On the extension of analytic solutions of a class of first-order q-difference equations

In this paper, we use the Banach fixed point theorem to examine the existence of meromorphic solutions to the following first-order $q$-difference equation \begin{align}\tag{†}\label{dagger} y(qz)=\frac{a_1(z)y(z)+a_2(z)y(z)^2+\dots+a_p(z)y(z)^p}{1+b_1(z)y(z)+\cdots +b_t(z)y(z)^t}, \end{align} where $q\in \mathbb{C},$ $a_1(z),\dots ,a_p(z);b_1(z),\dots ,b_t(z)$ are all meromorphic functions. We establish sufficient conditions ensuring the existence and uniqueness of meromorphic solutions that can be extended to the entire complex plane $\mathbb{C}\mathrm{.}$ More precisely, we have the following result. If $\mid q\mid \ge 3$ and $$\mathrm{\mid }{a}_1(z)\mathrm{\mid }=\underset{1\le j\le p}{\max }\mathrm{\mid }{a}_j(z)\mathrm{\mid }\le \frac{1}{\mathrm{\mid }z\mathrm{\mid }},\underset{1\le k\le t}{\max }\mathrm{\mid }{b}_k(z)\mathrm{\mid }\le \frac{1}{\mathrm{\mid }z\mathrm{\mid }},z\in \{\mathrm{\mid }\mathrm{\Re }(z)\mathrm{\mid }\ge \rho >0\},$$ and $y(0)\mathrm{\ne }\mathrm{\infty },$ then we prove that~\eqref{dagger} admits a unique meromorphic solution in $D(\rho ),$ which can be extended meromorphically to $\mathbb{C}\mathrm{.}$ Moreover, if $a_1(z)\equiv 0,$ the conclusion still holds. Furthermore, if $\mid q\mid \ge 6$ and \begin{gather*} |a_1(z)| \le \frac{1}{|q|}, \quad |a_j(z)| \le |q|^{|z|} \quad (2 \le j \le p), \quad |b_k(z)| \le |q|^{|z|} \quad (1 \le k \le t), \[4pt] z \in D(ρ,σ) = {, z : |\Re(z)| \le ρ,; |\Im(z)| \le σ, ,, ρ>0,,, σ>0 ,}, \end{gather*} and $y(0)\mathrm{\ne }\mathrm{\infty },$ then we prove that \eqref{dagger} admits a unique meromorphic solution in $D(\rho ,\sigma ),$ which can also be extended meromorphically to $\mathbb{C}\mathrm{.}$ This conclusion remains valid in the case where $a_1(z)\equiv 0.$