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Galerkin approximations in several parameter bifurcation problems

Published online by Cambridge University Press:  24 October 2008

J. C. Alexander  and
P. M. Fitzpatrick
J. C. Alexander
Affiliation:
University of Maryland, Maryland 20742
P. M. Fitzpatrick
Affiliation:
University of Maryland, Maryland 20742
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 3 , May 1980 , pp. 489 - 500
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Alexander, J. C.Bifurcation of zeros of parameterized functions. J. Functional Analysis, 29 (1978), 37–53.CrossRefGoogle Scholar
(2)Alexander, J. C.Calculating bifurcation invariants as elements for the homotopy of the general linear group: II. J. Pure Appl. Algebra (in Press).Google Scholar
(3)Alexander, J. C. and Fitzpatrick, P. M., Bifurcation of the fixed points of parameterized condensing functions. J. Functional Analysis (in Press).Google Scholar
(4)Alexander, J. C. and Auchmuty, J. F. G.Global bifurcation of waves, Manuscripta Math. 27 (1979), 159–166.CrossRefGoogle Scholar
(5)Alexander, J. C. and Yorke, James A.Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263–292.Google Scholar
(6)Alexander, J. C. and Yorke, James A.Calculating bifurcation invariants as elements in the homotopy of the general linear group, J. Pure Appl. Algebra 13 (1978), 1–8.Google Scholar
(7)Auchmuty, J. F. G. Bifurcating waves, Proc. Conf. on Bifurcation Theory and its Applications (New York Academy of Sciences, 1979).Google Scholar
(8)Browder, F. E. and Petryshyn, W. V.Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces. J. Functional Analysis 3 (1969), 217–245.CrossRefGoogle Scholar
(9)Crandall, M. G. An introduction to constructive aspects of Bifurcation and Implicit Function Theorem. Applications of Bifurcation Theory (Academic Press, 1977, pp. 1–35).Google Scholar
(10)Fitzpatrick, P. M. Linearizing bifurcation problems. (To appear.)Google Scholar
(11)Fitzpatrick, P. M. and Petryshyn, W. V.Galerkin methods in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations. Trans. Amer. Math. Soc. 238 (1978), 321–340.CrossRefGoogle Scholar
(12)Ize, J.Bifurcation theory for Fredholm operators. Memoirs of Amer. Math. Soc. 7, 174 (1976).Google Scholar
(13)Ize, J. Periodic solutions of nonlinear parabolic equations. (To appear.)Google Scholar
(14)Kuratowski, K.Topology, vol. ii (Warsaw, Panstwowe Wydawnictwo Naukowa; New York, Academic Press, 1968).Google Scholar
(15)Nussbaum, R. D.A Hopf global bifurcation theorem for retarded functional differential equations. Trans. Amer. Math. Soc. 238 (1978), 139–164.Google Scholar
(16)Nussbaum, R. D.The fixed point index for local condensing maps. Annali Mat. Pura Appl. 18 (1971), 217–258.Google Scholar
(17)Nussbaum, R. D. The fixed point index and fixed point theorems for k-set contractions. Ph.D. thesis, University of Chicago, 1969.Google Scholar
(18)Petryshyn, W. V.The approximation-solvability of equations involving A-proper and pseudo-A-proper mappings. Bull. Amer. Math. Soc. 81 (1975), 233–312.Google Scholar
(19)Petryshyn, W. V.Bifurcation and asymptotic bifurcation for equations involving A-proper mappings, with applications to differential equations, J. Differential Equations 28 (1978), 124–154.Google Scholar
(20)Rabinowitz, P. H.Some global results for nonlinear eigenvalue problems. J. Functional Analysis 7 (1971), 487–513.Google Scholar
(21)Toland, J. F.Global bifurcation via Galerkin's method. Report no. 69, Fluid Mechanics Research Institute, University of Essex.Google Scholar
(22)Webb, J. R. L.Remarks on k-set contractions, Boll. Un. Mat. Ital. 4 (1971), 614–629.Google Scholar