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

Given a set T ⊂ ( 0 , + ∞ ) , intervals I ⊂ ( 0 , + ∞ ) and J ⊂ R , as well as functions g t : I × J → J with t ’s running through the set T ∗ : = T ∪ { t - 1 : t ∈ T } ∪ { 1 } we study the simultaneous q -difference equations φ ( t x ) = g t x , φ ( x ) , t ∈ T ∗ , postulated for x ∈ I ∩ t - 1 I ; here the unknown function φ is assumed to map I into J . We prove an Extension theorem stating that if a solution φ is continuous [analytic] on a nontrivial subinterval of I , then it is continuous [analytic] provided g t , t ∈ T ∗ , 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 , + ∞ ] , characterising some density property of the set T ∗ . As an application of the Extension theorem we find the form of all continuous on a subinterval of I solutions φ : I → R of the simultaneous equations φ ( t x ) = φ ( x ) + c ( t ) x p , t ∈ T , where c : T → R is an arbitrary function, p is a given real number and sup I > R T inf I .