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We present two basic lemmas on exact and approximate solutions of inclusions and equations in general spaces. Its applications involve Ekeland's principle, characterize calmness, lower semicontinuity and the Aubin property of solution sets in some Hoeldertype setting and connect these properties with certain iteration schemes of descent type. In this way, the mentioned stability properties can be directly characterized by convergence of more or less abstract solution procedures. New stabilitydoi:10.1016/j.jmaa.2009.04.060 fatcat:x3j3ps3jd5g3nen4ib24mqn45u