A On the Existence of Solutions for Hybrid Caputo-Type $q$-Fractional Differential Equations in Banach Algebra

On the Existence of Solutions for Hybrid Caputo-Type $q$-Fractional Differential Equations in Banach Algebra

Abstract

In this paper, we investigate the existence of solutions for $q$-fractional hybrid differential equations in Caputo-type ($q$-FHDEs
\begin{equation*}\label{1.1}
\begin{cases}
^{c}\mathfrak{D}^{\alpha}_{q}[\frac{\phi(\eta)-g(\eta,\phi(\eta))}{h(\eta,\phi(\eta))}]=k(\eta,\varphi(\eta)), ~\eta\in\mathcal{J},\\
^{c}\mathfrak{D}^{\alpha}_{q}[\frac{\varphi(\eta)-g(\eta,\varphi(\eta))}{h(\eta,\varphi(\eta))}]=k(\eta, \phi(\eta)),~\eta\in\mathcal{J},~q\in(0,1),\\
\phi(0)=0,~~\varphi(0)=0,
\end{cases}
\end{equation*}
where $^{c}\mathfrak{D}^{\alpha}_{q}$ represents the fractional $q$-derivative in Caputo type with order $0<\alpha<1$, let $g:\mathcal{J}\times\mathbb{R}\rightarrow \mathbb{R}-\{0\}$,~$h:\mathcal{J}\times\mathbb{R}\rightarrow\mathbb{R},$~$g(0,0)=0$ and
$k:\mathcal{J}\times\mathbb{R}\rightarrow\mathbb{R}$ be the functions that satisfy certain prescribed conditions. The main results are established by employing a coupled fixed point theorem. An illustrative example is presented to demonstrate the applicability of the theoretical findings.