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8.—Bifurcation and Asymptotic Bifurcation for Non-compact Nonsymmetric Gradient Operators
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The first part of this paper is devoted to a study of the classical bifurcation problem in a Hilbert space, under the assumption that the operators involved are gradient operators, but not necessarily compact as mentioned in this paper.Abstract:
The first part of this paper is devoted to a study of the classical bifurcation problem in a Hilbert space, under the assumption that the operators involved are gradient operators, but not necessarily compact. Our approach to the problem was introduced by Krasnosel'skii, but here we show that his assumption about the compactness of the operators can be replaced by a much weaker Lipschitz type condition, without affecting the generality of his conclusions. The rest of the paper is concerned with the analogous problem when the operator is knownto be asymptotically linear rather than Frechet differentiable. Indeed, we show that this question can always be reduced to the first case, after some manipulation. After this manipulation the new operator is found to be a Frechet differentiable gradient operator, and so we can invoke the results of the first part. This manipulation is in the spirit of that of [11] but is necessarily different.read more
Citations
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Notes on the bifurcation theorem
Kung-Ching Chang,Zhi-Qiang Wang +1 more
TL;DR: In this article, a unified approach to bifurcations both at the origin and at infinity is presented based on the Conley index theory, which provides improvements of the existing theory.
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Asymptotic bifurcation and second order elliptic equations on RN
TL;DR: In this paper, Masson et al. considered the problem of asymptotic bifurcation for a second order non-linear elliptic equation on R-N.
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A global continuation theorem and bifurcation from infinity for infinite-dimensional dynamical systems
TL;DR: In this paper, a general continuation theorem for isolated sets in infinite-dimensional dynamical systems is proved for a class of semiflow systems, which is then used to prove the existence of continua of full bounded solutions bifurcating from infinity for systems of reaction-diffusion equations.
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16.—A Remark on a Paper by J. F. Toland and some Applications to Unilateral Problems
João-Paulo Dias,Jesús Hernández +1 more
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Bifurcation from infinity for an asymptotically linear Schrödinger equation
TL;DR: In this paper, the authors considered the Schrodinger equation and showed that if λ n is an isolated eigenvalue for the linearization at infinity, then under some additional conditions there exists a sequence (u_n, \lambda_n) of solutions such that λn n \rightarrow \infty} is a solution.
References
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Some global results for nonlinear eigenvalue problems
TL;DR: In this paper, the structure of the solution set for a large class of nonlinear eigenvalue problems in a Banach space is investigated, and the existence of continua, i.e., closed connected sets, of solutions of these equations is demonstrated.
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