Extension theorem for simultaneous q-difference equations and some its consequences

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