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We give a criterion for maps on ultrametric spaces to be surjective and topreserve spherical completeness. We show how Hensel's Lemma and themulti-dimensional Hensel's Lemma follow from our result. We give an easy proofthat the latter holds in every henselian field. We also prove a basicinfinite-dimensional Implicit Function Theorem. Further, we apply the criterionto deduce various versions of Hensel's Lemma for polynomials in severaladditive operators, and to give a criterion for the existence of integrationand solutions of certain differential equations on spherically complete valueddifferential fields, for both valued D-fields in the sense of Scanlon, anddifferentially valued fields in the sense of Rosenlicht. We modify the approachso that it also covers logarithmic-exponential power series fields. Finally, wegive a criterion for a sum of spherically complete subgroups of a valuedabelian group to be spherically complete. This in turn can be used to determineelementary properties of power series fields in positive characteristic.

Maps on ultrametric spaces, Hensel's Lemma, and differential equations over valued fields