
doi: 10.1090/proc/14116
The authors present \(q\)-versions of Massera's theorem for different types of \(q\)-difference equations. Let \(f^{\Delta}(t)\) denote the usual \(q\)-derivative (also known as Jackson derivative). For the linear \(q\)-difference equation \(f^{\Delta}(t)=a(t)f(t)+b(t)/t\) they derive a periodicity result using Brouwer's fixed point theorem. Precisely, such an \(\omega\)-periodic linear \(q\)-difference equation has an \(\omega\)-periodic solution iff it has a \(q\)-bounded solution. The second main result is a version of Massera's theorem for nonlinear \(q\)-difference equations of the form \(f^{\Delta}(t)=g(t,tf(t))\): under certain assumptions they show that if such an equation has a \(q\)-bounded solution, then it has an \(\omega\)-periodic solution.
\(q\)-calculus and related topics, Difference equations, scaling (\(q\)-differences), Stability theory for difference equations, Massera's theorem, $q$-difference equation, Periodic solutions of difference equations
\(q\)-calculus and related topics, Difference equations, scaling (\(q\)-differences), Stability theory for difference equations, Massera's theorem, $q$-difference equation, Periodic solutions of difference equations
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