{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T04:13:21Z","timestamp":1762056801086,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,25]],"date-time":"2022-06-25T00:00:00Z","timestamp":1656115200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CIDMA-Center for Research and Development in Mathematics and Applications","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with and without boundary conditions, and with additional restrictions like isoperimetric and holonomic. Herglotz\u2019s variational problem and when in presence of time delays are also considered.<\/jats:p>","DOI":"10.3390\/fractalfract6070356","type":"journal-article","created":{"date-parts":[[2022,6,25]],"date-time":"2022-06-25T10:39:13Z","timestamp":1656153553000},"page":"356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Minimization Problems for Functionals Depending on Generalized Proportional Fractional Derivatives"],"prefix":"10.3390","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1305-2411","authenticated-orcid":false,"given":"Ricardo","family":"Almeida","sequence":"first","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,25]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier Science B.V.. North-Holland Mathematics Studies, 204."},{"key":"ref_2","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives, Gordon and Breach. Translated from the 1987 Russian Original."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1186\/s13662-020-02576-2","article-title":"On Hilfer fractional difference operator","volume":"2020","author":"Haider","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1616","DOI":"10.1016\/j.camwa.2012.01.009","article-title":"Existence and uniqueness for a problem involving Hilfer fractional derivative","volume":"64","author":"Furati","year":"1012","journal-title":"Comput. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.cnsns.2016.09.006","article-title":"A Caputo fractional derivative of a function with respect to another function","volume":"44","author":"Almeida","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1137\/0501026","article-title":"Fractional derivatives of a composite function","volume":"1","author":"Osler","year":"1970","journal-title":"SIAM J. Math. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.cnsns.2018.01.005","article-title":"On the g-Hilfer fractional derivative","volume":"60","author":"Sousa","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3351","DOI":"10.1016\/j.camwa.2012.01.073","article-title":"Generalized fractional calculus with applications to the calculus of variations","volume":"64","author":"Odzijewicz","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"871912","DOI":"10.1155\/2012\/871912","article-title":"Fractional calculus of variations in terms of a generalized fractional integral with applications to physics","volume":"2012","author":"Odzijewicz","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Malinowska, A.B., Odzijewicz, T., and Torres, D.F.M. (2015). Advanced Methods in the Fractional Calculus of Variations, Springer. Springer Briefs in AppliedSciences and Technology.","DOI":"10.1007\/978-3-319-14756-7"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Malinowska, A.B., and Torres, D.F.M. (2012). Introduction to the Fractional Calculus of Variations, Imperial College Press.","DOI":"10.1142\/p871"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1016\/S0022-247X(02)00180-4","article-title":"Formulation of Euler\u2013Lagrange equations for fractional variational problems","volume":"272","author":"Agrawal","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s11071-007-9296-0","article-title":"On fractional Euler\u2013Lagrange and Hamilton equations and the fractional generalization of total time derivative","volume":"53","author":"Baleanu","year":"2008","journal-title":"Nonlinear Dynam."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"021102","DOI":"10.1115\/1.2833586","article-title":"Fractional constrained systems and Caputo derivatives","volume":"3","author":"Baleanu","year":"2008","journal-title":"J. Comput. Nonlinear Dynam."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1023\/A:1013378221617","article-title":"Fractional sequential mechanics\u2013models with symmetric fractional derivative","volume":"51","author":"Klimek","year":"2001","journal-title":"Czech. J. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6287","DOI":"10.1088\/1751-8113\/40\/24\/003","article-title":"Fractional variational calculus in terms of Riesz fractional derivatives","volume":"40","author":"Agrawal","year":"2007","journal-title":"J. Phys. A"},{"key":"ref_17","first-page":"1","article-title":"Optimality conditions for fractional variational problems with free terminal time","volume":"11","author":"Almeida","year":"2018","journal-title":"Discrete Contin. Dyn. Syst. S"},{"key":"ref_18","first-page":"2617","article-title":"Generalized fractional isoperimetric problem of several variables","volume":"19","author":"Odzijewicz","year":"2014","journal-title":"Discrete Contin. Dyn. Syst. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1813","DOI":"10.1140\/epjst\/e2013-01966-0","article-title":"Fractional calculus of variations of several independent variables","volume":"222","author":"Odzijewicz","year":"2013","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1186\/s13662-020-02792-w","article-title":"On Hilfer generalized proportional fractional derivative","volume":"2020","author":"Ahmed","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","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. Top."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"82","DOI":"10.3934\/math.2022005","article-title":"On \u03c8-Hilfer generalized proportional fractional operators","volume":"7","author":"Mallah","year":"2022","journal-title":"AIMS Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Agarwal, R., Hristova, S., and O\u2019Regan, D. (2022). Stability of generalized proportional Caputo fractional differential equations by lyapunov functions. Fractal Fract., 2022.","DOI":"10.1186\/s13661-022-01595-0"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Almeida, R., Agarwal, R.P., Hristova, S., and O\u2019Regan, D. (2022). Stability of gene regulatory networks modeled by generalized proportional Caputo fractional differential equations. Entropy, 24.","DOI":"10.3390\/e24030372"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1186\/s13661-022-01595-0","article-title":"Stability for generalized Caputo proportional fractional delay integro-differential equations","volume":"2022","author":"Bohner","year":"2022","journal-title":"Bound. Value Probl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6316477","DOI":"10.1155\/2021\/6316477","article-title":"Langevin equations with generalized proportional Hadamard\u2013Caputo fractional derivative","volume":"2021","author":"Barakat","year":"2021","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1186\/s13662-020-02767-x","article-title":"More properties of the proportional fractional integrals and derivatives of a function with respect to another function","volume":"2020","author":"Jarad","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1515\/math-2020-0014","article-title":"On more general forms of proportional fractional operators","volume":"18","author":"Jarad","year":"2020","journal-title":"Open Math."},{"key":"ref_29","first-page":"3703","article-title":"Some new bounds analogous to generalized proportional fractional integral operator with respect to another function","volume":"14","author":"Rashid","year":"2021","journal-title":"Discret. Contin. Dyn. Syst. S"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"4593","DOI":"10.3934\/dcds.2015.35.4593","article-title":"Variational problems of Herglotz type with time delay: Dubois\u2013Reymond condition and Noether\u2019s first theorem","volume":"35","author":"Santos","year":"2015","journal-title":"Discret. Contin. Dyn. Syst."}],"container-title":["Fractal and Fractional"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2504-3110\/6\/7\/356\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:38:30Z","timestamp":1760139510000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2504-3110\/6\/7\/356"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,25]]},"references-count":30,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["fractalfract6070356"],"URL":"https:\/\/doi.org\/10.3390\/fractalfract6070356","relation":{},"ISSN":["2504-3110"],"issn-type":[{"type":"electronic","value":"2504-3110"}],"subject":[],"published":{"date-parts":[[2022,6,25]]}}}