Extension Theorem for Simultaneous q-Difference Equations and Some Its Consequences

Given a set \(T \subset (0, +\infty )\) $T\subset (0,+\infty )$ , intervals \(I\subset (0, +\infty )\) $I\subset (0,+\infty )$ and \(J\subset \mathbb {R}\) $J\subset \mathbb{R}$ , as well as functions \(g_t:I\times J\rightarrow J\) $g_t:I\times J\to J$ with t’s running through the set \(\begin{aligned} T^{*}:=T \cup \big \{t^{-1}:t \in T\big \}\cup \{1\} \end{aligned}\) $$\begin{array}{c}\hfill T_{}^{\ast }:=T\cup \{t^{-1}:t\in T\}\cup \{1\}\end{array}$$ we study the simultaneous q-difference equations \( \varphi (tx)=g_t\left( x,\varphi (x)\right) , \qquad t \in T^{*}, \) $$\phi (tx)=g_t\left(x,,,\phi ,(,x,)\right),t\in T_{}^{\ast },$$ postulated for \(x \in I\cap t^{-1}I\) $x\in I\cap t^{-1}I$ ; here the unknown function \(\varphi \) $\phi$ is assumed to map I into J. We prove an Extension theorem stating that if a solution \(\varphi \) $\phi$ is continuous [analytic] on a nontrivial subinterval of I, then it is continuous [analytic] provided \(g_t, t \in T^{*}\) $g_t,t\in T_{}^{\ast }$ , are continuous [analytic]. The crucial assumption of the Extension theorem is formulated with the help of the so-called limit ratio \(R_T\) $R_T$ which is a uniquely determined number from \([1,+\infty ]\) $[1,+\infty ]$ , characterising some density property of the set \(T^{*}\) $T_{}^{\ast }$ . As an application of the Extension theorem we find the form of all continuous on a subinterval of I solutions \(\varphi :I \rightarrow \mathbb {R}\) $\phi :I\to \mathbb{R}$ of the simultaneous equations \( \varphi (tx)=\varphi (x)+c(t)x^p, \qquad t\in T, \) $$\phi (tx)=\phi (x)+c(t)x^p,t\in T,$$ where \(c:T\rightarrow \mathbb {R}\) $c:T\to \mathbb{R}$ is an arbitrary function, p is a given real number and \(\sup I > R_T \inf I\) $supI>R_{T}infI$ .