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 F : X → → Y , where X and Y are Banach spaces, be metrically regular at x for ȳ and with the property that the mapping whose graph is the restriction of the graph of the inverse F−1 to a neighborhood of (ȳ, x) is convex and closed valued. Then for any function G : X → Y with lipG(x) · regF (x | ȳ)) < 1, the mapping (F +G)−1 has a continuous local selection x(·) around (ȳ +G(x), x) which is also calm.