{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:07:48Z","timestamp":1760242068069,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,11,26]],"date-time":"2018-11-26T00:00:00Z","timestamp":1543190400000},"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 develop the existence criteria for solutions of Liouville\u2013Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are also presented. It is imperative to mention that our results correspond to the ones for a symmetric second-order nonlocal multipoint integral boundary value problem under suitable conditions (see the last section).<\/jats:p>","DOI":"10.3390\/sym10120667","type":"journal-article","created":{"date-parts":[[2018,11,27]],"date-time":"2018-11-27T03:31:33Z","timestamp":1543289493000},"page":"667","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Generalized Liouville\u2013Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3452-8922","authenticated-orcid":false,"given":"Ahmed","family":"Alsaedi","sequence":"first","affiliation":[{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"given":"Madeaha","family":"Alghanmi","sequence":"additional","affiliation":[{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5350-2977","authenticated-orcid":false,"given":"Bashir","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-2118","authenticated-orcid":false,"given":"Sotiris K.","family":"Ntouyas","sequence":"additional","affiliation":[{"name":"Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"},{"name":"Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,26]]},"reference":[{"key":"ref_1","unstructured":"Magin, R.L. (2006). Fractional Calculus in Bioengineering, Begell House."},{"key":"ref_2","unstructured":"Zaslavsky, G.M. (2008). Hamiltonian Chaos and Fractional Dynamics, Oxford University Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1016\/j.ecolmodel.2015.06.016","article-title":"Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system","volume":"318","author":"Javidi","year":"2015","journal-title":"Ecol. Model."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Fallahgoul, H.A., Focardi, S.M., and Fabozzi, F.J. (2017). Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application, Elsevier\/Academic Press.","DOI":"10.1016\/B978-0-12-804248-9.50002-4"},{"key":"ref_5","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). North-Holland Mathematics Studies. Theory and Applications of Fractional Differential Equations, Elsevier Science B.V."},{"key":"ref_6","unstructured":"Lakshmikantham, V., Leela, S., and Devi, J.V. (2009). Theory of Fractional Dynamic Systems, Cambridge Academic Publishers."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag.","DOI":"10.1007\/978-3-642-14574-2"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ahmad, B., Alsaedi, A., Ntouyas, S.K., and Tariboon, J. (2017). Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer.","DOI":"10.1007\/978-3-319-52141-1"},{"key":"ref_9","first-page":"398","article-title":"On some simple generalizations of linear elliptic boundary problems","volume":"10","author":"Bitsadze","year":"1969","journal-title":"Russ. Acad. Sci. Dokl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"253","DOI":"10.15388\/NA.17.3.14054","article-title":"Numerical approximation of one model of the bacterial self-organization","volume":"17","author":"Ciegis","year":"2012","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1038\/nmat1300","article-title":"Slow relaxation and compaction of granular system","volume":"4","author":"Richard","year":"2005","journal-title":"Nat. Mater."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Quezada, J.C., Sagnol, L., and Chazallon, C. (2017). Shear Test on Viscoelastic Granular Material Using Contact Dynamics Simulations. EPJ Web of Conferences, Powders &amp, Grains.","DOI":"10.1051\/epjconf\/201714008009"},{"key":"ref_13","unstructured":"Pereira, F.L., de Sousa, J.B., and de Matos, A.C. (1995, January 13\u201315). An Algorithm for Optimal Control Problems Based on Differential Inclusions. Proceedings of the 34th Conference on Decision & Control, New Orleans, LA, USA."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2944","DOI":"10.1137\/130914565","article-title":"Convex computation of the maximum controlled invariant set for polynomial control systems","volume":"52","author":"Korda","year":"2014","journal-title":"SIAM J. Control Optim."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1007\/s00419-014-0837-y","article-title":"Study of a driven and braked wheel using maximal monotone differential inclusions: Applications to the nonlinear dynamics of wheeled vehicles","volume":"84","author":"Bastien","year":"2014","journal-title":"Arch. Appl. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kisielewicz, M. (2013). Stochastic Differential Inclusions and Applications, Springer.","DOI":"10.1007\/978-1-4614-6756-4"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2065","DOI":"10.1007\/s11071-014-1577-9","article-title":"Synchronization of piecewise continuous systems of fractional order","volume":"78","author":"Danca","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1186\/1687-2770-2011-36","article-title":"Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions","volume":"2011","author":"Ahmad","year":"2011","journal-title":"Bound. Val. Prob."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1334","DOI":"10.1016\/j.mcm.2011.04.004","article-title":"Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval","volume":"54","author":"Liang","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1016\/j.camwa.2011.12.078","article-title":"Existence and multiplicity of positive solutions for singular fractional boundary value problems","volume":"63","author":"Bai","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1002\/mana.201000043","article-title":"Positive solutions for mixed problems of singular fractional differential equations","volume":"285","author":"Agarwal","year":"2012","journal-title":"Math. Nachr."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1155\/2013\/320415","article-title":"A study of nonlinear fractional differential equations of arbitrary order with Riemann\u2013Liouville type multistrip boundary conditions","volume":"2013","author":"Ahmad","year":"2013","journal-title":"Math. Probl. Eng."},{"key":"ref_23","first-page":"1","article-title":"Existence results for higher order fractional differential inclusions with multi-strip fractional integral boundary conditions","volume":"2013","author":"Ahmad","year":"2013","journal-title":"Electron. J. Qual. Theory Differ. Equat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1007\/s11071-012-0443-x","article-title":"Fractional boundary value problems with singularities in space variables","volume":"71","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2820","DOI":"10.1016\/j.cnsns.2014.01.003","article-title":"Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter","volume":"19","author":"Zhai","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"499","DOI":"10.2478\/s13540-014-0182-4","article-title":"Existence and uniqueness of solutions for a fractional boundary value problem on a graph","volume":"17","author":"Graef","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Wang, G., Liu, S., and Zhang, L. (2014). Eigenvalue problem for nonlinear fractional differential equations with integral boundary conditions. Abstr. Appl. Anal., 2014.","DOI":"10.1155\/2014\/916260"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"872","DOI":"10.2478\/s13540-014-0202-4","article-title":"Eigenvalue comparison for fractional boundary value problems with the Caputo derivative","volume":"17","author":"Henderson","year":"2014","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1017\/S0004972714000550","article-title":"Successive iterations for positive extremal solutions of nonlinear fractional differential equations on a half line","volume":"91","author":"Zhang","year":"2015","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1186\/s13661-015-0403-8","article-title":"Nonexistence of positive solutions for a system of coupled fractional boundary value problems","volume":"2015","author":"Henderson","year":"2015","journal-title":"Bound. Val. Probl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.amc.2015.05.036","article-title":"On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions","volume":"266","author":"Ntouyas","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1016\/j.indag.2015.05.004","article-title":"Existence and uniqueness of solutions for nonlinear general fractional differential equations in Banach spaces","volume":"26","author":"Mei","year":"2015","journal-title":"Indagat. Math."},{"key":"ref_33","first-page":"5","article-title":"Existence results for fractional differential inclusions with Erdelyi-Kober fractional integral conditions","volume":"25","author":"Ahmad","year":"2017","journal-title":"Ann. Univ. Ovidius Constanta-Seria Mat."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"560","DOI":"10.1080\/10652469.2017.1317248","article-title":"Remarks on some families of fractional-order differential equations","volume":"28","author":"Srivastava","year":"2017","journal-title":"Integral Transforms Spec. Funct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.cam.2018.04.062","article-title":"Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line","volume":"343","author":"Wang","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1016\/j.amc.2018.07.025","article-title":"Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions","volume":"339","author":"Ahmad","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"860","DOI":"10.1016\/j.amc.2011.03.062","article-title":"New approach to a generalized fractional integral","volume":"218","author":"Katugampola","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_38","first-page":"1","article-title":"A new approach to generalized fractional derivatives","volume":"6","author":"Katugampola","year":"2014","journal-title":"Bull. Math. Anal. Appl."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2607","DOI":"10.22436\/jnsa.010.05.27","article-title":"On the generalized fractional derivatives and their Caputo modification","volume":"10","author":"Jarad","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Granas, A., and Dugundji, J. (2003). Fixed Point Theory, Springer-Verlag.","DOI":"10.1007\/978-0-387-21593-8"},{"key":"ref_41","first-page":"123","article-title":"Two remarks on the method of successive approximations","volume":"10","author":"Krasnoselskii","year":"1955","journal-title":"Uspekhi Mat. Nauk"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Deimling, K. (1992). Multivalued Differential Equations, Walter De Gruyter.","DOI":"10.1515\/9783110874228"},{"key":"ref_43","first-page":"781","article-title":"An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations","volume":"13","author":"Lasota","year":"1965","journal-title":"Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/BF02771543","article-title":"Multivalued contraction mappings in generalized metric spaces","volume":"8","author":"Covitz","year":"1970","journal-title":"Israel J. Math."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Castaing, C., and Valadier, M. (1977). Convex Analysis and Measurable Multifunctions, Springer-Verlag.","DOI":"10.1007\/BFb0087685"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/S0375-9601(00)00201-2","article-title":"Fractional quantum mechanics and Levy path integrals","volume":"268","author":"Laskin","year":"2000","journal-title":"Phys. Lett. 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