This section introduces some related definitions and necessary lemmas of fractional calculus theory.
Definition 2.1
For given \(\gamma > 0\) and the function \(y:(0,\infty )\rightarrow \mathbb{R}\), define the left and right Riemann–Liouville fractional integrals respectively as follows:
$$ \begin{aligned} & I_{0+}^{\gamma } y(x)= \frac{1}{\Gamma (\gamma )} \int _{0}^{x}(x s)^{ \gamma  1}y(s)\,ds, \\ & I_{1}^{\gamma } y(x)=\frac{1}{\Gamma (\gamma )} \int _{x}^{1}(s x)^{ \gamma  1}y(s)\,ds. \end{aligned} $$
Definition 2.2
For given \(\gamma > 0\), \(\gamma \in (n, n+ 1)\) and the function \(y:(0,\infty )\rightarrow \mathbb{R}\), define the left Riemann–Liouville fractional derivative and the right Caputo fractional derivative respectively as follows:
$$ \begin{aligned} & D_{0+}^{\gamma } y(x)= \frac{d^{n}}{dx^{n}} \frac{1}{\Gamma (n \gamma )} \int _{0}^{x}(x s)^{n \gamma  1}y(s)\,ds, \\ & {}^{c} D_{1}^{\gamma } y(x)=(1)^{n} \frac{1}{\Gamma (n \gamma )} \int _{x}^{1}(s x)^{n \gamma  1}y^{(n)}(s) \,ds. \end{aligned} $$
Let \(E= C[0,1]\), whose norm \(\Vert \cdot \Vert \) is the maximum norm. Given \(\phi \in C[0,1]\) and the constants \(r_{1},r_{2}\in \mathbb{R}\), discuss the following fractionalorder mixed pLaplace boundary value problem:
$$ \textstyle\begin{cases} {}^{c} D_{1}^{\gamma }( \varphi _{p} (D_{0+}^{\delta } y(x)))= \phi (x), \\ y(0)= 0, \quad \quad y(1)= r_{1}y(\mu ), \\ D_{0+}^{\delta } y(1)= 0, \quad \quad \varphi _{p}( D_{0+}^{\delta } y(0))= r_{2} \varphi _{p}( D_{0+}^{\delta } y(\eta )), \end{cases} $$
(2.1)
where \(1< \gamma , \delta \leq 2\), \(0<\mu ,\eta < 1\), \(0\leq r_{1}< \frac{1}{\mu ^{\delta  1}}\), \(0\leq r_{2} < \frac{1}{1 \eta }\).
Lemma 2.1
The unique solution of the fractionalorder mixed pLaplace boundary value problem (2.1) is equivalent to
$$\begin{aligned}& y(x)= \int _{0}^{1} G_{1}(x, \tau ) \varphi _{q} \biggl( \int _{0}^{1} G_{2}(\tau , s) \phi (s) \,ds \biggr) \,d\tau , \end{aligned}$$
(2.2)
where
$$ G_{1}(x, \tau ) = \frac{1}{\Gamma (\delta )} \textstyle\begin{cases} \Lambda _{1} [(1  \tau )^{\delta  1}  r_{1} (\mu  \tau )^{\delta  1}]  (x  \tau )^{\delta  1},& 0 < \tau \leq \min \{x,\mu \}; \\ \Lambda _{1} [(1  \tau )^{\delta  1}  r_{1} (\mu  \tau )^{\delta  1}], & x \leq \tau \leq \mu ; \\ \Lambda _{1} (1  \tau )^{\delta  1}  (x  \tau )^{ \delta  1}, & \mu \leq \tau \leq x; \\ \Lambda _{1} (1  \tau )^{\delta  1},& \max \{x,\mu \} \leq \tau < 1, \end{cases} $$
and
$$ G_{2}(\tau , s) = \frac{1}{\Gamma (\gamma )} \textstyle\begin{cases} \Lambda _{2}s^{\gamma  1}, & 0 < s \leq \min \{\tau ,\eta \}; \\ \Lambda _{2}s^{\gamma  1} (s\tau )^{\gamma  1}, & \tau \leq s \leq \eta ; \\ \Lambda _{2}[s^{\gamma  1} r_{2}(s\eta )^{\gamma  1}], & \eta \leq s \leq \tau ; \\ \Lambda _{2}[s^{\gamma  1} r_{2}(s\eta )^{\gamma  1}] (s\tau )^{ \gamma  1},& \max \{\tau ,\eta \} \leq s < 1, \end{cases} $$
with
$$\begin{aligned}& \Lambda _{1}=\frac{x^{\delta  1}}{1r_{1}\mu ^{\delta  1}}, \qquad \Lambda _{2}= \frac{1x}{1r_{2}(1\eta )}. \end{aligned}$$
Moreover, \(G_{1}(x, \tau )>0\), \(G_{2}(\tau , s)>0\) for \(x,\tau ,s\in (0,1)\).
Proof
Let \(\varphi _{p} (D_{0+}^{\delta } y(x))= k(x)\), then the mixed boundary value problem (2.1) changes to
$$ \textstyle\begin{cases} {}^{c} D_{1}^{\gamma } k(x)= \phi (x), \\ k(1)=0, \quad \quad k(0)= r_{2}k(\eta ), \end{cases} $$
(2.3)
and
$$ \textstyle\begin{cases} D_{0+}^{\delta } y(x)= \varphi _{q}(k(x)), \\ y(0)=0, \quad \quad y(1)= r_{1}y(\mu ). \end{cases} $$
(2.4)
Reduce the equation \({}^{c} D_{1}^{\gamma } k(x)= \phi (x)\) as an equivalent equation
$$\begin{aligned}& k(x)= \frac{1}{\Gamma (\gamma )} \int _{x}^{1} (sx) ^{\gamma 1} \phi (s)\,ds+ c_{1}+ c_{2}(1x). \end{aligned}$$
Using the condition \(k(1)=0\) yields \(c_{1}= 0\). Since \(k(0)= r_{2}k(\eta )\), then
$$\begin{aligned}& k(0)= \frac{1}{\Gamma (\gamma )} \int _{0}^{1} s ^{\gamma 1}\phi (s)\,ds+ c_{2} \end{aligned}$$
and
$$\begin{aligned}& k(\eta )= \frac{1}{\Gamma (\gamma )} \int _{\eta }^{1} (s\eta ) ^{ \gamma 1}\phi (s) \,ds+c_{2}(1\eta ). \end{aligned}$$
By calculation, we can get
$$\begin{aligned}& c_{2}= \frac{1}{\Gamma (\gamma )[1r_{2}(1\eta )]} \biggl( \int _{0}^{1} s^{\gamma 1}\phi (s)\,ds \int _{\eta }^{1} r_{2} (s\eta ) ^{\gamma 1} \phi (s)\,ds \biggr). \end{aligned}$$
So,
$$ \begin{aligned} k(x) &= \frac{1}{\Gamma (\gamma )} \int _{x}^{1} (sx) ^{ \gamma 1}\phi (s)\,ds \\ & \quad {} + \frac{1x}{\Gamma (\gamma )[1  r_{2}(1  \eta )]} \biggl( \int _{0}^{1} s^{\gamma 1}\phi (s)\,ds  \int _{\eta }^{1} r_{2} (s\eta ) ^{\gamma 1}\phi (s)\,ds \biggr) \\ &= \int _{0}^{1} G_{2}(x,s)\phi (s)\,ds. \end{aligned} $$
Similarly, the solution of boundary value problem (2.4) is given by
$$\begin{aligned}& y(x)= \int _{0}^{1} G_{1}(x,s)\varphi _{q} \bigl(k(s) \bigr)\,ds. \end{aligned}$$
Consequently, boundary value problem (2.1) is equivalent to (2.2).
From the monotonicity, for \(x,\tau ,s\in (0,1)\), \(G_{1}(x, \tau ),G_{2}(\tau , s)>0\) are verified easily. The proof is completed. □
Lemma 2.2
For given \(\phi (s) \in C[0,1]\), set
$$\begin{aligned}& w(\tau )= \varphi _{q} \biggl( \int _{0}^{1} G_{2}(\tau , s) \phi (s) \,ds \biggr), \quad\quad y(x)= \int _{0}^{1} G_{1}(x, \tau ) w(\tau ) \,d \tau , \\& M_{1}= \max_{0\leq x\leq 1} \int _{0}^{1} G_{1}(x, \tau ) \,d \tau , \quad\quad M_{2}= \max_{0\leq \tau \leq 1} \int _{0}^{1} G_{2}( \tau , s) \,ds. \end{aligned}$$
Then
$$ \begin{aligned} \Vert w \Vert \leq M_{2}^{q1} \Vert \phi \Vert ^{q1}, \qquad \Vert y \Vert \leq M_{1}M_{2}^{q1} \Vert \phi \Vert ^{q1}. \end{aligned} $$
Proof
Since \(\frac{1}{p}+ \frac{1}{q}= 1\) and \(\varphi _{p}\) is increasing, then
$$ \begin{aligned} w(\tau ) &= \varphi _{q} \biggl( \int _{0}^{1} G_{2}( \tau , s) \phi (s) \,ds \biggr) \\ &\leq \varphi _{q} \biggl( \int _{0}^{1} G_{2}(\tau , s) \,ds \Vert \phi \Vert \biggr) \\ &\leq M_{2}^{q1} \Vert \phi \Vert ^{q1}. \end{aligned} $$
So, \(\Vert w \Vert \leq M_{2}^{q1} \Vert \phi \Vert ^{q1}\). Similarly, \(\Vert y \Vert \leq M_{1}M_{2}^{q1} \Vert \phi \Vert ^{q1}\). The proof is completed. □
Lemma 2.3
([7, 13])
The following relations of pLaplace operator hold:

(i)
There are \(1< p\leq 2\), \(\vert k_{1} \vert , \vert k_{2} \vert \geq n> 0\), and \(k_{1}k_{2}> 0\) such that
$$\begin{aligned}& \bigl\vert \varphi _{p}(k_{2}) \varphi _{p}(k_{1}) \bigr\vert \leq (p 1)n^{p2} \vert k_{2}k_{1} \vert ; \end{aligned}$$

(ii)
There are \(p> 2\) and \(\vert k_{1} \vert , \vert k_{2} \vert \leq N\) such that
$$\begin{aligned}& \bigl\vert \varphi _{p}(k_{2}) \varphi _{p}(k_{1}) \bigr\vert \leq (p 1)N^{p2} \vert k_{2}k_{1} \vert . \end{aligned}$$