In this section, we present the notations, definitions, and lemmas used in the main results. Let \(q\in(0,1)\), \(\omega>0\) and define
$$\begin{aligned}{} [n]_{q}:=\frac{1-q^{n}}{1-q}=q^{n-1}+\cdots+q+1 \quad\text{and}\quad [n]_{q}!:=\prod_{k=1}^{n} \frac{1-q^{k}}{1-q},\quad n\in\mathbb{R}. \end{aligned}$$
The q-analogue of the power function \((a-b)_{q}^{\underline {n}}\) with \(n\in\mathbb{N}_{0}:=[0,1,2,\ldots]\) is
$$(a-b)_{q}^{\underline{0}}:=1, \qquad (a-b)_{q}^{\underline{n}}:= \prod_{k=0}^{n-1} \bigl(a-bq^{k} \bigr),\quad a,b\in\mathbb{R}. $$
The \(q,\omega\)-analogue of the power function \((a-b)_{q,\omega}^{\underline{n}}\) with \(n\in\mathbb {N}_{0}:=[0,1,2,\ldots]\) is
$$(a-b)_{q,\omega}^{\underline{0}}:=1, \qquad (a-b)_{q,\omega}^{\underline {n}}:= \prod_{k=0}^{n-1} \bigl[ a- \bigl(bq^{k}+\omega[k]_{q}\bigr)\bigr],\quad a,b\in \mathbb{R}. $$
More generally, if \(\alpha\in\mathbb{R}\), we have
$$\begin{aligned} &(a-b)_{q}^{\underline{\alpha}}= a^{\alpha}\prod _{n=0}^{\infty} \frac {1-(\frac{b}{a} )q^{n}}{1-(\frac{b}{a} )q^{\alpha+n}},\quad a\neq0, \\ &(a-b)_{q,\omega}^{\underline{\alpha}}= (a-\omega_{0})^{\alpha}\prod_{n=0}^{\infty} \frac{1-(\frac{b-\omega_{0}}{a-\omega_{0}} )q^{n}}{1-(\frac{b-\omega_{0}}{a-\omega_{0}} )q^{\alpha+n}}= \bigl((a-\omega_{0})-(b-\omega_{0})\bigr)_{q}^{\underline{\alpha}},\quad a\neq\omega_{0}. \end{aligned}$$
Note that \(a_{q}^{\underline{\alpha}} = a^{\alpha}\) and \((a-\omega_{0})_{q,\omega}^{\underline{\alpha}} = (a-\omega _{0})^{\alpha}\). We also use the notation \((0)_{q}^{\underline{\alpha }}=({\omega_{0}})_{q,\omega}^{\underline{\alpha}}=0\) for \(\alpha>0\). The q-gamma and q-beta functions are defined by
$$\begin{aligned} &\Gamma_{q}(x):=\frac{(1-q)_{q}^{\underline{x-1}}}{(1-q)^{x-1}},\quad x\in \mathbb{R}\setminus \{0,-1,-2,\ldots\}, \\ &B_{q}(x,s):= \int_{0}^{1} t^{x-1}(1-qt)_{q}^{\underline{s-1}}\,d_{q}t= \frac {\Gamma_{q}(x)\Gamma_{q}(s)}{\Gamma_{q}(x+s)}. \end{aligned}$$
Definition 2.1
For \(q\in(0,1)\), \(\omega >0\) and f defined on an interval \(I\subseteq{\mathbb{R}}\) which contains \(\omega_{0}:=\frac{\omega}{1-q}\), the Hahn difference of f is defined by
$$ D_{q,\omega}f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega} \quad\text{for } t\neq\omega_{0}, $$
and \(D_{q,\omega}f(\omega_{0})=f'(\omega_{0})\). Provided that f is differentiable at \(\omega_{0}\), we call \(D_{q,\omega}f\) the \(q,\omega \)-derivative of f, and say that 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)
\(D_{q,\omega}[ \dfrac{f(t)}{g(t)}] =\dfrac {g(t)D_{q,\omega}f(t)-f(t)D_{q,\omega}g(t)}{g(t)g(qt+\omega)}\).
Letting \(a,b\in I \subseteq\mathbb{R} \) with \(a<\omega_{0}<b\) and \([k]_{q}=\frac{1-q^{k}}{1-q}, k\in{\mathbb{N}}_{0}:={\mathbb{N}}\cup\{0\} \), we define the \(q,\omega\)-interval by
$$\begin{aligned} {[a,b]}_{q,\omega}&:=\bigl\{ q^{k} a+\omega[k]_{q}: k \in{\mathbb {N}}_{0} \bigr\} \cup\bigl\{ q^{k} b+ \omega[k]_{q}: k\in{\mathbb{N}}_{0} \bigr\} \cup\{ \omega_{0}\} \\ &={[a,\omega_{0}]}_{q,\omega} \cup{[\omega_{0},b]}_{q,\omega} \\ &={(a,b)}_{q,\omega} \cup\{a,b\} = {[a,b)}_{q,\omega} \cup\{b\} = {(a,b]}_{q,\omega} \cup\{a\}. \end{aligned}$$
We observe that 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 \(\omega_{0}\).
In addition, we define the forward jump operator \(\sigma ^{k}_{q,\omega}(t):=q^{k}t+\omega[k]_{q} \) and the backward jump operator \(\rho^{k}_{q,\omega}(t):=\frac{t-\omega[k]_{q} }{q^{k}} \) for \(k\in {\mathbb{N}}\).
Definition 2.2
Let I be any closed interval of \(\mathbb{R}\) that contains \(a,b\), 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
$$\int_{\omega_{0}}^{x} f(t)\,d_{q,\omega}t:=\bigl[x(1-q)- \omega\bigr]\sum_{k=0}^{\infty} q^{k} f\bigl(xq^{k}+\omega[k]_{q}\bigr),\quad x\in I, $$
and the series converges at \(x=a\) and \(x=b\). We call f
\(q,\omega\)-integrable on \([a,b]\), and the sum to the right-hand side of the above equation is called the Jackson–Nörlund sum.
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.
Lemma 2.1
([17])
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
([26])
Let
\(q\in(0,1)\), \(\omega >0\)
and
\(f:I\rightarrow\mathbb{R}\)
be continuous at
\(\omega_{0}\). Then
$$\begin{aligned} \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{r} x(s) \,d_{q,\omega}s d_{q,\omega }r= \int_{\omega_{0}}^{t} \int_{qs+\omega}^{t} x(s) \,d_{q,\omega}r d_{q,\omega}s. \end{aligned}$$
Lemma 2.3
([26])
Let
\(q\in(0,1)\)
and
\(\omega>0\). Then
$$\begin{aligned} \int_{\omega_{0}}^{t} \,d_{q,\omega}s=t- \omega_{0}\quad \textit{and}\quad \int_{\omega_{0}}^{t} \bigl[t-\sigma_{q,\omega}(s)\bigr] \,d_{q,\omega}s=\frac {(t-\omega_{0})^{2}}{1+q}. \end{aligned}$$
Particulary, we introduce fractional Hahn integral and fractional Hahn difference of Riemann–Liouville type as follows.
Definition 2.3
For \(\alpha,\omega>0, q\in(0,1)\) and f defined on \([\omega _{0},T]_{q,\omega}\), the fractional Hahn integral is defined by
$$\begin{aligned} {\mathcal{I}}_{q,\omega}^{\alpha}f(t)&:=\frac{1}{\Gamma_{q}(\alpha)} \int_{\omega_{0}}^{t} \bigl(t-\sigma_{q,\omega}(s) \bigr)_{q,\omega }^{\underline{\alpha-1}}f(s) \,d_{q,\omega}s \\ &=\frac{[t (1-q)-\omega]}{\Gamma_{q}(\alpha)} \sum_{n=0}^{\infty }q^{n} \bigl(t-\sigma_{q,\omega}^{n+1}(t)\bigr)_{q,\omega}^{\underline {\alpha-1}} f\bigl(\sigma_{q,\omega}^{n}(t)\bigr), \end{aligned}$$
and \(({\mathcal{I}}^{0}_{q,\omega} f)(t) = f(t)\).
Definition 2.4
For \(\alpha,\omega>0, q\in(0,1)\), and f defined on \([\omega _{0},T]_{q,\omega}\), the fractional Hahn difference of the Riemann–Liouville type of order α is defined by
$$\begin{aligned} D_{q,\omega}^{\alpha}f(t)&:=\bigl(D_{q,\omega}^{N} { \mathcal{I}}_{q,\omega }^{N-\alpha} f\bigr) (t) \\ &=\frac{1}{\Gamma_{q}(-\alpha)} \int_{\omega_{0}}^{t} \bigl(t-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{-\alpha-1}}f(s) \,d_{q,\omega}s, \end{aligned}$$
and \(D^{0}_{q,\omega}f(t) = f(t)\), where N is the smallest integer that is greater than or equal to α.
Lemma 2.4
([33])
Letting
\(\alpha>0,q\in (0,1),\omega>0\), and
\(f:I_{q,\omega}^{T}\rightarrow{\mathbb{R}}\), we get
$$\begin{aligned} {\mathcal{I}}_{q,\omega}^{\alpha}D_{q,\omega}^{\alpha}f(t) =f(t)+C_{1}(t-\omega_{0})^{\alpha-1}+ \cdots+C_{N}(t-\omega_{0})^{\alpha-N} \end{aligned}$$
for some
\(C_{i}\in{\mathbb{R}},i={\mathbb{N}}_{1,N} \)
and
\(N-1<\alpha\leq N,N\in{\mathbb{N}}\).
Next, we give some auxiliary lemmas used for simplifying calculations.
Lemma 2.5
([33])
Letting
\(\alpha,\beta >0, p,q\in(0,1)\), and
\(\omega>0\), we have
$$\begin{aligned} &\int_{\omega_{0}}^{t}\bigl( t-\sigma_{q,\omega}(s) \bigr)_{q,\omega }^{\underline{\alpha-1}} (s-\omega_{0})_{q,\omega}^{\underline{\beta }} \,d_{q,\omega}s =(t-\omega_{0})^{\alpha+\beta}B_{q}( \beta+1,\alpha ), \\ &\int_{\omega_{0}}^{t} \int_{\omega_{0}}^{x}\bigl( t-\sigma_{p,\omega }(x) \bigr)_{p,\omega}^{\underline{\alpha-1}} \bigl( x-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\beta-1}} \,d_{q,\omega }s \,d_{p,\omega}x = \frac{(t-\omega_{0})^{\alpha+\beta}}{[\beta ]_{q}}B_{p}(\beta+1,\alpha). \end{aligned}$$
The following lemma deals with the linear variant of problem (1.7) and gives a representation of the solution.
Lemma 2.6
Let
\(\alpha\in(1,2], \beta\in(0,1]\), \(\omega>0, p,q\in(0,1), p=q^{m}, m \in{\mathbb{N}}\), \(\theta=\omega(\frac{1-p}{1-q} )\); \(\lambda_{1},\lambda_{2},\mu_{1},\mu_{2} \in{{\mathbb{R}}^{+}}\); \(h \in C(I_{q,\omega}^{T},\mathbb{R})\)
is a given function; \(\phi_{1},\phi_{2}: C(I_{q,\omega}^{T},\mathbb{R})\rightarrow \mathbb{R}\)
are given functionals. Then the problem
$$\begin{aligned} &D^{\alpha}_{q,\omega} u(t)=h(t),\quad t\in I^{T}_{q,\omega}, \\ &\lambda_{1}u(\eta)+\lambda_{2}D_{p,\theta}^{\beta}u(\eta)=\phi_{1}(u),\quad \eta\in I^{T}_{q,\omega} - \{ \omega_{0},T\}, \\ &\mu_{1}u(T)+\mu_{2}D_{p,\theta}^{\beta}u(T)= \phi_{2}(u) \end{aligned}$$
(2.1)
has the unique solution
$$\begin{aligned} u(t) = {}&\frac{1}{\Gamma_{q}(\alpha)} \int_{\omega_{0}}^{t} \bigl(t-\sigma_{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(s) \,d_{q,\omega}s \\ &{}-\frac{(t-\omega_{0})^{\alpha-1}}{\Omega} \bigl\{ \mathbf{B}_{T}{\Phi _{\eta}[\phi_{1},h]} -{\mathbf{B}}_{\eta}{ \Phi_{T}[\phi_{2},h]} \bigr\} \\ &{}+\frac{(t-\omega_{0})^{\alpha-2}}{\Omega} \bigl\{ \mathbf{A}_{T}{\Phi _{\eta}[\phi_{1},h]} -{\mathbf{A}}_{\eta}{ \Phi_{T}[\phi_{2},h]} \bigr\} , \end{aligned}$$
(2.2)
where the functionals
\({\Phi_{\eta}[\phi_{1},h]}, {\Phi _{T}[\phi_{2},h]}\)
are defined by
$$\begin{aligned} {\Phi_{\eta}[\phi_{1},h]}:= {}&\phi_{1}(u)- \frac{\lambda_{1}}{\Gamma_{q}(\alpha)} \int_{\omega_{0}}^{\eta} \bigl(\eta-\sigma_{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha -1}} h(s) \,d_{q,\omega}s-\frac{\lambda_{2}}{\Gamma_{q}(\alpha)\Gamma _{p}(-\beta)} \\ &{}\times \int_{\omega_{0}}^{\eta} \int_{\omega_{0}}^{x}\bigl(\eta-\sigma _{p,\theta}(s) \bigr)_{p,\theta}^{\underline{-\beta-1}} \bigl(x-\sigma_{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(s) \,d_{q,\omega}s \,d_{p,\theta}x, \end{aligned}$$
(2.3)
$$\begin{aligned} {\Phi_{T}[\phi_{2},h]}:= {}&\phi_{2}(u)- \frac{\mu_{1}}{\Gamma_{q}(\alpha)} \int_{\omega_{0}}^{T} \bigl(T-\sigma_{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(s) \,d_{q,\omega}s-\frac{\mu_{2}}{\Gamma_{q}(\alpha)\Gamma_{p}(-\beta )} \\ &{}\times \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{x}\bigl(T-\sigma_{p,\theta }(s) \bigr)_{p,\theta}^{\underline{-\beta-1}} \bigl(x-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(s) \,d_{q,\omega}s \,d_{p,\theta}x, \end{aligned}$$
(2.4)
and the constants
\(\mathbf{A}_{\eta},\mathbf{A}_{T},\mathbf{B}_{\eta },\mathbf{B}_{T}\), and Ω are defined by
$$\begin{aligned} &\mathbf{A}_{\eta}:=\lambda_{1}(\eta- \omega_{0})^{\alpha-1}+\frac {\lambda_{2}}{\Gamma_{p}(-\beta)} \int_{\omega_{0}}^{\eta}\bigl(\eta-\sigma_{p,\theta}(s) \bigr)_{p,\theta}^{\underline{-\beta-1}} (s-\omega_{0})^{\alpha-1} \,d_{p,\theta}s \\ &\phantom{\mathbf{A}_{\eta}}=(\eta-\omega_{0})^{\alpha-1} \biggl( \lambda_{1}+\frac{\lambda_{2}(\eta-\omega_{0})^{-\beta}\Gamma _{p}(\alpha)}{\Gamma_{p}(\alpha-\beta)}\biggr), \end{aligned}$$
(2.5)
$$\begin{aligned} &\mathbf{A}_{T}:= \mu_{1}(T- \omega_{0})^{\alpha-1}+\frac{\mu_{2}}{\Gamma _{p}(-\beta)} \int_{\omega_{0}}^{T}\bigl(T-\sigma_{p,\theta}(s) \bigr)_{p,\theta }^{\underline{-\beta-1}} (s-\omega_{0})^{\alpha-1} \,d_{p,\theta }s \\ &\phantom{\mathbf{A}_{\eta}}=(T-\omega_{0})^{\alpha-1} \biggl( \mu_{1}+\frac{\mu_{2}(T-\omega_{0})^{-\beta}\Gamma_{p}(\alpha)}{\Gamma _{p}(\alpha-\beta)}\biggr), \end{aligned}$$
(2.6)
$$\begin{aligned} &\mathbf{B}_{\eta}:= \lambda_{1}(\eta- \omega_{0})^{\alpha-2}+\frac {\lambda_{2}}{\Gamma_{p}(-\beta)} \int_{\omega_{0}}^{\eta}\bigl(\eta-\sigma_{p,\theta}(s) \bigr)_{p,\theta}^{\underline{-\beta-1}} (s-\omega_{0})^{\alpha-2} \,d_{p,\theta}s \\ &\phantom{\mathbf{A}_{\eta}}=(\eta-\omega_{0})^{\alpha-2} \biggl( \lambda_{1}+\frac{\lambda_{2}(\eta-\omega_{0})^{-\beta}\Gamma _{p}(\alpha-1)}{\Gamma_{p}(\alpha-\beta-1)}\biggr), \end{aligned}$$
(2.7)
$$\begin{aligned} &\mathbf{B}_{T}:= \mu_{1}(T- \omega_{0})^{\alpha-2}+\frac{\mu_{2}}{\Gamma _{p}(-\beta)} \int_{\omega_{0}}^{T}\bigl(T-\sigma_{p,\theta}(s) \bigr)_{p,\theta }^{\underline{-\beta-1}} (s-\omega_{0})^{\alpha-2} \,d_{p,\theta }s \\ &\phantom{\mathbf{A}_{\eta}}=(T-\omega_{0})^{\alpha-2} \biggl( \mu_{1}+\frac{\mu_{2}(T-\omega_{0})^{-\beta}\Gamma_{p}(\alpha -1)}{\Gamma_{p}(\alpha-\beta-1)}\biggr), \end{aligned}$$
(2.8)
$$\begin{aligned} &\Omega:= \mathbf{A}_{T}\mathbf{B}_{\eta}- \mathbf{A}_{\eta }\mathbf{B}_{T}. \end{aligned}$$
(2.9)
Proof
Taking fractional Hahn \(q,\omega\)-integral of order α for (2.1), we obtain
$$\begin{aligned} u(t)={} &C_{1}(t-\omega_{0})^{\alpha-1}+C_{2}(t- \omega_{0})^{\alpha -2}+{\mathcal{I}}^{\alpha}_{q,\omega}h(t) \\ = {}&C_{1}(t-\omega_{0})^{\alpha-1}+C_{2}(t- \omega_{0})^{\alpha-2}+\frac {1}{\Gamma_{q}(\alpha)} \int_{\omega_{0}}^{t}\bigl(t-\sigma_{q,\omega }(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(x) \,d_{q,\omega }s. \end{aligned}$$
(2.10)
Then we take fractional Hahn \(p,\theta\)-difference of order β for (2.10) to get
$$\begin{aligned} &D_{p,\theta}^{\beta}u(t) \\ &\quad =\frac{1}{\Gamma_{q}(\alpha)} \int_{\omega _{0}}^{t}\bigl(t-\sigma_{q,\omega}(s) \bigr)_{q,\omega}^{\underline {\alpha-1}} \bigl[C_{1}(s- \omega_{0})^{\alpha-1}+C_{2}(s-\omega _{0})^{\alpha-2}\bigr] \,d_{q,\omega}s \\ &\qquad{}+\frac{1}{\Gamma_{q}(\alpha)\Gamma_{p}(-\beta)} \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{x}\bigl(t-\sigma_{q,\omega }(x) \bigr)_{p,\theta}^{\underline{-\beta-1}}\bigl(x-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(s) \,d_{q,\omega}s \,d_{p,\theta}x. \end{aligned}$$
(2.11)
Substituting \(t=\eta\) into (2.10) and (2.11) and employing the first condition of (2.1), we have
$$\begin{aligned} {\mathbf{A}}_{\eta}C_{1}+{\mathbf{B}}_{\eta}C_{2} = \Phi_{\eta}[\phi_{1},h]. \end{aligned}$$
(2.12)
Taking \(t=T\) into (2.10) and (2.11) and employing the second condition of (2.1), we have
$$\begin{aligned} {\mathbf{A}}_{T }C_{1}+{ \mathbf{B}}_{T }C_{2} = \Phi_{T}[ \phi_{2},h]. \end{aligned}$$
(2.13)
The constants \(C_{1}\) and \(C_{2}\) are revealed from solving the system of equations (2.12)–(2.13) as
$$\begin{aligned} C_{1}= \frac{{\mathbf{B}}_{\eta} {\Phi_{T} } - {\mathbf{B}}_{T } {\Phi}_{\eta} }{\Omega} \quad\text{and}\quad C_{2}= \frac{ {\mathbf{A}}_{T } {\Phi}_{\eta} - {\mathbf{A}}_{\eta} {\Phi_{T} } }{\Omega}. \end{aligned}$$
Substituting the constants \(C_{1},C_{2}\) into (2.10), we obtain (2.2).
On the other hand, it is easy to show that (2.2) is the solution of problem (2.1). By taking fractional Hahn \(q,\omega\)-difference of order α for (2.2), we obtain (2.1). This completes the proof. □
We next introduce Schauder’s fixed point theorem used to prove the existence of a solution of problem (1.7).
Lemma 2.7
([36] Arzelá–Ascoli theorem)
A set of functions in
\(C[a,b]\)
with the sup norm is relatively compact if and only if it is uniformly bounded and equicontinuous on
\([a,b]\).
Lemma 2.8
([36])
If a set is closed and relatively compact, then it is compact.
Lemma 2.9
([37] Schauder’s fixed point theorem)
Let
\((D,d)\)
be a complete metric space, U
be a closed convex subset of
D, and
\(T: D\rightarrow D\)
be the map such that the set
\(Tu:u\in U\)
is relatively compact in
D. Then the operator
T
has at least one fixed point
\(u^{*}\in U\): \(Tu^{*}=u^{*}\).