Quantum Calculus

The main objective of this book is to extend the scope of the q-calculus based on the definition of q-derivative [Jackson (1910)] to make it applicable to dense domains. As a matter of fact, Jackson's definition of q-derivative fails to work for impulse points while this situation does not arise for impulsive equations on q-time scales as the domains consist of isolated points covering the case of consecutive points. In precise terms, we study quantum calculus on finite intervals.

In the first part, we discuss the concepts of q_k-derivative and q_k-integral, and establish their basic properties. As applications, we study initial and boundary value problems of impulsive q_k-difference equations and inclusions equipped with different kinds of boundary conditions. We also transform some classical integral inequalities and develop some new integral inequalities for convex functions in the context of q_k-calculus. In the second part, we develop fractional quantum calculus in relation to a new q_k-shifting operator and establish some existence and q_k uniqueness results for initial and boundary value problems of impulsive fractional q_k-difference equations.