# American Institute of Mathematical Sciences

October  2017, 10(5): 1043-1050. doi: 10.3934/dcdss.2017055

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

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

Received  August 2016 Revised  January 2017 Published  June 2017

Fund Project: 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.

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.

Citation: Qiang Li. A kind of generalized transversality theorem for $C^r$ mapping with parameter. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1043-1050. doi: 10.3934/dcdss.2017055
##### References:
 [1] K. C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005. [2] J. P. Ma, (1-2) Inverses of operators between banach spaces and local conjugacy theorem, Chinese Ann. Math. Ser. B, 20 (1999), 57-62.  doi: 10.1142/S0252959999000084. [3] J. P. Ma, A generalized preimage theorem in global analysis, Sci. China. Ser. A, 44 (2001), 299-303.  doi: 10.1007/BF02878710. [4] J. P. Ma, A generalized transversality in global analysis, Pacific J.Math., 236 (2008), 357-371.  doi: 10.2140/pjm.2008.236.357. [5] M. Z. Nashed, Generalized Inverses and Applications, New York-San Francisco-London: Academic Pr. , 1976. [6] E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer Verlag, New York-Berline, 1988. doi: 10.1007/978-1-4612-4838-5.

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##### References:
 [1] K. C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005. [2] J. P. Ma, (1-2) Inverses of operators between banach spaces and local conjugacy theorem, Chinese Ann. Math. Ser. B, 20 (1999), 57-62.  doi: 10.1142/S0252959999000084. [3] J. P. Ma, A generalized preimage theorem in global analysis, Sci. China. Ser. A, 44 (2001), 299-303.  doi: 10.1007/BF02878710. [4] J. P. Ma, A generalized transversality in global analysis, Pacific J.Math., 236 (2008), 357-371.  doi: 10.2140/pjm.2008.236.357. [5] M. Z. Nashed, Generalized Inverses and Applications, New York-San Francisco-London: Academic Pr. , 1976. [6] E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer Verlag, New York-Berline, 1988. doi: 10.1007/978-1-4612-4838-5.
$f(s,t)=(s,s^3,t)$ is generalized transversal to $P=\{(0,0,z)\mid z\in \mathbb{R}\}$ mod $\mathbb{R}^3$
$F(u,s)=(e^{u^2+s^2}-e,u^2+s^2-1)$ is generalized transversal to $P=\{\theta\}$ mod $\mathbb{R}^2$
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