{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:32:44Z","timestamp":1760401964867,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,5,1]],"date-time":"2020-05-01T00:00:00Z","timestamp":1588291200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we study the existence of solutions for nonlocal single and multi-valued boundary value problems involving right-Caputo and left-Riemann\u2013Liouville fractional derivatives of different orders and right-left Riemann\u2013Liouville fractional integrals. The existence of solutions for the single-valued case relies on Sadovskii\u2019s fixed point theorem. The first existence results for the multi-valued case are proved by applying Bohnenblust-Karlin\u2019s fixed point theorem, while the second one is based on Martelli\u2019s fixed point theorem. We also demonstrate the applications of the obtained results.<\/jats:p>","DOI":"10.3390\/axioms9020050","type":"journal-article","created":{"date-parts":[[2020,5,4]],"date-time":"2020-05-04T03:29:39Z","timestamp":1588562979000},"page":"50","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals"],"prefix":"10.3390","volume":"9","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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abrar","family":"Broom","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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1016\/j.camwa.2013.03.012","article-title":"Fractional differential equations and related exact mechanical models","volume":"66","author":"Paola","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"052002","DOI":"10.1063\/1.5032165","article-title":"Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel","volume":"30","author":"Ahmed","year":"2018","journal-title":"Phys. Fluids"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer.","DOI":"10.1007\/978-3-642-14003-7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.chaos.2016.12.012","article-title":"Logistic map with memory from economic model","volume":"95","author":"Tarasova","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1515\/fca-2015-0024","article-title":"On a system of fractional differential equations with coupled integral boundary conditions","volume":"18","author":"Henderson","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_6","first-page":"458","article-title":"Bifurcation from interval and positive solutions of the three-point boundary value problem for fractional differential equations","volume":"257","author":"Peng","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_7","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_8","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1216\/RMJ-2018-48-4-1043","article-title":"Nonlocal initial value problems for Hadamard-type fractional differential equations and inclusions","volume":"48","author":"Ahmad","year":"2018","journal-title":"Rocky Mt. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"31","DOI":"10.15388\/NA.2018.1.3","article-title":"New uniqueness results for boundary value problem of fractional differential equation","volume":"23","author":"Cui","year":"2018","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1515\/fca-2019-0034","article-title":"A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions","volume":"22","author":"Ahmad","year":"2019","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"113120","DOI":"10.1016\/j.aml.2018.12.006","article-title":"Extremal solutions for generalized Caputo fractional differential equations with Steiltjes-type fractional integro-initial conditions","volume":"91","author":"Alsaedi","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s11590-019-01437-6","article-title":"Fractional differential equation approach for convex optimization with convergence rate analysis","volume":"14","author":"Liang","year":"2020","journal-title":"Optim. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.aml.2013.12.014","article-title":"The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative","volume":"31","author":"Zhang","year":"2014","journal-title":"Appl. Math. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Khaldi, R., and Guezane-Lakoud, A. (2017). Higher order fractional boundary value problems for mixed type derivatives. J. Nonlinear Funct. Anal.","DOI":"10.23952\/jnfa.2017.30"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1186\/s13662-017-1226-y","article-title":"Existence of solutions for a mixed fractional boundary value problem","volume":"2017","author":"Lakoud","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","first-page":"937","article-title":"Existence theory for nonlocal boundary value problems involving mixed fractional derivatives","volume":"24","author":"Ahmad","year":"2019","journal-title":"Nonlinear Anal. Model. Control"},{"key":"ref_17","first-page":"139","article-title":"On a differential equation with left and right fractional derivatives","volume":"10","author":"Atanackovic","year":"2007","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"S789","DOI":"10.2298\/TSCI16S3789Y","article-title":"On nonlocal fractional Volterra integro-differential equations in fractional steady heat transfer","volume":"20","author":"Yang","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s11232-009-0029-z","article-title":"Fractional integro-differential equations for electromagnetic waves in dielectric media","volume":"158","author":"Tarasov","year":"2009","journal-title":"Theor. Math. Phys."},{"key":"ref_20","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_21","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_22","doi-asserted-by":"crossref","unstructured":"Kisielewicz, M. (2013). Stochastic Differential Inclusions and Applications, Springer.","DOI":"10.1007\/978-1-4614-6756-4"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.aml.2018.08.010","article-title":"Infinitely many nonnegative solutions for a fractional differential inclusion with oscillatory potential","volume":"88","author":"Yue","year":"2019","journal-title":"Appl. Math. Lett."},{"key":"ref_25","first-page":"3713","article-title":"Mild solutions to the time fractional Navier-Stokes delay differential inclusions","volume":"24","author":"Wang","year":"2019","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1186\/s13662-019-2026-3","article-title":"Oscillation and nonoscillation for Caputo-Hadamard impulsive fractional differential inclusions","volume":"2019","author":"Benchohra","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"269","DOI":"10.24193\/fpt-ro.2017.1.22","article-title":"On semilinear fractional order differential inclusions in Banach spaces","volume":"18","author":"Kamenskii","year":"2017","journal-title":"Fixed Point Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"960","DOI":"10.1515\/fca-2018-0053","article-title":"Existence and controllability for nonlinear fractional differential inclusions with nonlocal boundary conditions and time-varying delay","volume":"21","author":"Cheng","year":"2018","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2479","DOI":"10.3934\/cpaa.2018118","article-title":"Coupled systems of Hilfer fractional differential inclusions in Banach spaces","volume":"17","author":"Abbas","year":"2018","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ntouyas, S.K., Alsaedi, A., and Ahmed, B. (2019). Existence theorems for mixed Riemann-Liouville and Caputo fractional differential equations and inclusions with nonlocal fractional integro-differential boundary conditions. Fractal Fract., 3.","DOI":"10.3390\/fractalfract3020021"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2210","DOI":"10.1080\/00207179.2018.1433331","article-title":"Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille-Yosida operators","volume":"92","author":"Vijayakumar","year":"2019","journal-title":"Internat. J. Control"},{"key":"ref_32","first-page":"1","article-title":"Coupled systems of fractional differential inclusions with coupled boundary conditions","volume":"2019","author":"Ahmad","year":"2019","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_33","first-page":"169","article-title":"On inclusion problems involving Caputo and Hadamard fractional derivatives","volume":"89","author":"Ahmad","year":"2020","journal-title":"Acta Math. Univ. Comenian. (N.S.)"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ahmad, B., Broom, A., Alsaedi, A., and Ntouyas, S.K. (2020). Nonlinear integro-differential equations involving mixed right and left fractional derivatives and integrals with nonlocal boundary data. Mathematics, 8.","DOI":"10.3390\/math8030336"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/0377-0427(89)90320-8","article-title":"A class of nonlocal boundary value problems for partial differential equations and its applications in numerical analysis","volume":"28","author":"Li","year":"1989","journal-title":"J. Comput. Appl. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"494720","DOI":"10.1155\/2009\/494720","article-title":"Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations","volume":"2009","author":"Ahmad","year":"2009","journal-title":"Abstr. Appl. Anal."},{"key":"ref_37","first-page":"155","article-title":"On a theorem of Ville","volume":"Volume I","author":"Bohnenblust","year":"1950","journal-title":"Contributions to the Theory of Games"},{"key":"ref_38","first-page":"70","article-title":"A Rothe\u2019s theorem for non compact acyclic-valued maps","volume":"4","author":"Martelli","year":"1975","journal-title":"Boll. Un. Mat. Ital."},{"key":"ref_39","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_40","unstructured":"Granas, A., and Dugundji, J. (2005). Fixed Point Theory, Springer-Verlag."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Zeidler, E. (1986). Nonlinear Functional Analysis and Its Application: Fixed Point-Theorems, Springer-Verlag.","DOI":"10.1007\/978-1-4612-4838-5"},{"key":"ref_42","first-page":"74","article-title":"On a fixed point principle","volume":"1","author":"Sadovskii","year":"1967","journal-title":"Funct. Anal. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Deimling, K. (1992). Multivalued Differential Equations, De Gruyter.","DOI":"10.1515\/9783110874228"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1006\/jmaa.2000.6789","article-title":"Fixed point theorems for set-valued maps and existence principles for integral inclusions","volume":"245","author":"Precup","year":"2000","journal-title":"J. Math. Anal. Appl."},{"key":"ref_45","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."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/2\/50\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:25:06Z","timestamp":1760361906000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/2\/50"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,5,1]]},"references-count":45,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["axioms9020050"],"URL":"https:\/\/doi.org\/10.3390\/axioms9020050","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,5,1]]}}}