Quantum Calculus

New Concepts, Impulsive IVPs and BVPs, Inequalities

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 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.

Contents:

• Preliminaries
• Quantum Calculus on Finite Intervals
• Initial Value Problems for Impulsive qk-Difference Equations and Inclusions
• Boundary Value Problems for First-Order Impulsive qk-Integro-Difference Equations and Inclusions
• Impulsive qk-Difference Equations with Different Kinds of Boundary Conditions
• Nonlinear Second-Order Impulsive qk-Difference Langevin Equation with Boundary Conditions
• Quantum Integral Inequalities on Finite Intervals
• Impulsive Quantum Difference Systems with Boundary Conditions
• New Concepts of Fractional Quantum Calculus and Applications to Impulsive Fractional qk-Difference Equations
• Integral Inequalities via Fractional Quantum Calculus
• Nonlocal Boundary Value Problems for Impulsive Fractional qk-Difference Equations
• Existence Results for Impulsive Fractional qk-Difference Equations with Anti-periodic Boundary Conditions
• Impulsive Fractional qk-Integro-Difference Equations with Boundary Conditions
• Impulsive Hybrid Fractional Quantum Difference Equations

Readership: Mathematics and physics researchers.Key Features:

• This is the first book dealing with quantum calculus on finite intervals
• The material is new and will attract many researchers working on this topic
• This book contains enough material as a guideline for further work on the topic. In particular, it has a special attraction for the researchers who focus on impulsive and fractional differential equations to exploit the ideas presented in this book to enhance their work in the context of quantum calculus