We present some important lemmas which assist in proving our main results. Consider the linear generalized fractional boundary value problem associated with (9)
Lemma 3.1
For \(\mathrm{w} \in C(J)\), the integral solution of (14) is given by
$$\begin{aligned} \mathrm{q}(\tau ) & = \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \tau , \xi ) \mathrm{w}(\xi ) \,{\mathrm {d}}\xi + \mu \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} (1 - \mu ) } \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} +\lambda \biggl( \frac{\tau ^{ \uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}( 1 - \mu ) } \biggr) + F_{\circ} \bigl( \mathrm{w} (\grave{a}) \bigr) \end{aligned}$$
(15)
for \(\tau , \xi \in J\), where
$$ \mathcal{G}_{1}(\tau , \xi ) = \textstyle\begin{cases} \frac{1}{ \Gamma (\sigma _{1} - 1 ) } ( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}} ) ( \frac{\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1} } )^{ \sigma _{1} - 2} \xi ^{\uprho _{1} - 1 } \\ \quad {} - \frac{1 }{ \Gamma ( \sigma _{1})} ( \frac{\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} )^{ \sigma _{1} - 1 } \xi ^{\uprho _{1} -1}, & \xi \leq \tau , \\ {\frac{1}{ \Gamma ( \sigma _{1} - 1 ) } ( \frac{ \tau ^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}} ) ( \frac{ \grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} )^{ \sigma _{1} - 2 } \xi ^{\uprho _{1}-1},} & \tau \leq \xi , \end{cases} $$
(16)
and
$$ \mathcal{G}_{2}(\tau , \xi ) = \textstyle\begin{cases} \frac{1}{ \Gamma (\sigma _{1} - 1 ) } ( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}} ) ( \frac{\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1} } )^{ \sigma _{1} - 2} \xi ^{\uprho _{1} - 1 } \\ \quad {} - \frac{1}{ \Gamma ( \sigma _{1} - 1)} ( \frac{ \tau ^{ \uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} )^{\sigma _{1} - 2 } \xi ^{\uprho _{1}-1}, & \xi \leq \tau , \\ \frac{1}{ \Gamma (\sigma _{1} - 1 ) } ( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}} ) ( \frac{\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1} } )^{ \sigma _{1} - 2} \xi ^{\uprho _{1} - 1 },& \tau \leq \xi . \end{cases} $$
(17)
Proof
By applying (12), equation (14) becomes
$$\begin{aligned} \mathrm{q}(\tau ) & = -l_{0} - l_{1} \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr) - l_{2} \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{2} \\ &\quad {} - \frac{ \uprho _{1}^{ 1 - \sigma _{1}}}{ \Gamma (\sigma _{1})} \int _{ \grave{a}}^{\tau} \bigl( \grave{a}^{ \uprho _{1}} - \xi ^{\uprho _{1}}\bigr)^{ \sigma _{1}-1} \xi ^{\uprho _{1} - 1 } \mathrm{w}( \xi ) \,{\mathrm {d}} \xi \end{aligned}$$
for some arbitrary constants . From the boundary conditions of (14) we get
$$\begin{aligned} \mathrm{q}(\tau ) & = F_{\circ } \bigl(\mathrm{w}(\grave{a}) \bigr) + \lambda \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{\uprho _{1} (1 -\mu ) } \biggr) \\ &\quad {} - \frac{1}{\Gamma (\sigma _{1})} \int _{\grave{a}}^{\tau} \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1} - 1 } \xi ^{\uprho _{1} - 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} + \biggl( \frac{ \tau ^{\uprho _{1}} - \grave{a}^{ \uprho _{1}} }{ \uprho _{1}} \biggr) \frac{1}{ (1 - \mu ) \Gamma ( \sigma _{1} - 1 ) } \biggl[ \int _{\grave{a}}^{\grave{\iota}} \biggl( \frac{\grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1}-2} \xi ^{\uprho _{1} - 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ & \quad {} -\mu \int _{\grave{a}}^{ \tau} \biggl( \frac{ \tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{ \sigma _{1}-2} \xi ^{\uprho _{1}} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr] \\ &= F_{\circ } \bigl(\mathrm{w}(\grave{a}) \bigr) + \lambda \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} (1 -\mu ) } \biggr) \\ &\quad {} - \frac{1}{\Gamma (\sigma _{1})} \int _{\grave{a}}^{\tau} \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1} } \biggr)^{ \sigma _{1}-1} \xi ^{\uprho _{1} - 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} + \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{\uprho _{1}} \biggr) \biggl[ \frac{1}{ \Gamma (\sigma _{1} -1)} \\ &\quad {} + \frac{\mu}{ \Gamma (\sigma _{1} -1) (1 - \mu )} \biggr] \int _{\grave{a}}^{\grave{\iota}} \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{\uprho _{1}} \biggr)^{ \sigma _{1}-2} \xi ^{ \uprho _{1} - 1 } \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} - \biggl( \frac{ \tau ^{ \uprho _{1}} - \grave{a}^{ \uprho _{1}}}{ \uprho _{1}} \biggr) \frac{\mu}{ (1 - \mu ) \Gamma ( \sigma _{1}-1)} \int _{ \grave{a}}^{\tau} \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{ \uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1}- 2 } \xi ^{\uprho _{1} - 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi . \end{aligned}$$
Splitting the second integral in two parts permits us to write
$$\begin{aligned} \mathrm{q}(\tau ) & = \frac{1}{\Gamma (\sigma _{1})} \biggl[(\sigma _{1}-1) \biggl( \frac{ \tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr) \int _{\grave{a}}^{\tau} \biggl( \frac{\grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1} - 2 } \xi ^{\sigma _{1} - 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} - \int _{\grave{a}}^{\tau} \biggl( \frac{\tau ^{ \uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{ \sigma _{1}-1} \xi ^{\uprho _{1} - 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} + (\sigma _{1} -1) \biggl( \frac{\tau ^{ \uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr) \int _{\tau}^{\grave{\iota}} \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1}-2} \xi ^{\uprho _{1} - 1 } \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr] \\ &\quad {} + \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} (1 - \mu )} \biggr) \frac{1}{ \Gamma (\sigma _{1} -1)}\biggl[ \mu \biggl( \int _{ \grave{a}}^{ \tau} \biggl( \frac{\grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1} - 2} \xi ^{\uprho _{1} - 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} + \int _{\tau}^{\grave{\iota}} \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1}- 2} \xi ^{\uprho _{1}-1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr) \\ &\quad {} - \mu \int _{\grave{a}}^{\tau} \biggl( \frac{ \tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{ \uprho -2} \xi ^{\uprho _{1}- 1} \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr] \\ &\quad {} + F_{\circ} \bigl(\mathrm{w}(\grave{a}) \bigr) + \lambda \biggl( \frac{ \tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} (1 -\mu ) } \biggr) \\ & = \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \tau , \xi ) \mathrm{w}(\xi )+ \mu \biggl( \frac{\tau ^{\uprho _{1}} -\grave{a}^{\uprho _{1}}}{( 1 - \mu ) \uprho _{1}} \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \\ &\quad {} +\lambda \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{(1 - \mu ) \uprho _{1}} \biggr) + F_{\circ } \bigl( \mathrm{w}(\grave{a}) \bigr). \end{aligned}$$
The converse follows by direct computation. The proof is completed. □
Now, consider the generalized p-Laplacian fractional boundary value problem associated with (9)
Lemma 3.2
For \(\mathrm{w}(\tau ) \in C^{+}(J)\), fractional boundary value problem (18) has a unique solution
$$\begin{aligned} \mathrm{q}(\tau ) & = \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,{\mathrm {d}}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \mu \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} - \mu \uprho _{1}} \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,{\mathrm {d}}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \lambda \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{\uprho _{1} - \mu \uprho _{1}} \biggr) + F_{\circ } \biggl( \upphi _{\bar{p}} \biggl( \int _{ \grave{a}}^{\grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w} ( \xi ) \,{\mathrm {d}}\xi \biggr) \biggr), \end{aligned}$$
(19)
where
$$ \mathcal{H}(\tau , \xi ) = \textstyle\begin{cases} \frac{1}{\Gamma (\sigma _{2}-1)} ( \frac{\grave{\iota}^{\uprho _{2}} - \tau ^{\uprho _{2}} }{ \uprho _{2}} ) ( \frac{\xi ^{\uprho _{2}} - \grave{a}^{\uprho _{2}}}{\uprho _{2}} )^{\sigma _{2} - 2} \xi ^{\uprho _{2}-1} \\ \quad {} - \frac{1}{ \Gamma ( \sigma _{2})} ( \frac{\xi ^{\uprho _{2}} - \tau ^{\uprho _{2}}}{ \uprho _{2}} )^{ \sigma _{2} -1} \xi ^{\uprho _{2}-1},& \tau \leq \xi , \\ \frac{1}{\Gamma (\sigma _{2}-1)} ( \frac{\grave{\iota}^{\uprho _{2}} - \tau ^{\uprho _{2}}}{ \uprho _{2}} ) ( \frac{\xi ^{\uprho _{2}} - \grave{a}^{\uprho _{2}}}{ \uprho _{2}} )^{\sigma _{2}- 2} \xi ^{\uprho _{2}-1},& \xi \leq \tau ,\end{cases} $$
(20)
\(\mathcal{G}_{1}(\tau , \xi )\), \(\mathcal{G}_{2}(\tau , \xi )\) are defined in Lemma 3.1and \(\bar{p} = \frac{p}{p-1}\).
Proof
From Lemma 2.6, equation (18) is equivalent to the equation
for some constants . Using the second boundary condition, we get
Consequently,
Thus, problem (18) can be written as
which, according to Lemma 3.1, has a unique solution of the form (19). □
Lemma 3.3
The functions \(\mathcal{G}_{1}\), \(\mathcal{G}_{2}\), and \(\mathcal{H}\), equations (16), (17), and (20) satisfy the following:
-
(i)
\(\mathcal{G}_{1}(\tau , \xi )\), \(\mathcal{G}_{2}(\tau , \xi )\), and \(\mathcal{H}(\tau , \xi )\) are continuous on \([\grave{a}, \grave{\iota}] \times [\grave{a}, \grave{\iota}]\).
-
(ii)
For all \((\tau , \xi ) \in [\grave{a}, \grave{\iota}]\times [\grave{a}, \grave{\iota}]\),
$$\begin{aligned}& \begin{aligned} \mathcal{G}_{1} (\tau , \xi ) & \leqslant \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}} \biggr)^{\sigma _{1} -1 } \frac{ \grave{\iota}^{\uprho _{1} - 1 }}{ \Gamma (\sigma _{1} -1)} \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \tau , \xi ) \,{\mathrm {d}}\xi \\ & = \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \Gamma (\sigma _{1}) \uprho _{1}} \biggr) \biggl( \frac{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}}{ \uprho _{1}} \biggr)^{ \sigma _{1} -1} \\ &\quad {} - \frac{1}{\Gamma (\sigma _{1} + 1 ) } \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1}}, \end{aligned} \\& \begin{aligned} \mathcal{G}_{2}(\tau , \xi ) & \leqslant \biggl( \frac{\grave{\iota }^{\uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}} \biggr)^{\sigma _{1} -2 } \frac{\grave{\iota}^{ \uprho _{1}- 1} }{ \Gamma (\sigma _{1} -1)} \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \,{\mathrm {d}}\xi \\ & = \frac{1}{ \Gamma (\sigma _{1})} \biggl( \biggl( \frac{\grave{ \iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1} -1} - \biggl( \frac{ \tau ^{ \uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1} - 1} \biggr), \end{aligned} \\& \begin{aligned} \mathcal{H} (\tau , \xi ) & \leqslant \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}} }{ \uprho _{2}} \biggr)^{ \sigma _{2} -1 } \frac{ \grave{\iota}^{ \uprho _{2} - 1}}{ \Gamma (\sigma _{2} -1)} \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\tau , \xi ) \,{\mathrm {d}}\xi \\ & = \frac{\grave{\iota}^{\uprho _{2}} - \xi ^{\uprho _{2}}}{\uprho _{2} \Gamma (\sigma _{2})} \biggl( \biggl( \frac{ \grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}} }{ \uprho _{2}} \biggr)^{\sigma _{2} -1} - \frac{1}{\sigma _{2}} \biggl( \frac{ \grave{\iota}^{\uprho _{2}} - \xi ^{\uprho _{2}}}{ \uprho _{2} } \biggr)^{\sigma _{2} -1} \biggr). \end{aligned} \end{aligned}$$
-
(iii)
For all \((\tau , \xi ) \in [\grave{a}, \grave{\iota}]^{2} : \mathcal{G}_{1}( \tau , \xi ) \geqslant 0\), \(\mathcal{G}_{2}(\tau , \xi ) \geqslant 0\), \(\mathcal{H}(\tau , \xi )\geqslant 0\).
-
(iv)
For all \(\xi \in J\), the function \(\tau \to \mathcal{G}_{1}(\tau , \xi )\) is increasing and \(\tau \to \mathcal{H}(\tau , \xi )\) is decreasing. In addition, \(\forall (\tau , \xi ) \in (\grave{a}, \grave{\iota})^{2}\) we have
$$ \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} -1} \mathcal{G}_{1}( \grave{\iota}, \xi ) \leqslant \mathcal{G}_{1}(\tau , \xi ) $$
and
$$ \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \tau ^{ \uprho _{2}}}{ \grave{\iota}^{\uprho _{2}} - \grave{a}^{ \uprho _{2}}} \biggr)^{\sigma _{2}-1} \mathcal{H}(\grave{a}, \xi ) \leqslant \mathcal{H}(\tau , \xi ). $$
-
(v)
For all \((\tau , \xi ) \in (\grave{a}, \grave{\iota})^{2}\), we have
$$\begin{aligned} &\frac{\tau ^{\uprho _{1}-1} \uprho _{1}}{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggl[ 1 - \biggl( \frac{\tau}{ \grave{\iota}} \biggr)^{\uprho _{1}( \sigma _{1} -2)} \biggr] \mathcal{G}_{1}(\grave{\iota}, \xi ) \\ &\quad \leqslant {\mathcal{G}_{1}^{\prime}}_{\tau }(\tau , \xi ) \leqslant \frac{\sigma _{1}-1}{\sigma _{1} -2 } \frac{\tau ^{\uprho _{1}-1} \uprho _{1}}{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}} \mathcal{G}_{1}( \grave{\iota}, \xi ). \end{aligned}$$
Proof
Using the definitions of \(\mathcal{G}_{1}\), \(\mathcal{G}_{2}\), and \(\mathcal{H}\), (i) and (ii) are obtained straightforwardly. For property (iii), we only consider the case \(\xi \leq \tau \) as the other case is straightforward. When \(\xi \leq \tau \), we have
$$\begin{aligned} \mathcal{G}_{1}(\tau , \xi ) & \geqslant \frac{1}{ \Gamma (\sigma _{1} -1) } \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr) \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{ \uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1} - 2 } \grave{a}^{\uprho _{1}-1} \\ &\quad {} - \frac{1}{ \Gamma ( \sigma _{1})} \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{ \sigma _{1} -1} \grave{a}^{\uprho _{1} -1} \\ &\geqslant \biggl( \frac{ \tau ^{ \uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1} -1} \grave{a}^{\uprho _{1} - 1 } \biggl[ \frac{1}{ \Gamma (\sigma _{1} -1)} - \frac{1}{ \Gamma (\sigma _{1})} \biggr]\geqslant 0, \end{aligned}$$
because \(\Gamma (\sigma _{1} -1) \leqslant \Gamma (\sigma _{1})\) for \(2 < \sigma _{1}\leq 3\). Similarly, we can easily prove that \(\mathcal{G}_{2}(\tau , \xi )\geqslant 0\) and \(\mathcal{H}(\tau , \xi )\geqslant 0\), \(\forall (\tau , \xi ) \in J^{2}\). Now, for property (iv), we first check that \(\mathcal{G}_{1}(\tau , \xi )\) is nondecreasing w.r.t. \(\tau \in J\).
$$ \frac{\partial \mathcal{G}_{1}}{\partial \tau} (\tau , \xi )= \textstyle\begin{cases} \frac{\tau ^{\uprho _{1}-1}}{\Gamma (\sigma _{1} - 1) } ( \frac{\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} )^{ \sigma _{1} - 2 } \xi ^{\uprho _{1}-1} & \\ \quad {} - \frac{ \tau ^{\uprho _{1} - 1}}{ \Gamma ( \sigma _{1} -1) } ( \frac{\tau ^{ \uprho _{1}} - \xi ^{\uprho _{1}} }{ \uprho _{1}} )^{\sigma _{1} - 2 } \xi ^{\uprho _{1} -1},& \xi \leqslant \tau , \\ \frac{\tau ^{\uprho _{1} - 1}}{ \Gamma (\sigma _{1} -1 )} ( \frac{ \grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}}{ \uprho _{1}} )^{\sigma _{1} - 2} \xi ^{\uprho _{1}-1},& \tau \leqslant \xi .\end{cases} $$
(22)
Thus, \(\mathcal{G}_{1}(\tau , \xi )\) is increasing with respect to \(\tau \in J\), and therefore \(\mathcal{G}_{1}(\tau , \xi ) \leqslant \mathcal{G}_{1}(\grave{\iota}, \xi )\) for \(\grave{a} \leqslant \tau \), \(\xi \leqslant \grave{\iota}\). Furthermore, for \(\tau \leqslant \xi \), we have
$$\begin{aligned} \frac{\partial \mathcal{H}(\tau , \xi ) }{\partial \tau} &= - \frac{\tau ^{\uprho _{2} - 1}}{ \Gamma (\sigma _{2} -1)} \biggl( \frac{\xi ^{\uprho _{2}} - \grave{a}^{\uprho _{2}}}{ \uprho _{2}} \biggr)^{\sigma _{2} - 2} \xi ^{\uprho _{2}-1} \\ &\quad {} + \frac{ ( \sigma _{2} -1 ) \tau ^{ \uprho _{2}-1}}{ \Gamma (\sigma _{2})} \biggl( \frac{ \xi ^{\uprho _{2}} - \tau ^{ \uprho _{2}}}{ \uprho _{2}} \biggr)^{\sigma _{2} -2} \xi ^{\uprho _{2}-1} \\ & = \frac{ \tau ^{\uprho _{2}-1}}{ \Gamma (\sigma _{2} - 1)} \xi ^{ \uprho _{2} -1} \biggl[ \biggl( \frac{\xi ^{\uprho _{2}} - \tau ^{\uprho _{2}}}{ \uprho _{2} } \biggr)^{\sigma _{2} -2} - \biggl( \frac{ \xi ^{\uprho _{2}} - \grave{a}^{\uprho _{2}}}{ \uprho _{2}} \biggr)^{\sigma _{2} -2} \biggr] \\ &\leqslant \frac{ \tau ^{ \uprho _{2}-1}}{ \Gamma (\sigma _{2}-1)} \xi ^{\uprho _{2} -1} \biggl[ \biggl( \frac{ \xi ^{\uprho _{2}} - \grave{a}^{\uprho _{2}}}{\uprho _{2}} \biggr)^{\sigma _{2} -2} - \biggl( \frac{\xi ^{\uprho _{2}} - \grave{a}^{\uprho _{2}}}{ \uprho _{2}} \biggr)^{\sigma _{2} -2} \biggr]= 0, \end{aligned}$$
and for \(\xi \leqslant \tau \), we have
$$\begin{aligned} \frac{ \mathcal{H}(\tau , \xi ) }{\partial \tau} &= \frac{ - \tau ^{ \uprho _{2}-1}}{ \Gamma (\sigma _{2} - 1)} \biggl( \frac{\xi ^{\uprho _{2}} - \grave{a}^{\uprho _{2}}}{ \uprho _{2}} \biggr)^{\sigma _{2} -2} \xi ^{\uprho _{2}-1} \leqslant 0. \end{aligned}$$
Thus, \(\mathcal{H}(\tau , \xi )\) is nonincreasing with respect to τ. Consequently, \(\mathcal{H}(\tau , \xi )\leqslant \mathcal{H}(\grave{a}, \xi )\), \(\forall \tau , \xi \in J\). On the other hand, when \(\tau \geqslant \xi \),
$$\begin{aligned} \frac{\mathcal{G}_{1}(\tau , \xi )}{\mathcal{G}_{1}(\grave{\iota}, \xi )} & = \frac{ (\sigma _{1} - 1) ( \tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}) (\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}})^{\sigma _{1}-2} - (\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}})^{ \sigma _{1} -1} }{ (\sigma _{1} -1) (\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}) (\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}})^{\sigma _{1}-2} - (\grave{\iota}^{\uprho _{1} } - \xi ^{\uprho _{1}})^{\sigma _{1} -1} } \\ & = \frac{1}{ (\sigma _{1} -1) (\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}) ( \grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}})^{\sigma _{1}-2} - ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}})^{\sigma _{1} -1} } \\ &\quad {} \times \biggl[ (\sigma _{1}-1) \bigl(\tau ^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}\bigr) \bigl(\grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}}\bigr)^{ \alpha -2} \\ &\quad {} - \bigl(\tau ^{ \uprho _{1}} - \xi ^{\uprho _{1}}\bigr)^{\sigma _{1} -1} \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}} }{ \tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} -1} \biggr]. \end{aligned}$$
As
$$ \biggl( \frac{\tau ^{\uprho} - \xi ^{\uprho}}{ \tau ^{\uprho} - \grave{a}^{\uprho}} \biggr)^{\sigma} \leq \biggl( \frac{\grave{\iota}^{\uprho} - \xi ^{\uprho}}{ \grave{ \iota}^{ \uprho} - \grave{a}^{\uprho}} \biggr)^{\sigma},$$
for \(\sigma >0\), we obtain
$$\begin{aligned} \frac{ \mathcal{G}_{1}( \tau , \xi )}{\mathcal{G}_{1} ( \grave{\iota}, \xi )} & \geqslant \frac{1}{ (\sigma _{1} -1) (\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}) ( \grave{\iota}^{\uprho _{1}} - \xi ^{\uprho _{1}})^{\sigma _{1}-2} - ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}})^{\sigma _{1} -1} } \\ &\quad {} \times \biggl[ (\sigma _{1}-1) \bigl(\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}\bigr) \bigl( \grave{\iota}^{\uprho _{1} } - \xi ^{ \uprho _{1}}\bigr)^{\sigma _{1}-2} \\ &\quad {} - \bigl(\tau ^{ \uprho _{1}} - \xi ^{\uprho _{1}}\bigr)^{\sigma _{1} -1} \biggl( \frac{\grave{ \iota}^{\uprho _{1}} - \xi ^{\uprho _{1}} }{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} -1} \biggr] \\ & \geqslant \frac{ ( \tau ^{ \uprho _{1}} - \grave{a}^{\uprho _{1}})^{\sigma _{1} - 1 } }{ (\grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} )^{\sigma _{1}-1} } \\ &\quad {} \times \frac{1}{ (\sigma _{1}-1 ) ( \grave{\iota}^{ \uprho _{1}} - \grave{a}^{ \uprho _{1}}) ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}} )^{\sigma _{1} -2} - ( \grave{ \iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}} )^{\sigma _{1} -1} } \\ &\quad {}\times \bigl[ (\sigma _{1} -1) \bigl( \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}} \bigr)^{\sigma _{1} -1} \bigl(\tau ^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}\bigr)^{2 - \sigma _{1}} \bigl( \grave{ \iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}}\bigr)^{\sigma _{1} - 2} \\ &\quad {} - \bigl( \grave{\iota}^{ \uprho _{1} } -\xi ^{ \uprho _{1} } \bigr)^{ \sigma _{1} -1} \bigr] \\ & \geqslant \frac{ (\tau ^{ \uprho _{1} } - \grave{a}^{\uprho _{1}})^{\sigma _{1} -1 }}{ ( \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} )^{\sigma _{1} -1} } \\ &\quad {} \times \frac{1}{ (\sigma _{1} - 1) ( \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}) ( \grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}})^{ \sigma _{1} -2} - ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}})^{\sigma _{1} -1} } \\ &\quad {} \times \biggl[ \biggl( \frac{ \grave{ \iota}^{ \uprho _{1}} - \grave{a}^{ \uprho _{1} } }{ \tau ^{ \uprho _{1} } - \grave{a}^{ \uprho _{1}}} \biggr)^{ \sigma _{1} -2} (\sigma _{1} -1) \bigl( \grave{\iota}^{ \uprho _{1}} - \grave{a}^{ \uprho _{1}} \bigr) \bigl( \grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}}\bigr)^{\sigma _{1} -2} \\ &\quad {} - \bigl( \grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}} \bigr)^{\sigma _{1} -1} \biggr] \\ & \geqslant \frac{ (\tau ^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} )^{ \sigma _{1} - 1 } }{ ( \grave{\iota}^{\uprho _{1}} - \grave{a}^{ \uprho _{1} } )^{\sigma _{1} -1} }. \end{aligned}$$
For \(\tau \leqslant \xi \), we have
$$ \frac{ \mathcal{G}_{1}(\tau , \xi )}{ ( \tau ^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} )^{\sigma _{1} -1}} = \frac{ \uprho _{1}^{\sigma _{1} -1} \xi ^{\uprho _{1} - 1}}{ \Gamma (\sigma _{1} - 1)} \bigl( \grave{ \iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}} \bigr)^{ \sigma _{1} -2} \frac{1}{ (t^{\uprho _{1}} - \grave{a}^{\uprho _{1} } )^{ \sigma _{1} -2}},$$
which is a nonincreasing function as \(\sigma _{1} \geq 0\). Consequently,
$$ \frac{\mathcal{G}_{1}(\tau , \xi )}{ ( \tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}} )^{\sigma _{1} -1}} \geq \frac{ \mathcal{G} (\grave{\iota}, \xi )}{ ( \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}} )^{\sigma _{1} - 1}},$$
which implies
$$ \mathcal{G}_{1}( \tau , \xi ) \geqslant \biggl( \frac{ \tau ^{ \uprho _{1} } - \grave{a}^{\uprho _{1}} }{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} } \biggr)^{\sigma _{1} -1} \mathcal{G}_{1} ( \grave{\iota}, \xi ).$$
Using similar techniques, one can prove that
$$ \mathcal{H}(\tau , \xi ) \geqslant \biggl( \frac{ \grave{\iota}^{\uprho _{2}} - \tau ^{ \uprho _{2}} }{ \grave{\iota}^{ \uprho _{2}} - \grave{a}^{ \uprho _{2}}} \biggr)^{ \sigma -1} \mathcal{H}(\grave{a}, \xi )$$
for \(\grave{a} \leqslant \xi , \tau < \grave{\iota}\). Therefore (iv) of Lemma 3.3 holds. Finally, for property (v), we can consider two cases. Nevertheless, we prove the results for the case \(\xi \leq \tau \) only. The simpler case \(\grave{a} \leq \tau \leq \xi < \grave{ \iota}\) can be treated with similar arguments. When \(\xi \leq \tau \), we have
$$ \frac{{\mathcal{G}_{1}}_{\tau}^{\prime}(\tau , \xi )}{ \mathcal{G}_{1}( \grave{\iota}, \xi ) } \frac{ ( \grave{\iota}^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}) }{\tau ^{\uprho _{1} - 1} \uprho _{1} (\sigma _{1} - 1 ) } = \frac{ ( \grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1} } )^{ \sigma _{1} - 2 } - ( \tau ^{ \uprho _{1}} - \xi ^{ \uprho _{1} } )^{\sigma _{1} -2}}{(\sigma _{1} - 1) ( \grave{\iota}^{\uprho _{1}} - \xi ^{ \uprho _{1} } )^{\sigma _{1} -2} - \frac{ ( \grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}} )^{\sigma _{1} -1} }{ ( \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}})}}. $$
Consequently,
$$\begin{aligned} \frac{{\mathcal{G}_{1}}_{\tau}^{\prime}(\tau , \xi )}{\mathcal{G}_{1}(\grave{\iota}, \xi )} \frac{(\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}) }{ \tau ^{\uprho _{1} - 1} \uprho _{1} (\sigma _{1} -1)} & \leq \frac{ ( \grave{ \iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}} )^{\sigma _{1} -2} }{ (\sigma _{1} -1) ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}} )^{\sigma _{1} -2} - \frac{ ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}} )^{ \sigma _{1} -1}}{ (\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}})}} \\ & \leq \frac{1}{(\sigma _{1}-1) - \frac{ (\grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}} )}{ ( \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}})}} \\ & \leq \frac{1}{(\sigma _{1} - 2)}. \end{aligned}$$
On the other hand,
$$\begin{aligned} \frac{{\mathcal{G}_{1}}_{\tau}^{\prime}(\tau , \xi )}{ \mathcal{G}_{1}( \grave{\iota}, \xi )} \frac{(\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}) }{\tau ^{\uprho _{1}-1} \uprho _{1} } & = \frac{(\sigma _{1} -1) [ ( \grave{\iota}^{ \uprho _{1}} - \xi ^{ \uprho _{1}} )^{ \sigma _{1} -2} - ( \tau ^{ \uprho _{1}} - \xi ^{\uprho _{1} } )^{\sigma _{1} -2} ]}{(\sigma _{1}-1) ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}} )^{\sigma _{1} -2} - \frac{ ( \grave{\iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}} )^{ \sigma _{1} -1}}{( \grave{\iota}^{\uprho _{1}} - \grave{a}^{ \uprho _{1}})}} \\ & \geq 1 - \frac{ ( \tau ^{ \uprho _{1}} - \xi ^{\uprho _{1}} )^{ \sigma _{1} -2}}{ ( \grave{ \iota}^{ \uprho _{1}} - \xi ^{\uprho _{1}} )^{ \sigma _{1} -2} } \\ & \geq 1 - \biggl( \frac{\tau}{ \grave{\iota}} \biggr)^{ \uprho _{1} ( \sigma _{1} -2)} \biggl( \frac{ 1 - ( \frac{\xi}{\tau} )^{ \uprho _{1}}}{ 1 - ( \frac{\xi}{ \grave{\iota}} )^{ \uprho _{1}}} \biggr)^{\sigma _{1} -2} \\ & \geq 1 - \biggl( \frac{\tau}{\grave{\iota}} \biggr)^{ \uprho _{1} ( \sigma _{1} -2)}. \end{aligned}$$
Thus, the proof is completed. □
Now, consider the Banach space . Suppose that
is continuous on J for all , then from Definition 2.6 and Lemma 2.4 we can define the norm on as follows:
in which
$$ \breve{M}_{1}=\max \Bigl\{ \max_{\tau \in J} \bigl\vert \mathrm{q}(\tau ) \bigr\vert , \max_{\tau \in J} \bigl\vert \delta _{\uprho _{1}}^{1} \mathrm{q}(\tau ) \bigr\vert , \max_{\tau \in J} \bigl\vert \delta _{ \uprho _{1}}^{2} \mathrm{q}(\tau ) \bigr\vert \Bigr\} ,$$
and the cone
Lemma 3.4
Assume (H2) and let q be the unique solution of fractional boundary value problem (18) associated with given \(\mathrm{w}(\tau )\in C^{+}(J)\). Then \(\mathrm{q}\in K\) and the following inequalities hold for \(\tau \in [\grave{a}_{\circ}, \grave{\iota}_{\circ}] \subset ( \grave{a}, \grave{\iota})\):
$$\begin{aligned}& \max_{\tau \in J} \bigl\vert \mathrm{q}(\tau ) \bigr\vert \leqslant \biggl( \biggl( \frac{ \grave{a}_{\circ}^{ \uprho _{1}} - \grave{a}^{ \uprho _{1}}}{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} -1 } \biggl( \frac{ \grave{ \iota}^{\uprho _{2}}- \grave{\iota}_{\circ}^{\uprho _{2}}}{\grave{\iota}^{\uprho _{2}} - \grave{a}^{ \uprho _{2}}} \biggr)^{ \sigma _{1} - 1} \biggr)^{-1} \mathrm{q}(\tau ), \end{aligned}$$
(23)
$$\begin{aligned}& \max_{\tau \in J} \bigl\vert \delta _{\uprho _{1}}^{1} \mathrm{q}(\tau ) \bigr\vert \leqslant \frac{\sigma _{1} -1}{ \sigma _{1} - 2 } \frac{\uprho _{1}}{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}}} \max_{ \tau \in J} \bigl\vert \mathrm{q}(\tau ) \bigr\vert , \end{aligned}$$
(24)
$$\begin{aligned}& \begin{aligned}[b] \max_{\tau \in J} \bigl\vert \delta _{\uprho _{1}}^{2} \mathrm{q}(\tau ) \bigr\vert & \leqslant \biggl( \biggl( \frac{ \grave{a}_{\circ}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \grave{\iota}^{ \uprho _{1}}- \grave{a}^{ \uprho _{1}}} \biggr)^{ \sigma _{1}-1} Z(\grave{ \iota}_{\circ}) \int ^{ \grave{ \iota}}_{ \grave{a}} \mathcal{G}_{1}( \grave{\iota}, \xi ) \,{\mathrm {d}}\xi \biggr)^{-1} \\ & \quad {}\times \frac{1}{ \Gamma (\sigma _{1} - 1) } \biggl( \frac{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{ \sigma _{1} -2} \max_{\tau \in J} \bigl\vert \mathrm{q}( \tau ) \bigr\vert , \end{aligned} \end{aligned}$$
(25)
$$\begin{aligned}& \min_{\tau \in [\grave{a}_{\circ}, \grave{\iota}_{\circ}] } \mathrm{q}(\tau ) \geqslant \biggl( \frac{ \grave{a}_{\circ}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{ \uprho _{1}}} \biggr)^{\sigma _{1}-1} \breve{M}_{2} \Vert \mathrm{q} \Vert , \end{aligned}$$
(27)
where
$$ Z(\tau ) = \upphi _{\bar{p}} \biggl( \biggl( \frac{ \grave{\iota}^{\uprho _{2}} - \tau ^{ \uprho _{2}}}{ \grave{\iota}^{ \uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{ \sigma _{2}-1} \biggr)$$
and
$$\begin{aligned} \breve{M}_{2} & = \min \biggl\lbrace 1, \frac{ \sigma _{1} -2}{\sigma _{1} -1} \biggl( \frac{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr), \\ &\quad \min \biggl\lbrace \Gamma (\sigma _{1} -1) \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{ 2 - \sigma _{1}}, 1 \biggr\rbrace \\ &\quad {} \times \biggl( \frac{ \grave{a}_{\circ}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}} \biggr)^{ \sigma _{1} -1} \times Z( \grave{\iota}_{\circ}) \int ^{ \grave{ \iota}}_{\grave{a}} \mathcal{G}_{1} ( \grave{\iota}, \xi ) \,{\mathrm {d}}\xi \biggr\rbrace . \end{aligned}$$
(28)
Proof
From Lemma 3.2, we have
$$\begin{aligned} \mathrm{q}(\tau ) & = \int _{\grave{a}}^{ \grave{\iota}} \mathcal{G}_{1}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{ \grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d} s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \mu \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} - \mu \uprho _{1}} \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \lambda \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{\uprho _{1} - \mu \uprho _{1}} \biggr) + F_{\circ } \biggl( \upphi _{\bar{p}} \biggl( \int _{ \grave{a}}^{\grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w}( \xi ) \,\mathrm{d}\xi \biggr) \biggr). \end{aligned}$$
-
(1)
The functions \(\mathcal{G}_{1}\), \(\mathcal{G}_{2}\), and \(\mathcal{H}\) are nonnegative (Lemma 3.3(iii)). In addition, \(F_{\circ}(v)\) is nonnegative for \(v\geq 0\) (thanks to (H2)). Thus, q is also nonnegative. Furthermore, as \(\mathcal{G}_{1}\) is increasing w.r.t. τ (Lemma 3.3(iv)), so it is the function q. To prove that q is \(\uprho _{1}\)-concave, we need to show that \(\delta _{ \uprho _{1}}^{1} \mathrm{q}(\tau )\) is decreasing on J (Remark 2.2), which can be obtained from the negativity of the derivative
$$\begin{aligned} \bigl( \delta _{\uprho _{1}}^{1} \mathrm{q}(\tau ) \bigr)^{\prime} & = - \frac{\tau ^{ \uprho _{1} -1}}{ \Gamma (\sigma _{1} -2 )} \int _{ \grave{a}}^{\tau} \biggl( \frac{ \tau ^{\uprho _{1}} - \xi ^{ \uprho _{1}}}{ \uprho _{1}} \biggr)^{ \sigma _{1} -3} \xi ^{ \uprho _{1}-1} \\ &\quad {} \times \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d} s \biggr) \,\mathrm{d} \xi \leq 0. \end{aligned}$$
-
(2)
As q is nonnegative and increasing, we have
$$\begin{aligned} \max_{\tau \in J } \bigl\vert \mathrm{q}(\tau ) \bigr\vert & = \mathrm{q}(\grave{\iota}) \\ & = \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \grave{\iota}, \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,\mathrm{d}\xi \\ &\quad {} + \mu \biggl( \frac{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} - \mu \uprho _{1}} \biggr) \int _{\grave{a}}^{ \grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{ \grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,\mathrm{d} \xi \\ &\quad {} + \lambda \biggl( \frac{\grave{ \iota}^{ \uprho _{1}} - \grave{a}^{ \uprho _{1}}}{ \uprho _{1} - \mu \uprho _{1}} \biggr) + F_{\circ} \biggl( \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w}(\xi ) \,\mathrm{d}\xi \biggr) \biggr). \end{aligned}$$
For \(\tau \in [\grave{a}_{\circ},\grave{\iota}_{\circ}]\), using (iv) of Lemma 3.3 and the fact that
$$ \biggl( \frac{ \grave{a}_{\circ}^{\uprho _{1}}- \grave{a}^{\uprho _{1}}}{ \grave{\iota}^{ \uprho _{1}}- \grave{a}^{\uprho _{1}}} \biggr) < 1,$$
we get
$$\begin{aligned} \mathrm{q}(\tau )& \geqslant \int _{\grave{a}}^{\grave{\iota}} \biggl( \frac{\grave{a}_{\circ}^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}}{ \grave{\iota}^{ \uprho _{1}}- \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} -1} \mathcal{G}_{1}(\grave{\iota}, \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} +\mu \biggl( \frac{ \grave{a}_{\circ}^{\uprho _{1}}- \grave{a}^{\uprho _{1}}}{\grave{\iota}^{\uprho _{1}}- \grave{a}^{ \uprho _{1}}} \biggr)^{\sigma _{1} -2} \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} - \mu \uprho _{1}} \biggr) \\ &\quad {} \times \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} +\lambda \biggl( \frac{\grave{a}_{\circ}^{\uprho _{1}}- \grave{a}^{\uprho _{1}}}{\grave{\iota}^{\uprho _{1}}- \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1}-2} \biggl( \frac{\tau ^{\uprho _{1}}-\grave{a}^{\uprho _{1}}}{\uprho _{1} - \mu \uprho _{1}} \biggr) \\ &\quad {} + \biggl( \frac{\grave{a}_{\circ}^{ \uprho _{1}}- \grave{a}^{\uprho _{1}}}{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}} \biggr)^{\sigma _{1}-1} F_{\circ} \biggl( \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}( \grave{a}, \xi ) \mathrm{w}(\xi ) \,\mathrm{d}\xi \biggr) \biggr). \end{aligned}$$
Consequently,
$$ \mathrm{q}(\tau ) \geqslant \biggl( \frac{\grave{a}_{\circ}^{\uprho _{1}}- \grave{a}^{\uprho _{1}}}{\grave{\iota}^{\uprho _{1}}- \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} -1}\max_{t \in J} \bigl\vert \mathrm{q}( \tau ) \bigr\vert , $$
and thus (23) holds.
-
(3)
We have
$$\begin{aligned} \delta _{\uprho _{1}}^{1} \mathrm{q}(\tau ) & = \tau ^{1 - \uprho _{1}} \int _{\grave{a}}^{\grave{\iota}} {\mathcal{G}_{1}}^{\prime}_{\tau}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \frac{\mu }{ (1-\mu )} \int _{ \grave{a}}^{ \grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi + \frac{\lambda }{(1-\mu )}. \end{aligned}$$
From Lemma 3.3 ((iii) and (v)), we can deduce that \(\delta _{\uprho _{1}}^{1} \mathrm{q}(\tau ) \geq 0\) and
$$\begin{aligned} \delta _{\uprho _{1}}^{1} \mathrm{q}(\tau ) & \leqslant \int _{ \grave{a}}^{\grave{\iota}} \frac{ \sigma _{1}-1}{\sigma _{1} - 2} \frac{ \uprho _{1}}{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}}} \mathcal{G}_{1}( \grave{\iota}, \xi )\upphi _{\bar{p}} \biggl( \int _{ \grave{a}}^{\grave{\iota}} \mathcal{H}( \xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \frac{\mu }{ (1-\mu )} \int _{\grave{a}}^{ \grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d} s \biggr) \,{\mathrm {d}}\xi + \frac{\lambda }{(1-\mu )} \\ & \leqslant \frac{ \sigma _{1}-1}{\sigma _{1} - 2} \frac{ \uprho _{1}}{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{ \uprho _{1}}} \biggl[ \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \grave{\iota}, \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d} s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \mu \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}-\mu \uprho _{1} } \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \mathrm{s} s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \lambda \biggl( \frac{\grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1} - \mu \uprho _{1} } \biggr) \biggr] \\ & \leqslant \frac{\sigma _{1}-1}{\sigma _{1}- 2} \frac{\uprho _{1}}{ \grave{\iota}^{ \uprho _{1}}-\grave{a}^{\uprho _{1}}} \biggl[ \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \grave{\iota}, \xi )\upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \lambda \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1}- \mu \uprho _{1} } \biggr) \\ &\quad {} + \mu \biggl( \frac{ \grave{\iota}^{ \uprho _{1}} - \grave{a}^{\uprho _{1}} }{ \uprho _{1} -\mu \uprho _{1} } \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{ \grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}} \xi \\ &\quad {} + F_{\circ } \biggl(\upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w}(\xi ) \,\mathrm{d}\xi \biggr) \biggr) \biggr] \\ & \leqslant \frac{\sigma _{1}-1}{\sigma _{1} - 2} \frac{\uprho _{1}}{\grave{\iota}^{\uprho _{1}}-\grave{a}^{\uprho _{1}}} \mathrm{q}( \grave{\iota}). \end{aligned}$$
Thus, we obtain (24).
-
(4)
A straightforward calculus gives
$$\begin{aligned} \delta _{\uprho _{1}}^{2} \mathrm{q}(\tau ) & = - \frac{1}{\Gamma (\sigma _{1} -2) } \int _{\grave{a}}^{\tau} \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{\uprho _{1}}}{\uprho _{1}} \biggr)^{ \sigma _{1} -3} \xi ^{\uprho _{1}-1} \\ & \quad {}\times \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,{\mathrm {d}}s \biggr) \,{\mathrm {d}}\xi . \end{aligned}$$
Then we get
$$\begin{aligned} \bigl\vert \delta _{\uprho _{1}}^{2} \mathrm{q}(\tau ) \bigr\vert & \leqslant \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w}(\xi ) \,\mathrm{d} \xi \biggr) \\ &\quad {} \times \frac{1}{\Gamma (\sigma _{1} -2)} \int _{ \grave{a}}^{ \tau} \biggl( \frac{\tau ^{\uprho _{1}} - \xi ^{ \uprho _{1}}}{ \uprho _{1}} \biggr)^{ \sigma _{1} -3} \xi ^{ \uprho _{1}-1} \,{\mathrm {d}}\xi \\ & \leqslant \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr) \frac{1}{\Gamma (\sigma _{1} -1)} \biggl( \frac{\tau ^{\uprho _{1}}-\grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1}-2}. \end{aligned}$$
Thus,
$$ \max_{\tau \in J} \bigl\vert \delta _{\uprho _{1}}^{2} \mathrm{q}(\tau ) \bigr\vert \leqslant \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr) \frac{1}{\Gamma (\sigma _{1} -1)} \biggl( \frac{\grave{\iota}^{\uprho _{1}}-\grave{a}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1}-2}. $$
By multiplying both sides of the previous inequality by
$$ \upphi _{\bar{p}} \biggl( \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \xi ^{ \uprho _{2}}}{ \grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{\sigma _{2} -1} \biggr),$$
we get
$$\begin{aligned} &\upphi _{\bar{p}} \biggl( \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \xi ^{\uprho _{2}}}{\grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{\sigma _{2}-1} \biggr) \max_{\tau \in J} \bigl\vert \delta _{ \uprho _{1}}^{2} \mathrm{q}(\tau ) \bigr\vert \\ &\quad \leqslant \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \xi ^{\uprho _{2}}}{\grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{\sigma _{2} -1} \mathcal{H}(\grave{a},\xi ) \mathrm{w}(\xi ) \,{\mathrm {d}} \xi \biggr) \\ &\qquad {} \times \frac{1}{\Gamma (\sigma _{1} -1)} \biggl( \frac{\grave{\iota}^{\uprho _{1}}-\grave{\grave{a}}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1} -2}, \end{aligned}$$
using Lemma 3.3(iv), we get
$$\begin{aligned}& \upphi _{\bar{p}} \biggl( \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \xi ^{\uprho _{2}}}{\grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{\sigma _{2} -1} \biggr) \max_{\tau \in J} \bigl\vert \delta _{ \uprho _{1}}^{2}\mathrm{q}(\tau ) \bigr\vert \\& \quad \leqslant \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\tau , \xi ) \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr) \frac{1}{\Gamma (\sigma _{1}-1)} \biggl( \frac{\grave{\iota}^{\uprho _{1}}-\grave{\grave{a}}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1} -2}. \end{aligned}$$
(29)
Multiplying both sides by \(\mathcal{G}_{1}(\tau , \xi )\) and integrating over J w.r.t. ξ, we get
$$\begin{aligned}& \max_{\tau \in J } \bigl\vert \delta _{\uprho _{1}}^{2} \mathrm{q}( \tau ) \bigr\vert \int ^{\grave{\iota}}_{\grave{a}} \mathcal{G}_{1}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \xi ^{\uprho _{2}}}{ \grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{\sigma _{2} -1} \biggr) \,{\mathrm {d}}\xi \\& \quad \leqslant \frac{1}{\Gamma (\sigma _{1} -1)} \biggl( \frac{\grave{ \iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1} -2} \int ^{\grave{\iota}}_{\grave{a}} \mathcal{G}_{1}( \tau , \xi ) \\& \qquad {} \times \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d} s \biggr) \,{\mathrm {d}}\xi \\& \quad \leqslant \frac{1}{\Gamma (\sigma _{1} -1)} \biggl( \frac{\grave{\iota}^{\uprho _{1}}-\grave{\grave{a}}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1} -2} \biggl[ \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \tau , \xi ) \\& \qquad {} \times \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi +\lambda \biggl( \frac{\tau ^{\uprho _{1}}-\grave{a}^{\uprho _{1}}}{\uprho _{1} - \mu \uprho _{1}} \biggr) \\& \qquad {} + \mu \biggl( \frac{\tau ^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} - \mu \uprho _{1}} \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \mathrm{w}(s) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\& \qquad {} + F_{\circ} \biggl( \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \mathrm{w}(\xi ) \,{\mathrm {d}}\xi \biggr) \biggr) \biggr] \\& \quad =\frac{1}{\Gamma (\alpha -1)} \biggl( \frac{\grave{\iota}^{\uprho _{1}}-\grave{\grave{a}}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1} -2} \mathrm{q}(\tau ) \\& \quad \leqslant \frac{1}{\Gamma (\sigma _{1} -1)} \biggl( \frac{\grave{\iota}^{\uprho _{1}}-\grave{\grave{a}}^{\uprho _{1}}}{\uprho _{1}} \biggr)^{\sigma _{1} -2} \max_{\tau \in J} \bigl\vert \mathrm{q}( \tau ) \bigr\vert . \end{aligned}$$
Furthermore, for \(\tau \in [\grave{a}_{\circ },\grave{\iota}_{\circ}] \),
$$\begin{aligned}& \int ^{\grave{\iota}}_{\grave{a}} \mathcal{G}_{1}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \biggl( \frac{\grave{\iota}^{\uprho _{2}} -\xi ^{\uprho _{2}}}{\grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{\sigma _{2} -1} \biggr) \,\mathrm{d}\xi \\& \quad \geqslant \biggl( \frac{\tau ^{\uprho _{1}}- \grave{a}^{\uprho _{1}}}{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggr)^{\alpha -1} Z(\grave{ \iota}_{\circ}) \int ^{\grave{\iota}}_{ \grave{a}} \mathcal{G}_{1}( \grave{\iota},\xi ) \,\mathrm{d}\xi \end{aligned}$$
and
$$\begin{aligned}& \max_{\tau \in J} \bigl\vert \delta _{\uprho _{1}}^{2} \mathrm{q}(\tau ) \bigr\vert \int ^{\grave{\iota}}_{\grave{a}} \mathcal{G}_{1}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \biggl( \frac{\grave{\iota}^{\uprho _{2}} - \xi ^{\uprho _{2}}}{\grave{\iota}^{\uprho _{2}} - \grave{a}^{\uprho _{2}}} \biggr)^{\sigma _{2} -1} \biggr) \,\mathrm{d}\xi \\& \quad \geqslant \biggl( \frac{\tau ^{\uprho _{1}}- \grave{a}^{\uprho _{1}}}{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1}-1} Z( \grave{\iota}_{\circ}) \int ^{ \grave{\iota}}_{\grave{a}} \mathcal{G}_{1}( \grave{\iota},\xi ) \,\mathrm{d}\xi \max_{\tau \in J} \bigl\vert \delta _{\uprho _{1}}^{2} \mathrm{q}(\tau ) \bigr\vert . \end{aligned}$$
Thus, we obtain (25).
-
(5)
From the first equation in (21), one can see that
Thus,
As in (2), we can deduce (26).
-
(6)
Equation (27) is a direct consequence of the previous results.
□
Then, for given \([\grave{a}_{\circ},\grave{\iota}_{\circ}] \subset (\grave{a}, \grave{\iota})\), we define the cone
$$ \Upsilon = \biggl\lbrace \mathrm{q}\in K : \min_{\tau \in [ \grave{a}_{\circ}, \grave{\iota}_{\circ}] } \mathrm{q}(\tau ) \geqslant \biggl( \frac{\grave{a}_{\circ}^{\uprho _{1}}- \grave{a}^{\uprho _{1}}}{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} -1} \breve{M}_{2} \Vert \mathrm{q} \Vert \biggr\rbrace ,$$
and the integral operator is defined for \(\tau \in [\grave{a}_{\circ}, \grave{\iota}_{\circ}]\) by
$$\begin{aligned} \mathcal{N}_{\lambda }(\mathrm{q}) (\tau ) & = \int _{\grave{a}}^{ \grave{\iota}} \mathcal{G}_{1}( \tau , \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\xi , s) \hslash (s) \wp \bigl( \mathrm{q}( s) \bigr) \,\mathrm{d}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \mu \biggl( \frac{ \tau ^{\uprho _{1}} -\grave{a}^{\uprho _{1}}}{ \uprho _{1}-\mu \uprho _{1} } \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \\ &\quad {} \times \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\xi , s) \hslash (s) \wp \bigl( \mathrm{q}(s) \bigr) \,{\mathrm {d}}s \biggr) \,{\mathrm {d}}\xi + \lambda \biggl( \frac{ \tau ^{\uprho _{1}} -\grave{a}^{\uprho _{1}}}{\uprho _{1}-\mu \uprho _{1} } \biggr) \\ &\quad {} + F_{\circ} \biggl(\upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \hslash (\xi ) \wp \bigl( \mathrm{q}(\xi ) \bigr) \,{\mathrm {d}}\xi \biggr) \biggr). \end{aligned}$$
(31)
When (H2) holds, we have \(\mathcal{N}_{\lambda} (\Upsilon ) \subset \Upsilon \), and the fixed points of \(\mathcal{N}_{\lambda}\) are the solutions of (9). To use some fixed point theorems, we need to show that \(\mathcal{N}_{\lambda}\) is completely continuous.
Lemma 3.5
([19])
Let \(c, s>0\). For any \(x,y \in [0,c]\), the following propositions hold:
-
(1)
If \(s>1\), then \(|x^{s} - y^{s}|\leqslant s c^{s -1} |x - y| \);
-
(2)
If \(0< s \leqslant 1\), then \(|x^{s} - y^{s}|\leqslant |x - y|^{s}\).
Lemma 3.6
Assume (H2) is true. Then \(\mathcal{N}_{\lambda }: \Upsilon \to \Upsilon \) is continuous and compact.
Proof
The continuity of \(\mathcal{N}_{\lambda}\) is a consequence of the continuity and positiveness of \(\mathcal{G}_{1}\), \(\mathcal{G}_{2}\), \(\mathcal{H}\), ℏ, and ℘. To prove that \(\mathcal{N}_{\lambda}\) is compact, let us consider a bounded subset \(\Omega \subset \Upsilon \). Then there exists \(L > 0\) such that for any \(\mathrm{q} \in \Omega \) we have \(|\wp (\mathrm{q}( \tau ))|\leqslant L\). For any \(\mathrm{q} \in \Omega \), as \(\mathcal{N}_{\mathrm{q}}\) is positive and \(\mathcal{G}_{1}\) is increasing w.r.t. τ, we have
$$ \max_{\tau \in J} \bigl| \mathcal{N}_{\lambda}\bigl( \mathrm{q}(\tau )\bigr)\bigr| = \mathcal{N}_{\lambda}\bigl( \mathrm{q}( \grave{\iota})\bigr).$$
Consequently, using the previous inequality and hypothesis (H2), we get
$$\begin{aligned} \max_{\tau \in J} \bigl| \mathcal{N}_{\lambda} \bigl(\mathrm{q}(\tau )\bigr)\bigr| & \leq \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{1}( \grave{\iota}, \xi ) \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\grave{a}, s)\hslash (s)L \,{\mathrm {d}}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} + \mu \biggl( \frac{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{\uprho _{1}-\mu \uprho _{1} } \biggr) \int _{\grave{a}}^{\grave{\iota}} \mathcal{G}_{2}( \tau , \xi ) \\ &\quad {} \times \upphi _{\bar{p}} \biggl( \int _{\grave{a}}^{ \grave{\iota}} \mathcal{H}(\grave{a}, s) \hslash (s) L \,{\mathrm {d}}s \biggr) \,{\mathrm {d}}\xi \\ &\quad {} +\lambda \biggl( \frac{ \grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1} - \mu \uprho _{1} } \biggr) + A \int _{\grave{a}}^{\grave{\iota}} \mathcal{H}(\grave{a}, \xi ) \hslash (\xi )L \,{\mathrm {d}}\xi =: \bar{L}. \end{aligned}$$
(32)
Then, as in Lemma 3.4, we obtain \(\Vert \mathcal{N}_{\lambda }\mathrm{q} \Vert \leqslant \breve{M}_{3} \bar{L}\), where
$$\begin{aligned} \breve{M}_{3} & = \max \biggl\lbrace 1, \frac{\sigma _{1} -1}{\sigma _{1} - 2 } \biggl( \frac{\uprho _{1}}{\grave{\iota}^{\uprho _{1}} - \grave{a}^{ \uprho _{1}}} \biggr), \\ &\quad {} \max \biggl\lbrace \frac{1}{\Gamma (\sigma _{1} -1)} \biggl( \frac{\grave{\iota}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \uprho _{1}} \biggr)^{\sigma _{1} -2}, 1 \biggr\rbrace \\ &\quad {} \times \biggl[ \biggl( \frac{\grave{a}_{\circ}^{\uprho _{1}} - \grave{a}^{\uprho _{1}}}{ \grave{\iota}^{ \uprho _{1}}- \grave{a}^{\uprho _{1}}} \biggr)^{\sigma _{1} - 1 } Z(\grave{\iota}_{\circ}) \int ^{ \grave{\iota}}_{ \grave{a}} \mathcal{G}_{1}( \grave{\iota},\xi ) \,\mathrm{d}\xi \biggr]^{-1} \biggr\rbrace . \end{aligned}$$
Hence, \(\mathcal{N}_{\lambda}(\Omega )\) is uniformly bounded. Furthermore, by using Lemmas (3.2), (3.5), (3.3), and the Lebesgue dominated convergence theorem, we deduce the equicontinuity of \(\mathcal{N}_{\lambda}(\Omega )\). Therefore, \(\mathcal{N}_{\lambda }\) is completely continuous by the Arzelà–Ascoli theorem. □