Research
Open Access
On nonlocal Dirichlet boundary value problem for sequential Caputo fractional Hahn integrodifference equations
- Nichaphat Patanarapeelert1,
- Tanapat Brikshavana2 and
- Thanin Sitthiwirattham2Email authorView ORCID ID profile
Received: 29 July 2017
Accepted: 28 December 2017
Published: 11 January 2018
Abstract
The aim of this work is to study a nonlocal Dirichlet boundary value problem for sequential Caputo fractional Hahn integrodifference equation. The problem contains two fractional Hahn difference operators and a fractional Hahn integral with different numbers of order. We use the Banach fixed point theorem to prove the existence and uniqueness of the solution. In particular, the existence of at least one solution is presented by using the Schauder fixed point theorem.
Keywords
fractional Hahn integrodifference equationsboundary value problemsexistencefixed point theorem
MSC
39A1039A1339A70
1 Introduction
In this paper, we study the quantum calculus that is the calculus without considering limits based on the sets of non-differentiable functions. Some examples of quantum difference operators are the Jackson q-difference operator, the forward (delta) difference operator, and the backward (nabla) difference operator. The quantum difference operators have been employed in many applications [1–9] of mathematical areas such as the theory of relativity, particle physics, and quantum mechanics.
In this work, we focus on the Hahn difference operator that is the generalization of the forward difference operator and the Jackson q-difference operator.
The Hahn difference operator \(D_{q,\omega}\) is defined by [10]
Note that
This operator is used to construct families of orthogonal polynomials (see [11, 12] and the references therein).
$$D_{q,\omega}f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega},\quad t\neq \omega_{0}:= \frac{\omega}{1-q}. $$
$$\begin{gathered} D_{q,\omega}f(t)=\Delta_{\omega}f(t) \quad\text{whenever }q= 1,\qquad D_{q,\omega}f(t)=D_{q}f(t) \quad\text{whenever }\omega= 0,\quad\text{and} \\D_{q,\omega}f(t)=f'(t) \quad\text{whenever }q= 1,\omega \rightarrow0.\end{gathered} $$
In 2009, Aldwoah [13, 14] defined the right inverse of \(D_{q,\omega}\) in terms of the Jackson q-integral containing the right inverse of \(D_{q}\) [15] and the Nörlund sum containing the right inverse of \(\Delta_{\omega}\) [15].
Next, the Hahn quantum variational calculus was introduced by Malinowska and Torres [16, 17] in 2010, and the generalized transversality conditions for the Hahn quantum variational calculus were described by Malinowska and Martins [18] in 2013.
The studies of existence and uniqueness results for the initial value problems for Hahn difference equations were presented in 2013 by Hamza et al. [19, 20] by using the method of successive approximations. In addition, they proved Gronwall’s and Bernoulli’s inequalities with respect to the Hahn difference operator and also established the mean value theorems for this calculus.
Recently, Sitthiwirattham [21] proposed a nonlinear Hahn difference equation with nonlocal boundary value conditions of the form
where \(0< q<1\), \(0<\omega<T\), \(\omega_{0}:=\frac{\omega}{1-q}\), \(1\leq \lambda<\frac{T-\omega_{0}}{\eta-\omega_{0}}\), \(p=q^{m}\), \(m\in {\mathbb{N}}\), \(\theta=\omega (\frac{1-p}{1-q} )\), \(f: [\omega_{0},T]_{q,\omega}\times{\mathbb{R}}\times{\mathbb {R}}\rightarrow{\mathbb{R}}\) is a given function, and \(\varphi: C ([\omega_{0},T]_{q,\omega},{\mathbb{R}} )\rightarrow {\mathbb{R}}\) is a given functional.
$$\begin{gathered} D^{2}_{q,\omega} x(t)+f\bigl(t,x(t),D_{p,\theta}x(pt+\theta) \bigr)=0,\quad t\in[\omega_{0},T]_{q,\omega}, \\ x(\omega_{0})=\varphi(x), \\ x(T)=\lambda x(\eta),\quad\eta\in(\omega_{0},T)_{q,\omega },\end{gathered} $$
(1.1)
In 2017, Sriphanomwan et al. [22] considered a nonlocal boundary value problem for second-order nonlinear Hahn integro-difference equation with integral boundary condition of the form
where \(0< q<1\), \(0<\omega<T\), \(\omega_{0}:=\frac{\omega}{1-q}\), \(\mu \int_{\omega_{0}}^{T} g(r) \,d_{q,\omega}r\neq1\), \(\mu\in\mathbb {R}\), \(p=q^{m}\), \(m\in{\mathbb{N}}\), \(\theta=\omega (\frac {1-p}{1-q} )\), \(f\in C([\omega_{0},T]_{q,\omega}\times{\mathbb{R}}\times{\mathbb {R}}\times{\mathbb{R}},{\mathbb{R}})\), and \(g\in C([\omega _{0},T]_{q,\omega},{\mathbb{R}}^{+})\) are given functions, and for \(\varphi\in C([\omega_{0},T]_{q,\omega}\times[\omega_{0},T]_{q,\omega },[0,\infty))\),
$$\begin{gathered} D^{2}_{q,\omega} x(t)=f\bigl(t,x(t),D_{p,\theta}x(pt+ \theta),\varPsi _{p,\theta}x(pt+\theta)\bigr),\quad t\in[\omega_{0},T]_{q,\omega }, \\ x(\omega_{0})=x(T), \qquad x(\eta)=\mu \int_{\omega_{0}}^{T}g(s)x(s)\, d_{q,\omega}s,\quad\eta \in(\omega_{0},T)_{q,\omega},\end{gathered} $$
(1.2)
$$ \varPsi_{p,\theta} x(t):= \int_{\omega_{0}}^{t}\varphi(t,ps+\theta )x(ps+\theta) \,d_{p,\theta} s. $$
For the fractional Hahn difference operators, Čermák and Nechvátal [23] introduced the fractional \((q,h)\)-difference operator and the fractional \((q,h)\)-integral for \(q>1\). Next, Čermák et al. [24] presented discrete Mittag-Leffler functions in linear fractional difference equations for \(q>1\). On discrete time scales for \(q>1\), Rahmat [25, 26] considered the \((q,h)\)-Laplace transform and some \((q,h)\)-analogues of integral inequalities. In 2016, Du et al. [27] studied the monotonicity and convexity for nabla fractional \((q,h)\)-difference for \(q>0\), \(q\neq1\).
We observed from the above research that these operators are not fractional Hahn operators because the conditions are not satisfied with \(0< q<1\). To fill this gap, Brikshavana and Sitthiwirattham [28] introduced the fractional Hahn operators.
In this paper, we study the boundary value problem for fractional Hahn difference equation containing a sequential Caputo fractional Hahn integrodifference equation with nonlocal Dirichlet boundary conditions. The governing problem is given by
where \([\omega_{0},T]_{q,\omega}=I^{T}_{q,\omega}:=\{q^{k}T+\omega [k]_{q}:k \in{\mathbb{N}}_{0}\}\cup\{\omega_{0}\}\); \(\omega>0\), \(q\in (0,1)\); \(\alpha,\beta,\gamma\in(0,1]\); the shift operator \(E_{\sigma_{q,\omega}}u(t):=u ( \sigma_{q,\omega}(t) )\); \(F \in C (I_{q,\omega}^{T}\times\mathbb{R}\times\mathbb {R},\mathbb{R} )\) is a given function; \(\phi,\psi: C (I_{q,\omega}^{T},\mathbb{R} )\rightarrow\mathbb{R}\) are given functionals; and for \(\varphi\in C (I_{q,\omega}^{T}\times I_{q,\omega}^{T},[0,\infty) )\), we define
$$ \begin{gathered} {}^{C}D^{\alpha}_{q,\omega}{}^{C}D^{\beta}_{q,\omega} \biggl[\frac {E_{\sigma_{q,\omega}}}{\rho_{q,\omega}(t)}+qD_{q,\omega} \biggr] u(t)=F \bigl( t,u(t), \Psi^{\gamma}_{q,\omega}u(t) \bigr),\quad t\in [\omega_{0},T]_{q,\omega}, \\ u(\omega_{0})=\phi(u), \\ \rho_{q,\omega}(T)u(T)=\rho_{q,\omega}(\eta)u(\eta)=\psi (u),\quad\eta \in(\omega_{0},T)_{q,\omega},\end{gathered} $$
(1.3)
$$\Psi^{\gamma}_{q,\omega} u(t):= \bigl({\mathcal{I}}_{q,\omega}^{\gamma}\varphi u \bigr) (t)=\frac{1}{\Gamma_{q,\omega}(\gamma)} \int _{\omega_{0}}^{t} \bigl( t-\sigma_{q,\omega}(s) \bigr) ^{\underline {\gamma-1}}_{q,\omega}\varphi(t,s) u(s)\,d_{q,\omega}s. $$
We give some basic definitions and lemmas in the next section. In Section 3, we present the main results of the study. The existence and uniqueness of a solution to problem (1.3) are proved by using the Banach fixed point theorem. In addition, the existence of at least one solution to problem (1.3) is proved by using the Schauder fixed point theorem in Section 4. We end up with some example in the last section.
2 Preliminaries
In this section, we introduce notations, definitions, and lemmas used in the main results. For \(q\in(0,1)\) and \(\omega>0\), we define
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
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
where \((a-b)_{q,\omega}^{\underline{0}}:=1\). Generally, if \(\alpha \in\mathbb{R}\), we have
$$ [n]_{q}:=\frac{1-q^{n}}{1-q}=q^{n-1}+\cdots+q+1\quad\text{and} \quad [n]_{q}!:=\prod_{k=1}^{n} \frac{1-q^{k}}{1-q},\quad n\in\mathbb{R}. $$
$$(a-b)_{q}^{\underline{n}}:= \prod_{k=0}^{n-1} \bigl(a-bq^{k}\bigr), \quad a,b\in \mathbb{R}, $$
$$(a-b)_{q,\omega}^{\underline{n}}:= \prod_{k=0}^{n-1} \bigl[ a-\bigl(bq^{k}+\omega[k]_{q}\bigr) \bigr],\quad a,b \in\mathbb{R}, $$
$$\begin{gathered} (a-b)_{q}^{\underline{\alpha}}= a^{\alpha}\prod _{n=0}^{\infty} \frac {1- (\frac{b}{a} )q^{n}}{1- (\frac{b}{a} )q^{\alpha+n}},\quad a\neq0, \\(a-b)_{q,\omega}^{\underline{\alpha}}= (a-\omega_{0})^{\alpha}\prod_{n=0}^{\infty} \frac{1- (\frac{b-\omega_{0}}{a-\omega_{0}} )q^{n}}{1- (\frac{b-\omega_{0}}{a-\omega_{0}} )q^{\alpha+n}}= \bigl((a-\omega_{0})-(b-\omega_{0}) \bigr)_{q}^{\underline{\alpha}},\quad a\neq\omega_{0}.\end{gathered} $$
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
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.
$$ D_{q,\omega}f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega}\quad\text{for } t\neq \omega_{0}. $$
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
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 \(\omega_{0}\).
$$\begin{aligned} {[a,b]}_{q,\omega}&:= \bigl\{ q^{k} a+\omega[k]_{q} : k \in{\mathbb {N}}_{0} \bigr\} \cup \bigl\{ q^{k} b+ \omega[k]_{q} : k\in{\mathbb{N}}_{0} \bigr\} \cup \{ \omega_{0} \} \\ &={[a,\omega_{0}]}_{q,\omega} \cup{[\omega_{0},b]}_{q,\omega} \\ &={(a,b)}_{q,\omega} \cup \{a,b \} = {[a,b)}_{q,\omega} \cup \{b \} = {(a,b]}_{q,\omega} \cup \{a \}.\end{aligned} $$
We 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
where
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.
$$ \int_{a}^{b} f(t)\,d_{q,\omega}t:= \int_{\omega_{0}}^{b} f(t)\,d_{q,\omega }t- \int_{\omega_{0}}^{a} f(t)\,d_{q,\omega}t, $$
$$\int_{\omega_{0}}^{x} f(t)\,d_{q,\omega}t:= \bigl[x(1-q)-\omega \bigr]\sum_{k=0}^{\infty} q^{k} f \bigl(xq^{k}+\omega[k]_{q} \bigr),\quad x\in I. $$
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
Then
F
is continuous at
\(\omega_{0}\). Furthermore, \(D_{q,\omega _{0}}F(x)\)
exists for every
\(x\in I\)
and
Conversely,
$$F(x):= \int_{\omega_{0}}^{x} f(t)\,d_{q,\omega}t,\quad x\in I. $$
$$D_{q,\omega}F(x)=f(x). $$
$$\int_{a}^{b} D_{q,\omega}F(t) \,d_{q,\omega}t=F(b)-F(a) \quad\textit{for all }a,b\in I. $$
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
where \(({\mathcal{I}}^{0}_{q,\omega} f)(t) = f(t)\).
$$\begin{aligned} {\mathcal{I}}_{q,\omega}^{\alpha}f(t)&:=\frac{1}{\Gamma_{q}(\alpha)} \int_{\omega_{0}}^{t} \bigl(t-\sigma_{q,\omega}(s) \bigr)_{q,\omega }^{\underline{\alpha-1}}f(s) \,d_{q,\omega}s \\ &=\frac{[t (1-q)-\omega]}{\Gamma_{q}(\alpha)} \sum_{n=0}^{\infty }q^{n} \bigl(t-\sigma_{q,\omega}^{n+1}(t) \bigr)_{q,\omega}^{\underline {\alpha-1}} f \bigl(\sigma_{q,\omega}^{n}(t) \bigr),\end{aligned} $$
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
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.
$$\begin{aligned} {}^{C}D_{q,\omega}^{\alpha}f(t)&:=\bigl({ \mathcal{I}}_{q,\omega}^{N-\alpha } D_{q,\omega}^{N} f\bigr) (t) \\ &=\frac{1}{\Gamma_{q}(N-\alpha)} \int_{\omega_{0}}^{t} \bigl(t-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{N-\alpha-1}} D_{q,\omega}^{N}f(s) \,d_{q,\omega}s,\end{aligned} $$
Lemma 2.3
([28])
Let
\(\alpha>0\), \(q\in (0,1)\), \(\omega>0\), and
\(f:I_{q,\omega}^{T}\rightarrow{\mathbb{R}}\). Then
for some
\(C_{i}\in{\mathbb{R}}\), \(i={\mathbb{N}}_{0,N-1} \)
and
\(N-1<\alpha\leq N\), \(N\in{\mathbb{N}}\).
$$ {\mathcal{I}}_{q,\omega}^{\alpha}{}^{C}D_{q,\omega}^{\alpha}f(t) =f(t)+C_{0}+C_{1}(t-\omega_{0})+\cdots+C_{N-1}(t- \omega_{0})^{N-1} $$
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
has the unique solution
where
the functionals
\({\mathcal{A^{*}}}(h,u), {\mathcal{B^{*}}}(h,u)\)
are defined by
and
$$ \begin{gathered} {}^{C}D^{\alpha}_{q,\omega}{}^{C}D^{\beta}_{q,\omega} \biggl[\frac {E_{\sigma_{q,\omega}}}{\rho_{q,\omega}(t)}+qD_{q,\omega} \biggr] u(t)=h(t),\quad t\in I_{q,\omega}^{T}, \\ u(\omega_{0})=\phi(u), \\ \rho_{q,\omega}(T)u(T)=\rho_{q,\omega}(\eta)u(\eta)=\psi (u),\quad\eta \in I_{q,\omega}^{T}- \lbrace\omega_{0},T \rbrace\end{gathered} $$
(2.1)
$$ \begin{aligned}[b] u(t)={} &\frac{1}{q^{2}\rho_{p,\omega}(t)} \biggl\{ q^{2} \omega_{0}\phi (u)\\ &+\frac{1}{\Omega} \biggl[{\mathcal{A^{*}}}(h,u) \int_{\omega _{0}}^{t}(s-\omega)\,d_{q,\omega}s -{ \mathcal{B^{*}}}(h,u) \int_{\omega_{0}}^{t}(s-\omega) ( s-\omega_{0}) \, d_{q,\omega}s \biggr] \\ +& \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s} \int_{\omega_{0}}^{x} \frac{(s-\omega) (s-\sigma_{q,\omega}(x) )_{q,\omega }^{\underline{\beta-1}} (x-\sigma_{q,\omega}(z) )_{q,\omega}^{\underline{\alpha-1}} }{\Gamma_{q}(\alpha)\Gamma _{q}(\beta)} h(z) \,d_{q,\omega}z\,d_{q,\omega}x\,d_{q,\omega}s \biggr\} ,\end{aligned} $$
(2.2)
$$ \begin{aligned}[b]\Omega:= {}& \int_{\omega_{0}}^{T}(s-\omega)\,d_{q,\omega}s \int _{\omega_{0}}^{\eta}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s - \int_{\omega_{0}}^{\eta}(s-\omega)\,d_{q,\omega}s \int_{\omega _{0}}^{T}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s \\ = {}&(\eta-T) (T-\omega_{0}) (\eta-\omega_{0}) \biggl[ \frac{(T-\omega_{0})(\eta-\omega_{0}) }{(q+1)(q^{2}+q+1)} + \frac{\omega _{0}q(T+\eta-2\omega_{0})}{q^{2}+q+1} +\frac{\omega_{0}^{2} q^{2}}{q+1} \biggr],\end{aligned} $$
(2.3)
$$\begin{aligned}& {\mathcal{A^{*}}}(h,u):= {\mathcal{A}}(u)+{\mathcal {A}}_{1}(h)-{ \mathcal{A}}_{2}(h), \end{aligned}$$
(2.4)
$$\begin{aligned}& {\mathcal{B^{*}}}(h,u):={\mathcal{B}}(u)+{\mathcal {B}}_{1}(h)-{ \mathcal{B}}_{2}(h) , \end{aligned}$$
(2.5)
$$\begin{aligned}& {\mathcal{A}}(u):= \bigl( \omega_{0} \phi(u)-\psi(u) \bigr) \int _{\eta}^{T}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s, \end{aligned}$$
(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
Next, we take fractional \(q,\omega\)-integral of order β for (2.12). Thus,
or
Finally, we take \(q,\omega\)-integral for (2.13) to obtain
By substituting \(t=\omega_{0}\) into (2.14) and employing the first condition of (2.1), we have
Therefore,
$$\begin{aligned} &{}^{C}D^{\beta}_{q,\omega} \biggl[ \frac{E_{\sigma_{q,\omega}}}{\rho _{q,\omega}(t)}+qD_{q,\omega} \biggr] u(t)=C_{1}+{ \mathcal{I}}^{\alpha}_{q,\omega}h(t). \end{aligned}$$
(2.12)
$$\begin{gathered} \frac{u ( \sigma_{q,\omega}(t) ) }{\rho_{q,\omega }(t)}+qD_{q,\omega} u(t)= C_{2}+C_{1}(t- \omega_{0})+{\mathcal{I}}^{\beta}_{q,\omega}{ \mathcal{I}}^{\alpha}_{q,\omega}h(t), \\ \begin{aligned} \frac{u ( \sigma_{q,\omega}(t) )}{q}+\rho_{q,\omega }(t)D_{q,\omega}u(t)={} &C_{2}\frac{\rho_{q,\omega}(t)}{q}+C_{1}\frac {\rho_{q,\omega}(t)}{q}(t- \omega_{0})\\ &+\frac{\rho_{q,\omega }(t)}{q\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s} \bigl(t-\sigma_{q,\omega }(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} \bigl(s-\sigma _{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} h(x)\, d_{q,\omega}x\,d_{q,\omega}s,\end{aligned}\end{gathered} $$
$$ \begin{aligned} [b]D_{q,\omega} \bigl[\rho_{q,\omega}(t)u(t) \bigr]= {}& \frac {C_{2}}{q^{2}}(t-\omega)+\frac{C_{1}}{q^{2}}(t-\omega) (t-\omega_{0})\\ &+ \frac {1}{q^{2}\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s}(s-\omega) \bigl(t-\sigma _{q,\omega}(s) \bigr)_{q,\omega}^{\underline{\alpha-1}} \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} h(x) \,d_{q,\omega}x\,d_{q,\omega}s.\end{aligned} $$
(2.13)
$$ \begin{aligned}[b]\rho_{q,\omega}(t)u(t)= {}&C_{3}+C_{2} \frac{1}{q^{2}} \int_{\omega _{0}}^{t}(s-\omega)\,d_{q,\omega}s + \frac{C_{1}}{q^{2}} \int_{\omega_{0}}^{t}(s-\omega) (s-\omega_{0}) \, d_{q,\omega}s\\ &+\frac{1}{q^{2}\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega_{0}}^{t} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} h(z) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s.\end{aligned} $$
(2.14)
$$C_{3} = \omega_{0}\phi(u). $$
$$ \begin{aligned}[b]\rho_{q,\omega}(t)u(t)= {}&\omega_{0} \phi(u)+C_{2}\frac{1}{q^{2}} \int _{\omega_{0}}^{t}(s-\omega) \,d_{q,\omega}+ \frac{C_{1}}{q^{2}} \int _{\omega_{0}}^{t}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s \\ & +\frac{1}{q^{2}\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega _{0}}^{t} \int_{\omega_{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta -1}}\bigl(x-\sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha -1}} h(z) \,d_{q,\omega}z\,d_{q,\omega}x\,d_{q,\omega}s.\end{aligned} $$
(2.15)
Further, by letting \(t=\eta,T\) into (2.15) and employing the second and third conditions of (2.1), we have
where Ω is defined in (2.3).
$$\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}$$
To accomplish solution (2.2), we substitute the constants \(C_{1}\), \(C_{2}\) into (2.15). □
3 Existence and uniqueness of solution
In this section, we present the existence and uniqueness of solution for problem (1.3). Let \({\mathcal{C}}=C (I_{q,\omega}^{T}, {\mathbb{R}} )\) be a Banach space of all function u with the norm defined by
where \(\omega>0\), \(q\in(0,1)\). Define an operator \({\mathcal{F}}:{\mathcal{C}}\rightarrow{\mathcal{C}}\) by
where \(\Omega\neq0\) is defined in (2.3), the functionals \({\mathcal{A^{**}}}(F,u)\), \({\mathcal{B^{**}}}(F,u)\) are defined by
and
with functionals \({\mathcal{A}}(u)\), \({\mathcal{B}}(u) \) defined in (2.6), (2.9), respectively.
$$\|u\|_{\mathcal{C}}=\max_{t\in I_{q,\omega}^{T}}\bigl\{ |u|\bigr\} , $$
$$\begin{aligned} ({\mathcal{F}}u) (t):= {}&\frac{1}{q^{2}\rho_{q,\omega}(t)} \biggl\{ q^{2}\omega_{0}\phi(u)+\frac{1}{\Omega} \biggl[{ \mathcal{A^{**}}}(F,u) \int_{\omega_{0}}^{t}(s-\omega)\,d_{q,\omega}s \\ &-{\mathcal {B^{**}}}(F,u) \int_{\omega_{0}}^{t}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s \biggr] \\ &+ \frac{1 }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \\ &\times \int_{\omega _{0}}^{t} \int_{\omega_{0}}^{s} \int_{\omega_{0}}^{x} (s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x-\sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha -1}} \\ &\times F \bigl( z,u(z),\Psi^{\gamma}_{q,\omega}u(z) \bigr)\,d_{q,\omega }z \,d_{q,\omega}x\,d_{q,\omega}s \biggr\} , \end{aligned}$$
(3.1)
$$\begin{aligned}& {\mathcal{A^{**}}}(F,u):= {\mathcal{A}}(u)+{\mathcal {A}}^{*}_{1}(F)-{\mathcal{A}}_{2}^{*}(F), \end{aligned}$$
(3.2)
$$\begin{aligned}& {\mathcal{B^{**}}}(F,u):= {\mathcal{B}}(u)+{\mathcal {B}}_{1}^{*}(F)-{\mathcal{B}}_{2}^{*}(F) , \end{aligned}$$
(3.3)
$$\begin{aligned}& \begin{aligned}[b] {\mathcal{A}}_{1}^{*}(F):={} & \frac{\int_{\omega_{0}}^{T}(s-\omega )(s-\omega_{0}) \,d_{q,\omega}s }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \int_{\omega_{0}}^{\eta} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}}\bigl(x-\sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha -1}} \\ &\times F \bigl( z,u(z),\Psi^{\gamma}_{q,\omega}u(z) \bigr)\,d_{q,\omega }z \,d_{q,\omega}x\,d_{q,\omega}s,\end{aligned} \end{aligned}$$
(3.4)
$$\begin{aligned}& \begin{aligned}[b]{\mathcal{A}}_{2}^{*}(F):={} & \frac{ \int_{\omega_{0}}^{\eta}(s-\omega )(s-\omega_{0})\, d_{q,\omega}s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta )} \int_{\omega_{0}}^{T} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}}\bigl(x-\sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha -1}} \\ & \times F \bigl( z,u(z),\Psi^{\gamma}_{q,\omega}u(z) \bigr)\,d_{q,\omega }z \,d_{q,\omega}x\,d_{q,\omega}s,\end{aligned} \end{aligned}$$
(3.5)
$$\begin{aligned}& \begin{aligned}[b]{\mathcal{B}}_{1}^{*}(F):={} & \frac{\int_{\omega_{0}}^{T}(s-\omega) \, d_{q,\omega}s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \int_{\omega _{0}}^{\eta} \int_{\omega_{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta -1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline {\alpha-1}} \\ &\times F \bigl( z,u(z),\Psi^{\gamma}_{q,\omega}u(z) \bigr) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s,\end{aligned} \end{aligned}$$
(3.6)
$$\begin{aligned}& \begin{aligned}[b]{\mathcal{B}}_{2}^{*}(F):={} & \frac{\int_{\omega_{0}}^{\eta}(s-\omega) \, d_{q,\omega}s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \int_{\omega _{0}}^{T} \int_{\omega_{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta -1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline {\alpha-1}} \\ &\times F \bigl( z,u(z),\Psi^{\gamma}_{q,\omega}u(z) \bigr) \,d_{q,\omega}z\, d_{q,\omega}x\,d_{q,\omega}s, \end{aligned} \end{aligned}$$
(3.7)
We find that problem (1.3) has solutions if and only if the operator \(\mathcal{F}\) has fixed points.
Theorem 3.1
Assume that the following conditions hold:
Then problem (1.3) has a unique solution.
- \((H_{1})\) :
-
There exist constants \(\lambda_{1},\lambda_{2}>0\) such that, for each \(t\in I^{T}_{q,\omega}\) and \(u,v\in{\mathbb{R}}\),$$\begin{aligned} \big|F \bigl( t,u,\Psi^{\gamma}_{r,\omega}u \bigr)-F \bigl( t,v,\Psi ^{\gamma}_{q,\omega}v \bigr) \big| \leq\lambda_{1} |u-v |+ \lambda_{2} \big|\Psi^{\gamma}_{q,\omega }u-\Psi^{\gamma}_{q,\omega}v \big|. \end{aligned}$$
- \((H_{2})\) :
-
There exist constants \(\ell_{1},\ell_{2}>0\) such that, for each \(u, v\in{\mathcal{C}}\),$$\big|\phi(u)-\phi(v)\big| \leq\ell_{1}\|u-v\|_{\mathcal{C}} \quad\textit{and}\quad\big| \psi(u)-\psi(v)\big| \leq\ell_{2}\|u-v\|_{\mathcal{C}}. $$
- \((H_{3})\) :
-
\(\mathcal{L} [ \Theta_{1}+\Theta_{2} ] < q^{2}\omega_{0}\), where$$\begin{aligned}& {\mathcal{L}} := \max \biggl\{ \ell, \lambda_{1}+\lambda_{2} \varphi _{0}\frac{(T-\omega_{0})_{q,\omega}^{\underline{\gamma}}}{\Gamma _{r}(\gamma+1)} \biggr\} ,\quad \ell=|\omega_{0} \ell_{1}-\ell_{2}| \end{aligned}$$(3.8)$$\begin{aligned}& \Theta_{1} := \omega_{0}+\frac{(T-\omega_{0})^{2}}{|\Omega|} \bigl(\Lambda_{11}+(T-\omega_{0})\Lambda_{12} \bigr) , \end{aligned}$$(3.9)and$$\begin{aligned}& \Theta_{2} := \frac{(T-\omega_{0})^{2}}{|\Omega|} \bigl(\Lambda _{21}+(T-\omega_{0}) [ \Lambda_{22}+{ \mathcal{O}} ] \bigr) +(T-\omega_{0}){\mathcal{O}}, \end{aligned}$$(3.10)$$\begin{aligned}& \Lambda_{11} := (T-\eta)\bigg| \frac{T^{2}+T\eta+\eta^{2}-3\omega _{0}(T+\eta-\omega_{0}) }{q^{2}+q+1}+\omega_{0} q \frac{(T+\eta-2\omega_{0})}{1+q} \bigg|, \end{aligned}$$(3.11)$$\begin{aligned}& \Lambda_{12} := \frac{(\eta-\omega_{0})^{3}(T-\omega_{0})^{3}}{\Gamma _{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ) \big|(T-\omega_{0})^{\alpha+\beta-1}-(\eta- \omega_{0})^{\alpha+\beta -1} \big|, \end{aligned}$$(3.12)$$\begin{aligned}& \Lambda_{21} := \frac{T-\eta}{1+q} \big| T+\eta- \omega_{0} \bigl(2-q-q^{2}\bigr) \big|, \end{aligned}$$(3.13)$$\begin{aligned}& \Lambda_{22} := \frac{(\eta-\omega_{0})^{2}(T-\omega_{0})^{2}}{\Gamma _{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ) \big|(T-\omega_{0})^{\alpha+\beta}-(\eta- \omega_{0})^{\alpha+\beta } \big|, \end{aligned}$$(3.14)$$\begin{aligned}& {\mathcal{O}} := \frac{(T -\omega_{0})^{\alpha+\beta+2}}{\Gamma _{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ). \end{aligned}$$(3.15)
Proof
In order to prove that our problem has a unique solution, we show that F is a contraction. Since
for each \(t\in I^{T}_{q,\omega}\) and \(u,v\in{\mathcal{C}}\), we obtain
Similarly, we have
In addition, we have
and
Based on (3.16)-(3.21), we have
where \(\Lambda_{11}\), \(\Lambda_{12}\), \(\Lambda_{21}\), \(\Lambda_{22}\), and \({\mathcal{O}}\) are defined on (3.11)-(3.15), respectively.
$${\mathcal{H}}|u-v|(t):= \big|F \bigl( t,u(t),\Psi^{\gamma}_{q,\omega }u(t) \bigr)-F \bigl( t,v(t),\Psi^{\gamma}_{q,\omega}v(t) \bigr) \big|, $$
$$ \begin{aligned}[b] &\big|{\mathcal{A}}(u)-{\mathcal{A}}(v)\big| \\ &\quad= \bigl(\omega_{0} \big|\phi(u)-\phi(v)\big|-\big|\psi(u)-\psi(v)\big| \bigr) \bigg| \int_{\eta}^{T} (s-\omega) (s-\omega_{0})\, d_{q,\omega}s \bigg| \\ &\quad\leq\ell\|u-v\|_{\mathcal{C}} \bigg| \frac{(T-\omega_{0})^{3}-(\eta -\omega_{0})^{3}}{q^{2}+q+1}+ \omega_{0} q\frac{(T-\omega_{0})^{2}-(\eta-\omega _{0})^{2}}{1+p} \bigg| \\ &\quad= \ell\|u-v\|_{\mathcal{C}} (T-\eta)\bigg| \frac{T^{2}+T\eta+\eta ^{2}-3\omega_{0}(T+\eta-\omega_{0}) }{q^{2}+q+1}+ \omega_{0} q\frac{(T+\eta -2\omega_{0})}{1+q} \bigg|.\end{aligned} $$
(3.16)
$$ \big|{\mathcal{B}} (u)-{\mathcal{B}}(v)\big| \leq\ell\|u-v\|_{\mathcal{C}} \frac{T-\eta}{1+q} \big| T+\eta -\omega_{0} \bigl(2-q-q^{2}\bigr) \big|. $$
(3.17)
$$\begin{aligned} &\big|{\mathcal{A}}_{1}^{*}\bigl(F(u)\bigr)-{\mathcal{A}}_{1}^{*} \bigl(F(v)\bigr)\big| \\ &\quad= \frac{\int_{\omega_{0}}^{T}(s-\omega)(s-\omega_{0})\,d_{q,\omega }s }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)} \int_{\omega_{0}}^{\eta} \int _{\omega_{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s-\sigma _{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} \\ &\qquad\times{\mathcal{H}}|u-v|(s)\,d_{q,\omega}z\,d_{q,\omega}x \,d_{q,\omega }s \\ &\quad\leq \frac{\frac{( T-\omega_{0})^{2} }{1+q+q^{2}} \vert T-\omega _{0} ( \frac{1-q^{2}-q^{3}}{1+q} ) \vert }{\Gamma_{q}(\alpha )\Gamma_{q}(\beta)} \bigl(\lambda_{1}|u-v|+ \lambda_{2} \bigl\vert \Psi _{q,\omega}^{\gamma}u- \Psi_{q,\omega}^{\gamma}v \bigr\vert \bigr) \\ &\qquad\times \int_{\omega_{0}}^{\eta} \int_{\omega_{0}}^{s} \int_{\omega _{0}}^{x}(s-\omega) \bigl(s- \sigma_{q,\omega}(x) \bigr)_{q,\omega }^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} \,d_{q,\omega}z \,d_{q,\omega}x\,d_{q,\omega}s \\ &\quad\leq \bigl(\lambda_{1}|u-v|+\lambda_{2} \bigl\vert \Psi_{q,\omega}^{\gamma}u-\Psi_{q,\omega}^{\gamma}v \bigr\vert \bigr) \biggl\{ \frac{(\eta-\omega_{0})(T-\omega_{0})^{3}}{\Gamma_{q}(\alpha )\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ) \int_{\omega _{0}}^{\eta}s^{\alpha+\beta} \,d_{q,\omega}s \biggr\} \\ &\quad\leq \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma_{q}(\gamma +1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u-v\| _{\mathcal{C}} \biggl\{ \frac{(\eta-\omega_{0})^{\alpha+\beta+2}(T-\omega _{0})^{3}}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ) \biggr\} , \end{aligned}$$
(3.18)
$$\begin{aligned}& \begin{aligned}[b] &\big|{\mathcal{A}}_{2}^{*}\bigl(F(u)\bigr)-{\mathcal{A}}_{2}^{*} \bigl(F(v)\bigr)\big| \\ &\quad\leq \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma_{q}(\gamma +1)}(T-\omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u-v\| _{\mathcal{C}} \biggl\{ \frac{(\eta-\omega_{0})^{3}(T-\omega_{0})^{\alpha+\beta+2} }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ) \biggr\} ,\end{aligned} \end{aligned}$$
(3.19)
$$\begin{aligned}& \begin{aligned}[b] &\big|{\mathcal{B}}_{1}^{*}\bigl(F(u)\bigr)-{ \mathcal{B}}_{1}^{*}\bigl(F(v)\bigr)\big| \\ & \quad\leq\biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma_{q}(\gamma +1)}(T-\omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u-v\| _{\mathcal{C}} \biggl\{ \frac{(\eta-\omega_{0})^{\alpha+\beta+2}(T-\omega_{0})^{2} }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ) \biggr\} ,\end{aligned} \end{aligned}$$
(3.20)
$$\begin{aligned}& \begin{aligned}[b]&\big|{\mathcal{B}}_{2}^{*}\bigl(F(u)\bigr)-{ \mathcal{B}}_{2}^{*}\bigl(F(v)\bigr)\big| \\ &\quad\leq \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma_{q}(\gamma +1)}(T-\omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u-v\| _{\mathcal{C}} \biggl\{ \frac{(\eta-\omega_{0})^{2}(T-\omega_{0})^{\alpha+\beta+2} }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \beta+1,\alpha ) \biggr\} .\end{aligned} \end{aligned}$$
(3.21)
$$\begin{aligned}& \begin{aligned}[b] &\big|{\mathcal{A^{**}}}(F,u)-{\mathcal{A^{**}}}(F,v)\big| \\&\quad\leq\ell \|u-v\| _{\mathcal{C}}\Lambda_{11}+ \biggl(\lambda_{1}+ \frac{\lambda_{2}\varphi _{0}}{\Gamma_{q}(\gamma+1)}(T-\omega_{0})_{q,\omega}^{\underline{\gamma +1}} \biggr) \|u-v\|_{\mathcal{C}} \Lambda_{12},\end{aligned} \end{aligned}$$
(3.22)
$$\begin{aligned}& \begin{aligned}[b]&\big|{\mathcal{B^{**}}}(F,u)-{\mathcal{B^{**}}}(F,v)\big|\\ &\quad\leq\ell\|u-v\| _{\mathcal{C}}\Lambda_{21}+ \biggl( \lambda_{1}+\frac{\lambda_{2}\varphi _{0}}{\Gamma_{q}(\gamma+1)}(T-\omega_{0})_{q,\omega}^{\underline{\gamma +1}} \biggr)\|u-v\|_{\mathcal{C}} [ \Lambda_{22}+{\mathcal {O}} ],\end{aligned} \end{aligned}$$
(3.23)
Consider
where \({\mathcal{L}}\), \(\Theta_{1}\), and \(\Theta_{2}\) are defined on (3.8)-(3.10), respectively.
$$\begin{aligned}[b] &\big| ({\mathcal{F}}u)(t)-({\mathcal{F}}v) (t) \big|\\ &\quad\leq\frac{1}{q^{2}\rho_{q,\omega}(\omega_{0})} \biggl\{ \omega_{0}\ell\| u-v \|_{\mathcal{C}}\\ &\qquad+\frac{1}{|\Omega|} \bigl[(T-\omega _{0})^{2}\big|{ \mathcal{A^{**}}}(F,u)-{\mathcal{A^{**}}}(F,v)\big| \\ &\qquad+(T-\omega_{0})^{3}\big|{\mathcal{B^{**}}}(F,u)-{ \mathcal{B^{**}}}(F,v)\big| \bigr] \\ &\qquad+ \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma_{q}(\gamma +1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u-v\| _{\mathcal{C}} (T-\omega_{0}){\mathcal{O}} \biggr\} \\ &\quad\leq\ell\|u-v\|_{\mathcal{C}}\frac{\Theta_{1}}{q^{2}\omega_{0}} + \biggl(\lambda_{1}+ \frac{\lambda_{2}\varphi_{0}}{\Gamma_{q}(\gamma+1)}(T-\omega _{0})_{q,\omega}^{\underline{\gamma+1}} \biggr) \|u-v\|_{\mathcal {C}}\frac{\Theta_{2}}{q\omega_{0}} \\ &\quad\leq\frac{{\mathcal{L}}}{q^{2}\omega_{0}}\|u-v\|_{\mathcal{C}} [\Theta_{1}+ \Theta_{2} ],\end{aligned} $$
(3.24)
From \((H_{3})\), this implies that \({\mathcal{F}}\) is a contraction. Therefore, by using the Banach fixed point theorem, we can conclude that \({\mathcal{F}}\) has a fixed point which is a unique solution of problem (1.3) on \(t\in I^{T}_{q,\omega}\). □
4 Existence of at least one solution
In this section, we also deduce the existence of a solution to (1.3) by using the following Schauder fixed point theorem.
Lemma 4.1
([29] (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 4.3
([30] (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 \(u^{*}\in U\): \(Tu^{*}=u^{*}\).
Theorem 4.1
Suppose that \((H_{1})\)-\((H_{2})\) hold. Then (1.3) has at least one solution.
Proof
We divide the proof into three steps as follows.
Step I. Verify that \({\mathcal{F}}\) maps bounded sets to bounded sets in \(B_{R} = \{u \in \mathcal{C}: \|u\|_{\mathcal{C}} \leq R\}\). Let \(B_{R}=\{u\in C(I^{T}_{q,\omega}) :\|u\|_{\mathcal{C}}\leq R\}\), \(\max_{t\in I^{T}_{q,\omega}}|F(t,0,0)|=K\), \(\sup_{u\in{\mathcal {C}}}|\phi(u)|=M\), \(\sup_{u\in{\mathcal{C}}}|\psi(u)|=N\), and choose a constant
We note that \(|\mathcal{S}(t,u,0) |= |F (t,u(t),\Psi_{q,\omega}^{\gamma}u(t) )-F (t,0,0 ) |+ |F (t,0,0 ) |\).
$$ R\geq\frac{\Theta_{1} | \omega_{0}M-N |+\Theta_{2}K }{ q^{2}\omega _{0}- [ \Theta_{1}\ell+\Theta_{2} (\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma_{q}(\gamma +1)}(T-\omega_{0})_{q,\omega}^{\underline{\gamma+1}} ) ] }. $$
(4.1)
For each \(u\in B_{R}\), we obtain
Similarly,
In addition,
Similarly,
From (4.2)-(4.7), we obtain
where \(\Lambda_{11}\), \(\Lambda_{12}\), \(\Lambda_{21}\), \(\Lambda_{22}\), and \({\mathcal{O}}\) are defined in (3.11)-(3.15), respectively.
$$\begin{aligned}[b] \big|{\mathcal{A}}(u)\big| ={} & \big| \omega_{0} \bigl(\big|\phi(u)-\phi(0)\big|+\big|\phi(0)\big| \bigr) - \bigl( \big|\psi(u)-\psi(0)\big|+\big|\psi(0)\big| \bigr) \big| \\ &\times\bigg| \int_{\eta}^{T}(s-\omega) (s-\omega_{0}) \,d_{q,\omega}s \bigg| \\ \leq{}& \bigl(\ell\|u\|_{\mathcal{C}}+| \omega_{0}M-N | \bigr) (T- \eta ) \\ &\times \bigg| \frac{T^{2}+T\eta+\eta^{2}-3\omega_{0}(T+\eta-\omega_{0}) }{q^{2}+q+1}+\omega_{0} q\frac{(T+\eta-2\omega_{0})}{1+q}\bigg|.\end{aligned} $$
(4.2)
$$ \big|{\mathcal{B}}(u)\big| \leq \bigl(\ell\|u\|_{\mathcal{C}}+| \omega_{0}M-N | \bigr) \frac {T-\eta}{1+q} \big| T+\eta-\omega_{0} \bigl(2-q-q^{2}\bigr) \big|. $$
(4.3)
$$ \begin{aligned}[b] &\big|{\mathcal{A}}_{1}^{*}\bigl(F(u)\bigr)\big| \\ &\quad= \frac{\int_{\omega_{0}}^{T}(s-\omega)(s-\omega_{0})\,d_{q,\omega }s}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)}\\ &\qquad \times \int_{\omega_{0}}^{\eta} \int _{\omega_{0}}^{s} \int_{\omega_{0}}^{x}(s-\omega) \bigl(s-\sigma _{q,\omega}(x) \bigr)_{q,\omega}^{\underline{\beta-1}} \bigl(x- \sigma_{q,\omega}(z) \bigr)_{q,\omega}^{\underline{\alpha-1}} \big|{\mathcal{S}}(t,u,0) \big|\,d_{q,\omega}z\,d_{q,\omega}x\, d_{q,\omega}s \\ &\quad\leq \biggl[ \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma _{q}(\gamma+1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u\|_{\mathcal{C}} +K \biggr]\\ &\qquad\times \biggl\{ \frac{(\eta-\omega_{0})^{\alpha+\beta+2}(T-\omega _{0})^{3}}{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \alpha,\beta +1 ) \biggr\} .\end{aligned} $$
(4.4)
$$\begin{aligned}& \begin{aligned}[b] \big|{\mathcal{A}}_{2}^{*}\bigl(F(u)\bigr)\big| \leq{}& \biggl[ \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma _{q}(\gamma+1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u\|_{\mathcal{C}} +K \biggr]\\ &\times \biggl\{ \frac{(\eta-\omega_{0})^{3}(T-\omega_{0})^{\alpha+\beta+2} }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \alpha,\beta +1 ) \biggr\} ,\end{aligned} \end{aligned}$$
(4.5)
$$\begin{aligned}& \begin{aligned}[b]\big|{\mathcal{B}}_{1}^{*}\bigl(F(u)\bigr)\big| \leq{}& \biggl[ \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma _{q}(\gamma+1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u\|_{\mathcal{C}} +K \biggr]\\ &\times \biggl\{ \frac{(\eta-\omega_{0})^{\alpha+\beta+2}(T-\omega_{0})^{2} }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \alpha,\beta +1 ) \biggr\} ,\end{aligned} \end{aligned}$$
(4.6)
$$\begin{aligned}& \begin{aligned}[b]\big|{\mathcal{B}}_{2}^{*}\bigl(F(u)\bigr)\big| \leq{}& \biggl[ \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma _{q}(\gamma+1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u\|_{\mathcal{C}} +K \biggr]\\ &\times \biggl\{ \frac{(\eta-\omega_{0})^{2}(T-\omega_{0})^{\alpha+\beta+2} }{\Gamma_{q}(\alpha)\Gamma_{q}(\beta+1)}B_{q} ( \alpha,\beta +1 ) \biggr\} .\end{aligned} \end{aligned}$$
(4.7)
$$\begin{aligned}& \begin{aligned}[b] \big|{\mathcal{A^{**}}}(F,u)\big| \leq{}& \bigl(\ell\|u\|_{\mathcal{C}}+| \omega_{0}M-N | \bigr) \Lambda_{11} \\ &+ \biggl[ \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma _{q}(\gamma+1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u\|_{\mathcal{C}} +K \biggr] \Lambda_{12},\end{aligned} \end{aligned}$$
(4.8)
$$\begin{aligned}& \begin{aligned}[b]\big|{\mathcal{B^{**}}}(F,u)\big| \leq{}& \bigl(\ell\|u\|_{\mathcal{C}}+| \omega_{0}M-N | \bigr) \Lambda_{21} \\ &+ \biggl[ \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma _{q}(\gamma+1)}(T- \omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr)\|u\|_{\mathcal{C}} +K \biggr] [ \Lambda_{22}+{\mathcal {O}} ],\end{aligned} \end{aligned}$$
(4.9)
From (4.8) and (4.9), we have
where \(\Theta_{1}\) and \(\Theta_{2}\) are defined in (3.8)-(3.10), respectively.
$$ \begin{aligned}[b]\big| ({\mathcal{F}}u) (t) \big| \leq{}&\frac{R}{q^{2}\omega_{0}} \biggl[\Theta_{1}\ell +\Theta_{2} \biggl(\lambda_{1}+\frac{\lambda_{2}\varphi_{0}}{\Gamma _{q}(\gamma+1)}(T-\omega_{0})_{q,\omega}^{\underline{\gamma+1}} \biggr) \biggr] \\ &+\frac{1}{q^{2}\omega_{0}} \bigl[ | \omega_{0}M-N | \Theta_{1} +K\Theta_{2} \bigr],\end{aligned} $$
(4.10)
Therefore, \(\|{\mathcal{F}}u\|_{\mathcal{C}}\leq R\). This implies that \({\mathcal{F}}\) is uniformly bounded.
Step II. Show that \({\mathcal{F}}\) is continuous on \(B_{R}\). From the continuity of F, we can conclude that the operator \(\mathcal{F}\) is continuous on \(B_{R}\).
Step III. Examine that \({\mathcal{F}}\) is equicontinuous with \(B_{R}\). For any \(t_{1},t_{2}\in I^{T}_{q,\omega}\) with \(t_{1}< t_{2}\), we have
$$ \begin{aligned}[b]& \big| ({\mathcal{F}}u) (t_{2})- ({\mathcal{F}}u) (t_{1}) \big| \\ &\quad\leq\omega_{0} M \biggl\vert \frac{1}{t_{2}-\omega}- \frac{1}{t_{1}-\omega} \biggr\vert + \biggl\vert \frac{{\mathcal{A^{**}}}}{\Omega} \biggr\vert \\ &\qquad\times\bigg| \frac {1}{(t_{2}-\omega)} \int_{\omega_{0}}^{t_{2}}(s-\omega)\,d_{q,\omega}s - \frac{1}{(t_{1}-\omega)} \int_{\omega_{0}}^{t_{1}}(s-\omega)\, d_{q,\omega}s \bigg| \\ &\quad\quad+ \biggl\vert \frac{{\mathcal{B^{**}}}}{\Omega} \biggr\vert \bigg| \frac {1}{(t_{2}-\omega)} \int_{\omega_{0}}^{t_{2}}(s-\omega) (s-\omega_{0}) \, d_{q,\omega}s -\frac{1}{(t_{1}-\omega)} \int_{\omega_{0}}^{t_{1}}(s-\omega) (s-\omega _{0}) \,d_{q,\omega}s \bigg| \\ &\quad\quad+\|F\| \bigg|\frac{1}{t_{2}-\omega} \int_{\omega_{0}}^{t_{2}} \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)} \,d_{q,\omega}z \,d_{q,\omega }x\,d_{q,\omega}s \\ & \quad\quad-\frac{1}{t_{1}-\omega} \int_{\omega_{0}}^{t_{1}} \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)}\,d_{q,\omega}z \,d_{q,\omega}x\, d_{q,\omega}s \bigg| \\ &\quad< \frac{\omega_{0} M}{t_{1}-\omega} \vert t_{1}-t_{2} \vert + \biggl\vert \frac{{\mathcal{A^{**}}}}{\Omega(t_{1}-\omega)} \biggr\vert \int _{t_{1}}^{t_{2}}(s-\omega)\,d_{q,\omega}s\\ &\qquad+ \biggl\vert \frac{{\mathcal {B^{**}}}}{\Omega(t_{1}-\omega)} \biggr\vert \int_{t_{1}}^{t_{2}}(s-\omega ) (s-\omega_{0}) \,d_{q,\omega}s \\ &\quad\quad+\frac{\|F\|}{t_{1}-\omega} \int_{t_{1}}^{t_{2}} \int_{\omega_{0}}^{s} \int _{\omega_{0}}^{x}\frac{(s-\omega) (s-\sigma_{p,\omega}(x) )_{q,\omega}^{\underline{\beta-1}} (x-\sigma_{q,\omega}(z) )_{q,\omega}^{\underline{\alpha-1}} }{\Gamma_{q}(\alpha)\Gamma _{p}(\beta)}\,d_{q,\omega}z \,d_{p,\omega}x\,d_{q,\omega}s.\end{aligned} $$
When \(|t_{2}-t_{1}|\rightarrow0\), the right-hand side of the above inequality tends to be zero. Therefore, \({\mathcal{F}}\) is relatively compact on \(B_{R}\).
Thus, the set \({\mathcal{F}}(B_{R})\) is an equicontinuous set. As a consequence of Steps I to III together with the Arzelá-Ascoli theorem, \(\mathcal{F}:{\mathcal{C}}\rightarrow{\mathcal{C}}\) is completely continuous. By the Schauder fixed point theorem mentioned above, we can conclude that problem (1.3) has at least one solution. The proof is completed. □
5 Example
In this section, we present a boundary value problem for fractional Hahn difference equation to illustrate our results as follows:
where \(\varphi(t,s)=\frac{e^{-2|s-t|}}{2000\pi^{3}} \) and \(C_{i}\) are given constants with \(\frac{1}{(100\pi)^{4}}\leq\sum_{i=0}^{\infty }C_{i}\leq \frac{e}{(100\pi)^{4}}\).
$$ \begin{gathered}\begin{aligned}&{}^{C}D^{\frac{1}{2}}_{\frac{1}{2},2}{}^{C}D^{\frac{2}{5}}_{\frac{1}{2},2} \biggl[\frac{E_{\sigma_{\frac{1}{2},2}}}{\rho_{\frac {1}{2},2}(t)}+\frac{1}{2}D_{\frac{1}{2},2} \biggr] u(t) \\ &\quad= \frac{e^{-10t}}{\pi t^{2}}\cdot \frac{|u|+1}{1+\cos^{2}u} \\ &\qquad+\frac{\arctan ( \cos^{2}\pi t ) }{(t+100)^{4}}{\mathcal{I}}^{\frac{2}{3}}_{\frac{1}{2},2}( \varphi u) (t),\quad t\in \biggl[ \frac{67}{16},10 \biggr]_{\frac{1}{2},2},\end{aligned} \\ u \biggl(\frac{67}{16} \biggr)=\frac{|u|\sin^{2}|\pi u|}{(200e)^{6}}, \\ \rho_{\frac{1}{2},2}(10)u (10 )=\rho_{\frac {1}{2},2} \biggl( \frac{19}{4} \biggr)u \biggl( \frac{19}{4} \biggr) =\sum _{i=0}^{\infty}\frac{C_{i}|u(t_{i})|}{1+|u(t_{i})|},\quad t_{i}=10 \biggl(\frac{1}{30} \biggr)^{i}+2[i]_{\frac{1}{30}},\end{gathered} $$
(5.1)
We let \(\alpha=\frac{1}{2}\), \(\beta=\frac{2}{5}\), \(\gamma=\frac {2}{3}\), \(q=\frac{1}{2}\), \(\omega=2\), \(\omega_{0}=\frac{\omega}{1-\chi }=\frac{60}{29}\), \(T=10\), \(\eta=10 (\frac{1}{30} )^{3}+2[3]_{\frac{1}{30}}=\frac {5587}{2700}\), \(\phi(u)=\frac{|u|\sin^{2}|\pi u|}{(200e)^{6}}\), \(\psi (u)=\sum_{i=0}^{\infty}\frac{C_{i}|u(t_{i})|}{1+|u(t_{i})|} \), and \(F ( t,u(t),\Psi^{\gamma}_{r,\omega}u(t) ) = \frac {e^{-10t}}{\pi t^{2}}\cdot \frac{|u|+1}{1+\cos^{2}u}+\frac{\arctan ( \cos^{2}\pi t ) }{(t+100)^{4}}{\mathcal{I}}^{\frac{2}{3}}_{\frac{4}{5},2}( \varphi u)(t)\).
We find that
For all \(t\in [\frac{60}{29},10 ]_{\frac{1}{30},2} \) and \(u, v\in{\mathbb{R}}\), we have
Thus, \((H_{1})\) holds with \(\gamma_{1}=7.69\times10^{-11}\), \(\gamma _{2}=7.236\times10^{-9} \), and \(\varphi_{0}= [ \frac{1}{2000\pi ^{3}} ]\).
$$\begin{gathered} |\Omega|=0.644, \qquad\Lambda_{11}=15\mbox{,}130.184, \qquad\Lambda _{12}=198\mbox{,}828.485, \qquad\Lambda_{21}=14\mbox{,}966.095, \\ \Lambda_{22}=78\mbox{,}845.778, \quad\text{and} \quad{\mathcal{O}}=60.577. \end{gathered} $$
$$ \bigl\vert F \bigl( t,u,\Psi^{\gamma}_{r,\omega}u \bigr)-F \bigl( t,v,\Psi^{\gamma}_{r,\omega}v \bigr) \bigr\vert \leq \frac{1}{e^{\frac{600}{29}}\pi ( \frac{60}{29} )^{2} }|u-v|+\frac{\pi}{4 (\frac{60}{29}+100 )^{4} } \bigl\vert \Psi^{\gamma}_{r,\omega}u- \Psi^{\gamma}_{r,\omega}v \bigr\vert . $$
For all \(u, v\in{\mathcal{C}}\),
So, \((H_{2})\) holds with \(\ell_{1}= 3.873 \times10^{-17} \) and \(\ell _{2}=2.791\times10^{-10}\).
$$\begin{gathered} \big|\phi(u)-\phi(v)\big| = \frac{1}{(200e)^{6}}\|u-v\|_{\mathcal{C}}, \\ \big|\psi(u)-\psi(v)\big| = C_{i}\sum _{i=0}^{\infty}\big|u(t_{i})-v(t_{i})\big| \leq \frac{e}{(100\pi)^{4}}\|u-v\|_{\mathcal{C}}.\end{gathered} $$
Hence, from Theorem 4.1, problem (5.1) has at least one solution. □
In addition, we find that
Therefore, \((H_{3})\) holds with
Hence, problem (5.1) has a unique solution by Theorem 3.1. □
$$\begin{gathered} {\mathcal{L}} = \max \biggl\{ |\omega_{0}\ell_{1}- \ell_{2}|, \lambda _{1}+\lambda_{2} \varphi_{0}\frac{(T-\omega_{0})_{r,\omega}^{\underline {\gamma}}}{\Gamma_{r}(\gamma+1)} \biggr\} = 2.791\times10^{-10}, \\ \Theta_{1} = \omega_{0}+\frac{(T-\omega_{0})^{2}}{|\Omega|} \bigl(\Lambda _{11}+(T-\omega_{0})\Lambda_{12} \bigr) = 1.961\times10^{8}, \\ \Theta_{2} = \frac{(T-\omega_{0})^{2}}{|\Omega|} \bigl(\Lambda _{21}+(T-\omega_{0}) [ \Lambda_{22}+{ \mathcal{O}} ] \bigr) +T{\mathcal{O}} = 7.891.\end{gathered} $$
$$0.055= \mathcal{L} [ \Theta_{1}+\Theta_{2} ] < q^{2}\omega_{0} = 1.034. $$
Declarations
Acknowledgements
This research was funded by King Mongkut’s University of Technology North Bangkok. Contract No. KMUTNB-GEN-59-68.
Authors’ contributions
The authors declare that they carried out all the work in this manuscript and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
(1)
Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand
(2)
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, Thailand
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