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Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero and of A-Proper Maps with Applications

In: Approximation and Computation

Author

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  • P. S. Milojević

    (New Jersey Institute of Technology)

Abstract

In Part I, we develop a nonlinear Fredholm alternative theory involving k-ball and k-set perturbations of general homeomorphisms as well as of homeomorphisms that are nonlinear Fredholm maps of index zero. Various generalized first Fredholm theorems are given and finite solvability of general (odd) Fredholm maps of index zero is also studied. We apply these results to the unique and finite solvability of potential and semilinear problems with strongly nonlinear boundary conditions and to quasilinear elliptic equations. The results of Sect. 1.1 are based on the Browder and Banach-Mazur homeomorphism theorems. The basic tools used in Sect. 1.2 are the recent degree theories for nonlinear C 1-Fredholm maps of index zero and their perturbations.In Part II, we discuss a number of constructive homeomorphism results for nonlinear A-proper maps and their perturbations. Error estimates are given under various conditions. We prove that k-ball and k-set perturbations of these homeomorphisms are again homeomorphisms or that the corresponding equations are finitely solvable. The theory presented in this paper is based on recent work by the author.

Suggested Citation

  • P. S. Milojević, 2010. "Homeomorphisms and Fredholm Theory for Perturbations of Nonlinear Fredholm Maps of Index Zero and of A-Proper Maps with Applications," Springer Optimization and Its Applications, in: Walter Gautschi & Giuseppe Mastroianni & Themistocles M. Rassias (ed.), Approximation and Computation, pages 339-363, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-6594-3_22
    DOI: 10.1007/978-1-4419-6594-3_22
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