Implicit multifunction theorems with positively homogeneous maps

We extend Robinson’s and Ledyaev and Zhu’s implicit multifunction theorems using the language of generalized derivatives with positively homogeneous maps, allowing us to obtain results that more closely resemble the classical (single-valued) implicit function theorem. We highlight that using linear openness instead of metric regularity gives simpler proofs and stronger results. As part of our analysis, we study perturbations of generalized linear openness and metric regularity. Finally, we discuss how our methods may also be adapted to study generalized calmness.