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We provide a systematic, unified and insightful analysis of the hypotheses that ensure their weak, strong and linear convergence, either matching or improving previous results obtained by analysing particular cases separately. We also show that these methods are robust with respect to different kinds of perturbations\u2013which may come from computational errors, intentional deviations, as well as regularisation or approximation schemes\u2013under surprisingly weak assumptions. Although we mostly focus on theoretical aspects, numerical illustrations in image inpainting and electricity production markets reveal possible trends in the behaviour of these types of methods.<\/jats:p>","DOI":"10.1007\/s10957-024-02600-5","type":"journal-article","created":{"date-parts":[[2025,1,22]],"date-time":"2025-01-22T12:02:20Z","timestamp":1737547340000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Krasnoselskii\u2013Mann Iterations: Inertia, Perturbations and Approximation"],"prefix":"10.1007","volume":"204","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3278-1716","authenticated-orcid":false,"given":"Daniel","family":"Cortild","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8551-0522","authenticated-orcid":false,"given":"Juan","family":"Peypouquet","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,1,22]]},"reference":[{"key":"2600_CR1","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1023\/A:1011253113155","volume":"9","author":"F Alvarez","year":"2001","unstructured":"Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. 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