Advanced Topics in Difference Equations

. The theory of difference equations, the methods used in their solutions and their wide applications have advanced beyond their adolescent stage to occupy a central position in Applicable Analysis. In fact, in the last five years, the proliferation of the subject is witnessed by hundreds of research articles and several monographs, two International Conferences and numerous Special Sessions, and a new Journal as well as several special issues of existing journals, all devoted to the theme of Difference Equations. Now even those experts who believe in the universality of differential equations are discovering the sometimes striking divergence between the continuous and the discrete. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. In 1992, the first author published a monograph on the subject entitled Difference Equations and Inequalities. This book was an in-depth survey of the field up to the year of publication. Since then, the subject has grown to such an extent that it is now quite impossible for a similar survey, even to cover just the results obtained in the last four years, to be written. In the present monograph, we have collected some of the results which we have obtained in the last few years, as well as some yet unpublished ones.

Zusammenfassung
`I think this remarkable monograph should be on the bookshelf of veryone having connection with mathematical analysis and its applications, concentrating on difference equations, numerical analysis, ordinary and partial differential equations.'
Acta Sci. Math., 64 (1998)

Inhalt

1. Periodic Solutions of First Order Linear Equations.- 2. Periodic Solutions of Generalized Difference Systems.- 3. Periodic Boundary Value Problems: First Order Systems.- 4. Periodic Boundary Value Problems: Second Order Systems.- 5. Periodic Solutions of Higher Order Partial Difference Equations.- 6. Nonlinear Variation of Parameter Methods: One Independent Variable.- 7. Nonlinear Variation of Parameter Methods: Several Independent Variables.- 8. Convergence to Equilibria.- 9. Asymptotic Behavior of Generalized Difference Systems.- 10. Stability of Generalized Difference Systems.- 11. Oscillation via Lyapunov's Second Method.- 12. Oscillation via Difference Inequalities.- 13. Oscillation via Summation Averages.- 14. Oscillation for Nonlinear Difference Equations.- 15. Oscillation for Quasilinear Difference Equations.- 16. Oscillations Generated by Deviating Arguments.- 17. Positive Monotone Solutions for Nonlinear Difference Equations.- 18. Oscillation for nth Order Nonlinear Difference Equations.- 19. Comparison Theorems.- 20. Oscillation for Second Order Neutral Delay Difference Equations.- 21. Oscillation for Second Order Neutral Delay Difference Equations (Contd.).- 22. Oscillation for Higher Order Neutral Delay Difference Equations.- 23. Oscillation for Partial Difference Equations I.- 24. Oscillation for Partial Difference Equations II.- 25. Oscillation for Partial Difference Equations III.- 26. Oscillation for Partial Difference Equations IV.- 27. A Three-Point Boundary Value Problem.- 28. A Singular Boundary Value Problem.- 29. Eigenvalues of Boundary Value Problems.- 30. Two Solutions of a Boundary Value Problem.- 31. (n, p) Boundary Value Problems.- 32. Boundary Value Problems Arising in Transport Phenomena.- 33. Solutions of m-Point Boundary ValueProblems.- 34. Two-Point Boundary Value Problems for Generalized Difference Systems.- 35. Multi-Point Boundary Value Problems for Generalized Difference Systems.- 36. Resonant Boundary Value Problems for Generalized Difference Systems.- 37. Boundary Value Problems in Abstract Spaces.- 38. Linear Square Optimal Control Problem.- 39. Symmetries of Difference Systems on Manifolds.- 40. Discrete Polar Coordinates.- References.