A kind of generalized transversality theorem for $C^r$ mapping with parameter

Qiang Li

doi:10.3934/dcdss.2017055

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2017, Volume 10, Issue 5: 1043-1050. Doi: 10.3934/dcdss.2017055

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A kind of generalized transversality theorem for $C^r$ mapping with parameter

Qiang Li^1,2, ,

1.
Department of Mathematics, Jilin University, Changchun, 130012, China
2.
School of Science, Qiqihar University, Qiqihar, 161006, China

Received: July 31, 2016

Revised: December 31, 2016

Published: October 2017

The author is supported by NSFC grant No.11671070, Science Foundation of Heilongjiang Province of China No.QC2016008, and the Fundamental Research Funds for Education Department of Heilongjiang Province No.135109234

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Abstract

The author considers a generalized transversality theorem of the mappings with parameter in infinite dimensional Banach space. If the mapping is generalized transversal to a single point set, and in the sense of exterior parameters, the mapping is a Fredholm operator, then there exists a residual set of parameter, such that the Fredholm operator is generalized transversal to the single point set.

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Mathematics Subject Classification: Primary: 46T05, 47A53; Secondary: 15A09.

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Figure 1. $f(s,t)=(s,s^3,t)$ is generalized transversal to $P=\{(0,0,z)\mid z\in \mathbb{R}\}$ mod $\mathbb{R}^3$

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Figure 2. $F(u,s)=(e^{u^2+s^2}-e,u^2+s^2-1)$ is generalized transversal to $P=\{\theta\}$ mod $\mathbb{R}^2$

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