Quantum Calculus on Finite Intervals and Applications to Impulsive Difference Equations

In Mathematics, quantum calculus is a version of calculus in which limits are not taken. This type of calculus plays important role both in theoretical and practical areas of mathematics. In quantum calculus, derivatives are differences and anti derivatives are sums. Quantum calculus is a theory where smoothness is no more needed. In this work, we study finite intervals in quantum calculus. We review and study the -derivative and -integral of a function and demonstrate their properties. We apply this concept to provide existence and uniqueness results for the initial value problems, namely for first and second order impulsive -difference equations. Keywords: -derivative, - integral, impulsive -difference equation, existence, uniqueness.