Focal Boundary Value Problems for Differential and Difference Equations
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Focal Boundary Value Problems for Differential and Difference Equations

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The last fifty years have witnessed several monographs and hundreds of research articles on the theory, constructive methods and wide spectrum of applications of boundary value problems for ordinary differential equations. In this vast field of research, the conjugate (Hermite) and the right focal point (Abei) types of problems have received the maximum attention. This is largely due to the fact that these types of problems are basic, in the sense that the methods employed in their study are easily extendable to other types of prob- lems. Moreover, the conjugate and the right focal point types of boundary value problems occur frequently in real world problems. In the monograph Boundary Value Problems for Higher Order Differential Equations published in 1986, we addressed the theory of conjugate boundary value problems. At that time the results on right focal point problems were scarce; however, in the last ten years extensive research has been done. In Chapter 1 of the mono- graph we offer up-to-date information of this newly developed theory of right focal point boundary value problems. Until twenty years ago Difference Equations were considered as the dis- cretizations of the differential equations. Further, it was tacitly taken for granted that the theories of difference and differential equations are parallel. However, striking diversities and wide applications reported in the last two decades have made difference equations one of the major areas of research.
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Product details

  • Hardback | 294 pages
  • 157.48 x 236.22 x 25.4mm | 181.44g
  • Dordrecht, Netherlands
  • English
  • 1998 ed.
  • X, 294 p.
  • 0792349784
  • 9780792349785

Table of contents

Preface. 1: Continuous Problems. 1.1. Introduction. 1.2. Abel-Gontscharoff Interpolation. 1.3. Solution of Linear Problems. 1.4. Existence and Uniqueness. 1.5. Picard's and Approximate Picard's Methods. 1.6. Quasilinearization and Approximate Quasilinearization. 1.7. Integro-Differential Equations. 1.8. Delay-Differential Equations. 1.9. Necessary and Sufficient Conditions for Right Disfocality. 1.10. Tests for Right and Eventual Disfocalities. 1.11. Green's Functions. 1.12. Monotone Convergence. 1.13. Uniqueness Implies Uniqueness. 1.14. Uniqueness Implies Existence. 1.15. Continuous Dependence and Differentiation with respect to Boundary Values. 1.16. Right Disfocality Implies Right Disfocality. 1.17. Right Disfocality Implies Existence. 1.18. Differential Inequalities Imply Existence. 1.19. Infinite Interval Problems. 1.20. Best Possible Results: Control Theory Methods. 1.21. Converse Theorems. 1.22. Focal Subfunctions. 1.23. Generalized Problem I. 1.24. Generalized Problem II. 1.25. A Singular Problem. 1.26. A Problem with Impulse Effects. Comments and Remarks. References. 2: Discrete Problems. 2.1. Introduction. 2.2. Discrete Abel-Gontscharoff Interpolation. 2.3. Existence and Uniqueness. 2.4. Picard's and Approximate Picard's Methods. 2.5. Quasilinearization and Approximate Linearization. 2.6. Necessary and Sufficient Conditions for Right Disfocality. 2.7. Tests for Right and Eventual Disfocalities. 2.8. Green's Functions. 2.9. Monotone Convergence. 2.10. Continuous Dependence and Differentiation with Respect to Initial and Boundary Values. 2.11. Differences with Respect to Boundary Points. 2.12. Uniqueness Implies Existence. 2.13. Generalized Problems. Comments and Remarks. References. Index.
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Review quote

`...will be useful for a graduate course concerned with boundary value problems for either differential equations or difference equations. It also would be an excellent book for mathematicians doing research in this area.'
Mathematical Reviews, 99b
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