{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:16:40Z","timestamp":1760235400859,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T00:00:00Z","timestamp":1629331200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.<\/jats:p>","DOI":"10.3390\/math9161979","type":"journal-article","created":{"date-parts":[[2021,8,19]],"date-time":"2021-08-19T04:13:54Z","timestamp":1629346434000},"page":"1979","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0634-2370","authenticated-orcid":false,"given":"Ravi P.","family":"Agarwal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA"},{"name":"Florida Institute of Technology, Distinguished University Professor of Mathematics, Melbourne, FL 32901, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4922-641X","authenticated-orcid":false,"given":"Snezhana","family":"Hristova","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria"}]},{"given":"Donal","family":"O\u2019Regan","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 TK33 Galway, Ireland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1305-2411","authenticated-orcid":false,"given":"Ricardo","family":"Almeida","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"10432","DOI":"10.1002\/mma.7419","article-title":"Non-instantaneous impulsive fractional integro-differential equations with proportional fractional derivatives with respect to another function","volume":"44","author":"Abbas","year":"2021","journal-title":"Math. 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Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Higher Education Press.","DOI":"10.1007\/978-3-642-14003-7"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1186\/s13662-019-2038-z","article-title":"Existence results for nonlinear fractional boundary value problem involving generalized proportional derivative","volume":"2019","author":"Shammakh","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3457","DOI":"10.1140\/epjst\/e2018-00021-7","article-title":"Generalized fractional derivatives generated by a class of local proportional derivatives","volume":"226","author":"Jarad","year":"2017","journal-title":"Eur. Phys. J. Spec. 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Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.apnum.2021.04.015","article-title":"Analysis and numerical solution of the generalized proportional fractional Cauchy problem","volume":"167","author":"Boucenna","year":"2021","journal-title":"Appl. Num. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hristova, S., and Abbas, M.I. (2021). Explicit solutions of initial value problems for fractional generalized proportional differential equations with and without impulses. Symmetry, 13.","DOI":"10.3390\/sym13060996"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Benchohra, M., Henderson, J., and Ntouyas, S.K. (2006). Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation.","DOI":"10.1155\/9789775945501"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S. (1989). 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