{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T11:03:14Z","timestamp":1777546994961,"version":"3.51.4"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,21]],"date-time":"2021-05-21T00:00:00Z","timestamp":1621555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use o(un) as a measure of approximation.<\/jats:p>","DOI":"10.3390\/sym13060918","type":"journal-article","created":{"date-parts":[[2021,5,21]],"date-time":"2021-05-21T02:13:46Z","timestamp":1621563226000},"page":"918","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Asymptotic Properties of Solutions to Discrete Volterra Monotone Type Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0855-6180","authenticated-orcid":false,"given":"Janusz","family":"Migda","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Uniwersytetu Pozna\u0144skiego 4, 61-614 Pozna\u0144, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3188-1173","authenticated-orcid":false,"given":"Ma\u0142gorzata","family":"Migda","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, Pozna\u0144 University of Technology, Piotrowo 3A, 60-965 Pozna\u0144, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8893-2205","authenticated-orcid":false,"given":"Ewa","family":"Schmeidel","sequence":"additional","affiliation":[{"name":"Institute of Computer Sciences, University of Bialystok, Cio\u0142kowskiego 1M, 15-245 Bia\u0142ystok, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,21]]},"reference":[{"key":"ref_1","unstructured":"Burton, T.A. (2005). Volterra Integral and Differential Equations, Elsevier B.V.. [2nd ed.]. Mathematics in Science and Engineering, 202."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wazwaz, A.M. (2011). Linear and Nonlinear Integral Equations. Methods and Applications, Springer.","DOI":"10.1007\/978-3-642-21449-3"},{"key":"ref_3","first-page":"17","article-title":"Integral equations. Volterra equations, and the remarkable resolvent: Contractions","volume":"2","author":"Burton","year":"2006","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_4","first-page":"1","article-title":"Existence and asymptotic behavior of solutions for some nonlinear integral equations on an unbounded interval","volume":"2016","year":"2016","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.15352\/bjma\/1337014661","article-title":"Convex majorants method in the theory of nonlinear Volterra equations","volume":"6","author":"Sidorov","year":"2012","journal-title":"Banach J. Math. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/j.jtusci.2015.12.009","article-title":"Qualitative properties in nonlinear Volterra integro-differential equations with delay","volume":"11","year":"2017","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Anh, P.T., Babiarz, A., Czornik, A., Niezabitowski, M., and Siegmund, S. (2019, January 26\u201329). On asymptotic properties of discrete Volterra equations of convolution type. Proceedings of the 2019 24th International Conference on Methods and Models in Automation and Robotics (MMAR), Mi\u0119dzyzdroje, Poland.","DOI":"10.1109\/MMAR.2019.8864658"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1080\/10236190600986594","article-title":"On exact convergence rates for solutions of linear systems of Volterra difference equations","volume":"12","author":"Appleby","year":"2006","journal-title":"J. Differ. Equ. 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