Differential and Integral Inequalities, Softcover reprint of the original 1st ed. 1970 Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge Series, Vol. 55

In 1964 the author's mono graph "Differential- und Integral-Un­ 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.

I Volterra.Integral Equations.- 1. Monotone Kernels.- 2. Remarks on the Existence Problem. Maximal and Minimal Solutions.- 3. Generalization of the Monotonicity Concept.- 4. Estimates and Uniqueness Theorems.- 5. Ordinary Differential Equations (in the Sense of Carathéodory).- 6. Systems of Integral Equations.- 7. Bounds for Systems Using K-Norms.- II Ordinary Differential Equations.- 8. Basic Theorems on Differential Inequalities.- 9. Estimates for the Initial Value Problem for an Ordinary Differential Equation of First Order.- 10. Uniqueness Theorems.- 11. Systems of Ordinary Differential Equations. Estimation by K-Norms.- 12. Systems of Differential Inequalities.- 13. Component-wise Bounds for Systems.- 14. Further Uniqueness Results for Systems.- 15. Differential Equations of Higher Order.- 16. Supplement.- III Volterra Integral Equations in Several Variables Hyperbolic Differential Equations.- 17. Monotone Operators.- 18. Existence Theorems.- 19. Estimates for Integral Equations.- 20. The Hyperbolic Differential Equations uxy= f (x, y, u).- 21. The Differential Equation uxy = f (x, y, u, ux, uy).- 22. Supplements. The Local Method of Proof.- IV Parabolic Differential Equations.- 23. Notation.- 24. The Nagumo-Westphal Lemma.- 25. The First Boundary Value Problem.- 26. The Maximum-Minimum Principle.- 27. The Shape of Profiles.- 28. Infinite Domains, Discontinuous Initial Values, Problems Without Initial Values.- 29. Heat Conduction as an Example.- 30. Application to Boundary Layer Theory.- 31. The Third Boundary Value Problem.- 32. Systems of Parabolic Differential Equations.- 33. Uniqueness Problems for Parabolic Systems.- 34. Generalizations and Supplements. The Nonstationary Boundary Layer Equations.- 35. The Line Method for Parabolic Equations.- 36. ExistenceTheorems Based on the Line Method.- Appendix Elliptic Differential Equations.- List of Symbols.- Author Index.