In 1964 the author's mono graph "Differential- und Integral-Un
gleichungen," with the subtitle "und ihre Anwendung bei
Abschatzungs 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."
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