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Bifurcation of A-proper mappings without transversality considerations

Published online by Cambridge University Press:  14 November 2011

Stewart C. Welsh
Affiliation:
Department of Mathematics, Texas A & M University, College Station, TX 77843, U.S.A.

Synopsis

We consider a nonlinear eigenvalue problem in the form F(x, λ) = AxT(λ)xR(x, λ) = 0 with F:X × ℝ →Y where X, Y are Banach spaces. We assume that F(0, λ) = 0 for all λ ∈ ℝ and seek bifurcation points; that is, values λ0 ∈ ℝ for which there are solutions to F(x, λ) = 0 with x ≠ 0 in any neighbourhood of (0, λ0). Corresponding to these bifurcation points we obtain global properties of the maximal connected subset of solutions to F(x, λ) = 0 containing (0, λ0).

Generalised topological degree techniques are employed in the proofs of our results without requiring a transversality condition. The operators involved belong to the general class of A-proper mappings which include compact and k-ball contractive perturbations of the identity operator, accretive mappings, and many more.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Adams, R. A.. Sobolev Spaces (London and New York: Academic Press, 1975).Google Scholar
2Alexander, J. C. and Fitzpatrick, P. M.. Galerkin approximations in several parameter bifurcation problems. Math. Proc. Cambridge Philos. Soc. 87 (1980), 489500.Google Scholar
3Browder, F. E.. On the spectral theory of elliptic differential operators. Math. Ann. 142 (1961), 22130.CrossRefGoogle Scholar
4Gaines, R. E. and Mawhin, J.. Coincidence degree, and nonlinear differential equations. Lecture Notes in Mathematics 568 (Berlin: Springer, 1977).Google Scholar
5Krasnosel'skii, M. A.. Topological Methods in the Theory of Nonlinear Integral Equations (New York: Pergamon, 1964).Google Scholar
6Lloyd, N. G.. Degree Theory (London: Cambridge University Press, 1978).Google Scholar
7Martin, R. H. Jr, Nonlinear Operators and Differential Equations in Banach Spaces (New York: John Wiley, 1976).Google Scholar
8Nussbaum, R. D.. The radius of the essential spectrum. Duke Math. J. 38 (1970), 473478.Google Scholar
9Petryshyn, W. V.. On projectional solvability and the Fredholm alternative for equations involving linear, A-proper operations. Arch. Rational Mech. Anal. 30 (1968), 331338.CrossRefGoogle Scholar
10Petryshyn, W. V.. On the approximation-solvability of equations involving A-proper and pseudo S-proper mappings. Bull. Amer. Math. Soc. 81 (1975), 223448.CrossRefGoogle Scholar
11Petryshyn, W. V.. Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations.J. Differential Equations 28 (1978), 124154.Google Scholar
12Petryshyn, W. V.. Using degree theory for densely defined A-proper maps in the solvability of semi-linear equations with unbounded and noninvertible linear part. Nonlinear Anal., Theory, Methods, Appl. 4 (1980), 259281.CrossRefGoogle Scholar
13Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971), 487513.Google Scholar
14Stuart, C. A.. Some bifurcation theory for k-set contractions. Proc. London Math. Soc. (3) 27 (1973), 531550.Google Scholar
15Stuart, C. A. and Toland, J.. A global result applicable to nonlinear Steklov problems. J. Differential Equations 15 (1974), 247268.Google Scholar
16Toland, J.. Global bifurcation theory via Galerkin's method. Nonlinear Anal., Theory, Methods Appl. 1 (1977), 305317.Google Scholar
17Webb, J. R. L.. Remarks on k-set contractions. Boll. Un. Mat. Ital. (4), 4 (1971), 614629.Google Scholar
18Webb, J. R. L.. Topological degree and A-proper operators. Linear Algebra Appl. (to appear).Google Scholar
19Welsh, S. C.. Global results for Fredholm maps of index zero with a transversality condition (to appear).Google Scholar