{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:43:41Z","timestamp":1760402621567,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,4,27]],"date-time":"2020-04-27T00:00:00Z","timestamp":1587945600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations.<\/jats:p>","DOI":"10.3390\/axioms9020047","type":"journal-article","created":{"date-parts":[[2020,4,28]],"date-time":"2020-04-28T05:05:32Z","timestamp":1588050332000},"page":"47","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1979-4344","authenticated-orcid":false,"given":"Davor","family":"Dragi\u010devi\u0107","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Rijeka, Radmile Matej\u010di\u0107 2, 51000 Rijeka, Croatia"}]},{"given":"Ciprian","family":"Preda","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timi\u015foara, V. P\u00e2rvan Blvd. No. 4, 300223 Timi\u015foara, Romania"},{"name":"Institute for Economic Forecasting, Romanian Academy, 050711 Bucharest, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Lyapunov, A. (1992). The General Problem of the Stability of Motion, Taylor & Francis.","DOI":"10.1080\/00207179208934253"},{"key":"ref_2","unstructured":"LaSalle, J., and Lefschetz, S. (1961). Stability by Liapunov\u2019s Direct Method, with Applications. Mathematics in Science and Engineering, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Hahn, W. (1967). Stability of Motion. Grundlehren der Mathematischen Wissenschaften, Springer.","DOI":"10.1007\/978-3-642-50085-5"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bhatia, N., and Szeg\u00f6, G. (1970). Stability Theory of Dynamical Systems. Grundlehren der Mathematischen Wissenschaften, Springer.","DOI":"10.1007\/978-3-642-62006-5"},{"key":"ref_5","unstructured":"Daleckij, J., and Krein, M. (1974). Stability of Differential Equations in Banach Spaces. Amer. Math. Soc."},{"key":"ref_6","first-page":"20","article-title":"On stability of solutions of systems of differential equations","volume":"51","author":"Maizel","year":"1954","journal-title":"TRudi Uralskogo Politekh. Inst. Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Coppel, W. (1978). Dichotomies in Stability Theory. Lecture Notes in Mathematics, Springer.","DOI":"10.1007\/BFb0067780"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/0022-0396(84)90134-7","article-title":"Dichotomies and Lyapunov functions","volume":"52","author":"Coppel","year":"1984","journal-title":"J. Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1090\/S0002-9947-1984-0737880-1","article-title":"Dichotomies and asymptotic behaviour for linear differential systems","volume":"283","author":"Muldowney","year":"1984","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1016\/0022-247X(90)90082-Q","article-title":"Dichotomies in terms of Lyapunov functions for linear difference equations","volume":"152","author":"Papaschinopoulos","year":"1990","journal-title":"J. Math. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.jmaa.2012.10.004","article-title":"Lyapunov functions for strong exponential dichotomies","volume":"399","author":"Barreira","year":"2013","journal-title":"J. Math. Anal. 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Springer Briefs in Mathematics, Springer.","DOI":"10.1007\/978-3-319-90110-7"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3147","DOI":"10.1016\/j.jde.2017.04.041","article-title":"Lyapunov type characterization of hyperbolic behavior","volume":"263","author":"Barreira","year":"2017","journal-title":"J. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1134\/S1560354717030017","article-title":"Nonuniform exponential dichotomies and Lyapunov functions","volume":"22","author":"Barreira","year":"2017","journal-title":"Regul. Chaotic Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"7461","DOI":"10.1090\/tran\/7923","article-title":"Lyapunov-type characterisation of exponential dichotomies with applications to the heat and Klein-Gordon equations","volume":"372","author":"Chen","year":"2019","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s10955-012-0524-8","article-title":"Lyapunov functions and cone families","volume":"148","author":"Barreira","year":"2012","journal-title":"J. Stat. Phys."},{"key":"ref_20","first-page":"17","article-title":"Exponential instability of skew-evolution semiflows in Banach spaces","volume":"53","author":"Megan","year":"2008","journal-title":"Stud. Univ. Babes Bolyai Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4786","DOI":"10.1016\/j.jde.2019.10.037","article-title":"On the asymptotic behavior of discrete dynamical systems\u2014An ergodic theory approach","volume":"268","author":"Sasu","year":"2020","journal-title":"J. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1007\/s00233-018-9985-7","article-title":"Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line","volume":"98","author":"Lupa","year":"2019","journal-title":"Semigroup Forum"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"143","DOI":"10.36045\/bbms\/1102715145","article-title":"On uniform exponential stability of linear skew-product semiflows in Banach spaces","volume":"9","author":"Megan","year":"2002","journal-title":"Bull. Belg. Math. Soc. Simon. Stevin."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/j.jde.2004.07.019","article-title":"Sch\u00e4ffer spaces and uniform exponential stability of linear skew-product semiflows","volume":"212","author":"Preda","year":"2005","journal-title":"J. Differ. Equ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1080\/10236190412331314178","article-title":"Stability and stabilizability for linear systems of difference equations","volume":"10","author":"Sasu","year":"2004","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"869608","DOI":"10.1186\/1687-1847-2010-869608","article-title":"Stability of difference equations and applications to robustness problems","volume":"2010","author":"Sasu","year":"2010","journal-title":"Adv. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Backes, L., and Dragi\u010devi\u0107, D. (2019). A Rolewicz-type characterization of nonuniform behaviour. Appl. Anal., in press.","DOI":"10.1080\/00036811.2019.1707190"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1080\/10236198.2017.1408609","article-title":"Datko-Pazy conditions for nonuniform exponential stability","volume":"24","year":"2018","journal-title":"J. Differ. Equ. Appl."},{"key":"ref_29","unstructured":"Lupa, N., and Popescu, L.H. (2016). Admissible Banach function spaces and nonuniform stabilities. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1016\/j.jmaa.2011.06.082","article-title":"A version of a theorem of R. Datko for nonuniform exponential contractions","volume":"385","author":"Preda","year":"2012","journal-title":"J. Math Anal Appl."},{"key":"ref_31","first-page":"205","article-title":"On Rolewicz-Zabczyk techniques in the stability theory of dynamical systems","volume":"13","author":"Sasu","year":"2012","journal-title":"Fixed Point Theory"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1165","DOI":"10.1080\/00036810410001724391","article-title":"Generalizations of a theorem of Rolewicz","volume":"84","author":"Sasu","year":"2005","journal-title":"Appl Anal."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.bulsci.2009.06.006","article-title":"Integral conditions for exponential dichotomy: A nonlinear approach","volume":"134","author":"Sasu","year":"2010","journal-title":"Bull. Sci. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.laa.2018.04.018","article-title":"A note on the nonuniform exponential stability and dichotomy for nonautonomous difference equations","volume":"552","year":"2018","journal-title":"Linear Algebra Appl."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Dragi\u010devi\u0107, D. (2019). On the exponential stability and hyperbolicity of linear cocycles. Linear Multilinear Algebra, in press.","DOI":"10.1080\/03081087.2019.1594668"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/2\/47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:51:26Z","timestamp":1760363486000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/2\/47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,27]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["axioms9020047"],"URL":"https:\/\/doi.org\/10.3390\/axioms9020047","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,4,27]]}}}