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Article

Keywords:
Nonlinear fractional Langevin equations; boundary value problems; existence; fixed point theorem
Summary:
In this paper, generalized boundary value problems for nonlinear fractional Langevin equations is studied. Some new existence results of solutions in the balls with different radius are obtained when the nonlinear term satisfies nonlinear Lipschitz and linear growth conditions. Finally, two examples are given to illustrate the results.
References:
[1] Baleanu, D., Machado, J. A. T., Luo, A. C.-J.: Fractional Dynamics and Control. Springer, Berlin, 2012. MR 2905887 | Zbl 1231.93003
[2] Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics Vol. 2014, dmlbpublisherSpringer, Berlin, 2010. MR 2680847 | Zbl 1215.34001
[3] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. Vol. 204, Elsevier Science, 2006. MR 2218073 | Zbl 1092.45003
[4] Lakshmikantham, V., Leela, S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, 2009. Zbl 1188.37002
[5] Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, 1993. MR 1219954
[6] Michalski, M. W.: Derivatives of Noninteger Order and Their Applications. Dissertationes Mathematicae 328, Inst. Math., Polish Acad. Sci., 1993. MR 1247113 | Zbl 0880.26007
[7] Podlubny, I.: Fractional Differential Equations. Academic Press, 1999. MR 1658022 | Zbl 0924.34008
[8] Tarasov, V. E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. dmlbpublisherSpringer, 2011. MR 2796453
[9] Staněk, S.: Limit properties of positive solutions of fractional boundary value problems. Appl. Math. Comput. 219 (2012), 2361–2370. DOI 10.1016/j.amc.2012.09.008 | MR 2988118 | Zbl 1308.34104
[10] Staněk, S.: Two-point boundary value problems for the generalized Bagley–Torvik fractional differential equation. Cent. Eur. J. Math. 11 (2013), 574–593. DOI 10.2478/s11533-012-0141-4 | MR 3016324 | Zbl 1262.34008
[11] O’Regan, D., Staněk, S.: Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 71 (2013), 641–652. DOI 10.1007/s11071-012-0443-x | MR 3030127 | Zbl 1268.34023
[12] Agarwal, R. P., O’Regan, D., Staněk, S.: Positive solutions for mixed problems of singular fractional differential equations. Mathematische Nachrichten 11 (2011), 1–15.
[13] Agarwal, R. P., O’Regan, D., Staněk, S.: Positive solutions for Dirichlet problems of singular nonlinear frcational differentil equations. J. Math. Anal. Appl. 37 (2010), 57–68. DOI 10.1016/j.jmaa.2010.04.034
[14] Agarwal, R. P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta. Appl. Math. 109 (2010), 973–1033. DOI 10.1007/s10440-008-9356-6 | MR 2596185 | Zbl 1198.26004
[15] Ahmad, B., Nieto, J. J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray–Schauder degree theory. Topol. Methods Nonlinear Anal. 35 (2010), 295–304. MR 2676818 | Zbl 1245.34008
[16] Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal., TMA 72 (2010), 916–924. MR 2579357 | Zbl 1187.34026
[17] Benchohra, M., Henderson, J., Ntouyas, S. K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338 (2008), 1340–1350. DOI 10.1016/j.jmaa.2007.06.021 | MR 2386501 | Zbl 1209.34096
[18] Mophou, G. M., N’Guérékata, G. M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216 (2010), 61–69. DOI 10.1016/j.amc.2009.12.062 | MR 2596132 | Zbl 1191.34098
[19] Wang, J., Fec̆kan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dynam. Part. Differ. Eq. 8 (2011), 345–361. DOI 10.4310/DPDE.2011.v8.n4.a3 | MR 2901608 | Zbl 1264.34014
[20] Wang, J., Zhou, Y.: Analysis of nonlinear fractional control systems in Banach spaces. Nonlinear Anal., TMA 74 (2011), 5929–5942. MR 2833364 | Zbl 1223.93059
[21] Zhang, S.: Existence of positive solution for some class of nonlinear fractional differential equations. J. Math. Anal. Appl. 278 (2003), 136–148. DOI 10.1016/S0022-247X(02)00583-8 | MR 1963470 | Zbl 1026.34008
[22] Zhou, Y., Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., RWA 11 (2010), 4465–4475. MR 2683890 | Zbl 1260.34017
[23] Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for $p$-type fractional neutral differential equations. Nonlinear Anal., TMA 71 (2009), 2724–2733. MR 2532797 | Zbl 1175.34082
[24] Lutz, E.: Fractional Langevin equation. Phys. Rev. E 64, 051106 (2001), 1–4. DOI 10.1103/PhysRevE.64.051106
[25] Fa, K. S.: Generalized Langevin equation with fractional derivative and long-time correlation function. Phys. Rev. E 73, 061104 (2006), 1–4. DOI 10.1103/PhysRevE.73.061104
[26] Fa, K. S.: Fractional Langevin equation and Riemann–CLiouville fractional derivative. Eur. Phys. J. E 24 (2007), 139–143. DOI 10.1140/epje/i2007-10224-2
[27] Kobolev, V., Romanov, E.: Fractional Langevin equation to describe anomalous diffusion. Prog. Theor. Phys. Suppl. 139 (2000), 470–476. DOI 10.1143/PTPS.139.470
[28] Picozzi, S., West, B.: Fractional Langevin model of memory in financial markets. Phys. Rev. E 66, 046118 (2002), 1–12. DOI 10.1103/PhysRevE.66.046118 | MR 1935186
[29] Bazzani, A., Bassi, G., Turchetti, G.: Diffusion and memory effects for stochastic processes and fractional Langevin equations. Physica A 324 (2003), 530–550. DOI 10.1016/S0378-4371(03)00073-6 | MR 1982904 | Zbl 1050.82029
[30] Lim, S. C., Li, M., Teo, L. P.: Langevin equation with two fractional orders. Phys. Lett. A 372 (2008), 6309–6320. DOI 10.1016/j.physleta.2008.08.045 | MR 2462401 | Zbl 1225.82049
[31] Ahmad, B., Nieto, J. J.: Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions. Int. J. Difference Equ. 2010, ID 649486 (2010), 1–10. MR 2575288 | Zbl 1207.34007
[32] Ahmad, B., Eloe, P.: A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices. Commun. Appl. Nonlinear Anal. 17 (2010), 69–80. MR 2721923 | Zbl 1275.34005
[33] Ahmad, B., Nieto, J. J., Alsaedi, A., El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., RWA 13 (2012), 599–606. MR 2846866 | Zbl 1238.34008
[34] Chen, A., Chen, Y.: Existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value conditions. Bound. Value Probl. 2011, ID 516481 (2011), 1–17. MR 2783108 | Zbl 1228.34016
[35] Ibrahim, R. W.: Existence of nonlinear Lane–Emden equation of fractional order. Math. Notes, Miskolc 13 (2012), 39–52. MR 2954543 | Zbl 1265.34216
[36] Sandev, T., Tomovski, Ž., Dubbeldam, J. L. A.: Generalized Langevin equation with a three parameter Mittag–Leffler noise. Physica A 390 (2011), 3627–3636. DOI 10.1016/j.physa.2011.05.039
[37] Sandev, T., Metzler, R., Tomovski, Ž.: Velocity and displacement correlation functions for fractional generalized Langevin equations. Fract. Calc. Appl. Anal. 15 (2012), 426–450. DOI 10.2478/s13540-012-0031-2 | MR 2944109 | Zbl 1274.82045
[38] Smart, D. R.: Fixed Point Theorems. Cambridge University Press, Cambridge, 1980. MR 0467717 | Zbl 0427.47036
[39] Wang, J., Dong, X., Zhou, Y.: Analysis of nonlinear integral equations with Erdélyi–Kober fractional operator. Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 3129–3139. DOI 10.1016/j.cnsns.2011.12.002 | MR 2904208 | Zbl 1298.45011
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