Description
Book SynopsisThe 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 qk-derivative and qk-integral, and establish their basic properties. As applications, we study initial and boundary value problems of impulsive qk-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 qk-calculus. In the second part, we develop fractional quantum calculus in relation to a new qk-shifting operator and establish some existence and qk uniqueness results for initial and boundary value problems of impulsive fractional qk-difference equations.
Table of ContentsPreliminaries; Quantum Calculus on Finite Intervals; Initial Value Problems for Impulsive qk-Difference Equations and Inclusions; First-Order Impulsive qk-Difference Equations and Inclusions; Impulsive qk-Difference Equations with Boundary Conditions; Nonlinear Impulsive Langevin Equation; Quantum Integral Inequalities; Impulsive Quantum Difference Systems; New Concepts of Fractional Quantum Calculus; Integral Inequalities via Fractional Quantum Calculus; Nonlocal Boundary Value Problems for Impulsive Fractional q-Difference Equations; Impulsive Fractional q-Difference Equations with Boundary Conditions; Impulsive Fractional q-Integro-Difference Equations with Boundary Conditions; Impulsive Hybrid Fractional Quantum Difference Equations;
