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 $F:X \rightrightarrows Y$, where $X$ and $Y$ are Banach spaces, be metrically regular at $\bar x$ for $\bar y$ and with the property that the mapping whose graph is the restriction of the graph of the inverse $F^{-1}$ to a neighborhood of $(\bar y, \bar x)$ is convex and closed valued. Then for any function $G:X\to Y$ with $\operatorname{lip} G(\bar x)\cdot \operatorname{reg} F(\bar x\,|\,\bar y)) < 1$, the mapping $(F+G)^{-1}$ has a continuous local selection $x(\cdot)$ around $(\bar y+G(\bar x),\bar x)$ which is also calm.