On primal regularity estimates for single-valued mappings

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• 作者：Radek Cibulka ; Marián Fabian…
• 关键词：49J53 ; 49J52 ; 49K40 ; 90C31 ; Metric regularity ; linear openness ; tangential approximation ; contingent tangent cone ; strict prederivative ; measure of noncompactness ; Clarke’s inverse function theorem ; cofinal family ; renorming ; nonsmooth analysis
• 刊名：Journal of Fixed Point Theory and Applications
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：17
• 期：1
• 页码：187-208
• 全文大小：832 KB
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• 作者单位：Radek Cibulka (1)
  Marián Fabian (2)
  Alexander D. Ioffe (3)
  
  1. NTIS -New Technologies for the Information Society, Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 22, 306 14, Pilsen, Czech Republic
  2. Mathematical Institute of Czech Academy of Sciences, ?itná 25, 115 67, Praha 1, Czech Republic
  3. Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Mathematical Methods in Physics
  
• 出版者：Birkh盲user Basel
• ISSN：1661-7746

文摘

Based on a primal regularity criterion we provide lower bounds for the regularity modulus of a nonlinear single-valued mapping F from a Banach space X into another Banach space Y . We focus on the case when F is defined on a proper (closed convex) subset of X only rather than on the whole of X. Three possible ways of approximating F around the reference point are considered. First, we use a tangential approximation by set-valued mappings associated with the Bouligand’s tangent cone to the graph of F. Then we move on to approximations by positively homogeneous set-valued mappings whose graphs contain the graph of F, for example, by the strict prederivative. Finally, we use an approximation by bunches of continuous linear operators. In the first two cases finding approximating objects is relatively easy while in the third case the approximating object is very convenient to work with. On examples, we illustrate that these approaches are different and neither of them implies the other, unless the spaces in question are finite dimensional. Keywords Metric regularity linear openness tangential approximation contingent tangent cone strict prederivative measure of noncompactness Clarke’s inverse function theorem cofinal family renorming nonsmooth analysis