A Local Selection Theorem for Metrically Regular Mappings

We prove the following extension of a classical theorem due to Bartle and Graves. Let a set-valued mapping

, where

X X

and

Y Y

are Banach spaces, be metrically regular at

\bar x x ˉ

for

\bar y y ˉ

and with the property that the mapping whose graph is the restriction of the graph of the inverse

F^{-1} F − 1

to a neighborhood of

(\bar y, \bar x) ( y ˉ , x ˉ )

is convex and closed valued. Then for any function

G:X\to Y G : X → Y

with

\operatorname{lip} G(\bar x)\cdot \operatorname{reg} F(\bar x\,|\,\bar y)) < 1 lip ⁡ G ( x ˉ ) ⋅ reg ⁡ F ( x ˉ   ∣   y ˉ ) ) < 1

, the mapping

(F+G)^{-1} ( F + G ) − 1

has a continuous local selection

x(\cdot) x ( ⋅ )

around

(\bar y+G(\bar x),\bar x) ( y ˉ + G ( x ˉ ) , x ˉ )

which is also calm.