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Galerkin approximations in several parameter bifurcation problems

  • J. C. Alexander (a1) and P. M. Fitzpatrick (a1)

The purpose of this paper is to prove a theorem giving conditions yielding global bifurcation of the solutions of a family of parameterized nonlinear equations, the domain and the range of which lie in Banach spaces, where the parameter is allowed to be a vector in , k a positive integer. The basic contribution is that the parameter is vector valued and that the nonlinearities allowed are very general; however, even for scalar parameters, our results extend those of previous authors.

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(1) J. C. Alexander Bifurcation of zeros of parameterized functions. J. Functional Analysis, 29 (1978), 3753.

(4) J. C. Alexander and J. F. G. Auchmuty Global bifurcation of waves, Manuscripta Math. 27 (1979), 159166.

(5) J. C. Alexander and James A. Yorke Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263292.

(6) J. C. Alexander and James A. Yorke Calculating bifurcation invariants as elements in the homotopy of the general linear group, J. Pure Appl. Algebra 13 (1978), 18.

(8) F. E. Browder and W. V. Petryshyn Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces. J. Functional Analysis 3 (1969), 217245.

(11) P. M. Fitzpatrick and W. V. Petryshyn Galerkin methods in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations. Trans. Amer. Math. Soc. 238 (1978), 321340.

(15) R. D. Nussbaum A Hopf global bifurcation theorem for retarded functional differential equations. Trans. Amer. Math. Soc. 238 (1978), 139164.

(18) W. V. Petryshyn The approximation-solvability of equations involving A-proper and pseudo-A-proper mappings. Bull. Amer. Math. Soc. 81 (1975), 233312.

(19) W. V. Petryshyn Bifurcation and asymptotic bifurcation for equations involving A-proper mappings, with applications to differential equations, J. Differential Equations 28 (1978), 124154.

(20) P. H. Rabinowitz Some global results for nonlinear eigenvalue problems. J. Functional Analysis 7 (1971), 487513.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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