Theorem 3.1
Let
\(\psi_{1} \in C[0,1]\)
be an integrable function satisfying (1.1). Then the solution of
$$ \textstyle\begin{cases} \mathcal{D}_{0_{+}}^{\beta_{1}} (\phi_{p}(\mathcal{D}_{0_{+}}^{\alpha _{1}}u(t)) )+\psi_{1}(t,v(t))=0, \\ (\phi_{p}(\mathcal{D}_{0_{+}}^{\alpha_{1}}u(t)) )| _{t=1}=\mathcal {I}_{0_{+}}^{\beta_{1}-1} (\psi_{1}(t,v(t)) )| _{t=1},\\ (\phi_{p}(\mathcal{D}_{0_{+}}^{\alpha_{1}}u(t)) )'| _{t=1}=0= (\phi _{p}(\mathcal{D}_{0_{+}}^{\alpha_{1}}u(t)) )''| _{t=0},\\ u(0)=0=u''(0), \quad\quad u(1)=0, \end{cases} $$
(3.1)
is given by the integral equation
$$ \begin{aligned} u(x)&= \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds, \end{aligned} $$
(3.2)
where
\(\mathcal{G}^{\alpha_{1}}(t,s)\), \(\mathcal{G}^{\beta _{1}}(s,\theta)\)
are Green functions defined by
$$\begin{aligned}& \mathcal{G}^{\alpha_{1}}(t,s)= \textstyle\begin{cases} \frac{(t-s)^{\alpha_{1}-1}-t(1-s)^{\alpha_{1}-1}}{\Gamma(\alpha _{1})},& 0\leq s\leq t \leq1,\\ \frac{-t(1-s)^{\alpha_{1}-1}}{\Gamma(\alpha_{1})}, & 0\leq s\leq t \leq1, \end{cases}\displaystyle \end{aligned}$$
(3.3)
$$\begin{aligned}& \mathcal{G}^{\beta_{1}}(t,s)= \textstyle\begin{cases} \frac{-(t-s)^{\beta_{1}-1}+(1-s)^{\beta_{1}-1}}{\Gamma(\beta _{1})}+\frac{t(1-s)^{\beta_{1}-2}}{\Gamma(\beta_{1}-2)},& 0\leq s\leq t \leq1,\\ \frac{(1-s)^{\beta_{1}-1}}{\Gamma(\beta_{1})}+\frac{t(1-s)^{\beta _{1}-2}}{\Gamma(\beta_{1}-2)},& 0\leq s\leq t \leq1. \end{cases}\displaystyle \end{aligned}$$
(3.4)
Proof
Applying operator \(\mathcal{I}_{0_{+}}^{\beta_{1}}\) to (3.1) and using Lemma 2.3, we get the following equivalent integral form of (3.1):
$$\begin{aligned} \phi_{p} \bigl( \mathcal{D}_{0_{+}}^{\alpha_{1}}u(t) \bigr) =-\mathcal{I}_{0_{+}}^{\beta_{1}}\psi_{1} \bigl(t,v(t)\bigr)+c_{1}+c_{2}t+c_{3}t^{2}. \end{aligned}$$
(3.5)
The condition \((\phi_{p}( \mathcal{D}_{0_{+}}^{\alpha_{1}}u(t)) )''| _{t=0}=0\) results in \(c_{3}=0\). The idea that \((\phi_{p}( \mathcal {D}_{0_{+}}^{\alpha_{1}}u(t)) )'| _{t=1}=0\) implies
$$ c_{2}=\mathcal{I}_{0_{+}}^{\beta_{1}-1}\psi_{1} \bigl(t,v(t)\bigr)\big| _{t=1}=\frac{1}{\Gamma(\beta_{1}-1)} \int_{0}^{1}(1-s)^{\beta_{1}-2}\psi _{1}\bigl(s,v(s)\bigr)\,ds. $$
(3.6)
With condition \(( \phi_{p}(\mathcal{D}_{0_{+}}^{\alpha_{1}}u(t)) )| _{t=1}= \mathcal{I}_{0_{+}}^{\beta_{1}-1}\psi_{1}(t,v(t))| _{t=1}\), we have
$$ c_{1}=\mathcal{I}_{0_{+}}^{\beta_{1}}\psi_{1} \bigl(t,v(t)\bigr)\big| _{t=1}=\frac{1}{\Gamma(\beta_{1})} \int_{0}^{1}(1-s)^{\beta_{1}-1}\psi _{1}\bigl(s,v(s)\bigr)\,ds. $$
(3.7)
From the values of \(c_{i}\) for \(i=1,2,3\) and (3.5), we have
$$\begin{aligned} \phi_{p} \bigl(\mathcal{D}_{0_{+}}^{\alpha_{1}}u(t) \bigr) =&-\mathcal{I}_{0_{+}}^{\beta_{1}}\psi_{1} \bigl(t,v(t)\bigr)+\mathcal{I}_{0_{+}}^{\beta_{1}}\psi_{1} \bigl(t,v(t)\bigr)\big| _{t=1}+t\mathcal{I}^{\beta_{1}-1}\psi _{1}\bigl(t,v(t)\bigr) \\ =&\frac{-1}{\Gamma(\beta_{1})} \int_{0}^{t}(t-s)^{\beta_{1}-1}\psi _{1}\bigl(s,v(s)\bigr)\,ds+\frac{1}{\Gamma(\beta_{1})} \int_{0}^{1}(1-s)^{\beta_{1}-1}\psi _{1}\bigl(s,v(s)\bigr)\,ds \\ &{}+\frac{t}{\Gamma(\beta_{1})} \int_{0}^{1}(1-s)^{\beta_{1}-1}\psi _{1}\bigl(s,v(s)\bigr)\,ds \\ =& \int_{0}^{1}\mathcal{G}^{\beta_{1}}(t,s)\psi _{1}\bigl(s,v(s)\bigr)\,ds, \end{aligned}$$
(3.8)
where \(\mathcal{G}^{\beta_{1}}(t,s)\) is a Green function given in (3.4). From (3.8), we further have
$$\begin{aligned} \mathcal{D}_{0_{+}}^{\alpha_{1}}u(t) =&\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(t,s)\psi _{1}\bigl(s,v(s)\bigr)\,ds \biggr). \end{aligned}$$
(3.9)
Applying the fractional integral operator \(\mathcal{I}_{0_{+}}^{\alpha _{1}}\) on (3.9) and using Lemma 2.3 again, we have
$$\begin{aligned} u(t) =&\mathcal{I}_{0_{+}}^{\alpha_{1}} \biggl(\phi _{q}\biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(t,s)\psi _{1}\bigl(s,v(s)\bigr)\,ds\biggr) \biggr)+k_{1}+k_{2}t+k_{3}t^{2}. \end{aligned}$$
(3.10)
Using the condition \(u(0)=0=u''(0)\) in (3.10), we obtain \(k_{1}=0=k_{3}\). From the condition \(u(1)=0\), we have \(k_{2}=-\mathcal {I}_{0_{+}}^{\alpha_{1}} (\phi_{q}(\int_{0}^{1}\mathcal{G}^{\beta _{1}}(t,s)\psi_{1}(s,v(s))\,ds) )| _{t=1}\). Putting the values of \(k_{i}\) for \(i=1,2,3\) in (3.10), we get the solution \(u(t)\) in the following integral form:
$$\begin{aligned} u(t) =&\mathcal{I}_{0_{+}}^{\alpha_{1}} \biggl(\phi _{q}\biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(t,s)\psi _{1}\bigl(s,v(s)\bigr)\,ds\biggr) \biggr) \\ &{}-t\mathcal{I}^{\alpha_{1}} \biggl(\phi_{q}\biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(t,s)\psi _{1}\bigl(s,v(s)\bigr)\,ds\biggr) \biggr)\biggr| _{t=1} \\ =& \biggl( \int_{0}^{t}\frac{(t-s)^{\alpha_{1}-1}}{\Gamma(\alpha_{1})}-t \int_{0}^{1}\frac{(1-s)^{\alpha_{1}-1}}{\Gamma(\alpha_{1})} \biggr)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds \\ =& \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds, \end{aligned}$$
(3.11)
where \(\mathcal{G}^{\alpha_{1}}(t,s)\), \(\mathcal{G}^{\beta _{1}}(s,\theta)\) are Green functions defined by (3.3), (3.4), respectively. □
Theorem 3.1 implies that our problem (1.1) is equivalent to the following coupled system of Hammerstein-type integral equations:
$$\begin{aligned}& u(t)= \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds, \end{aligned}$$
(3.12)
$$\begin{aligned}& v(t)= \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{2}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds, \end{aligned}$$
(3.13)
where \(\mathcal{G}^{\alpha_{2}}(t,s)\), \(\mathcal{G}^{\beta_{2}}(t,s)\) are the following Green functions:
$$\begin{aligned}& \mathcal{G}^{\alpha_{2}}(t,s)= \textstyle\begin{cases} \frac{(t-s)^{\alpha_{2}-1}-t(1-s)^{\alpha_{2}-1}}{\Gamma(\alpha _{2})},&0\leq s\leq t \leq1,\\ \frac{-t(1-s)^{\alpha_{2}-1}}{\Gamma(\alpha_{2})}, & 0\leq s\leq t \leq1, \end{cases}\displaystyle \end{aligned}$$
(3.14)
$$\begin{aligned}& \mathcal{G}^{\beta_{2}}(t,s)= \textstyle\begin{cases} \frac{-(t-s)^{\beta_{2}-1}+(1-s)^{\beta_{2}-1}}{\Gamma(\beta _{2})}+\frac{t(1-s)^{\beta_{2}-2}}{\Gamma(\beta_{2}-2)},& 0\leq s\leq t \leq1,\\ \frac{(1-s)^{\beta_{2}-1}}{\Gamma(\beta_{2})}+\frac{t(1-s)^{\beta _{2}-2}}{\Gamma(\beta_{2}-2)}, & 0\leq s\leq t \leq1. \end{cases}\displaystyle \end{aligned}$$
(3.15)
Define \(\mathcal{T}^{*}_{i}:\mathcal{L}\rightarrow\mathcal{L}\) for (\(i=1,2\)) by
$$\begin{aligned}& \mathcal{T}^{*}_{1}u(t)= \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds, \end{aligned}$$
(3.16)
$$\begin{aligned}& \mathcal{T}^{*}_{2}v(t)= \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{2}\bigl(\theta,u(\theta)\bigr)\,d\theta\biggr)\,ds. \end{aligned}$$
(3.17)
By Theorem 3.1, the solution of the coupled system of the Hammerstein-type integral equations (3.12), (3.13) is equivalent to the fixed point, say \((u,v)\), of the operator equation
$$ (u,v)=\mathcal{T}^{*}(u,v)=\bigl(\mathcal{T}^{*}_{1}(u), \mathcal{T}^{*}_{2}(v)\bigr) (t), $$
(3.18)
for \(\mathcal{T}^{*}=(\mathcal{T}^{*}_{1},\mathcal{T}^{*}_{2})\).
To proceed, we introduce the following assumptions:
-
\((\mathcal{Q}_{1})\)
:
-
The functions \(\psi_{1}\), \(\psi_{2}\) satisfy the following growth conditions for the constants a, b, \(\mathbb{M}^{*}_{\psi_{1}}\), \(\mathbb{M}^{*}_{\psi_{2}}\):
$$\begin{aligned}& \bigl\vert \psi_{1}(x,u)\bigr\vert \leq a\vert u\vert + \mathbb{M}^{*}_{\psi_{1}}, \\& \bigl\vert \psi_{2}(x,v)\bigr\vert \leq b\vert v\vert + \mathbb{M}^{*}_{\psi_{2}}. \end{aligned}$$
-
\((\mathcal{Q}_{2})\)
:
-
There exist real valued constants \(\lambda _{\psi_{1}}\), \(\lambda_{\psi_{2}}\) such that, for all \(u, v, x, y \in\mathcal{L}\),
$$\begin{aligned}& \bigl\vert \psi_{1}(t,\upsilon)-\psi_{1}(t, x)\bigr\vert \leq\lambda_{\psi_{1}}\vert \upsilon- x\vert , \\& \bigl\vert \psi_{2}(t,\mu)-\psi_{2}(t, y)\bigr\vert \leq\lambda_{\psi_{2}}\vert \mu-y\vert . \end{aligned}$$
Theorem 3.2
With assumption
\((\mathcal{Q}_{1})\), the operator
\(\mathcal {T}^{*}:\omega^{*}\rightarrow\omega^{*}\)
is continuous and satisfies the following growth condition:
$$\begin{aligned} \bigl\Vert \mathcal{T}^{*}(u,v)\bigr\Vert \leq \mathcal{B} \bigl\Vert (u,v)\bigr\Vert +\mathbb{K}, \end{aligned}$$
(3.19)
where
\(\mathcal{B}=\Omega(a+b)\), \(\mathbb{K}=\Omega(\mathcal {M}_{1}^{*}+\mathcal{M}_{2}^{*})\), and
$$ \begin{aligned}[b] \Omega&=\max\biggl\{ \frac{2(q-1)\rho_{1}^{q-2}}{\Gamma (\alpha_{1}+1)} \biggl(\frac{1}{\Gamma(\beta_{1})}+ \frac{2}{\Gamma(\beta_{1}+1)} \biggr), \\ &\quad{} \frac{2(q-1)\rho _{2}^{q-2}}{\Gamma(\alpha_{2}+1)} \biggl(\frac{1}{\Gamma(\beta_{2})}+ \frac{2}{\Gamma(\beta_{2}+1)} \biggr) \biggr\} , \end{aligned} $$
(3.20)
for each
\((u,v)\in\wp_{r}\subset\omega^{*}\).
Proof
Consider the bounded set \(\wp_{r}= \{(u,v)\in\omega: \Vert (u,v)\Vert \leq r\}\) with sequence \(\{(u_{n},v_{n})\}\) converging to \((u,v)\) in \(\wp_{r}\). To show that \(\Vert \mathcal {T}^{*}(u_{n},v_{n})-\mathcal{T}^{*}(u,v)\Vert \rightarrow0\) as \(n\rightarrow\infty\), let us consider
$$\begin{aligned} \bigl\vert \mathcal{T}^{*}_{1}u_{n}(t)- \mathcal{T}^{*}_{1}u(t)\bigr\vert =&\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v_{n}(\theta)\bigr)\,d\theta\biggr)\,ds \\ &{}- \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds \biggr\vert \\ \leq&(q-1)\rho_{1}^{q-2} \biggl( \int_{0}^{1}\bigl\vert \mathcal{G}^{\alpha_{1}}(t,s) \bigr\vert \int_{0}^{1}\bigl\vert \mathcal{G}^{\beta_{1}}(s, \theta)\bigr\vert \\ &{}\times\bigl\vert \psi_{1}\bigl(\theta ,v_{n}(\theta)\bigr)-\psi_{1}\bigl(\theta,v(\theta) \bigr)\bigr\vert \,d\theta \,ds \biggr) \end{aligned}$$
(3.21)
and
$$\begin{aligned} \bigl\vert \mathcal{T}^{*}_{2}v_{n}(t)- \mathcal{T}^{*}_{2}v(t)\bigr\vert =&\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{2}\bigl(\theta,u_{n}(\theta)\bigr)\,d\theta\biggr)\,ds \\ &{}- \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{2}\bigl(\theta,u(\theta)\bigr)\,d\theta\biggr)\,ds\biggr\vert \\ \leq&(q-1)\rho_{2}^{q-2} \biggl( \int_{0}^{1}\bigl\vert \mathcal{G}^{\alpha_{2}}(t,s) \bigr\vert \int_{0}^{1}\bigl\vert \mathcal{G}^{\beta_{2}}(s, \theta)\bigr\vert \\ &{}\times \bigl\vert \psi_{2}\bigl(\theta ,u_{n}(\theta)\bigr)-\psi_{1}\bigl(\theta,u(\theta) \bigr)\bigr\vert \,d\theta \,ds \biggr). \end{aligned}$$
(3.22)
From (3.21) and (3.22), we have
$$\begin{aligned} \bigl\vert \mathcal{T}^{*}(u_{n},v_{n}) (t)-\mathcal{T}^{*}(u,v) (t)\bigr\vert \leq&(q-1)\rho _{1}^{q-2} \biggl( \int_{0}^{1}\bigl\vert \mathcal{G}^{\alpha_{1}}(t,s) \bigr\vert \int_{0}^{1}\bigl\vert \mathcal{G}^{\beta_{1}}(s, \theta)\bigr\vert \\ &{}\times\bigl\vert \psi_{1}\bigl(\theta,v_{n}(\theta)\bigr)- \psi_{1}\bigl(\theta,v(\theta)\bigr)\bigr\vert \,d\theta \,ds \biggr) \\ &{}+(q-1)\rho_{2}^{q-2} \biggl( \int_{0}^{1}\bigl\vert \mathcal{G}^{\alpha_{2}}(t,s) \bigr\vert \int_{0}^{1}\bigl\vert \mathcal{G}^{\beta_{1}}(s, \theta)\bigr\vert \\ & {}\times\bigl\vert \psi_{2}\bigl(\theta,u_{n}(\theta) \bigr)-\psi_{2}\bigl(\theta,u(\theta)\bigr)\bigr\vert \,d\theta \,ds \biggr). \end{aligned}$$
(3.23)
From the continuity of \(\psi_{1}\), \(\psi_{2}\) and (3.23), we have \(\vert \mathcal{T}^{*}(u_{n},v_{n})(t)-\mathcal {T}^{*}(u,v)(t)\vert \rightarrow0\), as \(n\rightarrow\infty\). Thus the operator \(\mathcal{T}^{*}\) is a continuous operator. Further, with the help of (3.16) and (3.17), we proceed as follows:
$$\begin{aligned}& \bigl\vert \mathcal{T}^{*}_{1}u(t)\bigr\vert =\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta \,ds \biggr)\biggr\vert \\& \hphantom{ \bigl\vert \mathcal{T}^{*}_{1}u(t)\bigr\vert }\leq(q-1)\rho_{1}^{q-2}\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s) \int_{0}^{1} \mathcal{G}^{\beta_{1}}(s,\theta )\psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta \,ds \biggr\vert \\& \hphantom{ \bigl\vert \mathcal{T}^{*}_{1}u(t)\bigr\vert }\leq(q-1)\rho_{1}^{q-2}\biggl\vert \biggl( \int_{0}^{t}\frac{(t-s)^{\alpha_{1}-1}}{\Gamma(\alpha_{1})}-t \int_{0}^{1}\frac{(1-s)^{\alpha_{1}-1}}{\Gamma(\alpha_{1})} \biggl( \frac{-1}{\Gamma(\beta_{1})} \int_{0}^{s}(s-\theta)^{\beta_{1}-1} \\& \hphantom{ \bigl\vert \mathcal{T}^{*}_{1}u(t)\bigr\vert }\quad {}+\frac{1}{\Gamma(\beta_{1})} \int_{0}^{1}(1-\theta)^{\beta_{1}-1} + \frac{s}{\Gamma(\beta_{1})} \int_{0}^{1}(1-\theta)^{\beta_{1}-1} \biggr)\,d\theta \,ds \biggr)\biggr\vert \bigl(a\vert v\vert +\mathbb {M}^{*}_{\psi_{1}} \bigr) \\& \hphantom{ \bigl\vert \mathcal{T}^{*}_{1}u(t)\bigr\vert } \leq\frac{2(q-1)\rho_{1}^{q-2}}{\Gamma(\alpha_{1}+1)} \biggl(\frac {1}{\Gamma(\beta_{1})}+\frac{2}{\Gamma(\beta_{1}+1)} \biggr) \bigl(a\vert v\vert +\mathbb{M}^{*}_{\psi_{1}}\bigr), \end{aligned}$$
(3.24)
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert \mathcal{T}^{*}_{2}v(t)\bigr\vert &=\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{1}\bigl(\theta,u(\theta)\bigr)\,d\theta\biggr)\,ds \biggr\vert \\ &\leq(q-1)\rho_{2}^{q-2}\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s) \biggl( \int_{0}^{1}| \mathcal{G}^{\beta_{2}}(s,\theta )\psi_{1}\bigl(\theta,u(\theta)\bigr)\,d\theta\biggr)\,ds \biggr\vert \\ &\leq(q-1)\rho_{2}^{q-2}\biggl\vert \biggl( \int_{0}^{t}\frac{(t-s)^{\alpha_{2}-1}}{\Gamma(\alpha_{2})}-t \int_{0}^{1}\frac{(1-s)^{\alpha_{2}-1}}{\Gamma(\alpha_{2})}\biggr) \biggl( \frac{-1}{\Gamma(\beta_{2})} \int_{0}^{s}(s-\theta)^{\beta_{2}-1} \\ &\quad {}+\frac{1}{\Gamma(\beta_{2})} \int_{0}^{1}(1-\theta)^{\beta_{2}-1} + \frac{s}{\Gamma(\beta_{2})} \int_{0}^{1}(1-\theta)^{\beta_{2}-1} \biggr)\,d\theta \,ds \biggr\vert \bigl(b\vert u\vert +\mathbb{M}^{*}_{\psi_{2}} \bigr) \\ &\leq\frac{2(q-1)\rho_{2}^{q-2}}{\Gamma(\alpha_{2}+1)} \biggl(\frac {1}{\Gamma(\beta_{2})}+\frac{2}{\Gamma(\beta_{2}+1)} \biggr) \bigl(b\vert u\vert +\mathbb{M}^{*}_{\psi_{2}}\bigr). \end{aligned} \end{aligned}$$
(3.25)
Consequently, we have
$$\begin{aligned} \bigl\vert \mathcal{T}^{*}(u,v) (t)\bigr\vert \leq &\Omega\bigl(a\vert v\vert +\mathbb{M}^{*}_{\psi_{1}}\bigr)+ \Omega\bigl(b\vert u\vert +\mathbb{M}^{*}_{\psi_{2}}\bigr) \\ \leq&\Omega(a+b) \bigl(\vert v\vert +\vert u\vert \bigr)+\Omega \bigl(\mathbb{M}^{*}_{\psi_{1}}+\mathbb{M}^{*}_{\psi_{2}} \bigr)=\mathcal{B}\bigl\Vert (u,v)\bigr\Vert +\mathbb{K} . \end{aligned}$$
(3.26)
This completes the proof. □
Theorem 3.3
Let assumption
\((\mathcal{Q}_{1})\)
hold. Then the operator
\(\mathcal {T}^{*}:\omega^{*} \rightarrow\omega^{*}\)
is compact and
ξ-Lipschitz with constant zero.
Proof
With the help of Theorem 3.2, we deduce that the operator \(\mathcal{T}^{*}: \omega\rightarrow\omega\) is bounded. Next, using assumption \((\mathcal{Q}_{1})\), Lemma 3.1, and equations (3.12), (3.13), for any \(t_{1},t_{2}\in[0,1]\), we have
$$\begin{aligned}& \bigl\vert \mathcal{T}^{*}_{1}u(t_{1})- \mathcal{T}^{*}_{1}u(t_{2})\bigr\vert =\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t_{1},s) \phi_{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds \\& \hphantom{\bigl\vert \mathcal{T}^{*}_{1}u(t_{1})- \mathcal{T}^{*}_{1}u(t_{2})\bigr\vert }\quad {}- \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t_{2},s) \phi_{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta \biggr)\,ds\biggr\vert \\& \hphantom{\bigl\vert \mathcal{T}^{*}_{1}u(t_{1})- \mathcal{T}^{*}_{1}u(t_{2})\bigr\vert } \leq(q-1)\rho_{1}^{q-2}\frac{\vert t_{1}^{\alpha _{1}}-t_{2}^{\alpha_{1}}\vert +\vert t_{1}-t_{2}\vert }{\Gamma(\alpha_{1}+1)} \biggl( \frac{1}{\Gamma(\beta_{1})}+\frac{2}{\Gamma(\beta_{1}+1)} \biggr) \\& \hphantom{\bigl\vert \mathcal{T}^{*}_{1}u(t_{1})- \mathcal{T}^{*}_{1}u(t_{2})\bigr\vert }\quad {}\times\bigl(a\vert v\vert +\mathbb{M}^{*}_{\psi_{1}}\bigr), \end{aligned}$$
(3.27)
$$\begin{aligned}& \begin{aligned}[b] \bigl\vert \mathcal{T}^{*}_{2}v(t_{1})- \mathcal{T}^{*}_{2}v(t_{2})\bigr\vert &=\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t_{1},s) \phi_{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{1}\bigl(\theta,u(\theta)\bigr)\,d\theta\biggr)\,ds \\ &\quad{} - \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t_{2},s) \phi_{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{1}\bigl(\theta,u(\theta)\bigr)\,d\theta\biggr)\biggr\vert \\ &\leq(q-1)\rho_{2}^{q-2}\frac{\vert t_{1}^{\alpha _{2}}-t_{2}^{\alpha_{2}}\vert +\vert t_{1}-t_{2}\vert }{\Gamma(\alpha_{2}+1)} \biggl( \frac{1}{\Gamma(\beta_{2})}+\frac{2}{\Gamma(\beta_{2}+1)} \biggr) \\ &\quad{}\times \bigl(a\vert u\vert +\mathbb{M}^{*}_{\psi_{2}}\bigr). \end{aligned} \end{aligned}$$
(3.28)
From (3.27), (3.28), we have
$$\begin{aligned}& \bigl\vert \mathcal{T}^{*}(u,v) (t_{1})- \mathcal{T}^{*}(u,v) (t_{2})\bigr\vert \\ & \quad \leq (q-1)\rho_{1}^{q-2}\frac{\vert t_{1}^{\alpha _{1}}-t_{2}^{\alpha_{1}}\vert +\vert t_{1}-t_{2}\vert }{\Gamma(\alpha_{1}+1)} \biggl( \frac{1}{\Gamma(\beta_{1})}+\frac{2}{\Gamma(\beta_{1}+1)} \biggr) \\ & \quad\quad{}\times \bigl(a\vert v\vert +\mathbb{M}^{*}_{\psi_{1}}\bigr) \\ & \quad\quad{} + (q-1)\rho_{2}^{q-2}\frac{\vert t_{1}^{\alpha _{2}}-t_{2}^{\alpha_{2}}\vert +\vert t_{1}-t_{2}\vert }{\Gamma(\alpha_{2}+1)} \biggl( \frac{1}{\Gamma(\beta_{2})}+\frac{2}{\Gamma(\beta_{2}+1)} \biggr) \\ & \quad\quad{}\times \bigl(a\vert u\vert +\mathbb{M}^{*}_{\psi_{2}}\bigr). \end{aligned}$$
(3.29)
As \(t_{1}\rightarrow t_{2}\), the right hand side of (3.29) approaches zero. Thus \(\mathcal{T}^{*}=(\mathcal{T}^{*}_{1},\mathcal {T}^{*}_{2})\) is an equicontinuous operator on D. By Arzela-Ascoli’s theorem, the operator \(\mathcal{T}^{*}(D)\) is compact. Hence D is ξ-Lipschitz with constant zero. □
Theorem 3.4
Let assumptions
\((\mathcal{Q}_{1})\), \((\mathcal{Q}_{2})\)
hold. Then the coupled system (1.1) has a unique solution provided that
\(\Omega ^{*} < 1\), where
$$\begin{aligned} \Omega^{*} =&\frac{2(p-1)\rho_{1}^{p-2}\lambda_{\psi _{1}}}{\Gamma(\alpha_{1}+1)} \biggl( \frac{1}{\Gamma(\beta_{1})}+\frac{2}{\Gamma(\beta_{1}+1)} \biggr) \\ &{}+\frac{2(q-1)\rho_{2}^{q-2}\lambda_{\psi_{2}}}{\Gamma(\alpha _{2}+1)} \biggl(\frac{1}{\Gamma(\beta_{2})}+\frac{2}{\Gamma(\beta _{2}+1)} \biggr). \end{aligned}$$
(3.30)
Proof
From (3.16), (3.17), and assumptions \((\mathcal{Q}_{1})\) and \((\mathcal{Q}_{2})\), we have
$$\begin{aligned} \bigl\vert \mathcal{T}^{*}_{1}u(t)- \mathcal{T}^{*}_{1}\bar{u}(t)\bigr\vert =&\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,v(\theta)\bigr)\,d\theta\biggr)\,ds \\ &{}- \int_{0}^{1}\mathcal{G}^{\alpha_{1}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{1}}(s,\theta) \psi_{1}\bigl(\theta,\bar{v}(\theta)\bigr)\,d\theta\biggr)\,ds\biggr\vert \\ \leq&\frac{2(q-1)\rho_{1}^{q-2}\lambda_{\psi_{1}}}{\Gamma(\alpha _{1}+1)} \biggl(\frac{1}{\Gamma(\beta_{1})}+\frac{2}{\Gamma(\beta _{1}+1)} \biggr) \bigl\vert v(t)-\bar{v}(t)\bigr\vert \end{aligned}$$
(3.31)
and
$$\begin{aligned} \bigl\vert \mathcal{T}^{*}_{2}v(t)- \mathcal{T}^{*}_{2}\bar{v}(t)\bigr\vert =&\biggl\vert \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{1}\bigl(\theta,u(\theta)\bigr)\,d\theta\biggr)\,ds \\ &{} - \int_{0}^{1}\mathcal{G}^{\alpha_{2}}(t,s)\phi _{q} \biggl( \int_{0}^{1}\mathcal{G}^{\beta_{2}}(s,\theta) \psi_{1}\bigl(\theta,\bar{u}(\theta)\bigr)\,d\theta\biggr)\,ds\biggr\vert \\ \leq&\frac{2(q-1)\rho_{1}^{q-2}\lambda_{\psi_{2}}}{\Gamma(\alpha _{2}+1)} \biggl(\frac{1}{\Gamma(\beta_{2})}+\frac{2}{\Gamma(\beta _{2}+1)} \biggr) \bigl\vert u(t)-\bar{u}(t)\bigr\vert . \end{aligned}$$
(3.32)
From (3.31), (3.32) we have
$$\begin{aligned} \bigl\vert \mathcal{T}^{*}(u,v) (t)- \mathcal{T}^{*}(\bar{u},\bar{v}) (t)\bigr\vert \leq& \frac{2(q-1)\rho_{1}^{q-2}\lambda_{\psi_{1}}}{\Gamma(\alpha _{1}+1)} \biggl(\frac{1}{\Gamma(\beta_{1})}+\frac{2}{\Gamma(\beta _{1}+1)} \biggr) \bigl( \bigl\vert v(t)-\bar{v}(t)\bigr\vert \bigr) \\ &{}+\frac{2(q-1)\rho_{1}^{q-2}\lambda_{\psi_{2}}}{\Gamma(\alpha _{2}+1)} \biggl(\frac{1}{\Gamma(\beta_{2})}+\frac{2}{\Gamma(\beta _{2}+1)} \biggr) \bigl(\bigl\vert u(t)-\bar{u}(t)\bigr\vert \bigr) \\ \leq& \biggl[\frac{2(q-1)\rho_{1}^{q-2}\lambda_{\psi_{1}}}{\Gamma (\alpha_{1}+1)} \biggl(\frac{1}{\Gamma(\beta_{1})}+\frac{2}{\Gamma (\beta_{1}+1)} \biggr) \\ &{}+\frac{2(q-1)\rho_{1}^{q-2}\lambda_{\psi_{2}}}{\Gamma(\alpha _{2}+1)} \biggl(\frac{1}{\Gamma(\beta_{2})}+\frac{2}{\Gamma(\beta _{2}+1)} \biggr) \biggr] \\ & {}\times\bigl(\bigl\Vert (u,v) (t)-(\bar{u},\bar{v}) (t)\bigr\Vert \bigr) \\ =&\Omega^{*}\bigl(\bigl\Vert (u,v) (t)-(\bar{u},\bar{v}) (t)\bigr\Vert \bigr). \end{aligned}$$
(3.33)
With the help of Banach’s FPT and our assumption \(\Omega^{*}<1\), the contraction \(\mathcal{T}^{*}\) has a unique fixed point. Thus, the coupled system of FDEs with p-Laplacian operator (1.1) has a unique solution. □