DOI QR코드

DOI QR Code

A FREDHOLM MAPPING OF INDEX ZERO

  • Published : 2008.03.31

Abstract

Sufficient conditions are given to assert that between any two Banach spaces over $\mathbb{K}$ Fredholm mappings share exactly N values in a specific open ball. The proof of the result is constructive and is based upon continuation methods.

References

  1. E. L. Allgower, A Survey of Homotopy Methods for Smooth Mappings, Numerical solution of nonlinear equations (Bremen, 1980), pp. 1-29, Lecture Notes in Math., 878, Springer, Berlin-New York, 1981
  2. E. L. Allgower, K. Glashoff, and H. Peitgen, Proceedings of the Conference on Numerical Solutions of Nonlinear Equations, Bremen, July 1980, Lecture Notes in Math. 878. Springer-Verlag, Berlin, 1981
  3. E. L. Allgower and K. Georg, Numerical Continuation Methods, An introduction. Springer Series in Computational Mathematics, 13. Springer-Verlag, Berlin, 1990
  4. J. C. Alexander and J. A. Yorke, The homotopy continuation method: numerically implementable topological procedures, Trans. Amer. Math. Soc. 242 (1978), 271-284 https://doi.org/10.2307/1997737
  5. S. Bernstein, Sur la generalisation du probleme de Dirichlet, Math. Ann. 69 (1910), no. 1, 82-136 https://doi.org/10.1007/BF01455154
  6. C. B. Garcia and T. Y. Li, On the number of solutions to polynomial systems of equations, SIAM J. Numer. Anal. 17 (1980), no. 4, 540-546 https://doi.org/10.1137/0717046
  7. C. B. Garcia and W. I. Zangwill, Determining all solutions to certain systems of non-linear equations, Math. Oper. Res. 4 (1979), no. 1, 1-14 https://doi.org/10.1287/moor.4.1.1
  8. J. M. Soriano, Existence of zeros for bounded perturbations of proper mappings, Appl. Math. Comput. 99 (1999), no. 2-3, 255-259 https://doi.org/10.1016/S0096-3003(97)10183-7
  9. J. M. Soriano, Global minimum point of a convex function, Appl. Math. Comput. 55 (1993), no. 2-3, 213-218 https://doi.org/10.1016/0096-3003(93)90022-7
  10. J. M. Soriano, Extremum points of a convex function, Appl. Math. Comput. 66 (1994), no. 2-3, 261-266 https://doi.org/10.1016/0096-3003(94)90121-X
  11. J. M. Soriano, On the existence of zero points, Appl. Math. Comput. 79 (1996), no. 1, 99-104 https://doi.org/10.1016/0096-3003(95)00246-4
  12. J. M. Soriano, On the number of zeros of a mapping, Appl. Math. Comput. 88 (1997), no. 2-3, 287-291 https://doi.org/10.1016/S0096-3003(96)00336-0
  13. J. M. Soriano, On the Bezout theorem real case, Comm. Appl. Nonlinear Anal. 2 (1995), no. 4, 59-66
  14. J. M. Soriano, On the Bezout theorem, Comm. Appl. Nonlinear Anal. 4 (1997), no. 2, 59-66
  15. J. M. Soriano, Mappings sharing a value on finite-dimensional spaces, Appl. Math. Comput. 121 (2001), no. 2-3, 391-395 https://doi.org/10.1016/S0096-3003(00)00023-0
  16. J. M. Soriano, Compact mappings and proper mappings between Banach spaces that share a value, Math. Balkanica (N.S.) 14 (2000), no. 1-2, 161-166
  17. J. M. Soriano, Zeros of compact perturbations of proper mappings, Comm. Appl. Nonlinear Anal. 7 (2000), no. 4, 31-37
  18. J. M. Soriano, A compactness condition, Appl. Math. Comput. 124 (2001), no. 3, 397-402 https://doi.org/10.1016/S0096-3003(00)00114-4
  19. J. M. Soriano, Open trajectories, Appl. Math. Comput. 124 (2001), no. 2, 235-240 https://doi.org/10.1016/S0096-3003(00)00092-8
  20. J. M. Soriano, Fredholm and compact mappings sharing a value, Chinese translation in Appl. Math. Mech. 22 (2001), no. 6, 609-612
  21. J. M. Soriano and V. G. Angelov, A zero of a proper mapping, Fixed Point Theory 4 (2003), no. 1, 97-104
  22. S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861-866 https://doi.org/10.2307/2373250
  23. E. Zeidler, Nonlinear Functional Analysis and Its Applications. III, Variational methods and optimization. Translated from the German by Leo F. Boron. Springer-Verlag, New York, 1985
  24. E. Zeidler, Applied Functional Analysis, Applied Mathematical Sciences, 109. Springer-Verlag, New York, 1995
  25. H. Cartan, Differential Calculus, Houghton Mifflin Co., Boston, Mass., 1971
  26. J. Leray and J. Shauder, Topologie et equations fonctionnelles, Ann. Sci. Ecole Norm. Sup. (3) 51 (1934), 45-78 https://doi.org/10.24033/asens.836
  27. J. M. Soriano, On the existence of zero points of a continuous function, Acta Math. Sci. Ser. B Engl. Ed. 22 (2002), no. 2, 171-177