Advances on Fractional Dynamic Inequalhb: Advances on Fractional Dynamic Inequalities on Time Scales

This book is devoted on recent developments of linear and nonlinear fractional Riemann-Liouville and Caputo integral inequalities on time scales. The book is intended for the use in the field of fractional dynamic calculus on time scales and fractional dynamic equations on time scales. It is also suitable for graduate courses in the above fields, and contains ten chapters. The aim of this book is to present a clear and well-organized treatment of the concept behind the development of mathematics as well as solution techniques. The text material of this book is presented in a readable and mathematically solid format. Contents Preface About the Authors 1. Elements of Time Scale Calculus 1.1 Introduction to Time Scales 1.2 Differentiation on Time Scales 1.3 Integration on Time Scales 1.4 The Taylor Formula 1.5 Definition of the Laplace Transform: Properties 1.6 Shifts and Convolutions 1.7 Investigation of the Shifting Problem 1.8 Convolutions 1.9 The Convolution Theorem 1.10 Advanced Practical Problems 2. Elements of Fractional Dynamic Calculus on Time Scales 2.1 The Δ-Power Function 2.2 Definition for the Riemann–Liouville Fractional Δ-Integral and the Riemann–Lioville Fractional Δ-Derivative 2.3 Properties of the Riemann–Liouville Fractional Δ-Integral and the Riemann–Liouville Fractional Δ-Derivative on Time Scales 2.4 Definition for the Caputo Fractional Δ-Derivative: Examples 2.5 Properties of the Caputo Fractional Δ-Derivative 2.6 Advanced Practical Problems 3. Linear Inequalities for Riemann–Liouville Fractional Delta Integral Operator 3.1 The Gronwall-Type Fractional Inequalities 3.2 Volterra-Type Fractional Inequalities 3.3 Simultaneous Fractional Inequalities 3.4 Pachpatte Fractional Inequalities 3.5 Existence and Uniqueness of the Solutions 3.6 The Dependency of the Solution upon the Initial Data 4. Fractional Young and Hölder Inequalities 4.1 Fractional Young Inequalities 4.2 Reverse Fractional Young Inequalities 4.3 Fractional Hölder Inequalities 4.4 Reverse Fractional Hölder Inequalities 5. Fractional Inequalities for Convex Functions 5.1 Inequalities for Exponentially Convex Functions 5.2 Examples 5.2.1 The time scale T = hZ, h > 0 5.2.2 The time scale T = rN0, r > 1 5.3 Ostrowski-Type Inequalities 5.4 Inequalities for Strongly r-Convex Functions 5.5 Inequalities for Exponentially s-Convex Functions 6. Opial-Type Inequalities 6.1 Opial-Type Inequalities for Riemann–Liouville Fractional Derivatives-I 6.2 Opial-Type Inequalities for Riemann–Liouville Fractional Derivatives-II 6.3 Poincaré-Type Inequalities 6.4 Ostrowski-Type Inequalities 7. Chebyshev-Type Inequalities 7.1 Chebyshev-Type Inequalities for Two Functions 7.2 Chebyshev-Type Inequalities for n Functions 8. Hardy-Type Fractional Inequalities 8.1 Copson-Type Fractional Inequalities 8.2 The Leindler Fractional Inequality 8.3 The Bennett Fractional Inequality 9. Reverse Hardy-Type Fractional Inequalities 9.1 Reverse Copson-Type Fractional Inequalities 9.2 The Reverse Leindler Fractional Inequality 9.3 The Reverse Bennett Fractional Inequality 10. Inequalities for Generalized Riemann–Liouville Fractional Integrals 10.1 Minkowski-Type Inequalit 10.2 Grüss-Type Inequalities 10.3 Some Other Inequalities Appendix A: Young and Hölder Inequalities A.1 Young Inequalities A.2 The Specht Ratio A.3 Refined Young Inequalities A.4 Reverse Young Inequalities A.5 The Hölder Inequality A.6 Reverse Hölder Inequalities Appendix B: Jensen Inequalities References Index