This book is intended as a self-contained exposition of hyperbolic functional dif­ ferential inequalities and their applications. Its aim is to give a systematic and unified presentation of recent developments of the following problems: (i) functional differential inequalities generated by initial and mixed problems, (ii) existence theory of local and global solutions, (iii) functional integral equations generated by hyperbolic equations, (iv) numerical method of lines for hyperbolic problems, (v) difference methods for initial and initial-boundary value problems. Beside classical solutions, the following classes of weak solutions are treated: Ca­ ratheodory solutions for quasilinear equations, entropy solutions and viscosity so­ lutions for nonlinear problems and solutions in the Friedrichs sense for almost linear equations. The theory of difference and differential difference equations ge­ nerated by original problems is discussed and its applications to the constructions of numerical methods for functional differential problems are presented. The monograph is intended for different groups of scientists. Pure mathemati­ cians and graduate students will find an advanced theory of functional differential problems. Applied mathematicians and research engineers will find numerical al­ gorithms for many hyperbolic problems. The classical theory of partial differential inequalities has been described exten­ sively in the monographs [138, 140, 195, 225). As is well known, they found applica­ tions in differential problems. The basic examples of such questions are: estimates of solutions of partial equations, estimates of the domain of the existence of solu­ tions, criteria of uniqueness and estimates of the error of approximate solutions.

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