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,…mehr
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.
1 Initial Problems on the Haar Pyramid.- 1.1 Introduction.- 1.2 Functional differential inequalities.- 1.3 Weak functional differential inequalities.- 1.4 Comparison theorems for classical solutions.- 1.5 Applications of comparison theorems.- 1.6 Kamke functions.- 1.7 Uniqueness of classical solutions.- 1.8 Nonlinear systems.- 1.9 Haar inequality for nonlinear systems.- 1.10 Uniqueness and continuous dependence.- 1.11 Chaplygin method for initial problems.- 2 Existence of Solutions on the Haar Pyramid.- 2.1 Introduction.- 2.2 Function spaces.- 2.3 Existence of classical solutions.- 2.4 Examples.- 2.5 Quasi - linear systems.- 2.6 Bicharacteristics of quasilinear systems.- 2.7 Integral operators for initial problems.- 2.8 Existence of Carathéodory solutions.- 2.9 Uniqueness of generalized solutions.- 3 Numerical Methods for Initial Problems.- 3.1 Introduction.- 3.2 Functional difference inequalities.- 3.3 Applications of functional difference inequalities.- 3.4 Almost linear problems.- 3.5 Error estimates of approximate solutions.- 3.6 Difference methods for nonlinear equations.- 3.7 Interpolating operators on Haar pyramid.- 3.8 The Euler method for the Cauchy problem.- 3.9 Error estimates for the Euler method.- 3.10 Difference methods for almost linear equations.- 4 Initial Problems on Unbounded Domains.- 4.1 Introduction.- 4.2 Bicharacteristics for quasilinear systems.- 4.3 Operator U? and its properties.- 4.4 Existence of weak solutions.- 4.5 Integral operators for quasilinear systems.- 4.6 Quasilinear systems in the second canonical form.- 4.7 Uniqueness of solutions.- 4.8 Function spaces.- 4.9 Bicharacteristics of nonlinear functional differential equations.- 4.10 Integral functional equations.- 4.11 The existence of the sequence of successive approximations.-4.12 Convergence of the sequence {z(m), u(m)}.- 4.13 The main theorem.- 4.14 Some noteworthy particular cases.- 5 Mixed Problems for Nonlinear Equations.- 5.1 Introduction.- 5.2 Functional differential inequalities.- 5.3 Comparison theorems for mixed problems.- 5.4 Chaplygin method for mixed problems.- 5.5 Approximate solutions.- 5.6 Difference methods for mixed problems.- 5.7 Functional difference equations with mixed conditions.- 5.8 Convergence of difference methods.- 5.9 Interpolating operators.- 5.10 The Euler method for mixed problems.- 5.11 Bicharacteristics for mixed problems.- 5.12 Functional integral equations.- 5.13 Bicharacteristics of nonlinear mixed problems.- 5.14 Integral functional equations.- 5.15 The existence of solutions of nonlinear mixed problems.- 5.16 Uniqueness of weak solutions of mixed problems.- 6 Numerical Method of Lines.- 6.1 Introduction.- 6.2 Comparison theorem.- 6.3 Existence theorem and stability.- 6.4 Convergence of the method of lines.- 6.5 Examples of the numerical methods of lines.- 6.6 Differential difference inequalities for mixed problems.- 6.7 Method of lines for mixed problem.- 6.8 Modified method of lines.- 7 Generalized Solutions.- 7.1 Introduction.- 7.2 Quasi - equicontinuous operators for semilinear systems.- 7.3 Existence of solutions.- 7.4 Functional differential inequalities.- 7.5 Extremal solutions of semilinear systems.- 7.6 Carathéodory solutions of functional differential inequalities.- 7.7 Existence of Carathéodory solutions.- 7.8 Functional differential problems with unbounded delay.- 7.9 Viscosity solutions of functional differential inequalities.- 8 Functional Integral Equations.- 8.1 Introduction.- 8.2 Properties of a comparison problem.- 8.3 The existence and uniqueness of solutions.- 8.4 Examples ofcomparison problems.- 8.5 A certain functional equation.- 8.6 Properties of the operator U.- 8.7 Nonlinear functional integral equations.- 8.8 Discretization of the Darboux problem.- 8.9 Solvability of difference problems.- 8.10 Nonlinear estimates.- 8.11 Implicit difference methods.
1 Initial Problems on the Haar Pyramid.- 1.1 Introduction.- 1.2 Functional differential inequalities.- 1.3 Weak functional differential inequalities.- 1.4 Comparison theorems for classical solutions.- 1.5 Applications of comparison theorems.- 1.6 Kamke functions.- 1.7 Uniqueness of classical solutions.- 1.8 Nonlinear systems.- 1.9 Haar inequality for nonlinear systems.- 1.10 Uniqueness and continuous dependence.- 1.11 Chaplygin method for initial problems.- 2 Existence of Solutions on the Haar Pyramid.- 2.1 Introduction.- 2.2 Function spaces.- 2.3 Existence of classical solutions.- 2.4 Examples.- 2.5 Quasi - linear systems.- 2.6 Bicharacteristics of quasilinear systems.- 2.7 Integral operators for initial problems.- 2.8 Existence of Carathéodory solutions.- 2.9 Uniqueness of generalized solutions.- 3 Numerical Methods for Initial Problems.- 3.1 Introduction.- 3.2 Functional difference inequalities.- 3.3 Applications of functional difference inequalities.- 3.4 Almost linear problems.- 3.5 Error estimates of approximate solutions.- 3.6 Difference methods for nonlinear equations.- 3.7 Interpolating operators on Haar pyramid.- 3.8 The Euler method for the Cauchy problem.- 3.9 Error estimates for the Euler method.- 3.10 Difference methods for almost linear equations.- 4 Initial Problems on Unbounded Domains.- 4.1 Introduction.- 4.2 Bicharacteristics for quasilinear systems.- 4.3 Operator U? and its properties.- 4.4 Existence of weak solutions.- 4.5 Integral operators for quasilinear systems.- 4.6 Quasilinear systems in the second canonical form.- 4.7 Uniqueness of solutions.- 4.8 Function spaces.- 4.9 Bicharacteristics of nonlinear functional differential equations.- 4.10 Integral functional equations.- 4.11 The existence of the sequence of successive approximations.-4.12 Convergence of the sequence {z(m), u(m)}.- 4.13 The main theorem.- 4.14 Some noteworthy particular cases.- 5 Mixed Problems for Nonlinear Equations.- 5.1 Introduction.- 5.2 Functional differential inequalities.- 5.3 Comparison theorems for mixed problems.- 5.4 Chaplygin method for mixed problems.- 5.5 Approximate solutions.- 5.6 Difference methods for mixed problems.- 5.7 Functional difference equations with mixed conditions.- 5.8 Convergence of difference methods.- 5.9 Interpolating operators.- 5.10 The Euler method for mixed problems.- 5.11 Bicharacteristics for mixed problems.- 5.12 Functional integral equations.- 5.13 Bicharacteristics of nonlinear mixed problems.- 5.14 Integral functional equations.- 5.15 The existence of solutions of nonlinear mixed problems.- 5.16 Uniqueness of weak solutions of mixed problems.- 6 Numerical Method of Lines.- 6.1 Introduction.- 6.2 Comparison theorem.- 6.3 Existence theorem and stability.- 6.4 Convergence of the method of lines.- 6.5 Examples of the numerical methods of lines.- 6.6 Differential difference inequalities for mixed problems.- 6.7 Method of lines for mixed problem.- 6.8 Modified method of lines.- 7 Generalized Solutions.- 7.1 Introduction.- 7.2 Quasi - equicontinuous operators for semilinear systems.- 7.3 Existence of solutions.- 7.4 Functional differential inequalities.- 7.5 Extremal solutions of semilinear systems.- 7.6 Carathéodory solutions of functional differential inequalities.- 7.7 Existence of Carathéodory solutions.- 7.8 Functional differential problems with unbounded delay.- 7.9 Viscosity solutions of functional differential inequalities.- 8 Functional Integral Equations.- 8.1 Introduction.- 8.2 Properties of a comparison problem.- 8.3 The existence and uniqueness of solutions.- 8.4 Examples ofcomparison problems.- 8.5 A certain functional equation.- 8.6 Properties of the operator U.- 8.7 Nonlinear functional integral equations.- 8.8 Discretization of the Darboux problem.- 8.9 Solvability of difference problems.- 8.10 Nonlinear estimates.- 8.11 Implicit difference methods.
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