New Book Announcement

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Differential Equations with Maxima

ISBN: 978-1-4398-6757-0

by

Drumi Bainov

and

Snezhana Hristova

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PURE AND APPLIED MATHEMATICS, vol. 298

CRC Press, Taylor&Francis Group, A CHAPMAN&HALL BOOK

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The current book is an introduction to the theory of differential equations with ``maxima''. Differential equations with ``maxima'' are a special type of differential equations that contain the maximum of the unknown function over a previous interval(s). Such equations adequately model real world processes whose present state significantly depends on the maximum value of the state on a past time interval. For example, in the theory of automatic control in various technical systems often the law of regulation depends on the maximum values of some regulated state parameters over certain time intervals and their behavior is modeled by differential equations with ``maxima''. Recently, the interest in differential equations with ``maxima'' has increased exponentially. The theoretical results and investigations of differential equations with ``maxima'' opens the door to enormous possibilities for their applications to real world processes and phenomena.

This book presents the qualitative theory and develops some approximate methods for differential equations with ``maxima''.

Chapter 1 gives an introduction to the mathematical apparatus of integral inequalities, involving maxima of unknown functions. Different types of linear and nonlinear integral inequalities with ``maxima'' are solved. Both cases of single integral inequalities and double integral inequalities are studied. Several direct applications of the solved inequalities are illustrated on various types of differential equations with ``maxima''.

In Chapter 2 are studied some general properties of the solutions of differential equations with ``maxima''. Several existence results for initial value problems and boundary value problems are presented.

In Chapter 3 several stability results for differential equations with ``maxima'' are given. The investigations are based on appropriate modifications of the Razumikhin technique by applying Lyapunov functions. Appropriate definitions about different types of stability are given and sufficient conditions are obtained.

Chapter 4 deals with the theory of oscillation for differential equations with ``maxima''. The asymptotic and oscillatory behavior of solutions of *n*-th order differential equations with ``maxima'' is studied. Several sufficient conditions for oscillation as well as almost oscillation are obtained.

Several differential equations with ``maxima'' and their corresponding delay differential equations are examined and the oscillatory properties of their solutions are studied. The influence of the presence of maxima function on the behavior of the solutions is demonstrated.

In Chapter 5 two approximate methods for solving differential equations with ``maxima'' are applied to initial value problems as well as boundary value problems for differential equations with ``maxima''. The considered methods combine the method of lower and upper solutions with appropriate monotone methods. Algorithms for constructing sequences of successive approximation to the solutions are introduced. Each term of the constructed sequences is a solution of an appropriately chosen linear equation.

In Chapter 6 a systematic development of the average method for differential equations with ``maxima'' is given. This method is applied to first-order differential equations with ``maxima'' and neutral differential equations with ``maxima''. Different schemes for averaging, such as ordinary averaging, partial averaging, partially-additive averaging, and partially-multiplicative averaging are suggested.

This book, being the first one in the field, gives a good overview of the entire field of differential equations with ``maxima'' and serves as a stimulating guide for the theoretical and applied researchers in mathematics. It is a helpful tool for further investigations and applications of these equations for better and more adequate studying of real world problems.

The current book is intended for a wide audience, including mathematicians, applied researchers and practitioners, whose interest extends beyond the boundaries of qualitative analysis of well known differential equations.