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

Given a set $T\subset (0,+\mathrm{\infty })$, intervals $I\subset (0,+\mathrm{\infty })$ and $J\subset \mathbb{R}$, as well as functions $g_t:I\times J\to J$ with $t$’s running through the set $$T^{\ast }:=T\cup \{t^{-1}\text{\hspace{0.17em}⁣}:t\in T\}\cup \{1\}$$ we study the simultaneous $q$-difference equations $$\phi (tx)=g_t(x,\phi (x)),t\in T^{\ast },$$ postulated for $x\in I\cap t^{-1}I$; here the unknown function $\phi$ is assumed to map $I$ into $J$. We prove an Extension theorem stating that if $\phi$ is continuous [analytic] on a nontrivial subinterval of $I$, then $\phi$ 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 $[1,+\mathrm{\infty }]$, 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 $\phi :I\to \mathbb{R}$ of the simultaneous equations $$\phi (tx)=\phi (x)+c(t)x^p,t\in T,$$ where $c:T\to \mathbb{R}$ is an arbitrary function, $p$ is a given real number and $\sup I>R_T\inf I$.