In this section, we introduce notations, definitions, and lemmas used in the main results. For \(q\in(0,1)\) and \(\omega>0\), we define
$$ [n]_{q}:=\frac{1-q^{n}}{1-q}=q^{n-1}+\cdots+q+1\quad\text{and} \quad [n]_{q}!:=\prod_{k=1}^{n} \frac{1-q^{k}}{1-q},\quad n\in\mathbb{R}. $$
The q-analogue of the power function \((a-b)_{q}^{\underline{n}}\) with \(n\in\mathbb{N}_{0}:=[0,1,2,\ldots]\) is defined by
$$(a-b)_{q}^{\underline{n}}:= \prod_{k=0}^{n-1} \bigl(a-bq^{k}\bigr), \quad a,b\in \mathbb{R}, $$
where \((a-b)_{q}^{\underline{0}}:=1\). The \(q,\omega\)-analogue of the power function \((a-b)_{q,\omega}^{\underline{n}}\) with \(n\in\mathbb {N}_{0}:={\mathbb{N}}\cup\{0\}\) is defined by
$$(a-b)_{q,\omega}^{\underline{n}}:= \prod_{k=0}^{n-1} \bigl[ a-\bigl(bq^{k}+\omega[k]_{q}\bigr) \bigr],\quad a,b \in\mathbb{R}, $$
where \((a-b)_{q,\omega}^{\underline{0}}:=1\). Generally, if \(\alpha \in\mathbb{R}\), we have
$$\begin{gathered} (a-b)_{q}^{\underline{\alpha}}= a^{\alpha}\prod _{n=0}^{\infty} \frac {1- (\frac{b}{a} )q^{n}}{1- (\frac{b}{a} )q^{\alpha+n}},\quad a\neq0, \\(a-b)_{q,\omega}^{\underline{\alpha}}= (a-\omega_{0})^{\alpha}\prod_{n=0}^{\infty} \frac{1- (\frac{b-\omega_{0}}{a-\omega_{0}} )q^{n}}{1- (\frac{b-\omega_{0}}{a-\omega_{0}} )q^{\alpha+n}}= \bigl((a-\omega_{0})-(b-\omega_{0}) \bigr)_{q}^{\underline{\alpha}},\quad a\neq\omega_{0}.\end{gathered} $$
Note that \(a_{q}^{\underline{\alpha}} = a^{\alpha}\) and \((a-\omega _{0})_{q,\omega}^{\underline{\alpha}} = (a-\omega_{0})^{\alpha}\). Here, we use the notation \((0)_{q}^{\underline{\alpha}}=({\omega _{0}})_{q,\omega}^{\underline{\alpha}}=0\) for \(\alpha>0\). The following are the definitions of q-gamma and q-beta functions, respectively:
$$\begin{gathered} \Gamma_{q}(x):=\frac{(1-q)_{q}^{\underline{x-1}}}{(1-q)^{x-1}},\quad x\in\mathbb{R}\setminus \{0,-1,-2,\ldots\}, \\ B_{q}(x,s):= \int_{0}^{1} t^{x-1}(1-qt)_{q}^{\underline{s-1}} \,d_{q}t=\frac {\Gamma_{q}(x)\Gamma_{q}(s)}{\Gamma_{q}(x+s)}.\end{gathered} $$
Definition 2.1
([10])
Assuming \(q\in (0,1)\), \(\omega>0\) and letting f be the function defined on an interval \(I\subseteq{\mathbb{R}}\) containing \(\omega_{0}:=\frac {\omega}{1-q}\), the Hahn difference of f is defined by
$$ D_{q,\omega}f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega}\quad\text{for } t\neq \omega_{0}. $$
In addition, \(D_{q,\omega}f(\omega_{0})=f'(\omega_{0})\) when f is differentiable at \(\omega_{0}\). We call \(D_{q,\omega}f\) the \(q,\omega \)-derivative of f and f is \(q,\omega\)-differentiable on I.
Remarks
-
(1)
\(D_{q,\omega}[f(t)+g(t)]=D_{q,\omega}f(t)+D_{q,\omega}g(t)\);
-
(2)
\(D_{q,\omega}[\alpha f(t)]=\alpha D_{q,\omega}f(t)\);
-
(3)
\(D_{q,\omega}[f(t)g(t)]=f(t)D_{q,\omega}g(t)+g(qt+\omega )D_{q,\omega}f(t)\);
-
(4)
\(D_{q,\omega} [ \frac{f(t)}{g(t)} ] =\frac {g(t)D_{q,\omega}f(t)-f(t)D_{q,\omega}g(t)}{g(t)g(qt+\omega)}\).
Letting \(a,b\in I \), where \(a<\omega_{0}<b\) and \([k]_{q}=\frac {1-q^{k}}{1-q}\), \(k\in{\mathbb{N}}_{0}\), we define the \(q,\omega\)-interval by
$$\begin{aligned} {[a,b]}_{q,\omega}&:= \bigl\{ q^{k} a+\omega[k]_{q} : k \in{\mathbb {N}}_{0} \bigr\} \cup \bigl\{ q^{k} b+ \omega[k]_{q} : k\in{\mathbb{N}}_{0} \bigr\} \cup \{ \omega_{0} \} \\ &={[a,\omega_{0}]}_{q,\omega} \cup{[\omega_{0},b]}_{q,\omega} \\ &={(a,b)}_{q,\omega} \cup \{a,b \} = {[a,b)}_{q,\omega} \cup \{b \} = {(a,b]}_{q,\omega} \cup \{a \}.\end{aligned} $$
For each \(s\in[a,b]_{q,\omega}\), the sequence \(\{\sigma _{q,\omega}^{k}(s) \}_{k=0}^{\infty}= \{q^{k} s+\omega[k]_{q} \} _{k=0}^{\infty}\) is uniformly convergent to \(\omega_{0}\).
We define the forward and backward jump operators as \(\sigma ^{k}_{q,\omega}(t):=q^{k}t+\omega[k]_{q} \) and \(\rho^{k}_{q,\omega }(t):=\frac{t-\omega[k]_{q} }{q^{k}} \) for \(k\in{\mathbb{N}}\), respectively.
Definition 2.2
([13])
Let I be any closed interval of \(\mathbb{R}\) containing a, b, and \(\omega_{0}\). Letting \(f:I\rightarrow\mathbb{R}\) be a given function, we define \(q,\omega\)-integral of f from a to b by
$$ \int_{a}^{b} f(t)\,d_{q,\omega}t:= \int_{\omega_{0}}^{b} f(t)\,d_{q,\omega }t- \int_{\omega_{0}}^{a} f(t)\,d_{q,\omega}t, $$
where
$$\int_{\omega_{0}}^{x} f(t)\,d_{q,\omega}t:= \bigl[x(1-q)-\omega \bigr]\sum_{k=0}^{\infty} q^{k} f \bigl(xq^{k}+\omega[k]_{q} \bigr),\quad x\in I. $$
Assuming that the series converges at \(x=a\) and \(x=b\), we say that f is \(q,\omega\)-integrable on \([a,b]\), and the sum to the right-hand side is called the Jackson-Nörlund sum.
We note that the actual domain of function f is defined on \([a,b]_{q,\omega}\subset I\).
We next introduce the fundamental theorem of Hahn calculus in the following lemma.
Lemma 2.1
([13])
Let
\(f:I\rightarrow\mathbb {R}\)
be continuous at
\(\omega_{0}\)
and define
$$F(x):= \int_{\omega_{0}}^{x} f(t)\,d_{q,\omega}t,\quad x\in I. $$
Then
F
is continuous at
\(\omega_{0}\). Furthermore, \(D_{q,\omega _{0}}F(x)\)
exists for every
\(x\in I\)
and
$$D_{q,\omega}F(x)=f(x). $$
Conversely,
$$\int_{a}^{b} D_{q,\omega}F(t) \,d_{q,\omega}t=F(b)-F(a) \quad\textit{for all }a,b\in I. $$
Lemma 2.2
([21])
Let
\(q\in(0,1)\), \(\omega >0\), and
\(f:I\rightarrow\mathbb{R}\)
be continuous at
\(\omega_{0}\). Then
$$\begin{gathered} \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{r} x(s)\,d_{q,\omega}s\, d_{q,\omega}r= \int_{\omega_{0}}^{t} \int_{qs+\omega}^{t} h(s)\, d_{q,\omega}r \,d_{q,\omega}s, \\ \int_{\omega_{0}}^{t}\,d_{q,\omega}s=t- \omega_{0}, \\ \int_{\omega_{0}}^{t} \bigl[t-\sigma_{q,\omega}(s)\bigr] \,d_{q,\omega}s=\frac { (t-\omega_{0} )^{2}}{1+q}.\end{gathered} $$
In the sequel, we define fractional Hahn integral, fractional Hahn difference of Riemann-Liouville and Caputo types.
Definition 2.3
([28])
For \(\alpha,\omega>0\), \(q\in(0,1)\) and f defined on \([\omega _{0},T]_{q,\omega}\), the fractional Hahn integral is defined by
$$\begin{aligned} {\mathcal{I}}_{q,\omega}^{\alpha}f(t)&:=\frac{1}{\Gamma_{q}(\alpha)} \int_{\omega_{0}}^{t} \bigl(t-\sigma_{q,\omega}(s) \bigr)_{q,\omega }^{\underline{\alpha-1}}f(s) \,d_{q,\omega}s \\ &=\frac{[t (1-q)-\omega]}{\Gamma_{q}(\alpha)} \sum_{n=0}^{\infty }q^{n} \bigl(t-\sigma_{q,\omega}^{n+1}(t) \bigr)_{q,\omega}^{\underline {\alpha-1}} f \bigl(\sigma_{q,\omega}^{n}(t) \bigr),\end{aligned} $$
where \(({\mathcal{I}}^{0}_{q,\omega} f)(t) = f(t)\).
Definition 2.4
([28])
For \(\alpha,\omega>0\), \(q\in(0,1)\) and f defined on \([\omega _{0},T]_{q,\omega}\), the fractional Hahn difference of the Caputo type of order α is defined by
$$\begin{aligned} {}^{C}D_{q,\omega}^{\alpha}f(t)&:=\bigl({ \mathcal{I}}_{q,\omega}^{N-\alpha } D_{q,\omega}^{N} f\bigr) (t) \\ &=\frac{1}{\Gamma_{q}(N-\alpha)} \int_{\omega_{0}}^{t} \bigl(t-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{N-\alpha-1}} D_{q,\omega}^{N}f(s) \,d_{q,\omega}s,\end{aligned} $$
and \(D^{0}_{q,\omega}f(t) ={}^{C}D^{0}_{q,\omega}f(t) = f(t)\), where N is the smallest integer that is greater than α.
Lemma 2.3
([28])
Let
\(\alpha>0\), \(q\in (0,1)\), \(\omega>0\), and
\(f:I_{q,\omega}^{T}\rightarrow{\mathbb{R}}\). Then
$$ {\mathcal{I}}_{q,\omega}^{\alpha}{}^{C}D_{q,\omega}^{\alpha}f(t) =f(t)+C_{0}+C_{1}(t-\omega_{0})+\cdots+C_{N-1}(t- \omega_{0})^{N-1} $$
for some
\(C_{i}\in{\mathbb{R}}\), \(i={\mathbb{N}}_{0,N-1} \)
and
\(N-1<\alpha\leq N\), \(N\in{\mathbb{N}}\).
The following lemma is used to simplify the calculations in this study.
Lemma 2.4
([28])
Letting
\(\alpha,\beta >0\), \(p,q\in(0,1)\), and
\(\omega>0\),
$$\begin{gathered} \int_{\omega_{0}}^{t} \bigl( t-\sigma_{q,\omega}(s) \bigr)_{q,\omega }^{\underline{\alpha-1}} (s-\omega_{0})_{q,\omega}^{\underline{\beta }} \,d_{q,\omega}s =(t-\omega_{0})^{\alpha+\beta}B_{q}( \beta+1,\alpha ), \\ \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{x} \bigl( t-\sigma_{p,\omega }(x) \bigr)_{p,\omega}^{\underline{\alpha-1}} \bigl( x-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\beta-1}} \,d_{q,\omega }s\,d_{p,\omega}x = \frac{(t-\omega_{0})^{\alpha+\beta}}{[\beta ]_{q}}B_{p}(\beta+1,\alpha).\end{gathered} $$
The next lemma presents the solution of the linear variant of problem (1.3).
Lemma 2.5
Let
\(\alpha,\beta\in(0,1)\), \(\omega>0\), \(q\in(0,1)\), \(h \in C (I_{q,\omega}^{T},\mathbb{R} )\)
be a given function, and
\(\phi,\psi: C (I_{q,\omega}^{T},\mathbb{R} )\rightarrow \mathbb{R}\)
be given functionals. Then the problem
$$ \begin{gathered} {}^{C}D^{\alpha}_{q,\omega}{}^{C}D^{\beta}_{q,\omega} \biggl[\frac {E_{\sigma_{q,\omega}}}{\rho_{q,\omega}(t)}+qD_{q,\omega} \biggr] u(t)=h(t),\quad t\in I_{q,\omega}^{T}, \\ u(\omega_{0})=\phi(u), \\ \rho_{q,\omega}(T)u(T)=\rho_{q,\omega}(\eta)u(\eta)=\psi (u),\quad\eta \in I_{q,\omega}^{T}- \lbrace\omega_{0},T \rbrace\end{gathered} $$
(2.1)
has the unique solution
$$ \begin{aligned}[b] u(t)={} &\frac{1}{q^{2}\rho_{p,\omega}(t)} \biggl\{ q^{2} \omega_{0}\phi (u)\\ &+\frac{1}{\Omega} \biggl[{\mathcal{A^{*}}}(h,u) \int_{\omega _{0}}^{t}(s-\omega)\,d_{q,\omega}s -{ \mathcal{B^{*}}}(h,u) \int_{\omega_{0}}^{t}(s-\omega) ( s-\omega_{0}) \, d_{q,\omega}s \biggr] \\ +& \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s} \int_{\omega_{0}}^{x} \frac{(s-\omega) (s-\sigma_{q,\omega}(x) )_{q,\omega }^{\underline{\beta-1}} (x-\sigma_{q,\omega}(z) )_{q,\omega}^{\underline{\alpha-1}} }{\Gamma_{q}(\alpha)\Gamma _{q}(\beta)} h(z) \,d_{q,\omega}z\,d_{q,\omega}x\,d_{q,\omega}s \biggr\} ,\end{aligned} $$
(2.2)
where
$$ \begin{aligned}[b]\Omega:= {}& \int_{\omega_{0}}^{T}(s-\omega)\,d_{q,\omega}s \int _{\omega_{0}}^{\eta}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s - \int_{\omega_{0}}^{\eta}(s-\omega)\,d_{q,\omega}s \int_{\omega _{0}}^{T}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s \\ = {}&(\eta-T) (T-\omega_{0}) (\eta-\omega_{0}) \biggl[ \frac{(T-\omega_{0})(\eta-\omega_{0}) }{(q+1)(q^{2}+q+1)} + \frac{\omega _{0}q(T+\eta-2\omega_{0})}{q^{2}+q+1} +\frac{\omega_{0}^{2} q^{2}}{q+1} \biggr],\end{aligned} $$
(2.3)
the functionals
\({\mathcal{A^{*}}}(h,u), {\mathcal{B^{*}}}(h,u)\)
are defined by
$$\begin{aligned}& {\mathcal{A^{*}}}(h,u):= {\mathcal{A}}(u)+{\mathcal {A}}_{1}(h)-{ \mathcal{A}}_{2}(h), \end{aligned}$$
(2.4)
$$\begin{aligned}& {\mathcal{B^{*}}}(h,u):={\mathcal{B}}(u)+{\mathcal {B}}_{1}(h)-{ \mathcal{B}}_{2}(h) , \end{aligned}$$
(2.5)
and
$$\begin{aligned}& {\mathcal{A}}(u):= \bigl( \omega_{0} \phi(u)-\psi(u) \bigr) \int _{\eta}^{T}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s, \end{aligned}$$
(2.6)
$$\begin{aligned}& \begin{aligned}[b] {\mathcal{A}}_{1}(h):= {}& \frac{\int_{\omega_{0}}^{T} (s-\omega )(s-\omega_{0}) \,d_{q,\omega}s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{\eta} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s,\end{aligned} \end{aligned}$$
(2.7)
$$\begin{aligned}& \begin{aligned}[b] {\mathcal{A}}_{2}(h):= {}& \frac{\int_{\omega_{0}}^{\eta} (s-\omega )(s-\omega_{0}) \,d_{q,\omega}s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ & \times \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s,\end{aligned} \end{aligned}$$
(2.8)
$$\begin{aligned}& {\mathcal{B}}(u):= \bigl(\omega_{0} \phi(u)- \psi(u) \bigr) \int _{\eta}^{T}s\,d_{q,\omega}s, \end{aligned}$$
(2.9)
$$\begin{aligned}& \begin{aligned}[b] {\mathcal{B}}_{1}(h):= {}& \frac{\int_{\omega_{0}}^{T}(s-\omega) \, d_{q,\omega}s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ & \times \int_{\omega_{0}}^{\eta} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s,\end{aligned} \end{aligned}$$
(2.10)
$$\begin{aligned}& \begin{aligned}[b] {\mathcal{B}}_{2}(h):= & \frac{ \int_{\omega_{0}}^{\eta}(s-\omega) \, d_{q,\omega}s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ & \times \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s.\end{aligned} \end{aligned}$$
(2.11)
Proof
We first take fractional \(q,\omega\)-integral of order α for (2.1) to obtain
$$\begin{aligned} &{}^{C}D^{\beta}_{q,\omega} \biggl[ \frac{E_{\sigma_{q,\omega}}}{\rho _{q,\omega}(t)}+qD_{q,\omega} \biggr] u(t)=C_{1}+{ \mathcal{I}}^{\alpha}_{q,\omega}h(t). \end{aligned}$$
(2.12)
Next, we take fractional \(q,\omega\)-integral of order β for (2.12). Thus,
$$\begin{gathered} \frac{u ( \sigma_{q,\omega}(t) ) }{\rho_{q,\omega }(t)}+qD_{q,\omega} u(t)= C_{2}+C_{1}(t- \omega_{0})+{\mathcal{I}}^{\beta}_{q,\omega}{ \mathcal{I}}^{\alpha}_{q,\omega}h(t), \\ \begin{aligned} \frac{u ( \sigma_{q,\omega}(t) )}{q}+\rho_{q,\omega }(t)D_{q,\omega}u(t)={} &C_{2}\frac{\rho_{q,\omega}(t)}{q}+C_{1}\frac {\rho_{q,\omega}(t)}{q}(t- \omega_{0})\\ &+\frac{\rho_{q,\omega }(t)}{q\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s} \bigl(t-\sigma_{q,\omega }(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} \bigl(s-\sigma _{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} h(x)\, d_{q,\omega}x\,d_{q,\omega}s,\end{aligned}\end{gathered} $$
or
$$ \begin{aligned} [b]D_{q,\omega} \bigl[\rho_{q,\omega}(t)u(t) \bigr]= {}& \frac {C_{2}}{q^{2}}(t-\omega)+\frac{C_{1}}{q^{2}}(t-\omega) (t-\omega_{0})\\ &+ \frac {1}{q^{2}\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s}(s-\omega) \bigl(t-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} h(x) \,d_{q,\omega}x\,d_{q,\omega}s.\end{aligned} $$
(2.13)
Finally, we take \(q,\omega\)-integral for (2.13) to obtain
$$ \begin{aligned}[b]\rho_{q,\omega}(t)u(t)= {}&C_{3}+C_{2} \frac{1}{q^{2}} \int_{\omega _{0}}^{t}(s-\omega)\,d_{q,\omega}s + \frac{C_{1}}{q^{2}} \int_{\omega_{0}}^{t}(s-\omega) (s-\omega_{0}) \, d_{q,\omega}s\\ &+\frac{1}{q^{2}\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s.\end{aligned} $$
(2.14)
By substituting \(t=\omega_{0}\) into (2.14) and employing the first condition of (2.1), we have
$$C_{3} = \omega_{0}\phi(u). $$
Therefore,
$$ \begin{aligned}[b]\rho_{q,\omega}(t)u(t)= {}&\omega_{0} \phi(u)+C_{2}\frac{1}{q^{2}} \int _{\omega_{0}}^{t}(s-\omega) \,d_{q,\omega}+ \frac{C_{1}}{q^{2}} \int _{\omega_{0}}^{t}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s \\ & +\frac{1}{q^{2}\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega _{0}}^{t} \int_{\omega_{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta -1}}\bigl(x-\sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha -1}} h(z) \,d_{q,\omega}z\,d_{q,\omega}x\,d_{q,\omega}s.\end{aligned} $$
(2.15)
Further, by letting \(t=\eta,T\) into (2.15) and employing the second and third conditions of (2.1), we have
$$\begin{aligned}& C_{1}=\frac{1}{\Omega} \biggl\{ q^{2} \bigl( \psi(u)- \omega_{0} \phi (u) \bigr) \biggl[ \int_{\omega_{0}}^{T}(s-\omega) \,d_{q,\omega} - \int_{\omega_{0}}^{\eta}(s-\omega)\,d_{q,\omega}s \biggr] \\& \phantom{C_{1}=} - \frac{\int_{\omega_{0}}^{T}(s-\omega) \,d_{q,\omega}s}{\Gamma _{q}(\alpha)\Gamma_{q}(\beta)} \\& \phantom{C_{1}=}\times \int_{\omega_{0}}^{\eta} \int_{\omega _{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega }(z) \bigr)_{q,\omega}^{\underline{\alpha-1}}h(z)\,d_{q,\omega}z\,d_{q,\omega}x\,d_{q,\omega}s\\& \phantom{C_{1}=}+ \frac{\int _{\omega_{0}}^{\eta}(s-\omega)\,d_{q,\omega}s}{\Gamma_{q}(\alpha )\Gamma_{q}(\beta)}\\& \phantom{C_{1}=}\times \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{s} \int _{\omega_{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} \bigl(x-\sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha -1}}h(z) \,d_{q,\omega}z\,d_{q,\omega}x\,d_{q,\omega}s \biggr\} ,\\& \begin{aligned}C_{2}={}&{-}\frac{1}{\Omega} \biggl\{ q^{2} \bigl( \psi(u)-\omega_{0} \phi (u) \bigr) \biggl[ \int_{\omega_{0}}^{T}(s-\omega) (s-\omega_{0})\, d_{q,\omega}s- \int_{\omega_{0}}^{\eta}(s-\omega) (s-\omega_{0})\, d_{q,\omega}s \biggr] \\ & - \frac{\int_{\omega_{0}}^{T}(s-\omega)(s-\omega_{0})\, d_{q,\omega }s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{\eta} \int _{\omega_{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s-\sigma _{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} \bigl(x-\sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z)\,d_{q,\omega}z\,d_{q,\omega}x\,d_{q,\omega}s\\ & +\frac{\int _{\omega_{0}}^{\eta}(s-\omega)(s-\omega_{0})\, d_{q,\omega}s}{\Gamma _{q}(\alpha)\Gamma_{q}(\beta)} \\ & \times \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s \biggr\} ,\end{aligned} \end{aligned}$$
where Ω is defined in (2.3).
To accomplish solution (2.2), we substitute the constants \(C_{1}\), \(C_{2}\) into (2.15). □