Walter W. Differential and Integral Inequalities

Berlin, Heidelberg, New York: Springer-Verlag, 1970. — 352 p.In 1964 the author's monograph "Differential- und Integral- gleichungen," with the subtitle "und ihre Anwendung bei Abschätzungs­ und Eindeutigkeitsproblemen" was published. The present volume grew out of the response to the demand for an English translation of this book. In the meantime the literature on differential and integral in­ equalities increased greatly. We have tried to incorporate new results as far as possible. As a matter of fact, the Bibliography has been almost doubled in size. The most substantial additions are in the field of existence theory. In Chapter I we have included the basic theorems on Volterra integral equations in Banach space (covering the case of ordinary differential equations in Banach space). Corresponding theorems on differential inequalities have been added in Chapter II. This was done with a view to the new sections; dealing with the line method, in the chapter on parabolic differential equations. Section 35 contains an exposition of this method in connection with estimation and convergence. An existence theory for the general nonlinear parabolic equation in one space variable based on the line method is given in Section 36. This theory is considered by the author as one of the most significant recent applications of in­ equality methods. We should mention that an exposition of Krzyzanski's method for solving the Cauchy problem has also been added. The numerous requests that the new edition include a chapter on elliptic differential equations have been satisfied to some extent.Volterra Integral Equations
Monotone Kerneis
Remarks on die Existence Problem. Maximal and Minimal Solutions
Generalization of the Monotonicity Concept
Estimates and Uniqueness Theorems
Ordinary Differential Equations (in the Sense of Caratheodory)
Systems of Integral Equations
Bounds for Systems Using K-Norms
Ordinary Differential Equations
Basic Theorems on Differential Inequalities
Estimates for the Initial Value Problem for an Ordinary Differential Equation of First Order
Uniqueness Theorems
Systems of Ordinary Differential Equations. Estimation by K-Norms
Systems of Differential Inequalities
Component-wise Bounds for Systems
Further Uniqueness Results for Systems
Differential Equations of Higher Order
Supplement
Volterra Integral Equations in Several Variables Hyperbolic Differential Equations
Monotone Operators
Existence Theorems
Estimates for Integral Equations
The Hyperbolic Differential Equations Uxy = f(x, y, u)
The Differential Equation Uxy = f(x, y, U, ux , uy)
Supplements. The Local Method of Proof
Parabolic Differential Equations
Notation
The Nagumo-Westphal Lemma
The First Boundary Value Problem
The Maximum-Minimum Principle
The Shape of Profiles
Infinite Domains, Discontinuous Initial Values, Problems Without Initial Values.
Reat Conduction as an Example.
Application to Boundary Layer Theory
The Third Boundary Value Problem
Systems of Parabolic Differential Equations
Uniqueness Problems for Parabolic Systems
Generalizations and Supplements. The Nonstationary Boundary Layer Equations.
The Line Method for Parabolic Equations
Existence Theorems Based on the Line Method
Appendix
Elliptic Differential Equations
List of SymbolsSubject Index
Author Index