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We obtain equivalences among the covering property at a positive-order rate of a multifunction, the metric regularity property of a positive order, and the Hölder-like continuity property of the inverse mapping. Our results develop some aspects of the preceding results of J.-P. Penot, J.M. Borwein and D.M. Zhuang, H. Frankowska, B.S. Mordukhovich, and L.I. Minchenko. Necessary conditions for having these properties are given in terms of positive-order variational coderivative, a concept used here for the first time. We also discuss some known sufficient conditions for the validity of the three properties in terms of the positive-order variation in the sense of H. Frankowska. Illustrative examples are considered.

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