Lyapunov Inequalities and Applications

✍ Scribed by Ravi P. Agarwal, Martin Bohner, Abdullah Özbekler

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 616
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This book provides an extensive survey on Lyapunov-type inequalities. It summarizes and puts order into a vast literature available on the subject, and sketches recent developments in this topic. In an elegant and didactic way, this work presents the concepts underlying Lyapunov-type inequalities, covering how they developed and what kind of problems they address.

This survey starts by introducing basic applications of Lyapunov’s inequalities. It then advances towards even-order, odd-order, and higher-order boundary value problems; Lyapunov and Hartman-type inequalities; systems of linear, nonlinear, and quasi-linear differential equations; recent developments in Lyapunov-type inequalities; partial differential equations; linear difference equations; and Lyapunov-type inequalities for linear, half-linear, and nonlinear dynamic equations on time scales, as well as linear Hamiltonian dynamic systems.

Senior undergraduate students and graduate students of mathematics, engineering, and science will benefit most from this book, as well as researchers in the areas of ordinary differential equations, partial differential equations, difference equations, and dynamic equations. Some background in calculus, ordinary and partial differential equations, and difference equations is recommended for full enjoyment of the content.

✦ Table of Contents

Preface
Contents
1 Lyapunov-Type Inequalities for Second-Order Linear Differential Equations
1.1 Introduction
1.2 Preliminaries
1.3 Basic Results
1.4 Number of Zeros
1.5 Distance between Zeros
1.6 Oscillation of Solutions
1.7 Disconjugacy and Disfocality
1.8 Eigenvalues of Sturm–Liouville Problems
1.9 An Inequality of Nehari
1.10 Notes and References
2 Lyapunov-Type Inequalities for Higher-Order Linear Differential Equations
2.1 Introduction
2.2 Even-Order Differential Equations
2.2.1 Dirichlet Boundary Value Problems
2.2.2 Lidstone Boundary Value Problems
2.2.3 Clamped-Free Boundary Value Problems
2.3 Odd-Order Differential Equations
2.3.1 Third-Order Differential Equations
2.3.2 General Odd-Order Differential Equations
2.4 General Higher-Order Differential Equations
2.4.1 Multiple-Point Boundary Value Problems
2.4.2 Conjugate Boundary Value Problems
2.5 Notes and References
3 Lyapunov-Type Inequalities for Half-Linear Differential Equations
3.1 Introduction
3.2 Second-Order Half-Linear Equations
3.2.1 Lower Bounds for Eigenvalues
3.3 Third-Order Half-Linear Equations
3.3.1 Lyapunov-Type Inequalities
3.3.2 Generalization
3.3.3 The Linear Case
3.3.4 Applications to Boundary Value Problems
3.4 Higher-Order Half-Linear Equations
3.5 Notes and References
4 Lyapunov-Type Inequalities for Nonlinear Differential Systems
4.1 Introduction
4.2 Nonlinear Systems
4.3 Quasilinear Systems
4.4 Dirichlet Quasilinear Systems Involving the (p1,…,pn)-Laplacian
4.5 Lower Bounds for Generalized Eigenvalues
4.6 Nonlinear Systems with Anti-periodic Boundary Conditions
4.7 Quasilinear Systems with Clamped-Free Boundary Conditions
4.8 Notes and References
5 Lyapunov-Type Inequalities for Fractional Differential Equations
5.1 Introduction
5.2 Linear FDEs with Dirichlet Boundary Conditions
5.3 Linear FDEs with Fractional Boundary Conditions
5.4 Linear FDEs with Robin Boundary Conditions
5.5 Notes and References
6 Lyapunov-Type Inequalities for Partial Differential Equations
6.1 Introduction
6.2 Linear PDEs with Neumann Boundary Conditions
6.2.1 Lyapunov-Type Inequalities
6.2.2 The Subcritical Case
6.2.3 The Supercritical Case
6.2.4 The Critical Case
6.2.5 Qualitative Properties of βp
6.2.6 Nonlinear Resonant Problems
6.3 Two-Dimensional Nonlinear Systems of PDEs
6.3.1 An Application
6.4 Multivariate Lyapunov Inequalities
6.5 Linear and Quasilinear Elliptic Differential Operators
6.5.1 Lyapunov-Type Inequality for p>N
6.5.2 Lyapunov-Type Inequality for p<N
6.5.3 Applications to Eigenvalue Problems
6.5.3.1 Optimality of the Bounds
6.5.3.2 Comparison with Other Estimates
6.5.4 Eigenvalues of the p-Laplacian
6.6 Notes and References
7 Lyapunov-Type Inequalities for Difference Equations
7.1 Introduction
7.2 Second-Order Linear Difference Equations
7.3 Lyapunov-Type Finite Difference Inequalities
7.4 Even-Order Difference Equations
7.5 Discrete Linear Hamiltonian Systems
7.5.1 A Disconjugacy Criterion
7.5.2 Stability Criteria
7.5.3 Concluding Remarks
7.6 Quasilinear Difference Systems
7.6.1 Some Applications
7.7 Discrete Nonlinear Systems
7.7.1 Some Applications
7.8 Partial Difference Systems
7.8.1 Nets and Discrete Harmonic Functions
7.8.2 Green's Functions and Lyapunov-Type Inequalities
7.8.3 Maxima of Green's Functions on Straight Nets
7.8.4 Maxima of Green's Functions on Circular Nets
7.8.5 Maxima of Green's Functions on Rectangular Nets
7.8.6 Final Remarks
7.9 Two-Dimensional Nonlinear Systems of Partial Difference Equations
7.10 Notes and References
8 Lyapunov-Type Inequalities for Dynamic Equations on TimeScales
8.1 Introduction
8.2 Preliminaries
8.3 Sturm–Liouville Equations
8.4 Linear Hamiltonian Systems
8.5 Higher-Order Dynamic Equations
8.6 Stability Theory for Hill's Equation
8.6.1 Auxiliary Propositions
8.6.2 Proofs
8.7 Linear Hamiltonian Systems
8.7.1 Proofs
8.7.2 A Disconjugacy Criterion
8.8 Planar Linear Hamiltonian Systems
8.8.1 Time Scales Exponential Function
8.8.2 Lyapunov-Type Inequalities
8.8.3 Disconjugacy Criteria
8.9 Nonlinear Dynamic Systems
8.9.1 Preliminaries and Auxiliary Results
8.9.2 Lyapunov-Type Inequalities
8.10 Notes and References
References
Index