To begin, we define the set of selections of
T for an arbitrary element
\(u\in\mathcal{X}\) which contains all
\(v \in L^{1}(\overline {J}, \mathbb{R})\) such that
\(v(t)\) belongs to the multifunction
\(\widetilde{T}(t, u(t))\) for each
\(t\in\overline{J}\) and is denoted by
\(S_{T, u}\), where
$$\begin{aligned} \widetilde{T} \bigl(t, u(t) \bigr)= {}&T \bigl( t, u(t), u'(t), u''(t), u'''(t), \varphi_{1} u(t), \varphi_{2} u(t), \\ & {}^{c}D_{q}^{\beta_{11}} u(t), {}^{c}D_{q}^{\beta_{12}} u(t), \dots, {}^{c}D_{q}^{\beta_{1k_{1}}} u(t), \\ & {}^{c}D_{q}^{\beta_{21}} u(t), {}^{c}D_{q}^{\beta_{22}} u(t), \dots, {}^{c}D_{q}^{\beta_{2k_{2}}} u(t), \\ & {}^{c}D_{q}^{ \beta_{31}} u(t), {}^{c}D_{q}^{\beta_{32}} u(t), \dots, {}^{c}D_{q}^{\beta_{3k_{3}}} u(t) \bigr). \end{aligned}$$
By considering the first property of the multifunction
T and using Theorem 1.3.5 in [
8], we know that
\(S_{T,u}\) is nonempty. Defining an operator
\(H: X \to P(X)\) on the set of all
\(h \in X\) for which there exists
\(v \in S_{T,u}\) such that
\(h(t) = T_{v} (t)\) for
\(t\in\overline{J}\) and denoting by
\(H(x)\) where
$$\begin{aligned} T_{v}(t) ={}& I_{q}^{\alpha}v(t) - \frac{a_{2}}{ a_{1}+ a_{2}} I_{q}^{\alpha}v(\delta) + B_{1}(t, \delta) _{q}^{\alpha- p_{1}} v(\delta) \\ & - B_{2}(t, \delta) I_{q}^{\alpha- p_{2}} v(\delta) - B_{3}(t, \delta) I_{q}^{\alpha- p_{3}} v(\delta), \end{aligned}$$
we claim that
\(H(x)\) is convex for all
\(u\in\mathcal{X}\). Assume that
\(h_{1}, h_{2} \in H(x)\) and
\(\tau\in[0,1]\). Choose
\(v_{1}, v_{2}\in S_{T,u}\) such that
$$\begin{aligned} h_{i}(t) ={}& I_{q}^{\alpha}v_{i}(t) - \frac{a_{2}}{ a_{1}+ a_{2}} I_{q}^{\alpha}v_{i}(\delta)+ B_{1}(t, \delta) _{q}^{\alpha- p_{1}} v_{i}( \delta) \\ & - B_{2}(t, \delta) I_{q}^{\alpha- p_{2}} v_{i}(\delta) - B_{3}(t, \delta ) I_{q}^{\alpha- p_{3}} v_{i}(\delta), \end{aligned}$$
for each
t in
J̅. Then, we obtain
$$\begin{aligned} \bigl[\tau h_{1} + ( 1 - \tau) h_{2} \bigr] (t) ={}& I_{q}^{\alpha}\bigl[ \tau v_{1} (t) + (1 - \tau) v_{2} (t) \bigr] \\ & - \frac{a_{2}}{ a_{1} + a_{2}} I_{q}^{\alpha}\bigl[\tau v_{1} (\delta) + ( 1 -\tau) v_{2} (\delta) \bigr] \\ & + B_{1}(t, \delta) I_{q}^{\alpha- p_{1}} \bigl[\tau v_{1} (\delta) + ( 1 - \tau) v_{2}(\delta) \bigr] \\ & - B_{2}(t, \delta) I_{q}^{\alpha- p_{2}} \bigl[\tau v_{1}(\delta) + ( 1 -\tau) v_{2}(\delta) \bigr] \\ & - B_{3}(t, \delta) I_{q}^{\alpha-p_{3}} \bigl[\tau v_{1} (\delta) + ( 1 -\tau) v_{2} (\delta) \bigr]. \end{aligned}$$
Since
T has convex values, by simple calculation, we can see that
\(S_{T,u}\) is convex and so
\(\tau h_{1} + ( 1 - \tau) h_{2} \in H(x)\). At present, we prove that
H maps bounded sets into bounded sets in
\(\mathcal{X}\). Suppose that
\(B_{r}\) is the set of all
\(u \in\mathcal{X}\) such that
\(\|u\|\) is less than or equal to
r,
\(u \in B_{r}\) and
\(h\in H(x)\). We select
\(v \in S_{T,u}\) such that
$$\begin{aligned} \bigl\vert h(t) \bigr\vert \leq{}& I_{q}^{\alpha}v (t)- \frac{a_{2}}{ a_{1} + a_{2}} I_{q}^{\alpha}v (\delta) + B_{1}(t, \delta) I_{q}^{\alpha- p_{1}} v (\delta) - B_{2}(t, \delta ) I_{q}^{\alpha- p_{2}} v(\delta) \\ & - B_{3}(t, \delta) I_{q}^{\alpha-p_{3}} v (\delta) \\ \leq{}& I_{q}^{\alpha}\Biggl[ g_{01} (t) \phi_{1} \bigl( \bigl\vert u(t) \bigr\vert \bigr) + g_{02} (t) \phi_{2} \bigl( \bigl\vert u'(t) \bigr\vert \bigr) + g_{03}(t) \phi_{3} \bigl( \bigl\vert u''(t) \bigr\vert \bigr) \\ & + g_{04}(t) \phi_{4} \bigl( \bigl\vert u'''(t) \bigr\vert \bigr) + g_{05}(t) \phi_{5} \bigl( \bigl\vert \varphi_{1} u(t) \bigr\vert \bigr) +g_{06}(t) \phi_{6} \bigl( \bigl\vert \varphi_{2} u(t) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{1}}g_{1j}(t) \psi_{1j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{1j}} u(t) \bigr\vert \bigr) + \sum_{j=1}^{k_{2}} g_{2j}(t) \psi_{2j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{2j}} u(t) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{3} } g_{3j}(t) \psi_{3j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{3j}} u(t) \bigr\vert \bigr) \Biggr] \\ & + \frac{ \vert a_{2} \vert }{ \vert a_{1} + a_{2} \vert } I_{q}^{\alpha}\Biggl[ g_{01} (\delta ) \phi_{1} \bigl( \bigl\vert u(\delta) \bigr\vert \bigr) + g_{02} (\delta) \phi_{2} \bigl( \bigl\vert u'(\delta) \bigr\vert \bigr) \\ & + g_{03}(\delta) \phi_{3} \bigl( \bigl\vert u''(\delta) \bigr\vert \bigr) + g_{04}( \delta) \phi_{4} \bigl( \bigl\vert u'''( \delta) \bigr\vert \bigr) \\ & + g_{05}(\delta) \phi_{5} \bigl( \bigl\vert \varphi_{1} u(\delta) \bigr\vert \bigr) + g_{06}(\delta ) \phi_{6} \bigl( \bigl\vert \varphi_{2} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{1}}g_{1j}( \delta) \psi_{1j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta _{1j}} u(\delta) \bigr\vert \bigr) + \sum_{j=1}^{k_{2}} g_{2j}(\delta) \psi_{2j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{2j}} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{3} } g_{3j}(t) \psi_{3j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{3j}} u(t) \bigr\vert \bigr) \Biggr] \\ & + \bigl\vert B_{1}(t, \delta) \bigr\vert I_{q}^{\alpha- p_{1}} \Biggl[ g_{01} (\delta) \phi_{1} \bigl( \bigl\vert u( \delta) \bigr\vert \bigr) + g_{02} (\delta) \phi_{2} \bigl( \bigl\vert u'(\delta) \bigr\vert \bigr) \\ & + g_{03}(\delta) \phi_{3} \bigl( \bigl\vert u''(\delta) \bigr\vert \bigr) + g_{04}( \delta) \phi _{4} \bigl( \bigl\vert u'''( \delta) \bigr\vert \bigr) \\ & + g_{05}(\delta) \phi_{5} \bigl( \bigl\vert \varphi_{1} u(\delta) \bigr\vert \bigr) + g_{06}(\delta ) \phi_{6} \bigl( \bigl\vert \varphi_{2} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{1}}g_{1j}( \delta) \psi_{1j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta _{1j}} u(\delta) \bigr\vert \bigr) + \sum_{j=1}^{k_{2}} g_{2j}(\delta) \psi_{2j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{2j}} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{3} } g_{3j}(t) \psi_{3j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{3j}} u(t) \bigr\vert \bigr) \Biggr] \\ & + \bigl\vert B_{2}(t, \delta) \bigr\vert I_{q}^{\alpha- p_{2}} \Biggl[ g_{01} (\delta) \phi_{1} \bigl( \bigl\vert u( \delta) \bigr\vert \bigr) + g_{02} (\delta) \phi_{2} \bigl( \bigl\vert u'(\delta) \bigr\vert \bigr) \\ & + g_{03}(\delta) \phi_{3} \bigl( \bigl\vert u''(\delta) \bigr\vert \bigr) + g_{04}( \delta) \phi _{4} \bigl( \bigl\vert u'''( \delta) \bigr\vert \bigr) \\ & + g_{05}(\delta) \phi_{5} \bigl( \bigl\vert \varphi_{1} u(\delta) \bigr\vert \bigr) + g_{06}(\delta ) \phi_{6} \bigl( \bigl\vert \varphi_{2} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{1}}g_{1j}( \delta) \psi_{1j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta _{1j}} u(\delta) \bigr\vert \bigr) + \sum_{j=1}^{k_{2}} g_{2j}(\delta) \psi_{2j} \bigl( \bigl\vert {} ^{c}D_{q}^{\beta_{2j}} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{3} } g_{3j}(t) \psi_{3j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{3j}} u(t) \bigr\vert \bigr) \Biggr] \\ & + \bigl\vert B_{3}(t, \delta) \bigr\vert \Biggl[ g_{01} (\delta) \phi_{1} \bigl( \bigl\vert u(\delta) \bigr\vert \bigr) + g_{02} (\delta) \phi_{2} \bigl( \bigl\vert u'(\delta) \bigr\vert \bigr) \\ & + g_{03}(\delta) \phi_{3} \bigl( \bigl\vert u''(\delta) \bigr\vert \bigr) + g_{04}( \delta) \phi _{4} \bigl( \bigl\vert u'''( \delta) \bigr\vert \bigr) \\ & + g_{05}(\delta) \phi_{5} \bigl( \bigl\vert \varphi_{1} u(\delta) \bigr\vert \bigr) + g_{06}(\delta ) \phi_{6} \bigl( \bigl\vert \varphi_{2} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{1}}g_{1j}( \delta) \psi_{1j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta _{1j}} u(\delta) \bigr\vert \bigr) + \sum_{j=1}^{k_{2}} g_{2j}(\delta) \psi_{2j} \bigl( \bigl\vert {} ^{c}D_{q}^{\beta_{2j}} u(\delta) \bigr\vert \bigr) \\ & + \sum_{j=1}^{k_{3} } g_{3j}(t) \psi_{3j} \bigl( \bigl\vert {}^{c}D_{q}^{\beta_{3j}} u(t) \bigr\vert \bigr) \Biggr] \\ \leq{}& \Biggl[ \sum_{j=1}^{4} \Vert g_{0j} \Vert _{1} \phi_{j}(r) + \Vert g_{05} \Vert _{1} \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum _{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\ & + {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma _{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\ & + \Vert g_{06} \Vert _{1} \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2}\eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma _{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\ & + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\ & + \sum_{j=1}^{k_{1}} \Vert g_{1j} \Vert _{1} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta_{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) \\ & + \sum_{j=1}^{k_{2}} \Vert g_{2j} \Vert _{1} \psi_{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\ & + \sum_{j=1}^{k_{3}} \Vert g_{3j} \Vert _{1} \psi_{3j} \biggl( \frac{ \delta ^{ 3 - \beta_{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \\ & \times \biggl[ \frac{( \vert a_{1} \vert + 2 \vert a_{2} \vert ) \delta^{ \alpha- 1 }}{ \vert a_{1} + a_{2} \vert \varGamma_{q}( \alpha)} + \frac{( \vert a_{1} \vert + 2 \vert a_{2} \vert ) \varGamma_{q}( 2 - p_{1}) \delta^{ \alpha- 1}}{ \vert a_{1} + a_{2} \vert \varGamma_{q}( \alpha- p_{1})} \\ & + \frac{( \vert a_{2} \vert p_{1} + \vert a_{1}+ a_{2} \vert ( 4 - p_{1})) \varGamma_{q}( 3 - p_{2}) \delta^{ \alpha- 1}}{ 2 \vert a_{1} + a_{2} \vert ( 2 - p_{1}) \varGamma_{q}( \alpha- p_{2})} \\ & + \frac{ \vert a_{2} \vert [6 ( p_{2} - p_{1}) + ( 2 -p_{1}) ( 3 - p_{1}) p_{2}] \varGamma_{q}( 4 -p_{3}) \delta^{\alpha-1}}{ 6 \vert a_{1} + a_{2} \vert ( 2 - p_{1})( 3 - p_{1})( 3 - p_{2}) \varGamma_{q}( \alpha- p_{3})} \\ & + \frac{ [6 ( p_{2} - p_{1}) + ( 2 -p_{1})( 3 -p_{1})( 6 -p_{2}) ] \varGamma _{q}( 4 -p_{3}) \delta^{\alpha-1}}{ 6 ( 2 -p_{1})( 3 -p_{1})( 3 -p_{2}) \varGamma_{q}( \alpha- p_{3})} \biggr], \end{aligned}$$
(18)
for any
\(t\in\overline{J}\). Thus, similarly as for inequality (
18), we get
$$\begin{aligned}& \bigl\vert h'(t) \bigr\vert \leq \Biggl[ \sum _{j=1}^{4} \Vert g_{0j} \Vert _{1} \phi_{j}(r) + \Vert g_{05} \Vert _{1} \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq}+ {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma _{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq} + \Vert g_{06} \Vert _{1} \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2}\eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma _{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq}+ {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq} + \sum_{j=1}^{k_{1}} \Vert g_{1j} \Vert _{1} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta_{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq} + \sum_{j=1}^{k_{2}} \Vert g_{2j} \Vert _{1} \psi_{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq} + \sum_{j=1}^{k_{3}} \Vert g_{3j} \Vert _{1} \psi_{3j} \biggl( \frac{ \delta ^{ 3 - \beta_{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq} \times \biggl[ \frac{\delta^{\alpha- 2}}{ \varGamma_{q}( \alpha- 1)} + \frac{ \varGamma_{q}( 2 -p_{1}) \delta^{ \alpha- 2}}{ \varGamma_{q}( \alpha-p_{1})} + \frac{ (3 - p_{1}) \varGamma_{q}( 3 -p_{2}) \delta^{ \alpha- 2}}{ ( 2- p_{1}) \varGamma_{q}( \alpha- p_{2})} \\& \phantom{\bigl\vert h'(t) \bigr\vert \leq} + \frac{ [ 2 (p_{2} -p_{1}) + ( 2 - p_{1}) ( 3- p_{1}) ( 5 - p_{2}) ] \varGamma_{q}( 4 - p_{3}) \delta^{ \alpha- 2}}{ 2 (2 - p_{1}) ( 3 - p_{1})( 3 -p_{2}) \varGamma_{q}( \alpha- p_{3})} \biggr], \end{aligned}$$
(19)
$$\begin{aligned}& \bigl\vert h''(t) \bigr\vert \leq \Biggl[ \sum _{j=1}^{4} \Vert g_{0j} \Vert _{1} \phi_{j}(r) + \Vert g_{05} \Vert _{1} \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\& \phantom{\bigl\vert h''(t) \bigr\vert \leq}+ {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma _{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h''(t) \bigr\vert \leq}+ \Vert g_{06} \Vert _{1} \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2}\eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma _{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\& \phantom{\bigl\vert h''(t) \bigr\vert \leq} + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h''(t) \bigr\vert \leq}+ \sum_{j=1}^{k_{1}} \Vert g_{1j} \Vert _{1} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta_{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) \\& \phantom{\bigl\vert h''(t) \bigr\vert \leq} + \sum_{j=1}^{k_{2}} \Vert g_{2j} \Vert _{1} \psi_{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\& \phantom{\bigl\vert h''(t) \bigr\vert \leq} + \sum_{j=1}^{k_{3}} \Vert g_{3j} \Vert _{1} \psi_{3j} \biggl( \frac{ \delta ^{ 3 - \beta_{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \\& \phantom{\bigl\vert h''(t) \bigr\vert \leq} \times \biggl[ \frac{ \delta^{\alpha- 3}}{ \varGamma_{q}( \alpha- 2)} + \frac{ \varGamma_{q}( 3 -p_{2})\delta^{\alpha- 3}}{\varGamma_{q}( \alpha-p_{2})} + \frac{ (4 - p_{2}) \varGamma_{q}( 4- p_{3}) \delta^{ \alpha- 3}}{( 3 -p_{2}) \varGamma_{q}( \alpha-p_{3})} \biggr], \end{aligned}$$
(20)
$$\begin{aligned}& \bigl\vert h'''(t) \bigr\vert \leq \Biggl[ \sum_{j=1}^{4} \Vert g_{0j} \Vert _{1} \phi_{j}(r) + \Vert g_{05} \Vert _{1} \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum _{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\& \phantom{\bigl\vert h'''(t) \bigr\vert \leq}+ {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma _{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h'''(t) \bigr\vert \leq} + \Vert g_{06} \Vert _{1} \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2}\eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma _{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\& \phantom{\bigl\vert h'''(t) \bigr\vert \leq} + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h'''(t) \bigr\vert \leq}+ \sum_{j=1}^{k_{1}} \Vert g_{1j} \Vert _{1} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta_{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) \end{aligned}$$
(21)
$$\begin{aligned}& \phantom{\bigl\vert h'''(t) \bigr\vert \leq}+ \sum_{j=1}^{k_{2}} \Vert g_{2j} \Vert _{1} \psi_{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\& \phantom{\bigl\vert h'''(t) \bigr\vert \leq}+ \sum_{j=1}^{k_{3}} \Vert g_{3j} \Vert _{1} \psi_{3j} \biggl( \frac{ \delta ^{ 3 - \beta_{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \\& \phantom{\bigl\vert h'''(t) \bigr\vert \leq}\times \biggl[ \frac{\delta^{\alpha- 4}}{ \varGamma_{q}( \alpha- 3)} + \frac{ \varGamma_{q}( 4 -p_{3}) \delta^{\alpha- 4}}{ \varGamma_{q}( \alpha- p_{3})} \biggr] . \end{aligned}$$
(22)
Thus, from inequalities (
18), (
19), (
20), and (
22), we obtain
$$\begin{aligned} \Vert h \Vert \leq{}&\varLambda_{2} \Biggl[ \sum _{j=1}^{4} \Vert g_{0j} \Vert _{1} \phi_{j}(r) + \Vert g_{05} \Vert _{1} \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\ & + {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma _{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\ & + \Vert g_{06} \Vert _{1} \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2}\eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma _{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\ & + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\ & + \sum_{j=1}^{k_{1}} \Vert g_{1j} \Vert _{1} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta_{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) \\ & + \sum_{j=1}^{k_{2}} \Vert g_{2j} \Vert _{1} \psi_{2j} \biggl( \frac{\delta ^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\ & + \sum_{j=1}^{k_{3}} \Vert g_{3j} \Vert _{1} \psi_{3j} \biggl( \frac{ \delta ^{ 3 - \beta_{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr]. \end{aligned}$$
Thus, we conclude that
H maps bounded sets into bounded sets in
\(\mathcal{X}\). Let
\(\tau_{1}, \tau_{2} \in\overline{J}\) with
\(\tau_{1} < \tau_{2}\),
\(u\in B_{r}\) and
\(h\in H(x)\). Then, we have
$$\begin{aligned} \bigl\vert h(\tau_{2}) - h(\tau_{1}) \bigr\vert \leq{}& \frac{1}{ \varGamma_{q}( \alpha) } \int _{0}^{\tau_{1}} \bigl[(\tau_{2} - qs)^{( \alpha-1)} - (\tau_{1} -qs)^{(\alpha -1)} \bigr] \bigl\vert v(s) \bigr\vert \, d_{q}s \\ & + \frac{1}{ \varGamma_{q}( \alpha)} \int_{\tau_{1}}^{\tau_{2}} (\tau_{2} - qs)^{(\alpha-1)} \bigl\vert v(s) \bigr\vert \, d_{q}s \\ & + (\tau_{2} - \tau_{1})\delta^{p_{1}-1} \varGamma_{q}( 2 -p_{1}) I_{q}^{\alpha - p_{1}} \bigl\vert v(\delta) \bigr\vert \\ & + \frac{ [ 2 \delta( \tau_{2} -\tau_{1}) + ( 2 -p_{1}) ( \tau_{2}^{2} -\tau_{1}^{2} ) ] \varGamma_{q} ( 3 -p_{2}) \delta^{ p_{2}-2} }{ 2 ( 2 - p_{1}) } I_{q}^{\alpha- p_{2}} \bigl\vert v(\delta) \bigr\vert \\ & + \biggl( \frac{( p_{2} -p_{1}) \delta^{2} ( \tau_{2} -\tau_{1}) \varGamma_{q}( 4 -p_{3}) \delta^{p_{3}-3} }{ ( 2 - p_{1}) (3 - p_{1}) (3 -p_{2}) } \\ & + \frac{ (2 -p_{1})( 3 - p_{1}) \varGamma_{q}( 4 -p_{3}) \delta^{p_{3}-3} }{6 ( 2 - p_{1}) (3 - p_{1}) (3 -p_{2}) } \\ & \times \bigl[ 3\delta \bigl(\tau_{2}^{2} - \tau_{1}^{2} \bigr) + ( 3 -p_{2}) \bigl( \tau_{2}^{3} -\tau_{1}^{3} \bigr) \bigr] \biggr) I_{q}^{\alpha-p_{3} } \bigl\vert v(\delta) \bigr\vert \\ \leq{}& \Biggl[ \sum_{j=1}^{4} \phi_{j}(r) + \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\ & + {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma_{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\ & + \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2} \eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma_{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\ & + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\ & + \sum_{j=1}^{k_{1}} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta _{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) + \sum_{j=1}^{k_{2}} \psi _{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\ & + \sum_{j=1}^{k_{3}} \psi_{3j} \biggl( \frac{ \delta^{ 3 - \beta _{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \\ & \times \Biggl[ \frac{1}{ \varGamma_{q}( \alpha)} \int_{0}^{\tau_{1}} \bigl[(\tau_{2} -qs)^{( \alpha- 1)} -(\tau_{1} -qs)^{(\alpha-1)} \bigr] \mathcal{G}(s)\, d_{q}s \\ & + \frac{1}{ \varGamma_{q}( \alpha) } \int_{\tau_{1}}^{\tau_{2}} (\tau_{2} - qs)^{(\alpha-1)} \mathcal{G}(s)\, d_{q}s \\ & + (\tau_{2} -\tau_{1}) \varGamma(2 -p_{1}) \delta^{p_{1}-1} I_{q}^{\alpha -p_{1}} \mathcal{G}(\delta) \\ & + \frac{ [ 2 \delta( \tau_{2} -\tau_{1}) + ( 2 -p_{1}) ( \tau_{2}^{2} -\tau_{1}^{2} ) ] \varGamma_{q} ( 3 -p_{2}) \delta^{ p_{2}-2} }{ 2 ( 2 - p_{1}) } I_{q}^{\alpha- p_{2}} \mathcal{G}(\delta) \\ & + \biggl( \frac{ ( p_{2} -p_{1}) \delta^{2} ( \tau_{2} -\tau_{1}) \varGamma _{q}( 4 -p_{3}) \delta^{p_{3}-3} }{ ( 2 - p_{1}) (3 - p_{1}) (3 -p_{2}) } \\ & + \frac{ (2 -p_{1})( 3 - p_{1})\varGamma_{q}( 4 -p_{3}) \delta^{p_{3}-3} }{6 ( 2 - p_{1}) (3 - p_{1}) (3 -p_{2}) } \\ & \times \bigl[ 3\delta \bigl(\tau_{2}^{2} - \tau_{1}^{2} \bigr) + ( 3 -p_{2}) \bigl(\tau _{2}^{3} -\tau_{1}^{3} \bigr) \bigr] \biggr) I_{q}^{\alpha-p_{3} } \mathcal{G}(\delta) \Biggr] , \end{aligned}$$
(23)
where
$$\begin{aligned} \mathcal{G}(z) &=\sum_{j=1}^{6} g_{0j} (z) + \sum_{j=1}^{k_{1}} g_{1j}(z) + \sum_{j=1}^{k_{2}} g_{2j} (z) + \sum_{j=1}^{k_{3}} g_{3j}(z). \end{aligned}$$
Similarly, from inequality (
23), we have
$$\begin{aligned}& \bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq \Biggl[ \sum_{j=1}^{4} \phi_{j}(r) + \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq}+ {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq}+ {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma_{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} + \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2} \eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma_{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq}+ {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} + \sum_{j=1}^{k_{1}} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta _{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) + \sum_{j=1}^{k_{2}} \psi _{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} + \sum_{j=1}^{k_{3}} \psi_{3j} \biggl( \frac{ \delta^{ 3 - \beta _{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \biggl[ \frac {1}{\varGamma_{q}(\alpha- 1)} \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} \times \int_{0}^{\tau_{1}} \bigl[(\tau_{2} -qs)^{(\alpha-2)} - ( \tau_{1} -qs)^{(\alpha-2) } \bigr] \mathcal{G}(s)\, d_{q}s \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} + \frac{1}{\varGamma_{q}(\alpha-1)} \int_{\tau_{1}}^{\tau_{2}} (\tau_{2} - qs)^{(\alpha-2)} \mathcal{G}(s) \, d_{q}s \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} + (\tau_{2} - \tau_{1}) \varGamma_{q}( 3 -p_{2}) \delta^{ p_{2} -2 } I_{q}^{\alpha-p_{2}} \mathcal{G}(\delta) \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} + \frac{ \varGamma_{q}( 4 -p_{3}) \delta^{ p_{3}-3}}{ 2 (3 - p_{2}) } \\& \phantom{\bigl\vert h' (\tau_{2} ) - h' ( \tau_{1}) \bigr\vert \leq} \times \bigl[2 \delta(\tau_{2} -\tau_{1}) + (3 - p_{2}) \bigl( t_{2}^{2} -\tau _{1}^{2} \bigr) \bigr] I_{q}^{\alpha- p_{3}} \mathcal{G}(\delta) \biggr], \end{aligned}$$
(24)
$$\begin{aligned}& \bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq \Biggl[ \sum_{j=1}^{4} \phi_{j}(r) + \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum _{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma_{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2} \eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma_{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + \sum_{j=1}^{k_{1}} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta _{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) + \sum_{j=1}^{k_{2}} \psi _{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + \sum_{j=1}^{k_{3}} \psi_{3j} \biggl( \frac{ \delta^{ 3 - \beta _{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \biggl[ \frac {1}{\varGamma_{q}(\alpha- 1)} \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} \times \int_{0}^{\tau_{1}} \bigl[(\tau_{2} -qs)^{(\alpha-2)} - ( \tau_{1} -qs)^{(\alpha-2) } \bigr] \mathcal{G}(s) \, d_{q}s \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + \frac{1}{\varGamma_{q}(\alpha-2)} \int_{\tau_{1}}^{\tau_{2}} (\tau_{2} - qs)^{(\alpha-3)} \mathcal{G}(s) \, d_{q}s \\& \phantom{\bigl\vert h''(\tau_{2} ) - h''(\tau_{1}) \bigr\vert \leq} + (\tau_{2} - \tau_{1}) \varGamma_{q}( 4 -p_{3}) \delta^{ p_{3} -3 } I_{q}^{\alpha-p_{3}} \mathcal{G}(\delta) \biggr], \end{aligned}$$
(25)
$$\begin{aligned}& \bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq \Biggl[ \sum_{j=1}^{4} \phi_{j}(r) + \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + r \gamma_{1}^{0} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq}+ {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1}\eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma_{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq} + \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + r \gamma_{2}^{0} \Biggl[ \sum_{j=1}^{4} {}_{2} \eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma_{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq} + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq} + \sum_{j=1}^{k_{1}} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta _{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } r \biggr) + \sum_{j=1}^{k_{2}} \psi _{2j} \biggl( \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} r \biggr) \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq}+ \sum_{j=1}^{k_{3}} \psi_{3j} \biggl( \frac{ \delta^{ 3 - \beta _{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} r \biggr) \Biggr] \biggl[ \frac {1}{\varGamma_{q}(\alpha- 1)} \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq} \times \int_{0}^{\tau_{1}} \bigl[(\tau_{2} -qs)^{(\alpha-2)} - ( \tau_{1} -qs)^{(\alpha-2) } \bigr] \mathcal{G}(s) \, d_{q}s \\ & \phantom{\bigl\vert h'''(\tau_{2} ) -h'''(\tau_{1}) \bigr\vert \leq} + \frac{1}{\varGamma_{q}(\alpha-3)} \int_{\tau_{1}}^{\tau_{2}} (\tau_{2} - qs)^{(\alpha-4)} \mathcal{G}(\delta) \, d_{q}s \biggr]. \end{aligned}$$
(26)
Therefore, since
\(u\in B_{r}\), when
\(t_{2} - t_{1} \to0\), the above inequalities (
23)–(
26) tend to zero. Therefore, by employing Arzelà–Ascoli theorem, we get that
\(H: \mathcal{X} \to P(\mathcal{X})\) is a compact multivalued map. Let
\(u_{n} \to u^{*}\),
\(h_{n} \in H(u_{n})\) for all
n and
\(h_{n}\to h^{*}\). We show that
\(h^{*} \in H(u^{*})\). Since
\(h_{n}\in H(u_{n})\) for all
n, there exists
\(v_{n} \in S_{T,u_{n}}\) such that
$$\begin{aligned} h_{n}(t) ={}& I_{q}^{\alpha}v_{n}(t) - \frac{a_{2}}{ a_{1}+ a_{2}} I_{q}^{\alpha}v_{n}(\delta) + B_{1}(t, \delta) I_{q}^{\alpha- p_{1}} v_{n}( \delta) \\ & - B_{2}(t, \delta) I_{q}^{\alpha- p_{2}} v_{n}( \delta) - B_{3}(t, \delta ) I_{q}^{\alpha- p_{3}} v_{n}(\delta), \end{aligned}$$
for all
\(t\in\overline{J}\). We claim that there exists
\(v^{*} \) belonging to
\(S_{T,u^{*}}\) such that
$$\begin{aligned} h^{*}(t) ={}& I_{q}^{\alpha}v^{*}(t) - \frac{a_{2}}{ a_{1}+ a_{2}} I_{q}^{\alpha}v^{*}(\delta) + B_{1}(t, \delta) I_{q}^{\alpha- p_{1}} v^{*}(\delta) \\ & - B_{2}(t, \delta) I_{q}^{\alpha- p_{2}} v^{*}(\delta) - B_{3}(t, \delta ) I_{q}^{\alpha- p_{3}} v^{*}(\delta), \end{aligned}$$
for each
t belonging to
J̅. In this case, we consider the linear operator
\(\varOmega: L^{1}(\overline{J}, \mathbb{R})\to\mathcal{X}\) defined by
\(v \mapsto\varOmega(v)(t)\), where
Ω is continuous and
$$\begin{aligned} \begin{aligned} \varOmega(v) (t) ={}& I_{q}^{\alpha}v(t) - \frac{a_{2}}{ a_{1}+ a_{2}} I_{q}^{\alpha}v(\delta) + B_{1}(t, \delta) I_{q}^{\alpha- p_{1}} v(\delta) \\ & - B_{2}(t, \delta) I_{q}^{\alpha- p_{2}} v(\delta) - B_{3}(t, \delta) I_{q}^{\alpha- p_{3}} v(\delta),\end{aligned} \end{aligned}$$
for all
t in
J̅. On the other hand,
Ω is a linear continuous map and by applying Lemma
7, we obtain
\(\varOmega \circ S_{T, u}\) is a closed-graph operator. Note that
\(h_{n}\in\varOmega \circ S_{T, u_{n}}\) for all
n. Since
\(u_{n}\to u^{*}\) and
\(h_{n}\to h^{*}\), there exists
\(v^{*}\in S_{T, u^{*}}\) such that
$$\begin{aligned} h^{*}(t) ={}& I_{q}^{\alpha}v^{*}(t) - \frac{a_{2}}{ a_{1}+ a_{2}} I_{q}^{\alpha}v^{*}(\delta) + B_{1}(t, \delta) I_{q}^{\alpha- p_{1}} v^{*}(\delta) \\ & - B_{2}(t, \delta) I_{q}^{\alpha- p_{2}} v^{*}(\delta) - B_{3}(t, \delta ) I_{q}^{\alpha- p_{3}} v^{*}(\delta). \end{aligned}$$
If
\(0<\kappa<1\) and
\(u\in\kappa H(x)\), then there exists
\(v \in S_{T,u}\) such that
$$\begin{aligned} u(t) ={}& \kappa I_{q}^{\alpha}v(t) - \frac{a_{2}}{ a_{1}+ a_{2}} \kappa I_{q}^{\alpha}v(\delta) + B_{1}(t, \delta) I_{q}^{\alpha- p_{1}} v(\delta) \\ & - B_{2}(t, \delta) \kappa I_{q}^{\alpha- p_{2}} v( \delta) - B_{3}(t, \delta) \kappa I_{q}^{\alpha- p_{3}} v( \delta), \end{aligned}$$
for any
\(t\in\overline{J}\). Hence,
$$\begin{aligned} \Vert u \Vert ={}& \sup_{t\in\overline{J}} \bigl\vert u(t) \bigr\vert + \sup_{t\in\overline{J}} \bigl\vert u'(t) \bigr\vert + \sup_{t\in\overline{J}} \bigl\vert u''(t) \bigr\vert + \sup_{t\in\overline{J}} \bigl\vert u'''(t) \bigr\vert \\ \leq{}&\varLambda_{2} \Biggl[ \sum_{j=1}^{4} \Vert g_{0j} \Vert _{1} \phi_{j} \bigl( \Vert u \Vert \bigr) + \Vert g_{05} \Vert _{1} \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + \Vert u \Vert \gamma_{1}^{0} \Biggl[ \Biggl( \sum _{j=1}^{4} {}_{1} \eta_{j} \Biggr) \\ & + {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 - \gamma_{13}}}{ \varGamma _{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\ & + \Vert g_{06} \Vert _{1} \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + \Vert u \Vert \gamma _{2}^{0} \Biggl[ \sum _{j=1}^{4} {}_{2}\eta_{j} + {}_{2}\eta_{5} \frac{\delta^{ 1 - \gamma_{21}}}{ \varGamma_{q}( 2 - \gamma_{21})} \\ & + {}_{2}\eta_{6} \frac{\delta^{ 2 - \gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2} \eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma_{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\ & + \sum_{j=1}^{k_{1}} \Vert g_{1j} \Vert _{1} \psi_{1j} \biggl( \frac{\delta^{ 1 - \beta_{1j}}}{ \varGamma_{q}( 2 - \beta_{1j}) } \Vert u \Vert \biggr) + \sum_{j=1}^{k_{2}} \Vert g_{2j} \Vert _{1} \psi_{2j} \biggl( \frac{\delta^{ 2 - \beta _{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} \Vert u \Vert \biggr) \\ & + \sum_{j=1}^{k_{3}} \Vert g_{3j} \Vert _{1} \psi_{3j} \biggl( \frac{ \delta ^{ 3 - \beta_{3j}}}{ \varGamma_{q}( 4 - \beta_{3j})} \Vert u \Vert \biggr) \Biggr] \\ = {}&\varLambda_{2} A \bigl( \Vert u \Vert \bigr). \end{aligned}$$
Indeed,
\(\|u\| \leq\varLambda_{2} A(\|u\|)\). On the other hand, the operator
\(\varPhi: \overline{D} \to P_{cp,c}(\mathcal{X})\) is upper semicontinuous and compact, where
\(D = \{u\in\mathcal{X}: \| u\| < \Delta\}\). By considering the choice of
D, there is no
\(u\in \partial D\) such that
\(u\in\kappa H(u)\) for some
\(\kappa\in(0,1)\) and so
H has a fixed point
\(u\in\overline{D}\) due to Theorem
10. Therefore
H satisfies the assumptions of the nonlinear alternative of the Leray–Schauder-type result. It is easy to check that each fixed point of
H is a solution of problem (
2). This completes the proof. □