Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
fractional Langevin equation; Caputo fractional derivative; integrable solution; existence; uniqueness; initial value problem; fixed point theorem
fractional Langevin equation; Caputo fractional derivative; integrable solution; existence; uniqueness; initial value problem; fixed point theorem
Summary:
We study the existence and uniqueness of integrable solutions to fractional Langevin equations involving two fractional orders with initial value problems. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem. Examples are provided to illustrate the main results.
References:
[1] Ahmad, B., Nieto, J. J., Alsaedi, A.: A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders. Adv. Difference Equ. 2012 (2012), Article ID 54, 16 pages. DOI 10.1186/1687-1847-2012-54 | MR 2944130 | Zbl 1291.34004
[2] Ahmed, B., Nieto, J. J., Alsaedi, A., El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 13 (2012), 599-606. DOI 10.1016/j.nonrwa.2011.07.052 | MR 2846866 | Zbl 1238.34008
[3] Ahmad, B., Ntouyas, S.: Existence of solutions for fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions. Math. Bohem. 139 (2014), 451-465. DOI 10.21136/MB.2014.143936 | MR 3269368 | Zbl 1340.34056
[4] Baghani, H.: Existence and uniqueness of solutions to fractional Langevin equations involving two fractional orders. J. Fixed Point Theory Appl. 20 (2018), Article ID 63, 7 pages. DOI 10.1007/s11784-018-0540-7 | MR 3777810 | Zbl 1397.34017
[5] Bhairat, S. P., Dhaigude, D.-B.: Existence of solutions of generalized fractional differential equation with nonlocal initial condition. Math. Bohem. 144 (2019), 203-220. DOI 10.21136/MB.2018.0135-17 | MR 3974188 | Zbl 07088846
[6] Benchohra, M., Hamani, S., Ntouyas, S. K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal., Theory Methods Appl., Ser. A 71 (2009), 2391-2396. DOI 10.1016/j.na.2009.01.073 | MR 2532767 | Zbl 1198.26007
[7] Benchohra, M., Souid, M. S.: $L^1$-solutions for implicit fractional order differential equations with nonlocal conditions. Filomat 30 (2016), 1485-1492. DOI 10.2298/FIL1606485B | MR 3530093 | Zbl 06749806
[8] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985). DOI 10.1007/978-3-662-00547-7 | MR 0787404 | Zbl 0559.47040
[9] El-Sayed, A. M. A., El-Salam, S. A. Abd: $L_p$-solution of weighted Cauchy-type problem of a diffre-integral functional equation. Int. J. Nonlinear Sci. 5 (2008), 281-288. MR 2410798 | Zbl 1230.34006
[10] El-Sayed, A. M. A., Hashem, H. H. G.: Integrable and continuous solutions of a nonlinear quadratic integral equation. Electron. J. Qual. Theory Differ. Equ. 2008 (2008), Article ID 25, 10 pages. DOI 10.14232/ejqtde.2008.1.25 | MR 2443206 | Zbl 1178.45008
[11] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). DOI 10.1016/s0304-0208(06)x8001-5 | MR 2218073 | Zbl 1092.45003
[12] Liu, Y.: Existence of solutions of impulsive boundary value problems for singular fractional differential systems. Math. Bohem. 142 (2017), 405-444. DOI 10.21136/MB.2017.0029-14 | MR 3739026 | Zbl 06819594
[13] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience Publication. John Wiley & Sons, New York (1993). MR 1219954 | Zbl 0789.26002
[14] Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering 198. Academic Press, San Diego (1999). MR 1658022 | Zbl 0924.34008
[15] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). MR 1347689 | Zbl 0818.26003
[16] Seba, D.: Nonlinear fractional differential inclusion with nonlocal fractional integrodifferential boundary conditions in Banach spaces. Math. Bohem. 142 (2017), 309-321. DOI 10.21136/MB.2017.0041-16 | MR 3695469 | Zbl 06770148
[17] Sontakke, B., Shaikh, A., Nisar, K.: Existence and uniqueness of integrable solutions of fractional order initial value equations. J. Math. Model. 6 (2018), 137-148. DOI 10.22124/JMM.2018.9971.1147 | MR 3886133 | Zbl 1413.34044
[18] Souid, M. S.: $L^1$-solutions of boundary value problems for implicit fractional order differential equations with integral conditions. Int. J. Adv. Research Math. 11 (2018), 8-17. DOI 10.18052/www.scipress.com/IJARM.11.8
[19] Yu, T., Deng, K., Luo, M.: Existence and uniqueness of solutions of initial value problems for nonlinear Langevin equation involving two fractional orders. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 1661-1668. DOI 10.1016/j.cnsns.2013.09.035 | MR 3144748 | Zbl 07172530
[20] Zhang, S.: Existence results of positive solutions to fractional differential equation with integral boundary conditions. Math. Bohem. 135 (2010), 299-317. DOI 10.21136/MB.2010.140706 | MR 2683641 | Zbl 1224.26025
[21] Zhou, Z., Qiao, Y.: Solutions for a class of fractional Langevin equations with integral and anti-periodic boundary conditions. Bound. Value Probl. 2018 (2018), Article ID 152, 10 pages. DOI 10.1186/s13661-018-1070-3 | MR 3859564
Search
Partner of
