{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T02:49:16Z","timestamp":1772678956852,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T00:00:00Z","timestamp":1616025600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.<\/jats:p>","DOI":"10.3390\/fractalfract5010024","type":"journal-article","created":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T22:19:36Z","timestamp":1616105976000},"page":"24","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter"],"prefix":"10.3390","volume":"5","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3535-3909","authenticated-orcid":false,"given":"Nat\u00e1lia","family":"Martins","sequence":"additional","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":[[2021,3,18]]},"reference":[{"key":"ref_1","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"542","DOI":"10.1121\/1.3268508","article-title":"A unifying fractional wave equation for compressional and shear waves","volume":"127","author":"Holm","year":"2010","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_5","first-page":"47","article-title":"Discrete-time fractional-order controllers","volume":"4","author":"Machado","year":"2001","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.amc.2014.12.136","article-title":"Heat conduction modeling by using fractional-order derivatives","volume":"257","author":"Terpxaxk","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_7","first-page":"357","article-title":"Fractional differential analysis for texture of digital image","volume":"1","author":"Pu","year":"2007","journal-title":"J. Alg. Comput. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1615\/CritRevBiomedEng.v32.10","article-title":"Fractional calculus in bioengineering","volume":"32","author":"Magin","year":"2004","journal-title":"Crit. Rev. Biomed. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.cam.2016.05.019","article-title":"Fractional order model for HIV dynamics","volume":"312","author":"Pinto","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"189","DOI":"10.2478\/s13540-013-0013-z","article-title":"Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping","volume":"16","author":"Chen","year":"2013","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.camwa.2010.10.030","article-title":"Optimal control of fractional diffusion equation","volume":"61","author":"Mophou","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Saeedian, M., Khalighi, M., Azimi\u2013Tafreshi, N., Jafari, G.R., and Ausloos, M. (2017). Memory effects on epidemic evolution: The susceptible-infected-recovered epidemic model. Phys. Rev. E, 95.","DOI":"10.1103\/PhysRevE.95.022409"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1322","DOI":"10.1016\/j.econmod.2012.03.019","article-title":"Modeling of the national economies in state-space: A fractional calculus approach","volume":"29","author":"Podlubny","year":"2012","journal-title":"Econ. Model."},{"key":"ref_14","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_15","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_16","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_17","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_18","doi-asserted-by":"crossref","unstructured":"Malinowska, A.B., Odzijewicz, T., and Torres, \u1e0a.F.M. (2015). Advanced Methods in the Fractional Calculus of Variations, SpringerBriefs in Applied Sciences and Technology, Springer.","DOI":"10.1007\/978-3-319-14756-7"},{"key":"ref_19","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_20","doi-asserted-by":"crossref","unstructured":"Atanackovi\u0107, T.M., Konjik, S., and Pilipovi\u0107, \u1e60 (2008). Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A, 41.","DOI":"10.1088\/1751-8113\/41\/9\/095201"},{"key":"ref_21","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_22","first-page":"3","article-title":"Existence of minimizers for fractional variational problems containing Caputo derivatives","volume":"8","author":"Bourdin","year":"2013","journal-title":"Adv. Dyn. Syst. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/s11071-009-9486-z","article-title":"Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations","volume":"58","author":"Herzallah","year":"2009","journal-title":"Nonlinear Dynam."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1023\/A:1021389004982","article-title":"Lagrangean and Hamiltonian fractional sequential mechanics","volume":"52","author":"Klimek","year":"2002","journal-title":"Czechoslov. J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1016\/S0022-247X(02)00180-4","article-title":"Formulation of Euler-Lagrange equations for fractional variational problems","volume":"272","author":"Agrawal","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_26","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_27","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. Ser. S"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1590\/S0101-74382013000100004","article-title":"A non-standard optimal control problem arising in an economics application","volume":"33","author":"Zinober","year":"2013","journal-title":"Pesqui. Oper."},{"key":"ref_29","first-page":"1357","article-title":"Stability results for constrained calculus of variations problems: An analysis of the twisted elastic loop","volume":"461","author":"Hoffman","year":"2005","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3110","DOI":"10.1016\/j.camwa.2010.02.032","article-title":"Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative","volume":"59","author":"Malinowska","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Martins, N. (2021). A non-standard class of variational problems of Herglotz type. Discrete Contin. Dyn. Syst., in press.","DOI":"10.3934\/dcdss.2021152"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Van Brunt, B. (2004). The Calculus of Variations, Universitext, Springer.","DOI":"10.1007\/b97436"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1177\/1077546307077472","article-title":"Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative","volume":"13","author":"Agrawal","year":"2007","journal-title":"J. Vib. Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/S0252-9602(12)60043-5","article-title":"Isoperimetric problems of the calculus of variations with fractional derivatives","volume":"32","author":"Almeida","year":"2012","journal-title":"Acta Math. Sci. Ser. 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