Nonlinear Interpolation and Boundary Value Problems

This book is devoted to the study of boundary value problems for nonlinear ordinary differential equations and focuses on questions related to the study of nonlinear interpolation. In 1967, Andrzej Lasota and Zdzisław Opial showed that, under suitable hypotheses, if solutions of a second-order nonlinear differential equation passing through two distinct points are unique, when they exist, then, in fact, a solution passing through two distinct points does exist. That result, coupled with the pioneering work of Philip Hartman on what was then called unrestricted n-parameter families, has stimulated 50 years of development in the study of solutions of boundary value problems as nonlinear interpolation problems.

The purpose of this book is two-fold. First, the results that have been generated in the past 50 years are collected for the first time to produce a comprehensive and coherent treatment of what is now a well-defined area of study in the qualitative theory of ordinary differential equations. Second, methods and technical tools are sufficiently exposed so that the interested reader can contribute to the study of nonlinear interpolation.

Contents:

• Uniqueness Implies Uniqueness
• Uniqueness Implies Existence
• Nonlocal Boundary Value Problems: Uniqueness and Existence
• Boundary Value Problems for Finite Difference Equations
• Boundary Value Problems for Dynamic Equations on Time Scales
• Postscript

Readership: Graduate students and researchers interested in boundary value problems for nonlinear ordinary differential equations.
Key Features:

• This book represents the only comprehensive treatment of nonlinear interpolation and boundary value problems
• There are no competing titles. Some of the material in the book is addressed in published survey articles or book chapters