Abstract

We prove the following extension of a classical theorem due to Bartle and Graves. Let a set-valued mapping F:X⇉YF:X \rightrightarrows Y, where XX and YY are Banach spaces, be metrically regular at xˉ\bar x for yˉ\bar y and with the property that the mapping whose graph is the restriction of the graph of the inverse F−1F^{-1} to a neighborhood of (yˉ,xˉ)(\bar y, \bar x) is convex and closed valued. Then for any function G:X→YG:X\to Y with lip⁡G(xˉ)⋅reg⁡F(xˉ ∣ yˉ))<1\operatorname{lip} G(\bar x)\cdot \operatorname{reg} F(\bar x\,|\,\bar y)) < 1, the mapping (F+G)−1(F+G)^{-1} has a continuous local selection x(⋅)x(\cdot) around (yˉ+G(xˉ),xˉ)(\bar y+G(\bar x),\bar x) which is also calm.

Author information

Contact details are reproduced from the original publication and may be historical.

A. L. Dontchev

Mathematical Reviews, Ann Arbor, MI 48107-8604, U.S.A.,

ald@ams.org

Keywords

  • set-valued mapping
  • metric regularity
  • continuous selection
  • Bartle-Graves theorem

Mathematics Subject Classification

49J53, 47H04, 54C65

References

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Suggested citation

A. L. Dontchev. “A Local Selection Theorem for Metrically Regular Mappings.” Journal of Convex Analysis 11 (2004), No. 1, 81–94. https://doi.org/10.68381/jca11006

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Published by Heldermann Verlag, 2004. Rights now held by Banach Press LLC.