Differential Equations with Maxima

by Drumi D. Bainov, Snezhana G. Hristova

Overview

Differential equations with "maxima"—differential equations that contain the maximum of the unknown function over a previous interval—adequately model real-world processes whose present state significantly depends on the maximum value of the state on a past time interval. More and more, these equations model and regulate the behavior of various technical systems on which our ever-advancing, high-tech world depends. Understanding and manipulating the theoretical results and investigations of differential equations with maxima opens the door to enormous possibilities for applications to real-world processes and phenomena.

Presenting the qualitative theory and approximate methods, Differential Equations with Maxima begins with an introduction to the mathematical apparatus of integral inequalities involving maxima of unknown functions. The authors solve various types of linear and nonlinear integral inequalities, study both cases of single and double integral inequalities, and illustrate several direct applications of solved inequalities. They also present general properties of solutions as well as existence results for initial value and boundary value problems.

Later chapters offer stability results with definitions of different types of stability with sufficient conditions and include investigations based on appropriate modifications of the Razumikhin technique by applying Lyapunov functions. The text covers the main concepts of oscillation theory and methods applied to initial and boundary value problems, combining the method of lower and upper solutions with appropriate monotone methods and introducing algorithms for constructing sequences of successive approximations. The book concludes with a systematic development of the averaging method for differential equations with maxima as applied to first-order and neutral equations. It also explores different schemes for averaging, partial averaging, partially additiv

Product Details

ISBN-13:	9780367382827
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Pages:	312
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Drumi D. Bainov, Medical University of Sofia, Bulgaria

Snezhana G. Hristova, Plovdiv University, Bulgaria

Table of Contents

Preface ix

1 Introduction 1

2 Integral Inequalities with Maxima 17

2.1 Linear Integral Inequalities with Maxima for Scalar Functions of One Variable 17

2.2 Nonlinear Integral Inequalities with Maxima for Scalar Functions of One Variable 29

2.3 Integral Inequalities with Maxima for Scalar Functions of Two Variables 40

2.4 Applications of the Integral Inequalities with Maxima 50

3 General Theory 61

3.1 Existence Theory for Initial Value Problems 61

3.2 Existence Theory for Boundary Value Problems 70

3.3 Differential Equations with "Maxima" via Weakly Picard Operator Theory 82

4 Stability Theory and Lyapunov Functions 91

4.1 Stability and Uniform Stability 92

4.1.1 Stability in Terms of Two Measures 93

4.1.2 Stability of Zero Solution 103

4.2 Integral Stability in Terms of Two Measures 109

4.3 Stability and Cone Valued Lyapunov Functions 118

4.4 Practical Stability on a Cone 127

4.4.1 Practical Stability 128

4.4.2 Eventual Practical Stability 133

5 Oscillation Theory 141

5.1 Differential Equations with "Maxima." versus Differential Equations with Delay 141

5.2 Oscillations of Differential Equations with "Maxima" and Delay 145

5.3 Oscillations of Forced n-th Order Differential Equations: with "Maxima" 160

5.4 Oscillations and Almost Oscillations of n-th Order Differential Equations with "Maxima" 174

5.5 Oscillations of Differential Inequalities with "Maxima" 186

6 Asymptotic Methods 193

6.1 Monotone-Iterative Technique for Initial Value Problems 193

6.1.1 Multidimensional Case 197

6.1.2 Scalar Case 202

6.2 Monotone-Iterative Technique for Periodic Boundary Value Problems 205

6.3 Monotone-Iterative Technique for Second Order Differential Equations with "Maxima" 215

6.4 Method of Quasilinearization for Initial Value Problems 223

6.5 Method of Quasilinearization for Periodic Boundary Value Problems 235

7 Averaging Method 243

7.1 Averaging Method for Initial Value Problems 243

7.2 Averaging Method for Multipoint Boundary Value Problems 248

7.3 Partial Averaging Method 253

7.4 Partially Additive and Partially Multiplicative Averaging Method 258

8 Notes and Comments 265

Bibliography 267

Index 293