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A note on continuation problems

Published online by Cambridge University Press:  18 May 2009

Rita Nugari
Rita Nugari
Affiliation:
Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende (CS)
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Recently M. Martelli [6] and M. Furi and M. P. Pera [1] proved some interesting results about the existence and the global topological structure of connected sets of solutions to problems of the form:

Lx = N(λ, x)

with L:E → F a bounded linear Fredholm operator of index zero (where E, F are real Banach spaces), and N:ℝ × E → F a nonlinear map satisfying suitable conditions.

While the existence of solution sets for this kind of problem follows from the Leray–Schauder continuation principle, it is our aim to show in this note that their global topological structure can be obtained as a consequence of the theory developed by J. Ize, I. Massabò, J. Pejsachowicz and A. Vignoli in [3, 4] about parameter dependent compact vector fields in Banach spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 55 - 61
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Furi, M. and Pera, M. P., Co-bifurcating branches of solutions for nonlinear eigenvalue problems in Banach spaces, Ann. Mat. Pura Appl. (4) 135 (1983), 119–131.CrossRefGoogle Scholar
2.Gaines, R. E. and Mawhin, J. L., Coincidence degree and nonlinear differential equations, Lecture Notes in Mathematics no. 568 (Springer, 1977).Google Scholar
3.Ize, J., Massabò, I., Pejsachowicz, J. and Vignoli, A., Nonlinear multiparameteric equations: structure and topological dimension of global branches of solutions, Rapporto no. 6 (1983), Dipartimento di Matematica, Università della Calabria.Google Scholar
4.Ize, J., Massabò, I., Pejsachowicz, J. and Vignoli, A., Structure and dimension of global branches of solutions to multiparametric nonlinear equations, to appear in the Proceedings of the AMS Conference held in Berkeley, 07 1983.Google Scholar
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7.Whyburn, G. T., Topological analysis (Princeton University Press, 1958).Google Scholar