Throughout this paper, \(\mathcal{U}_{\gamma }= \lbrace z_{\gamma }:z_{\gamma }, \mathfrak{D}^{\nu } z_{\gamma },z_{\gamma }^{\prime },z_{\gamma }^{ \prime \prime } \in C[0,1] \rbrace \) is a Banach space with norm
$$\begin{aligned} \Vert z_{\gamma } \Vert _{\mathcal{U}_{\gamma }}= \sup_{s \in {}[ 0,1]} \bigl\vert z_{\gamma }(s ) \bigr\vert + \sup_{s \in {}[ 0,1]} \bigl\vert \mathfrak{D}^{\nu } z_{\gamma }(s ) \bigr\vert +\sup _{s \in {}[ 0,1]} \bigl\vert z_{\gamma }^{\prime }(s ) \bigr\vert +\sup_{s \in {}[ 0,1]} \bigl\vert z_{\gamma }^{ \prime \prime }(s ) \bigr\vert \end{aligned}$$
for \(\gamma =1,2,\dots,18\). It is obvious that the product space \(\mathcal{U}=\mathcal{U}_{1}\times \mathcal{U}_{2}\times \cdots \times \mathcal{U}_{18}\) is a Banach space with norm
$$\begin{aligned} \Vert z = (z_{1},z_{2},\dots,z_{18})\Vert _{ \mathcal{U}}=\sum_{\gamma =1}^{18} \Vert z_{\gamma } \Vert _{ \mathcal{U}_{\gamma }}. \end{aligned}$$
Referring to Lemma 2.2, we introduce the operator \(\mathcal{S}:\mathcal{U}\rightarrow \mathcal{U}\) by
$$\begin{aligned} \mathcal{S} (z_{1},z_{2},\dots,z_{18}) (s ):= \bigl( \mathcal{S} _{1}(z_{1},z_{2}, \dots,z_{18}),\dots,\mathcal{S}_{18}(z_{1},z_{2}, \dots,z_{18}) (s ) \bigr), \end{aligned}$$
(3.1)
where
$$\begin{aligned} &\mathcal{S} _{\gamma }(z_{1},z_{2}, \dots,z_{18}) (s ) \\ &\quad= \int _{0}^{s } \frac{(s -\xi )^{\ell -1}}{\Gamma (\ell )} \mathcal{S}_{\gamma } \bigl( \xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{\gamma }^{ \prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d\xi \\ &\qquad{} + \frac{1}{V_{0}} \biggl(\frac{\varpi _{2}}{\varpi _{1}} + s \biggr) \\ &\qquad{} \times \biggl[ \varpi _{3} \int _{0}^{\theta } \int _{0}^{\xi } \mathcal{S}_{\gamma } \bigl( \tau,z_{\gamma }(\tau ),\mathfrak{D}^{\nu }z_{ \gamma }(\tau ),z_{\gamma }^{\prime }(\tau ),z_{\gamma }^{\prime \prime }( \tau ) \bigr) \,d\tau\, d\xi \\ &\qquad{} + \varpi _{2} \int _{0}^{1} \mathcal{S}_{\gamma } \bigl( \xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }(\xi ),z_{\gamma }^{ \prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d\xi \\ & \qquad{} - \varpi _{1} \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \biggr] \\ &\qquad{} + \frac{1}{\varpi _{1} V_{0}} ( V_{0} - \varpi _{2} - \varpi _{1}s ) \\ & \qquad{}\times \biggl[ \varpi _{2} \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \\ & \qquad{} + \varpi _{3} \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \biggr] \end{aligned}$$
(3.2)
for \(s \in {}[ 0,1]\) and \(z_{\gamma }\in \mathcal{U}_{\gamma } \).
To facilitate calculations, we use the following notation:
$$\begin{aligned} &V_{0} = \biggl[\varpi _{1} + \frac{\varpi _{2} ( \Gamma (3 - \ell ) - 1 ) }{ \Gamma (3 - \ell )} - \frac{\varpi _{3} \theta ^{3 - \ell }}{\Gamma (4 - \ell )} \biggr] \neq 0 , \end{aligned}$$
(3.3)
$$\begin{aligned} &V_{1} = \biggl[ \vert \varpi _{1} \vert + \frac{ \vert \varpi _{2} \vert \vert ( \Gamma (3 - \ell ) - 1 ) \vert }{ \Gamma (3 - \ell )} + \frac{ \vert \varpi _{3} \vert }{\Gamma (4 - \ell )} \biggr] \neq 0, \end{aligned}$$
(3.4)
$$\begin{aligned} &\mathcal{I}_{0}^{\ast } = \frac{1 }{\Gamma (\ell +1)} + \biggl( \frac{ \vert \varpi _{2} \vert + \vert \varpi _{1} \vert }{ \vert \varpi _{1} \vert V_{1}} \biggr) \biggl(\frac{ \vert \varpi _{3} \vert }{2} + \vert \varpi _{2} \vert + \frac{ \vert \varpi _{1} \vert }{\Gamma (\ell + 1)} \biggr) \\ &\phantom{\mathcal{I}_{0}^{\ast } = }{}+ \biggl( \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1} \vert }{ \vert \varpi _{1} \vert V_{1}} \biggr) \biggl(\frac{ \vert \varpi _{2} \vert }{\Gamma (\ell )} + \frac{ \vert \varpi _{3} \vert }{\Gamma (\ell - 1)} \biggr), \end{aligned}$$
(3.5)
$$\begin{aligned} &\mathcal{I}_{1}^{\ast } = \frac{1 }{\Gamma (\ell - \nu +1)} + \biggl( \frac{1}{V_{1} \Gamma (2 - \nu )} \biggr) \\ &\phantom{\mathcal{I}_{1}^{\ast } = }{} \times \biggl( \frac{ \vert \varpi _{3} (2+\Gamma (\ell -1) ) \vert }{2 \Gamma (\ell - 1)} + \frac{ \vert \varpi _{2} (1+\Gamma (\ell ) ) \vert }{ \Gamma (\ell )} + \frac{ \vert \varpi _{1} \vert }{\Gamma (\ell + 1)} \biggr), \end{aligned}$$
(3.6)
$$\begin{aligned} &\mathcal{I}_{2}^{\ast } = \frac{1}{\Gamma (\ell )} + \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{3} (2+\Gamma (\ell -1) ) \vert }{2 \Gamma (\ell - 1)} + \frac{ \vert \varpi _{2} (1+\Gamma (\ell ) ) \vert }{ \Gamma (\ell )} + \frac{ \vert \varpi _{1} \vert }{\Gamma (\ell + 1)} \biggr), \end{aligned}$$
(3.7)
$$\begin{aligned} &\mathcal{I}_{3}^{\ast } = \frac{1}{\Gamma (\ell - 1)}, \end{aligned}$$
(3.8)
$$\begin{aligned} &\mathcal{I}_{4}^{\ast } = \biggl( \frac{ \vert \varpi _{2} \vert + \vert \varpi _{1} \vert }{ \vert \varpi _{1} \vert V_{1}} \biggr) \biggl(\frac{ \vert \varpi _{3} \vert }{2} + \vert \varpi _{2} \vert + \frac{ \vert \varpi _{1} \vert }{\Gamma (\ell + 1)} \biggr) \\ &\phantom{\mathcal{I}_{4}^{\ast } =}{} + \biggl( \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1} \vert }{ \vert \varpi _{1} \vert V_{1}} \biggr) \biggl(\frac{ \vert \varpi _{2} \vert }{\Gamma (\ell )} + \frac{ \vert \varpi _{3} \vert }{\Gamma (\ell - 1)} \biggr), \end{aligned}$$
(3.9)
$$\begin{aligned} &\mathcal{I} _{5}^{\ast } = \biggl( \frac{1}{V_{1} \Gamma (2 - \nu )} \biggr) \biggl( \frac{ \vert \varpi _{3} (2+\Gamma (\ell -1) ) \vert }{2 \Gamma (\ell - 1)} + \frac{ \vert \varpi _{2} (1+\Gamma (\ell ) ) \vert }{ \Gamma (\ell )} + \frac{ \vert \varpi _{1} \vert }{\Gamma (\ell + 1)} \biggr), \end{aligned}$$
(3.10)
$$\begin{aligned} &\mathcal{I} _{6}^{\ast } = \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{3} (2+\Gamma (\ell -1) ) \vert }{2 \Gamma (\ell - 1)} + \frac{ \vert \varpi _{2} (1+\Gamma (\ell ) ) \vert }{ \Gamma (\ell )} + \frac{ \vert \varpi _{1} \vert }{\Gamma (\ell + 1)} \biggr). \end{aligned}$$
(3.11)
Theorem 3.1
Let \(\mathcal{S}_{1},\mathcal{S}_{2},\dots,\mathcal{S}_{18}:[0,1]\times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) be continuous functions, and let there exist constants \(M_{\gamma } > 0\), \(\gamma =1,2,\dots,18\), satisfying
$$\begin{aligned} \bigl\vert \mathcal{S} _{\gamma } ( s,z_{1},z_{2}, z_{3}, z_{4} ) \bigr\vert \leq M_{\gamma } \end{aligned}$$
for all \(z, z_{1},z_{2}, z_{3}, z_{4} \in \mathbb{R}\) and \(s \in {}[ 0,1]\). Then problem (1.4) has a solution.
Proof
It is obvious from (3.2) that the fixed points of the operator \(\mathcal{S} \) given in (3.1) exist if and only if (1.4) has a solution. To prove this, we first show that \(\mathcal{S} \) is completely continuous.
As \(\mathcal{S}_{1},\mathcal{S}_{2},\dots,\mathcal{S}_{18}\) are continuous, \(\mathcal{S}:\mathcal{U}\rightarrow \mathcal{U}\) is continuous too. Let \(\mathcal{V} \in \mathcal{U}\) be a bounded set, and let \(z=(z_{1},z_{2},\dots,z_{18})\in \mathcal{U}\). So for each \(s \in {}[ 0,1]\), we have
$$\begin{aligned} &\bigl\vert ( \mathcal{S} _{\gamma }z ) (s ) \bigr\vert \\ &\quad \leq \int _{0}^{s }\frac{(s -\xi )^{\ell -1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ), \mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{ \gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{2} \vert }{ \vert \varpi _{1} \vert } + s \biggr) \\ &\qquad{}\times [ \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{ \xi } \bigl\vert \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }( \tau ),z_{\gamma }^{\prime }(\tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\tau\, d\xi \\ &\qquad{} + \vert \varpi _{2} \vert \int _{0}^{1} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }(\xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi ] \\ &\qquad{} + \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1}s \vert }{ \vert \varpi _{1} \vert V_{1}} \\ &\qquad{} \times \biggl[ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \varpi _{3} \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ &\quad\leq M_{\gamma } \mathcal{I}_{0}^{\ast }, \end{aligned}$$
where \(\mathcal{I}_{0}^{\ast }\) is given in (3.5). Also,
$$\begin{aligned} &\bigl\vert \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{\gamma }z \bigr) (s ) \bigr\vert \\ &\quad \leq \int _{0}^{s } \frac{(s -\xi )^{\ell - \nu - 1}}{\Gamma (\ell - \nu )} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \biggl( \frac{s^{1-\nu }}{ V_{1} \Gamma (2 - \nu )} \biggr) \\ &\qquad{}\times \biggl[ \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{ \xi } \bigl\vert \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }( \tau ),z_{\gamma }^{\prime }(\tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\tau\, d\xi \\ &\qquad{} + \vert \varpi _{2} \vert \int _{0}^{1} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }(\xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & \qquad{} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ & \qquad{}+ \biggl( \frac{s^{1-\nu }}{ V_{1} \Gamma (2 - \nu )} \biggr) \\ &\qquad{}\times \biggl[ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & \qquad{} + \vert \varpi _{3} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ &\quad\leq M_{\gamma } \mathcal{I}_{1}^{\ast }, \\ &\bigl\vert \bigl( \mathcal{S} _{\gamma }^{\prime }z \bigr) (s ) \bigr\vert \\ &\quad \leq \int _{0}^{s } \frac{(s -\xi )^{\ell - 2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & \qquad{}+ \frac{1}{V_{1}} \biggl[ \vert \varpi _{3} \vert \int _{0}^{ \theta } \int _{0}^{\xi } \bigl\vert \mathcal{S}_{\gamma } \bigl(\tau,z_{ \gamma }(\tau ),\mathfrak{D}^{\nu }z_{\gamma }( \tau ),z_{\gamma }^{ \prime }(\tau ),z_{\gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\tau\, d\xi \\ &\qquad{} + \vert \varpi _{2} \vert \int _{0}^{1} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }(\xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & \qquad{} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ & \qquad{}+ \frac{1}{V_{1}} \biggl[ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \vert \varpi _{3} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ &\quad\leq M_{\gamma } \mathcal{I}_{2}^{\ast }, \end{aligned}$$
and
$$\begin{aligned} \bigl\vert \bigl( \mathcal{S} _{\gamma }^{\prime \prime }z \bigr) (s ) \bigr\vert & \leq \int _{0}^{s } \frac{(s -\xi )^{\ell - 3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\leq M_{\gamma } \mathcal{I}_{3}^{\ast } \end{aligned}$$
for all \(s \in {}[ 0,1]\), where \(\mathcal{I}_{1}^{\ast }\)–\(\mathcal{I}_{3}^{\ast }\) are defined in (3.6)–(3.8), respectively. Therefore
$$\begin{aligned} \bigl\Vert ( \mathcal{S} _{\gamma }z ) (s ) \bigr\Vert _{ \mathcal{U}_{\gamma }} \leq M_{\gamma } \bigl( \mathcal{I}_{0}^{\ast }+ \mathcal{I}_{1}^{\ast } + \mathcal{I}_{2}^{\ast } + \mathcal{I}_{3}^{ \ast } \bigr). \end{aligned}$$
Hence
$$\begin{aligned} \bigl\Vert ( \mathcal{S}z ) (s ) \bigr\Vert _{ \mathcal{U}} &=\sum _{\gamma =1}^{18} \bigl\Vert ( \mathcal{S}_{\gamma }z ) (s ) \bigr\Vert _{\mathcal{U}_{\gamma }} \\ &\leq \sum_{\gamma =1}^{18} M_{\gamma } \bigl( \mathcal{I}_{0}^{ \ast }+\mathcal{I}_{1}^{\ast } + \mathcal{I}_{2}^{\ast } + \mathcal{I}_{3}^{\ast } \bigr) \\ &< \infty, \end{aligned}$$
which reveals that \(\mathcal{S} \) is uniformly bounded.
Now we have to show that \(\mathcal{S} \) is equicontinuous. For this purpose, let \(z=(z_{1},z_{2},\dots,z_{18})\in \mathcal{V}\) and \(s _{1},s _{2} \in {}[ 0,1]\) with \(s _{1}< s _{2}\). Then we have
$$\begin{aligned} &\bigl\vert (\mathcal{S} _{\gamma }z) (s _{2})-(\mathcal{S} _{\gamma }z) (s_{1}) \bigr\vert \\ &\quad\leq \int _{0}^{s _{1}} \frac{(s _{2}-\theta )^{\ell -1}-(s _{1}-\theta )^{\ell -1}}{\Gamma (\ell )} \\ &\qquad{}\times \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ), \mathfrak{D}^{\nu }z_{\gamma }(\xi ),z_{\gamma }^{\prime }( \xi ),z_{ \gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{}+ \int _{s _{1}}^{s _{2}} \frac{(s _{2}-\theta )^{\ell -1}}{\Gamma ( \ell )} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \biggl( \frac{s_{2} - s_{1} }{V_{1}} \biggr) \\ &\qquad{}\times \biggl[ \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{ \xi } \bigl\vert \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }( \tau ),z_{\gamma }^{\prime }(\tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\tau\, d\xi \\ &\qquad{} + \vert \varpi _{2} \vert \int _{0}^{1} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }(\xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ &\qquad{} + \biggl( \frac{s_{2} - s_{1} }{V_{1}} \biggr) \\ &\qquad{} \times \biggl[ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{} + \vert \varpi _{3} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \biggr]. \end{aligned}$$
We can see that if \(s _{1}\rightarrow s _{2}\), then, independently, the right-hand side of the expression converges to zero. Also,
$$\begin{aligned} &\lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{\gamma }z \bigr) (s_{2} ) - \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{\gamma }z \bigr) (s_{1} ) \bigr\vert = 0, \\ &\lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathcal{S} _{ \gamma }^{\prime }z \bigr) (s_{2} ) - \bigl( \mathcal{S} _{\gamma }^{ \prime } z \bigr) (s_{1} ) \bigr\vert = 0, \\ &\lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathcal{S} _{ \gamma }^{\prime \prime }z \bigr) (s_{2} ) - \bigl( \mathcal{S} _{ \gamma }^{\prime \prime } z \bigr) (s_{1} ) \bigr\vert = 0. \end{aligned}$$
As a result, \(\Vert ( \mathcal{S} z ) (s _{2}) - ( \mathcal{S} z ) (s _{1}) \Vert _{\mathcal{U}} \rightarrow 0\) as \(s _{1}\rightarrow s _{2}\). This proves that \(\mathcal{S} \) is equicontinuous on \(\mathcal{U}=\mathcal{U}_{1}\times \mathcal{U}_{2}\times \cdots \times \mathcal{U}_{18}\). Now the Arzelà–Ascoli theorem implies the complete continuity of the operator.
Further, we define the subset Λ of \(\mathcal{U}\) as
$$\begin{aligned} \Lambda:= \bigl\{ (z_{1},z_{2},\dots,z_{18})\in \mathcal{U}:(z_{1},z_{2},\dots,z_{18}) = \vartheta \mathcal{S} (z_{1},z_{2}, \dots,z_{18}), \vartheta \in (0,1) \bigr\} . \end{aligned}$$
We will show that Λ is bounded. For this, let \((z_{1},z_{2},\dots,z_{18})\in \Lambda \). Then we can write
$$\begin{aligned} (z_{1},z_{2},\dots,z_{18}) = \vartheta \mathcal{S} (z_{1},z_{2}, \dots,z_{18}), \end{aligned}$$
and so
$$\begin{aligned} z_{\gamma }(s ) = \vartheta \mathcal{S} _{\gamma }(z_{1},z_{2}, \dots,z_{18}) \end{aligned}$$
for all \(s \in {}[ 0,1]\) and \(\gamma =1,2,\dots,18\). Thus
$$\begin{aligned} \bigl\vert z_{\gamma }(s ) \bigr\vert \leq {}& \vartheta \biggl[ \int _{0}^{s }\frac{(s -\xi )^{\ell -1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &{} + \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{2} \vert }{ \vert \varpi _{1} \vert } + s \biggr) \biggl\lbrace \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{ \xi } \bigl\vert \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }( \tau ),z_{\gamma }^{\prime }(\tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\tau\, d\xi \\ &{} + \vert \varpi _{2} \vert \int _{0}^{1} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }(\xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & {} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \biggr\rbrace \\ &{} + \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1}s \vert }{ \vert \varpi _{1} \vert V_{1}} \\ & {}\times \biggl\lbrace \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & {} + \vert \varpi _{3} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \biggr\rbrace \biggr] \\ \leq {}& \vartheta M_{\gamma }\mathcal{I}_{0}^{\ast }, \end{aligned}$$
and by similar computations we have
$$\begin{aligned} &\bigl\vert \mathfrak{D}^{\nu } z_{\gamma }(s ) \bigr\vert \leq \vartheta M_{\gamma }\mathcal{I}_{1}^{\ast }, \\ &\bigl\vert z_{\gamma }^{\prime }(s ) \bigr\vert \leq \vartheta M_{ \gamma }\mathcal{I}_{2}^{\ast }, \\ &\bigl\vert z_{\gamma }^{\prime \prime }(s ) \bigr\vert \leq \vartheta M_{\gamma }\mathcal{I}_{3}^{\ast }, \end{aligned}$$
where \(\mathcal{I}_{0}^{\ast }\)–\(\mathcal{I}_{3}^{\ast }\) are given in (3.5)–(3.8). Hence
$$\begin{aligned} \Vert z \Vert _{\mathcal{U}} &=\sum_{\gamma =1}^{18} \Vert z_{\gamma } \Vert _{\mathcal{U}_{\gamma }} \\ &\leq \vartheta \sum_{\gamma =1}^{18} M_{\gamma } \bigl( \mathcal{I}_{0}^{\ast }+ \mathcal{I}_{1}^{\ast } + \mathcal{I}_{2}^{ \ast } + \mathcal{I}_{3}^{\ast } \bigr) \\ &< \infty, \end{aligned}$$
which shows the boundedness of Λ. Now using Theorem 2.4 and Lemma 2.2, we see that \(\mathcal{S} \) has a fixed point in \(\mathcal{U}\). This demonstrates that (1.4) does indeed have a solution. □
We will now examine the solution of problem (1.4) by applying various conditions.
Theorem 3.2
Suppose that \(\mathcal{S}_{1},\mathcal{S}_{2},\dots,\mathcal{S}_{18}:[0,1] \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) are continuous functions and that there exist bounded continuous functions \(\mathcal{G}_{1},\mathcal{G} _{2},\dots,\mathcal{G} _{18}:[0,1] \rightarrow \mathbb{R}\), \(\mathcal{Z} _{1},\mathcal{Z} _{2},\dots, \mathcal{Z} _{18}:[0,1]\rightarrow [0,\infty ) \) and nondecreasing continuous functions \(\mathcal{L} _{1},\mathcal{L} _{2},\dots,\mathcal{L}_{18}:[0,1] \rightarrow [0,\infty ) \) such that
$$\begin{aligned} \bigl\vert \mathcal{S} _{\gamma } ( s,z_{1},z_{2}, z_{3}, z_{4} ) \bigr\vert \leq \mathcal{Z}_{\gamma }(s ) \mathcal{L}_{ \gamma } \bigl( \vert z_{1} \vert + \vert z_{2} \vert + \vert z_{3} \vert + \vert z_{4} \vert \bigr) \end{aligned}$$
and
$$\begin{aligned} &\bigl\vert \mathcal{S} _{\gamma } ( s,z_{1},z_{2},z_{3}, z_{4} ) - \mathcal{S} _{\gamma } ( s,\tilde{z}_{1}, \tilde{z}_{2}, \tilde{z}_{3}, \tilde{z}_{4} ) \bigr\vert \\ &\quad \leq \mathcal{G}_{ \gamma }(s ) \bigl( \vert z_{1}-\tilde{z}_{1} \vert + \vert z_{2}- \tilde{z}_{2} \vert + \vert z_{3}- \tilde{z}_{3} \vert + \vert z_{4}-\tilde{z}_{4} \vert \bigr) \end{aligned}$$
for all \(s \in {}[ 0,1]\), \(z_{1},z_{2},z_{3},z_{4}, \tilde{z}_{1}, \tilde{z}_{2},\tilde{z}_{3}, \tilde{z}_{4} \in \mathbb{R,}\) and \(\gamma =1,2,\dots,18\). If
$$\begin{aligned} \Delta:= \bigl(\mathcal{I} _{4}^{\ast }+\mathcal{I}_{5}^{\ast } + \mathcal{I}_{6}^{\ast } \bigr) \sum _{\gamma =1}^{18} \Vert \mathcal{G} _{\gamma } \Vert < 1, \end{aligned}$$
then (1.4) has a solution, where \(\Vert \mathcal{G} _{\gamma } \Vert = \sup_{s \in {}[ 0,1]} \vert \mathcal{G} _{\gamma }(s ) \vert \), and the constants \(\mathcal{I} _{4}^{\ast }\)–\(\mathcal{I} _{6}^{\ast }\) are given in (3.9)–(3.11), respectively.
Proof
Let \(\Vert \mathcal{Z}_{\gamma } \Vert =\sup_{s \in {}[ 0,1]} \vert \mathcal{Z}_{\gamma }(s ) \vert \). Suppose that for suitable constants \(\varepsilon _{\gamma }\), we have
$$\begin{aligned} \varepsilon _{\gamma }\geq \sum_{\gamma =1}^{18} \mathcal{L}_{ \gamma } \bigl( \Vert z_{\gamma } \Vert _{\mathcal{U}_{ \gamma }} \bigr) \Vert \mathcal{Z}_{\gamma } \Vert \bigl\{ \mathcal{I}_{0}^{\ast }+ \mathcal{I}_{1}^{\ast } +\mathcal{I}_{2}^{ \ast } +\mathcal{I}_{3}^{\ast } \bigr\} , \end{aligned}$$
(3.12)
where \(\mathcal{I}_{0}^{\ast }\)–\(\mathcal{I}_{3}^{\ast }\) are given in (3.5)–(3.8). We define the set
$$\begin{aligned} \mathcal{V}_{\varepsilon _{\gamma }}:= \bigl\{ z=(z_{1},z_{2}, \dots,z_{18}) \in \mathcal{U}: \Vert z \Vert _{\mathcal{U}}\leq \varepsilon _{\gamma } \bigr\} , \end{aligned}$$
where \(\varepsilon _{\gamma }\) is defined in (3.12). It is obvious that \(\mathcal{V}_{\varepsilon _{\gamma }}\) is a nonempty, closed, bounded, and convex subset of \(\mathcal{U}=\mathcal{U} _{1}\times \mathcal{U}_{2}\times \cdots \times \mathcal{U}_{18}\). Now we define \(\mathcal{S} _{1}\) and \(\mathcal{S} _{2}\) on \(\mathcal{O}_{\varepsilon _{\gamma }}\) by
$$\begin{aligned} &\mathcal{S} _{1} ( z_{1},z_{2},\dots, z_{18} ) (s ) := \bigl( \mathcal{S}_{1}^{(1)} ( z_{1},z_{2},\dots, z_{18} ) (s ),\dots, \mathcal{S}_{1}^{(18)} ( z_{1},z_{2}, \dots, z_{18} ) (s ) \bigr), \\ &\mathcal{S} _{2} ( z_{1},z_{2},\dots, z_{18} ) (s ) := \bigl( \mathcal{S}_{2}^{(1)} ( z_{1},z_{2},\dots, z_{18} ) (s ),\dots, \mathcal{S}_{2}^{(18)} ( z_{1},z_{2}, \dots, z_{18} ) (s ) \bigr), \end{aligned}$$
where
$$\begin{aligned} \bigl( \mathcal{S} _{1}^{(\gamma )} z \bigr) (s ) = \int _{0}^{s } \frac{(s -\xi )^{\ell -1}}{\Gamma (\ell )} \mathcal{S}_{\gamma } \bigl( \xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{\gamma }^{ \prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d\xi, \end{aligned}$$
(3.13)
and
$$\begin{aligned} &\bigl( \mathcal{S} _{2}^{(\gamma )} z \bigr) (s ) \\ &\quad = \frac{1}{V_{0}} \biggl(\frac{\varpi _{2} }{ \varpi _{1} } + s \biggr) \biggl[\varpi _{3} \int _{0}^{\theta } \int _{0}^{\xi } \mathcal{S}_{ \gamma } \bigl( \tau,z_{\gamma }(\tau ),\mathfrak{D}^{\nu }z_{\gamma }( \tau ),z_{\gamma }^{\prime }(\tau ),z_{\gamma }^{\prime \prime }(\tau ) \bigr) \,d\tau\, d\xi \\ & \qquad{} + \varpi _{2} \int _{0}^{1} \mathcal{S}_{\gamma } \bigl( \xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }(\xi ),z_{\gamma }^{ \prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d\xi \\ & \qquad{} + \varpi _{1} \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \biggr] \\ & \qquad{}+ \biggl( \frac{ V_{0} - \varpi _{2} - \varpi _{1}s }{ \varpi _{1} V_{0}} \biggr) \\ &\qquad{} \times \biggl[ \varpi _{2} \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \\ &\qquad{} + \vert \varpi _{3} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \biggr] \end{aligned}$$
(3.14)
for all \(s \in {}[ 0,1]\) and z= \((z_{1},z_{2},\dots,z_{18}) \in \mathcal{V}_{\varepsilon _{\gamma }}\).
Let \(\tilde{\mathcal{L}}_{\gamma }=\sup_{z_{\gamma }\in \mathcal{U}_{ \gamma }}\mathcal{L}_{\gamma } ( \Vert z_{\gamma } \Vert _{\mathcal{U}_{\gamma }} ) \). For all \(\tilde{z}=(\tilde{z}_{1},\tilde{z}_{2},\dots,\tilde{z}_{18}), z=(z_{1},z_{2}, \dots,z_{18})\in \mathcal{V}_{\varepsilon _{\gamma }}\), we have
$$\begin{aligned} &\bigl\vert \bigl( \mathcal{S} _{1}^{(\gamma )}\tilde{z} + \mathcal{S} _{2}^{(\gamma )}z \bigr) (s) \bigr\vert \\ &\quad\leq \int _{0}^{s }\frac{(s -\xi )^{\ell -1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,\tilde{z}_{\gamma }(\xi ), \mathfrak{D}^{\nu } \tilde{z}_{\gamma }(\xi ),\tilde{z}_{\gamma }^{\prime }( \xi ),\tilde{z}_{ \gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \\ & \qquad{}+ \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{2} \vert }{ \vert \varpi _{1} \vert } + s \biggr) \\ &\qquad{}\times \biggl[ \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{ \xi } \bigl\vert \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }( \tau ),z_{\gamma }^{\prime }(\tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\tau\, d\xi \\ & \qquad{} + \vert \varpi _{2} \vert \int _{0}^{1} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }(\xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & \qquad{} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ & \qquad{}+ \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1}s \vert }{ \vert \varpi _{1} \vert V_{1}} \\ &\qquad{} \times \biggl[ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ & \qquad{} + \vert \varpi _{3} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \biggr] \\ &\quad\leq \int _{0}^{s }\frac{(s -\xi )^{\ell -1}}{\Gamma (\ell )} \mathcal{Z}_{\gamma }(\xi )\mathcal{L}_{\gamma } \bigl( \bigl\vert \tilde{z}_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } \tilde{z}_{\gamma }(\xi ) \bigr\vert + \bigl\vert \tilde{z}_{\gamma }^{ \prime }( \xi ) \bigr\vert + \bigl\vert \tilde{z}_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr) \,d\xi \\ &\qquad{} + \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{2} \vert }{ \vert \varpi _{1} \vert } + s \biggr) \biggl[ \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{\xi } \mathcal{Z}_{\gamma }(\tau ) \\ & \qquad{}\times \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }(\tau ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } z_{\gamma }(\tau ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\tau ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\tau ) \bigr\vert \bigr) \,d \tau\, d\theta \\ & \qquad{} + \vert \varpi _{2} \vert \int _{0}^{1} \mathcal{Z}_{ \gamma }(\xi ) \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr) \,d\xi \\ &\qquad{} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \mathcal{Z}_{\gamma }( \xi ) \\ &\qquad{}\times \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr) \,d\xi \biggr] \\ & \qquad{}+ \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1}s \vert }{ \vert \varpi _{1} \vert V_{1}} \biggl[ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \mathcal{Z}_{\gamma }( \xi ) \\ &\qquad{}\times \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr) \,d\xi \\ & \qquad{}+ \vert \varpi _{3} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \mathcal{Z}_{\gamma }( \xi ) \\ &\qquad{}\times \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }( \xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr) \,d \xi \biggr] \\ &\quad\leq \Vert \mathcal{Z}_{\gamma } \Vert \tilde{\mathcal{L}}_{\tau } \mathcal{I}_{0}^{\ast }. \end{aligned}$$
By using similar computations we have
$$\begin{aligned} &\bigl\vert \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{1}^{(\gamma )} \tilde{z} \bigr) (s ) + \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{2}^{( \gamma )} z \bigr) (s ) \bigr\vert \leq \Vert \mathcal{Z}_{ \gamma } \Vert \tilde{\mathcal{L}}_{\tau } \mathcal{I}_{1}^{\ast }, \\ &\bigl\vert \bigl( \mathcal{S} _{1}^{(\gamma )} \tilde{z} \bigr)^{ \prime } (s ) + \bigl( \mathcal{S} _{2}^{(\gamma )} z \bigr)^{ \prime } (s ) \bigr\vert \leq \Vert \mathcal{Z}_{\gamma } \Vert \tilde{\mathcal{L}}_{\tau } \mathcal{I}_{2}^{\ast }, \end{aligned}$$
and
$$\begin{aligned} \bigl\vert \bigl( \mathcal{S} _{1}^{(\gamma )} \tilde{z} \bigr)^{ \prime \prime } (s ) + \bigl( \mathcal{S} _{2}^{(\gamma )} z \bigr)^{ \prime \prime } (s ) \bigr\vert \leq \Vert \mathcal{Z}_{ \gamma } \Vert \tilde{\mathcal{L}}_{\tau } \mathcal{I}_{3}^{\ast }. \end{aligned}$$
This yields that
$$\begin{aligned} \Vert \mathcal{S} _{1} \tilde{z} +\mathcal{S} _{2} z \Vert _{\mathcal{U}} &= \sum_{\gamma =1}^{18} \bigl\Vert \mathcal{S} _{1}^{(\gamma )} \tilde{z} + \mathcal{S}_{2}^{(\gamma )} z \bigr\Vert _{\mathcal{U}_{\gamma }} \\ &\leq \Vert \mathcal{Z}_{\gamma } \Vert \tilde{\mathcal{L}}_{\tau } \bigl( \mathcal{I}_{0}^{\ast }+ \mathcal{I}_{1}^{\ast } +\mathcal{I}_{2}^{\ast } +\mathcal{I}_{3}^{ \ast } \bigr) \\ &\leq \varepsilon _{\gamma }, \end{aligned}$$
and so \(\mathcal{S} _{1} \tilde{z} +\mathcal{S} _{2} z \in \mathcal{V}_{ \varepsilon _{\gamma }}\). Furthermore, the continuity of \(\mathcal{S} _{1}\) is implied by the continuity of the operator \(\mathcal{S} _{\gamma }\).
We will now demonstrate that \(\mathcal{S} _{1}\) is uniformly bounded. For this, we have
$$\begin{aligned} \bigl\vert \bigl( \mathcal{S} _{1}^{(\gamma )} z \bigr) (s ) \bigr\vert &\leq \int _{0}^{s } \frac{(s -\xi )^{\ell -1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \\ &\leq \frac{1}{\Gamma (\ell + 1)} \Vert \mathcal{Z}_{\gamma } \Vert \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu }z_{\gamma }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr) \end{aligned}$$
for all z∈ \(\mathcal{V}_{\varepsilon _{\gamma }}\). Also,
$$\begin{aligned} \bigl\vert \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{1}^{(\gamma )} z \bigr) (s ) \bigr\vert &\leq \int _{0}^{s } \frac{(s -\theta )^{\ell - \nu - 1}}{\Gamma (\ell - \nu )} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{ \nu }z_{\gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \\ &\leq \frac{1}{\Gamma (\ell - \nu + 1 )} \Vert \mathcal{Z}_{ \gamma } \Vert \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }( \xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu }z_{\gamma }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr), \end{aligned}$$
and
$$\begin{aligned} &\bigl\vert \bigl( \mathcal{S} _{1}^{(\gamma )}z \bigr)^{\prime } (s ) \bigr\vert \leq \frac{1}{\Gamma (\ell )} \Vert \mathcal{Z}_{\gamma } \Vert \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu }z_{ \gamma }( \xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }(\xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr), \\ &\bigl\vert \bigl( \mathcal{S} _{1}^{(\gamma )}z \bigr)^{\prime \prime } (s ) \bigr\vert \leq \frac{1}{\Gamma (\ell - 1 )} \Vert \mathcal{Z}_{\gamma } \Vert \mathcal{L}_{\gamma } \bigl( \bigl\vert z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{ \nu }z_{\gamma }( \xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime }( \xi ) \bigr\vert + \bigl\vert z_{\gamma }^{\prime \prime }(\xi ) \bigr\vert \bigr) \end{aligned}$$
for all z∈ \(\mathcal{V}_{\varepsilon _{\gamma }}\). Thus
$$\begin{aligned} \Vert \mathcal{S} _{1} z \Vert _{\mathcal{U}} &= \sum _{\gamma =1}^{18} \bigl\Vert \mathcal{S} _{1}^{(\gamma )} z \bigr\Vert _{\mathcal{U}_{\gamma }} \\ &\leq \biggl\{ \frac{\ell ^{2}}{\Gamma (\ell +1)}+ \frac{1}{\Gamma (\ell - \nu + 1)} \biggr\} \sum _{\gamma =1}^{18} \Vert \mathcal{Z}_{\gamma } \Vert \mathcal{L}_{\gamma } \bigl( \Vert z_{\gamma } \Vert _{\mathcal{U}_{\gamma }} \bigr), \end{aligned}$$
which shows that \(\mathcal{S} _{1}\) is uniformly bounded on \(\mathcal{V}_{\varepsilon _{\gamma }}\).
Now we will prove that \(\mathcal{S} _{1}\) is compact on \(\mathcal{V}_{\varepsilon _{\gamma }}\). For this, let \(s _{1},s _{2}\in {}[ 0,1]\) with \(s _{1}< s _{2}\). Then we have
$$\begin{aligned} &\bigl\vert \bigl( \mathcal{S} _{1}^{(\gamma )}z \bigr) (s _{2})- \bigl( \mathcal{S} _{1}^{(\gamma )}z \bigr) (s _{1}) \bigr\vert \\ &\quad \leq \biggl\vert \int _{0}^{s _{2}} \frac{(s _{2}-\theta )^{\ell - 1}}{\Gamma (\ell )} \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \\ &\qquad{} - \int _{0}^{s _{1}} \frac{(s _{1}-\theta )^{\ell - 1}}{\Gamma (\ell )} \mathcal{S}_{ \gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \biggr\vert \\ &\quad\leq \biggl\vert \int _{0}^{s _{1}} \frac{(s _{2}-\theta )^{\ell - 1}-(s _{1}-\theta )^{\ell - 1}}{\Gamma (\ell )} \\ & \qquad{}\times\mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ), \mathfrak{D}^{\nu }z_{\gamma }(\xi ),z_{\gamma }^{\prime }( \xi ),z_{ \gamma }^{\prime \prime }(\xi ) \bigr) \,d\xi \biggr\vert \\ &\qquad{}+ \biggl\vert \int _{s _{1}}^{s _{2}} \frac{(s_{2}-\theta )^{\ell - 1}}{\Gamma (\ell )} \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{\gamma }( \xi ),z_{ \gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }(\xi ) \bigr) \,d \xi \biggr\vert \\ &\qquad\leq \int _{0}^{s _{1}} \frac{(s _{2}-\theta )^{\ell - 1}-(s _{1}-\theta )^{\ell - 1}}{\Gamma (\ell )} \\ &\qquad{} \times \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ), \mathfrak{D}^{\nu }z_{\gamma }(\xi ),z_{\gamma }^{\prime }( \xi ),z_{ \gamma }^{\prime \prime }(\xi ) \bigr) \bigr\vert \,d\xi \\ &\qquad{}+ \int _{s _{1}}^{s _{2}} \frac{(s _{2}-\theta )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,z_{\gamma }(\xi ),\mathfrak{D}^{\nu }z_{ \gamma }( \xi ),z_{\gamma }^{\prime }(\xi ),z_{\gamma }^{\prime \prime }( \xi ) \bigr) \bigr\vert \,d\xi \\ &\quad\leq \biggl\{ \frac{s _{2}^{\ell } - s _{1}^{\ell } - (s _{2}-s _{1} ) ^{\ell }}{\Gamma (\ell + 1)} + \frac{ ( s _{2}-s _{1} ) ^{\ell }}{\Gamma (\ell + 1)} \biggr\} \Vert \mathcal{Z}_{\gamma } \Vert \mathcal{L}_{ \gamma } \bigl( \Vert z_{\gamma } \Vert _{\mathcal{U}_{ \gamma }} \bigr). \end{aligned}$$
Hence \(\vert ( \mathcal{S} _{1}^{(\gamma )}z ) (s _{2}) - (\mathcal{S} _{1}^{(\gamma )} z ) (s _{1}) \vert \rightarrow 0\) as \(s _{1}\rightarrow s _{2}\). Also, we have
$$\begin{aligned} &\lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{1}^{(\gamma )} z \bigr) (s_{2} ) - \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{1}^{(\gamma )} z \bigr) (s_{1} ) \bigr\vert = 0, \\ &\lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathcal{S} _{1}^{( \gamma )}z \bigr)^{\prime } (s_{2} ) - \bigl( \mathcal{S} _{1}^{( \gamma )} z \bigr) ^{\prime } (s_{1} ) \bigr\vert = 0, \\ &\lim_{s_{1} \rightarrow s_{2}} \bigl\vert \bigl( \mathcal{S} _{1}^{( \gamma )}z \bigr)^{\prime \prime } (s_{2} ) - \bigl( \mathcal{S} _{1}^{( \gamma )} z \bigr) ^{\prime \prime } (s_{1} ) \bigr\vert =0. \end{aligned}$$
Hence \(\Vert (\mathcal{S} _{1}z)(s_{2})- (\mathcal{S} _{1}z)(s _{1}) \Vert _{\mathcal{U}}\) tends to zero as \(s _{1}\rightarrow s _{2}\). Thus \(\mathcal{S} _{1}\) is equicontinuous, and therefore \(\mathcal{S} _{1}\) is relatively compact operator on \(\mathcal{V}_{\varepsilon _{\gamma }}\). So \(\mathcal{S} _{1}\) is compact on \(\mathcal{V}_{\varepsilon _{\gamma }}\) by the Arzelà–Ascoli theorem.
It remains to prove that \(\mathcal{S} _{2}\) is a contraction. To show this, letting \(\tilde{z},z\in \mathcal{V}_{\varepsilon _{\gamma }}\), we obtain
$$\begin{aligned} &\bigl\vert \bigl( \mathcal{S} _{2}^{(\gamma )} \tilde{z} \bigr) (s )- \bigl( \mathcal{S} _{2}^{(\gamma )} z \bigr) (s ) \bigr\vert \\ &\quad \leq \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{2} \vert }{ \vert \varpi _{1} \vert } + s \biggr) \\ &\qquad{}\times \biggl[ \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{ \xi } \bigl\vert \mathcal{S}_{\gamma } \bigl(\tau,\tilde{z}_{\gamma }( \tau ),\mathfrak{D}^{\nu } \tilde{z}_{\gamma }(\tau ),\tilde{z}_{\gamma }^{ \prime }(\tau ), \tilde{z}_{\gamma }^{\prime \prime }(\tau ) \bigr) \\ & \qquad{}- \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }(\tau ),z_{\gamma }^{\prime }( \tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\tau\, d\xi \\ &\qquad{} + \vert \varpi _{2} \vert \int _{0}^{1} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,\tilde{z}_{\gamma }(\xi ),\mathfrak{D}^{\nu } \tilde{z}_{\gamma }(\xi ),\tilde{z}_{\gamma }^{\prime }(\xi ), \tilde{z}_{ \gamma }^{\prime \prime }(\xi ) \bigr) \\ & \qquad{} - \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }(\tau ),z_{\gamma }^{\prime }( \tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\xi \\ & \qquad{}+ \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \bigl\vert \mathcal{S}_{ \gamma } \bigl(\xi,\tilde{z}_{\gamma }(\xi ), \mathfrak{D}^{\nu } \tilde{z}_{\gamma }(\xi ),\tilde{z}_{\gamma }^{\prime }( \xi ),\tilde{z}_{ \gamma }^{\prime \prime }(\xi ) \bigr) \\ & \qquad{} - \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }(\tau ),z_{\gamma }^{\prime }( \tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\xi \biggr] \\ & \qquad{}+ \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1}s \vert }{ \vert \varpi _{1} \vert V_{1}} \\ &\qquad{}\times \biggl[ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,\tilde{z}_{\gamma }(\xi ), \mathfrak{D}^{ \nu }\tilde{z}_{\gamma }(\xi ),\tilde{z}_{\gamma }^{\prime }( \xi ), \tilde{z}_{\gamma }^{\prime \prime }(\xi ) \bigr) \\ &\qquad{} - \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }(\tau ),z_{\gamma }^{\prime }( \tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\xi \\ & \qquad{}+ \varpi _{3} \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \bigl\vert \mathcal{S}_{\gamma } \bigl(\xi,\tilde{z}_{\gamma }(\xi ), \mathfrak{D}^{ \nu }\tilde{z}_{\gamma }(\xi ),\tilde{z}_{\gamma }^{\prime }( \xi ), \tilde{z}_{\gamma }^{\prime \prime }(\xi ) \bigr) \\ &\qquad{} - \mathcal{S}_{\gamma } \bigl(\tau,z_{\gamma }(\tau ), \mathfrak{D}^{\nu }z_{\gamma }(\tau ),z_{\gamma }^{\prime }( \tau ),z_{ \gamma }^{\prime \prime }(\tau ) \bigr) \bigr\vert \,d\xi \biggr] \\ &\quad\leq \frac{1}{V_{1}} \biggl( \frac{ \vert \varpi _{2} \vert }{ \vert \varpi _{1} \vert } + s \biggr) \biggl[ \vert \varpi _{3} \vert \int _{0}^{\theta } \int _{0}^{\xi } \mathcal{G}_{\gamma }(s ) \bigl( \bigl\vert \tilde{z}_{\gamma }(\tau )- z_{\gamma }(\tau ) \bigr\vert \\ &\qquad{} + \bigl\vert \mathfrak{D}^{\nu } \tilde{z}_{\gamma }( \tau )- \mathfrak{D}^{\nu } z_{\gamma }(\tau ) \bigr\vert + \bigl\vert \tilde{z}^{\prime }_{\gamma }(\tau )-z^{\prime }_{\gamma }( \tau ) \bigr\vert \\ &\qquad{} + \bigl\vert \tilde{z}^{\prime \prime }_{\gamma }(\tau )-z^{ \prime \prime }_{\gamma }(\tau ) \bigr\vert \bigr) \,d\tau\, d\xi + \vert \varpi _{2} \vert \int _{0}^{1} \mathcal{G}_{\gamma }(s ) \bigl( \bigl\vert \tilde{z}_{\gamma }(\xi )- z_{\gamma }(\xi ) \bigr\vert \\ & \qquad{} + \bigl\vert \mathfrak{D}^{\nu } \tilde{z}_{\gamma }( \xi )- \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \tilde{z}^{\prime }_{\gamma }(\xi )-z^{\prime }_{\gamma }( \xi ) \bigr\vert + \bigl\vert \tilde{z}^{\prime \prime }_{\gamma }(\xi )-z^{ \prime \prime }_{\gamma }(\xi ) \bigr\vert \bigr) \,d\xi \\ &\qquad{} + \vert \varpi _{1} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell - 1}}{\Gamma (\ell )} \mathcal{G}_{\gamma }(s ) \bigl( \bigl\vert \tilde{z}_{\gamma }(\xi )- z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } \tilde{z}_{\gamma }(\xi )- \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert \\ & \qquad{} + \bigl\vert \tilde{z}^{\prime }_{\gamma }(\xi )-z^{ \prime }_{\gamma }(\xi ) \bigr\vert + \bigl\vert \tilde{z}^{\prime \prime }_{\gamma }(\xi )-z^{\prime \prime }_{\gamma }(\xi ) \bigr\vert \bigr) \biggr] \,d\xi \\ & \qquad{}+ \frac{ \vert V_{1} - \varpi _{2} - \varpi _{1}s \vert }{ \vert \varpi _{1} \vert V_{1}} [ \vert \varpi _{2} \vert \int _{0}^{1} \frac{(1 - \xi )^{\ell -2}}{\Gamma (\ell - 1)} \mathcal{G}_{\gamma }(s ) \bigl( \bigl\vert \tilde{z}_{\gamma }(\xi )- z_{\gamma }(\xi ) \bigr\vert \\ & \qquad{} + \bigl\vert \mathfrak{D}^{\nu } \tilde{z}_{\gamma }( \xi )- \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \tilde{z}^{\prime }_{\gamma }(\xi )-z^{\prime }_{\gamma }( \xi ) \bigr\vert + \bigl\vert \tilde{z}^{\prime \prime }_{\gamma }(\xi )-z^{ \prime \prime }_{\gamma }(\xi ) \bigr\vert \bigr) \,d\xi \\ &\qquad{} + \varpi _{3} \int _{0}^{1} \frac{(1 - \xi )^{\ell -3}}{\Gamma (\ell - 2)} \mathcal{G}_{\gamma }(s ) \bigl( \bigl\vert \tilde{z}_{\gamma }(\xi )- z_{\gamma }(\xi ) \bigr\vert + \bigl\vert \mathfrak{D}^{\nu } \tilde{z}_{\gamma }(\xi )- \mathfrak{D}^{\nu } z_{\gamma }(\xi ) \bigr\vert \\ &\qquad{} + \bigl\vert \tilde{z}^{\prime }_{\gamma }(\xi )-z^{ \prime }_{\gamma }(\xi ) \bigr\vert + \bigl\vert \tilde{z}^{\prime \prime }_{\gamma }(\xi )-z^{\prime \prime }_{\gamma }(\xi ) \bigr\vert \bigr) ] \,d\xi \\ &\quad \leq \Vert \mathcal{G}_{\gamma } \Vert \mathcal{I} _{4}^{ \ast } \Vert \tilde{z}_{\gamma }-z_{\gamma } \Vert _{ \mathcal{U}_{\gamma }} \end{aligned}$$
for each \(\gamma = 1, 2, \dots, 18 \), where \(\mathcal{I} _{4}^{\ast }\) is given in (3.9). Also, by similar computations we have
$$\begin{aligned} &\sup_{s \in {}[ 0,1]} \bigl\vert \bigl( \mathfrak{D}^{\nu } \mathcal{S} _{2}^{(\gamma )}\tilde{z} \bigr) (s )- \bigl( \mathfrak{D}^{\nu }\mathcal{S} _{2}^{(\gamma )} z \bigr) (s) \bigr\vert \leq \Vert \mathcal{G}_{\gamma } \Vert \mathcal{I} _{5}^{\ast } \Vert \tilde{z}_{\gamma }-z_{\gamma } \Vert _{\mathcal{U}_{\gamma }}, \\ &\sup_{s \in {}[ 0,1]} \bigl\vert \bigl( \mathcal{S} _{2}^{( \gamma )} \tilde{z} \bigr) ^{\prime }(s )- \bigl( \mathcal{S} _{2}^{( \gamma )}z \bigr) ^{\prime }(s) \bigr\vert \leq \Vert \mathcal{G}_{\gamma } \Vert \mathcal{I} _{6}^{\ast } \Vert \tilde{z}_{\gamma } - z_{\gamma } \Vert _{\mathcal{U}_{\gamma }}, \\ &\sup_{s \in {}[ 0,1]} \bigl\vert \bigl( \mathcal{S} _{2}^{( \gamma )} \tilde{z} \bigr) ^{\prime \prime }(s )- \bigl( \mathcal{S} _{2}^{( \gamma )}z \bigr) ^{\prime \prime }(s) \bigr\vert \leq 0, \end{aligned}$$
where \(\mathcal{I} _{5}^{\ast }\) and \(\mathcal{I} _{6}^{\ast }\) are given in (3.10) and (3.11), respectively. Thus we have
$$\begin{aligned} \Vert \mathcal{S} _{2} \tilde{z} -\mathcal{S} _{2}z \Vert _{\mathcal{U}} &=\sum_{\gamma =1}^{18} \bigl\Vert \mathcal{S} _{2}^{(\gamma )}\tilde{z}- \mathcal{S}_{2}^{(\gamma )}z \bigr\Vert _{\mathcal{U}_{\gamma }} \\ &\leq \bigl( \mathcal{I}_{4}^{\ast }+\mathcal{I}_{5}^{\ast } + \mathcal{I}_{6}^{\ast } \bigr) \sum _{\gamma =1}^{18} \Vert \mathcal{G}_{\gamma } \Vert \Vert \tilde{z}_{\gamma }-z_{ \gamma } \Vert _{\mathcal{U}_{\gamma }}, \end{aligned}$$
and so
$$\begin{aligned} \Vert \mathcal{S} _{2} \tilde{z}-\mathcal{S} _{2}z \Vert _{ \mathcal{U}}\leq \Delta \Vert \tilde{z} - z \Vert _{ \mathcal{U}}. \end{aligned}$$
Since \(\Delta <1 \), \(\mathcal{S} _{2}\) is a contraction on \(\mathcal{V}_{\varepsilon _{\gamma }}\). As a result of Theorem 2.3, we infer that \(\mathcal{S} \) contains a fixed point, which is a solution to problem (1.4). □
To illustrate the significance of our results, we provide the following example.
Example 3.3
Consider the differential equations:
$$\begin{aligned} \textstyle\begin{cases} \mathfrak{D}^{2.01}z_{1}(s )= \frac{4s}{1000} \vert \arcsin z_{1}(s) \vert + \frac{8 \vert \mathfrak{D}^{0.2} z_{1}(s) \vert s}{2000 + 2000 \vert \mathfrak{D}^{0.2} z_{1}(s) \vert } + 0.004 s \vert \arcsin z_{1}^{\prime }(s) \vert \\ \phantom{\mathfrak{D}^{2.01}z_{1}(s )=}{} + \frac{12 s \vert \sin z_{1}^{\prime \prime } (s ) \vert }{3000( 1 + \vert \sin z_{1}^{\prime \prime }(s) \vert )}, \\ \mathfrak{D}^{2.01}z_{2}(s )= \frac{21 e^{s} \vert \sin z_{2} (s ) \vert }{3000( 1 + \vert \sin z_{2}(s) \vert )} + \frac{7 e^{s}}{1000} \vert \sin (\mathfrak{D}^{0.2}z_{2}(s) ) \vert + \frac{14 \vert \arctan z_{2}^{\prime } (s) \vert e^{s}}{2000 + 2000 \vert \arctan z_{2}^{\prime } (s) \vert } \\ \phantom{\mathfrak{D}^{2.01}z_{2}(s )=} {} + 0.007 e^{s} \vert \arcsin z_{2}^{\prime \prime }(s) \vert , \\ \mathfrak{D}^{2.01}z_{3}(s )= 0.011 \vert \arctan z_{3}(s) \vert s + \frac{44 s \vert \mathfrak{D}^{0.2} z_{3} (s ) \vert }{4000( 1 + \vert \mathfrak{D}^{0.2} z_{3}(s) \vert )} + \frac{11}{1000} \vert \arcsin z_{3}^{\prime }(s) \vert s \\ \phantom{\mathfrak{D}^{2.01}z_{3}(s )=}{} + \frac{22 \vert \sin z_{3}^{\prime \prime } (s ) \vert s}{2000 + 2000 \vert \sin z_{3}^{\prime \prime }(s) \vert }, \end{cases}\displaystyle \end{aligned}$$
(3.15)
associated with the boundary conditions
$$\begin{aligned} \textstyle\begin{cases} \frac{9}{11} z_{1} (0) = \frac{3}{14} \mathfrak{D}^{1} z_{1} (1) + \frac{6}{19} \mathfrak{D}^{2} z_{1} (1), \\ \frac{9}{11} z_{1} (1) = \frac{3}{14} \mathfrak{D}^{1.01} z_{1} (1) + \frac{6}{19} \int _{0}^{0.05} \mathfrak{D}^{1.01} z_{1} (\xi ) \,d\xi, \\ \frac{9}{11} z_{2} (0) = \frac{3}{14} \mathfrak{D}^{1} z_{2} (1) + \frac{6}{19} \mathfrak{D}^{2} z_{2} (1), \\ \frac{9}{11} z_{2} (1) = \frac{3}{14} \mathfrak{D}^{1.01} z_{2} (1) + \frac{6}{19} \int _{0}^{0.05} \mathfrak{D}^{1.01} z_{2} (\xi ) \,d\xi \\ \frac{9}{11} z_{3} (0) = \frac{3}{14} \mathfrak{D}^{1} z_{3} (1) + \frac{6}{19} \mathfrak{D}^{2} z_{3} (1), \\ \frac{9}{11} z_{3} (1) = \frac{3}{14} \mathfrak{D}^{1.01} z_{3} (1) + \frac{6}{19} \int _{0}^{0.05} \mathfrak{D}^{1.01} z_{3} (\xi ) \,d\xi, \end{cases}\displaystyle \end{aligned}$$
(3.16)
where \(\ell = 2.01, \nu =0.2, \varpi _{1}=\frac{9}{11},\varpi _{2}= \frac{3}{14},\varpi _{3}=\frac{6}{19}\), and \(\mathfrak{D}^{\ell }\), \(\mathfrak{D}^{\nu }\) represent the Caputo fractional derivatives of orders ℓ and ν, respectively. Let \(\mathcal{S}_{1},\mathcal{S}_{2}, \mathcal{S} _{3}: [0,1]\times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) be continuous functions given as
$$\begin{aligned} \textstyle\begin{cases} \mathcal{S}_{1} ( s,z_{1}(s ),z_{2}(s ), z_{3}(s ), z_{4}(s ) )\\ \quad = \frac{4s}{1000} \vert \arcsin z_{1}(s) \vert + \frac{8 \vert \mathfrak{D}^{0.2} z_{2}(s) \vert s}{2000 + 2000 \vert \mathfrak{D}^{0.2} z_{2}(s) \vert } \\ \qquad{} + 0.004 s \vert \arcsin z_{3}^{\prime }(s) \vert + \frac{12 s \vert \sin z_{4}^{\prime \prime } (s ) \vert }{3000( 1 + \vert \sin z_{4}^{\prime \prime }(s) \vert )} , \\ \mathcal{S}_{2} ( s,z_{1}(s ),z_{2}(s ), z_{3}(s ), z_{4}(s ) ) \\ \quad= \frac{21 e^{s} \vert \sin z_{1} (s ) \vert }{3000( 1 + \vert \sin z_{1}(s) \vert )} + \frac{7 e^{s}}{1000} \vert \sin (\mathfrak{D}^{0.2}z_{2}(s) ) \vert \\ \qquad{} + \frac{14 \vert \arctan z_{3}^{\prime } (s) \vert e^{s}}{2000 + 2000 \vert \arctan z_{3}^{\prime } (s) \vert } + 0.007 e^{s} \vert \arcsin z_{4}^{\prime \prime }(s) \vert , \\ \mathcal{S} _{3} ( s,z_{1}(s ),z_{2}(s ), z_{3}(s ), z_{4}(s ) ) \\ \quad= 0.011 \vert \arctan z_{1}(s) \vert s + \frac{44 s \vert \mathfrak{D}^{0.2} z_{2} (s ) \vert }{4000( 1 + \vert \mathfrak{D}^{0.2} z_{2}(s) \vert )} \\ \qquad{} + \frac{11}{1000} \vert \arcsin z_{3}^{\prime }(s) \vert s + \frac{22 \vert \sin z_{4}^{\prime \prime } (s ) \vert s}{2000 + 2000 \vert \sin z_{4}^{\prime \prime }(s) \vert } . \end{cases}\displaystyle \end{aligned}$$
Let \(z_{1},z_{2}, z_{3}, z_{4}, \tilde{z}_{1},\tilde{z}_{2}, \tilde{z}_{3}, \tilde{z}_{4} \in \mathbb{R}\). Then we have
$$\begin{aligned} &\bigl\vert \mathcal{S}_{1} \bigl( s,z_{1}(s ),z_{2}(s ), z_{3}(s ), z_{4}(s ) \bigr) - \mathcal{S}_{1} \bigl( s,\tilde{z}_{1}(s ), \tilde{z}_{2}(s ), \tilde{z}_{3}(s ), \tilde{z}_{4}(s ) \bigr) \bigr\vert \\ &\quad\leq \frac{4 s }{1000} \bigl( \bigl\vert \arcsin z_{1}(s ) - \arcsin \tilde{z}_{1}(s ) \bigr\vert + \bigl\vert z_{2}(s )- \tilde{z}_{2}(s) \bigr\vert \\ &\qquad{}+ \bigl\vert \sin z_{3}(s )- \sin \tilde{z}_{3}(s) \bigr\vert + \bigl\vert \sin z_{4}(s )- \sin \tilde{z}_{4}(s) \bigr\vert \bigr), \\ &\bigl\vert \mathcal{S}_{2} \bigl( s,z_{1}(s ),z_{2}(s ), z_{3}(s ), z_{4}(s ) \bigr) - \mathcal{S}_{2} \bigl( s,\tilde{z}_{1}(s ), \tilde{z}_{2}(s ), \tilde{z}_{3}(s ), \tilde{z}_{4}(s ) \bigr) \bigr\vert \\ &\quad\leq \frac{7 e^{s} }{1000} \bigl( \bigl\vert \sin z_{1}(s )- \sin \tilde{z}_{1}(s ) \bigr\vert + \bigl\vert \sin z_{2}(s )- \sin \tilde{z}_{2}(s) \bigr\vert \\ &\qquad{}+ \bigl\vert \arctan z_{3}(s )- \arctan \tilde{z}_{3}(s) \bigr\vert + \bigl\vert \arcsin z_{4}(s )- \arcsin \tilde{z}_{4}(s) \bigr\vert \bigr), \\ &\bigl\vert \mathcal{S}_{3} \bigl( s,z_{1}(s ),z_{2}(s ), z_{3}(s ), z_{4}(s ) \bigr) - \mathcal{S}_{3} \bigl( s,\tilde{z}_{1}(s ), \tilde{z}_{2}(s ), \tilde{z}_{3}(s ), \tilde{z}_{4}(s ) \bigr) \bigr\vert \\ &\quad\leq \frac{11 s }{1000} \bigl( \bigl\vert \arctan z_{1}(s )- \arctan \tilde{z}_{1}(s ) \bigr\vert + \bigl\vert z_{2}(s )- \tilde{z}_{2}(s) \bigr\vert \\ &\qquad{}+ \bigl\vert \arcsin z_{3}(s )- \arcsin \tilde{z}_{3}(s) \bigr\vert + \bigl\vert \sin z_{4}(s )- \sin \tilde{z}_{4}(s) \bigr\vert \bigr). \end{aligned}$$
Here \(\mathcal{G} _{1}(s )=\frac{4 s }{1000}, \mathcal{G} _{2}(s)= \frac{7 e^{s} }{1000}, \mathcal{G} _{3}(s)=\frac{11 s}{1000}\), where \(\Vert \mathcal{G} _{1} \Vert =\frac{4}{1000}, \Vert \mathcal{G} _{2} \Vert =\frac{7}{1000}, \Vert \mathcal{G} _{3} \Vert =\frac{11}{1000}\). Let \(\mathcal{L} _{1},\mathcal{L} _{2}, \mathcal{L} _{3}: [0,\infty ) \rightarrow \mathbb{R}\) be the identity functions. Then we obtain
$$\begin{aligned} \bigl\vert \mathcal{S}_{1} \bigl( s, z(s),\mathfrak{D}^{0.2 }z(s ),z^{ \prime }(s), z^{\prime \prime }(s) \bigr) \bigr\vert &\leq \frac{4 s }{1000} \bigl( \vert \arcsin z \vert + \vert \mathfrak{D}z \vert + \bigl\vert \sin z^{\prime } \bigr\vert + \bigl\vert \sin z^{\prime \prime } \bigr\vert \bigr) \\ &\leq \frac{4 s }{1000} \bigl( \vert z \vert + \vert \mathfrak{D}z \vert + \bigl\vert z^{\prime } \bigr\vert + \bigl\vert z^{\prime \prime } \bigr\vert \bigr). \end{aligned}$$
Also,
$$\begin{aligned} \bigl\vert \mathcal{S}_{2} \bigl( s, z(s),\mathfrak{D}^{0.2 }z(s ),z^{ \prime }(s), z^{\prime \prime }(s) \bigr) \bigr\vert &\leq \frac{7 e^{s} }{1000} \bigl( \vert \sin z \vert + \bigl\vert \sin ( \mathfrak{D}z) \bigr\vert + \bigl\vert \arctan z^{\prime } \bigr\vert + \bigl\vert \arcsin z^{\prime \prime } \bigr\vert \bigr) \\ &\leq \frac{7 e^{s} }{1000} \bigl( \vert z \vert + \vert \mathfrak{D}z \vert + \bigl\vert z^{\prime } \bigr\vert + \bigl\vert z^{\prime \prime } \bigr\vert \bigr) \end{aligned}$$
and
$$\begin{aligned} \bigl\vert \mathcal{S} _{3} \bigl( s, z(s),\mathfrak{D}^{0.2 }z(s ),z^{ \prime }(s), z^{\prime \prime }(s) \bigr) \bigr\vert &\leq \frac{11 s }{1000} \bigl( \vert \arctan z \vert + \vert \mathfrak{D}z \vert + \bigl\vert \arcsin z^{\prime } \bigr\vert + \bigl\vert \sin z^{\prime \prime } \bigr\vert \bigr) \\ &\leq \frac{11 s }{1000} \bigl( \vert z \vert + \vert \mathfrak{D}z \vert + \bigl\vert z^{\prime } \bigr\vert + \bigl\vert z^{\prime \prime } \bigr\vert \bigr), \end{aligned}$$
where the continuous functions \(\mathcal{Z} _{1},\mathcal{Z} _{2}, \mathcal{Z} _{3}:[0,1] \rightarrow \mathbb{R} \) are defined by
$$\begin{aligned} \mathcal{Z} _{1}(s )=\frac{4 s }{1000},\qquad \mathcal{Z} _{2}(s )=\frac{7 e^{s} }{1000},\qquad \mathcal{Z} _{3}(s )= \frac{11 s}{1000}. \end{aligned}$$
Also,
$$\begin{aligned} \mathcal{I} _{4}^{\ast }\simeq 0.9227,\qquad \mathcal{I} _{5}^{\ast } \simeq 1.2360\quad \text{and}\quad \mathcal{I} _{6}^{\ast }\simeq 1.1512, \end{aligned}$$
and so
$$\begin{aligned} \Delta:= \bigl( \mathcal{I} _{4}^{\ast } + \mathcal{I} _{5}^{\ast }+ \mathcal{I}_{6}^{\ast } \bigr) \bigl( \Vert \mathcal{G} _{1} \Vert + \Vert \mathcal{G} _{2} \Vert + \Vert \mathcal{G} _{3} \Vert \bigr) \simeq 0.0728< 1. \end{aligned}$$
Hence by Theorem 3.2 problem (3.15)–(3.16) has a solution.