Let be a unital -algebra with unitary group . Assume that are left Banach modules over and that is complete. Given , and let
The authors prove the following theorem. Let be such that there is some with
for all and all . Assume furthermore that there is some such that the mapping satisfies for all . Then there is a unique -linear mapping such that
for all .
By specializing the hypotheses, several corollaries are derived. The main tool for proving these results is a fixed point theorem in generalized metric spaces [cf. J. B. Diaz and B. Margolis, Bull. Am. Math. Soc. 74, 305–309 (1968; Zbl 0157.29904)].