega , \\ \frac{3}{2} \mathcal{D}^{0.04} w_{1} (1) + \frac{7}{4} \mathcal{D}^{0.08}w_{1}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{1} (\omega )\,d\omega , \\ \frac{3}{2} w_{2} (0) + \frac{7}{4} w_{2}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{2} (\omega )\,d\omega , \\ \frac{3}{2} \mathcal{D}^{0.04} w_{2} (1) + \frac{7}{4} \mathcal{D}^{0.08}w_{2}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{2} (\omega )\,d\omega , \\ \frac{3}{2} w_{3} (0) + \frac{7}{4} w_{3}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{3} (\omega )\,d\omega , \\ \frac{3}{2} \mathcal{D}^{0.04} w_{3} (1) + \frac{7}{4} \mathcal{D}^{0.08}w_{3}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{3} (\omega )\,d\omega , \\ \frac{3}{2} w_{4} (0) + \frac{7}{4} w_{4}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{4} (\omega )\,d\omega , \\ \frac{3}{2} \mathcal{D}^{0.04} w_{4} (1) + \frac{7}{4} \mathcal{D}^{0.08}w_{4}(1) = \frac{17}{6} \int _{0}^{1} \mathfrak{D}^{0.7} w_{4} (\omega )\,d\omega , \end{cases}\displaystyle \end{aligned}$$
(3.14)
where \(j =1.7\), \(k=0.04\), \(\eta _{1}=\frac{3}{2}\), \(\eta _{2}=\frac{7}{4}\), \(\eta _{3}= \frac{17}{6}\), and \(\mathfrak{D}^{j}\), \(\mathfrak{D}^{k}\) serve as the Caputo fractional derivative of orders j and k, respectively.
Let are continuous functions defined by
$$\begin{aligned}& \textstyle\begin{cases} \mathcal{H} _{1} ( s ,w(s ), \tilde{w}(s ), \tilde{\tilde{w}}(s ) ) = \frac{8e^{s} \vert w(s) \vert }{20{,}000(1 + \vert w(s) \vert )} +0.0004 e^{s} \vert \mathfrak{D}^{0.04} \tilde{w}(s) \vert \\ \hphantom{\mathcal{H} _{1} ( s ,w(s ), \tilde{w}(s ), \tilde{\tilde{w}}(s ) ) = }{}+ \frac{2 e^{s} \vert \arcsin \tilde{\tilde{w}} (s ) \vert }{5000 }, \\ \mathcal{H} _{2} ( s ,w(s ),\tilde{w}(s ), \tilde{\tilde{w}}(s ) ) = \frac{e^{s} \vert \arctan w(s) \vert }{5000}+0.0002 e^{s} \vert \sin ( \mathfrak{D}^{0.04} \tilde{w}(s) ) \vert \\ \hphantom{\mathcal{H} _{2} ( s ,w(s ),\tilde{w}(s ), \tilde{\tilde{w}}(s ) ) =} {}+ \frac{30 e^{s} \vert \tilde{\tilde{w}}(s ) \vert }{150{,}000 ( 1+ \vert \tilde{\tilde{w}}(s) \vert ) }, \\ \mathcal{H} _{3} ( s ,w(s ),\tilde{w}(s ), \tilde{\tilde{w}}(s ) ) = \frac{s \vert w_{3}(s) \vert }{6000(1 + \vert w_{3}(s) \vert )} + \frac{4 s \vert \mathfrak{D}^{0.04} w_{3}(s ) \vert }{24{,}000 } \\ \hphantom{\mathcal{H} _{3} ( s ,w(s ),\tilde{w}(s ), \tilde{\tilde{w}}(s ) ) = } {}+ \frac{12 s \vert \arcsin w_{3}^{\prime }(s ) \vert }{72{,}000 }, \\ \mathcal{H} _{4} ( s ,w(s ),\tilde{w}(s ), \tilde{\tilde{w}}(s ) ) = 0.0009s \vert \sin w(s) \vert + \frac{180s \vert \mathfrak{D}^{0.04} \tilde{w}(s) \vert }{200{,}000 + 200{,}000 \vert \mathfrak{D}^{0.04} \tilde{w}(s) \vert } \\ \hphantom{\mathcal{H} _{4} ( s ,w(s ),\tilde{w}(s ), \tilde{\tilde{w}}(s ) ) =} {}+ \frac{36s \vert \arcsin \tilde{\tilde{w}}(s ) \vert }{40{,}000 } . \end{cases}\displaystyle \end{aligned}$$
Let . Then, we have
$$\begin{aligned}& \bigl\vert \mathcal{H} _{1} \bigl( s ,w_{1}(s ), \tilde{w}_{1}(s ), \tilde{\tilde{w}}_{1}(s ) \bigr) - \mathcal{H} _{1} \bigl( s ,w_{2}(s ), \tilde{w}_{2}(s ), \tilde{\tilde{w}}_{2}(s ) \bigr) \bigr\vert \\& \quad \leq \frac{e^{s} }{2500} \bigl( \bigl\vert w_{1}(s )- w_{2}(s ) \bigr\vert + \bigl\vert \tilde{w}_{1}(s )- \tilde{w}_{2}(s) \bigr\vert + \bigl\vert \arcsin \tilde{ \tilde{w}}_{1}(s )- \arcsin \tilde{\tilde{w}}_{2}(s) \bigr\vert \bigr), \\& \bigl\vert \mathcal{H} _{2} \bigl( s ,w_{1}(s ), \tilde{w}_{1}(s ), \tilde{\tilde{w}}_{1}(s ) \bigr) - \mathcal{H} _{2} \bigl( s ,w_{2}(s ),\tilde{w}_{2}(s ), \tilde{\tilde{w}}_{2}(s ) \bigr) \bigr\vert \\& \quad \leq \frac{e^{s} }{5000} \bigl( \bigl\vert \arctan w_{1}(s )- \arctan w_{2}(s ) \bigr\vert + \bigl\vert \sin \tilde{w}_{1}(s )- \sin \tilde{w}_{2}(s) \bigr\vert + \bigl\vert \tilde{\tilde{w}}_{1}(s )- \tilde{\tilde{w}}_{2}(s) \bigr\vert \bigr), \\& \bigl\vert \mathcal{H} _{3} \bigl( s ,w_{1}(s ), \tilde{w}_{1}(s ), \tilde{\tilde{w}}_{1}(s ) \bigr) - \mathcal{H} _{3} \bigl( s ,w_{2}(s ),\tilde{w}_{2}(s ), \tilde{\tilde{w}}_{2}(s ) \bigr) \bigr\vert \\& \quad \leq \frac{s }{6000} \bigl( \bigl\vert w_{1}(s )- w_{2}(s ) \bigr\vert + \bigl\vert \tilde{w}_{1}(s )- \tilde{w}_{2}(s) \bigr\vert + \bigl\vert \arcsin \tilde{ \tilde{w}}_{1}(s )- \arcsin \tilde{\tilde{w}}_{2}(s) \bigr\vert \bigr), \\& \bigl\vert \mathcal{H} _{4} \bigl( s ,w_{1}(s ), \tilde{w}_{1}(s ), \tilde{\tilde{w}}_{1}(s ) \bigr) - \mathcal{H} _{4} \bigl( s ,w_{2}(s ),\tilde{w}_{2}(s ), \tilde{\tilde{w}}_{2}(s ) \bigr) \bigr\vert \\& \quad \leq \frac{9s }{10{,}000} \bigl( \bigl\vert \sin w_{1}(s )- \sin w_{2}(s ) \bigr\vert + \bigl\vert \tilde{w}_{1}(s )- \tilde{w}_{2}(s) \bigr\vert + \bigl\vert \arcsin \tilde{ \tilde{w}}_{1}(s )- \arcsin \tilde{\tilde{w}}_{2}(s) \bigr\vert \bigr). \end{aligned}$$
Here,
$$\begin{aligned}& \mathcal{S} _{1}(s ) =\frac{e^{s} }{2500},\qquad \mathcal{S} _{2}(s ) = \frac{e^{s} }{5000}, \qquad \mathcal{S} _{3}(s) = \frac{s}{6000}, \qquad \mathcal{S} _{4}(s) =\frac{9s}{10{,}000}, \\& \Vert \mathcal{S} _{1} \Vert =\frac{1}{2500}, \qquad \Vert \mathcal{S} _{2} \Vert =\frac{1}{5000}, \qquad \Vert \mathcal{S} _{3} \Vert =\frac{1}{6000}, \qquad \Vert \mathcal{S} _{4} \Vert =\frac{9}{10{,}000}. \end{aligned}$$
Let be identity functions. Then, we obtain
$$\begin{aligned} &\bigl\vert \mathcal{H} _{1} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert \leq \frac{e^{s} }{2500} \bigl( \vert w \vert + \vert \mathfrak{D}w \vert + \bigl\vert \arcsin w^{\prime } \bigr\vert \bigr) \\ &\hphantom{bigl\vert \mathcal{H} _{1} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert }\leq \frac{e^{s} }{2500} \bigl( \vert w \vert + \vert \mathfrak{D}w \vert + \bigl\vert w^{\prime } \bigr\vert \bigr), \\ &\bigl\vert \mathcal{H} _{2} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert \leq \frac{e^{s} }{5000} \bigl( \vert \arctan w \vert + \bigl\vert \sin (\mathfrak{D}w) \bigr\vert + \bigl\vert w^{\prime } \bigr\vert \bigr) \\ &\hphantom{\bigl\vert \mathcal{H} _{2} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert }\leq \frac{e^{s} }{5000} \bigl( \vert w \vert + \vert \mathfrak{D}w \vert + \bigl\vert w^{\prime } \bigr\vert \bigr), \\ &\bigl\vert \mathcal{H} _{3} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert \leq \frac{s }{6000} \bigl( \vert w \vert + \vert \mathfrak{D}w \vert + \bigl\vert \arcsin w^{\prime } \bigr\vert \bigr) \\ &\hphantom{\bigl\vert \mathcal{H} _{3} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert }\leq \frac{s }{6000} \bigl( \vert w \vert + \vert \mathfrak{D}w \vert + \bigl\vert w^{\prime } \bigr\vert \bigr), \\ &\bigl\vert \mathcal{H} _{4} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert \leq \frac{9s }{10{,}000} \bigl( \vert \sin w \vert + \vert \mathfrak{D}w \vert + \bigl\vert \arcsin w^{\prime } \bigr\vert \bigr) \\ &\hphantom{\bigl\vert \mathcal{H} _{4} \bigl(s, w(s),\mathfrak{D}^{0.04 }w(s ),w^{ \prime }(s) \bigr) \bigr\vert }\leq \frac{9s }{10{,}000} \bigl( \vert w \vert + \vert \mathfrak{D}w \vert + \bigl\vert w^{\prime } \bigr\vert \bigr), \end{aligned}$$
where the continuous function are defined by
$$ \mathcal{W} _{1}(s )=\frac{e^{s} }{2500},\qquad \mathcal{W} _{2}(s )=\frac{e^{s} }{5000},\qquad \mathcal{W} _{3}(s )=\frac{s }{6000}, \qquad \mathcal{W} _{4}(s )=\frac{9s }{10{,}000} . $$
Furthermore, \(\mathcal{V} _{0}^{\ast }\simeq 5.642\), \(\mathcal{V} _{1}^{\ast } \simeq 4.084\), and \(\mathcal{V} _{2}^{\ast }\simeq 4.019\), thus
$$ \mathcal{F}:= \bigl( \mathcal{V} _{0}^{\ast } + \mathcal{V} _{1}^{ \ast }+\mathcal{V}_{2}^{\ast } \bigr) \bigl( \Vert \mathcal{S} _{1} \Vert + \Vert \mathcal{S} _{2} \Vert + \Vert \mathcal{S} _{3} \Vert + \Vert \mathcal{S} _{4} \Vert \bigr) \simeq 0.023< 1. $$
Hence by Theorem 3.2, the proposed problem (3.13)–(3.14) has a solution.