Differential Equations and Computational Simulations III

Electron. J. Diff. Eqns., Conf. 01, 1997, pp. 223-235.

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Persistence of Crandall-Rabinowitz type bifurcations under small perturbations

Bettina E. Schmidt

**Abstract:**

We discuss a class of nonlinear operator equations in a Banach space setting
and present a generalization of the Crandall-Rabinowitz bifurcation theorem
that describes the effect of small perturbations of the operators involved on
the local structure of the solution set in the vicinity of a bifurcation
point of the unperturbed equation. The result is applied to a
parameter-dependent Neumann boundary-value problem with spatially homogeneous
source terms that exhibits infinitely many bifurcation points.
We obtain conditions for the persistence or nonpersistence of these
bifurcations under small, spatially inhomogeneous perturbations of the
source terms.
Published November 12, 1998.

Mathematics Subject Classifications: 34B15, 34C23, 46N20.

Key words and phrases: Neumann problem, nonlinear eigenvalue problem,
bifurcation from simple eigenvalues, Crandall-Rabinowitz theorem,
regular-singular points, perturbed bifurcation theory.

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Bettina E. Schmidt

Department of Mathematics,
Auburn University at Montgomery

Montgomery, AL 36124-4023, USA

Email address: bes@strudel.aum.edu

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