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

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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  • 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.

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

    Citation:

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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 $\mathbb{R}^3$

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

  •   K. C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.
      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.
      J. P. Ma , A generalized preimage theorem in global analysis, Sci. China. Ser. A, 44 (2001) , 299-303.  doi: 10.1007/BF02878710.
      J. P. Ma , A generalized transversality in global analysis, Pacific J.Math., 236 (2008) , 357-371.  doi: 10.2140/pjm.2008.236.357.
      M. Z. Nashed, Generalized Inverses and Applications, New York-San Francisco-London: Academic Pr. , 1976.
      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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