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

Abstract

Given a set $T\subset (0,+\infty )$, intervals $I\subset (0,+\infty )$ and $J\subset \mathbb{R}$, as well as functions $g_t:I\times J\to J$ 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 \[ φ(tx)=g_t\left(x,φ(x)\right), \qquad t \in T^{\ast}, \] 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 $[1,+\infty ]$, 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 \[ φ(tx)=φ(x)+c(t)x^p, \qquad t\in T, \] where $c:T\to \mathbb{R}$ is an arbitrary function, $p$ is a given real number and $\sup I>R_T\inf I$.