{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T07:45:11Z","timestamp":1768722311306,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"14","license":[{"start":{"date-parts":[[2023,7,21]],"date-time":"2023-07-21T00:00:00Z","timestamp":1689897600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Portuguese funds through the CIDMA","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}]},{"name":"Portuguese Foundation for Science and Technology","award":["UIDB\/04106\/2020"],"award-info":[{"award-number":["UIDB\/04106\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>The goal of this paper is to present the necessary and sufficient conditions that every extremizer of a given class of functionals, defined on the set C1[a,b], must satisfy. The Lagrange function depends on a generalized fractional derivative, on a generalized fractional integral, and on an antiderivative involving the previous fractional operators. We begin by obtaining the fractional Euler\u2013Lagrange equation, which is a necessary condition to optimize a given functional. By imposing convexity conditions over the Lagrange function, we prove that it is also a sufficient condition for optimization. After this, we consider variational problems with additional constraints on the set of admissible functions, such as the isoperimetric and the holonomic problems. We end by considering a generalization of the fundamental problem, where the fractional order is not restricted to real values between 0 and 1, but may take any positive real value. We also present some examples to illustrate our results.<\/jats:p>","DOI":"10.3390\/math11143208","type":"journal-article","created":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T01:12:28Z","timestamp":1690161148000},"page":"3208","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Euler\u2013Lagrange-Type Equations for Functionals Involving Fractional Operators and Antiderivatives"],"prefix":"10.3390","volume":"11","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, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,21]]},"reference":[{"key":"ref_1","unstructured":"Silverman, R.A. (1963). Calculus of Variations (Revised English Edition. Transl. from Russian), Prentice-Hall."},{"key":"ref_2","unstructured":"Ioffe, A.D., and Tihomirov, V.M. (1979). Theory of Extremal Problems (Transl. from Russian), Elsevier."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Mesterton-Gibbons, M. (2009). A Primer on the Calculus of Variations and Optimal Control Theory, American Mathematical Society.","DOI":"10.1090\/stml\/050"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/3779"},{"key":"ref_5","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_6","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives (Translated from the 1987 Russian Original), Gordon and Breach."},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1137\/0118059","article-title":"Leibniz Rule for Fractional Derivatives and an Application to Infinite Series","volume":"18","author":"Osler","year":"1970","journal-title":"SIAM J. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"7359","DOI":"10.3934\/math.2020471","article-title":"Properties of positive solutions for a fractional boundary value problem involving fractional derivative with respect to another function","volume":"5","author":"Yang","year":"2020","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"6749","DOI":"10.3934\/math.2021397","article-title":"Langevin equation with nonlocal boundary conditions involving a \u03c8-Caputo fractional operators of different orders","volume":"6","author":"Seemab","year":"2021","journal-title":"AIMS Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"10375","DOI":"10.1088\/0305-4470\/39\/33\/008","article-title":"Fractional variational calculus and the transversality conditions","volume":"39","author":"Agrawal","year":"2006","journal-title":"J. Phys. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1816","DOI":"10.1016\/j.aml.2009.07.002","article-title":"Calculus of variations with fractional derivatives and fractional integrals","volume":"22","author":"Almeida","year":"2009","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3097","DOI":"10.1016\/j.camwa.2011.03.098","article-title":"Fractional variational calculus for nondifferentiable functions","volume":"61","author":"Almeida","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_14","first-page":"1","article-title":"Optimality conditions for fractional variational problems with free terminal time","volume":"11","author":"Almeida","year":"2018","journal-title":"Discret. Contin. Dyn. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2513","DOI":"10.1177\/1077546320961685","article-title":"A new generalization of the fractional Euler\u2013Lagrange equation for a vertical mass-spring-damper","volume":"27","author":"Baleanu","year":"2021","journal-title":"J. Vib. Control"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1134\/S0005117913040012","article-title":"Fractional integro-differential calculus and its control-theoretical applications. I. Mathematical fundamentals and the problem of interpretation","volume":"74","author":"Butkovskii","year":"2013","journal-title":"Autom. Remote Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1134\/S0005117913050019","article-title":"Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: Modeling and hardware implementation","volume":"74","author":"Butkovskii","year":"2013","journal-title":"Autom. Remote Control"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Tenreiro Machado, J.A., and Luo, A.C.J. (2012). Fractional Dynamics and Control, Springer.","DOI":"10.1007\/978-1-4614-0457-6"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zhou, Y. (2016). Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press.","DOI":"10.1016\/B978-0-12-804277-9.50002-X"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"van Brunt, B. (2004). The Calculus of Variations, Universitext Springer.","DOI":"10.1007\/b97436"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3581","DOI":"10.1103\/PhysRevE.55.3581","article-title":"Mechanics with fractional derivatives","volume":"55","author":"Riewe","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1103\/PhysRevE.53.1890","article-title":"Nonconservative Lagrangian and Hamiltonian mechanics","volume":"53","author":"Riewe","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","unstructured":"Almeida, R., Pooseh, S., and Torres, D.F.M. (2015). Computational Methods in the Fractional Calculus of Variations, Imperial College Press.","DOI":"10.1142\/p991"},{"key":"ref_25","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_26","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 International Publishing. Springer Briefs in Applied Sciences and Technology.","DOI":"10.1007\/978-3-319-14756-7_3"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1016\/j.jmaa.2004.09.043","article-title":"Hamiltonian formulation of systems with linear velocities within Riemann\u2013Liouville fractional derivatives","volume":"304","author":"Muslih","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1016\/j.jmaa.2006.04.076","article-title":"The Hamilton formalism with fractional derivatives","volume":"327","author":"Rabei","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1016\/j.na.2011.02.028","article-title":"Fractional variational problems depending on indefinite integrals","volume":"75","author":"Almeida","year":"2012","journal-title":"Nonlinear Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"427","DOI":"10.4310\/MAA.2008.v15.n4.a2","article-title":"Generalizing variational theory to include the indefinite integral, higher derivatives, and a variety of means as cost variables","volume":"15","author":"Gregory","year":"2008","journal-title":"Methods Appl. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2424","DOI":"10.1016\/j.camwa.2011.02.022","article-title":"Generalizing the variational theory on time scales to include the delta indefinite integral","volume":"61","author":"Martins","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.1216\/RMJ-2019-49-8-2459","article-title":"Further properties of Osler\u2019s generalized fractional integrals and derivatives with respect to another function","volume":"49","author":"Almeida","year":"2019","journal-title":"Rocky Mountain J. Math."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/14\/3208\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:16:38Z","timestamp":1760127398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/11\/14\/3208"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,21]]},"references-count":32,"journal-issue":{"issue":"14","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["math11143208"],"URL":"https:\/\/doi.org\/10.3390\/math11143208","relation":{},"ISSN":["2227-7390"],"issn-type":[{"value":"2227-7390","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,21]]}}}