Generalized Boundary Value Problems for Nonlinear Fractional Langevin Equations

Xuezhu Li; Milan Medveď; Jin Rong Wang

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2014)

  • Volume: 53, Issue: 2, page 85-100
  • ISSN: 0231-9721

Abstract

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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.

How to cite

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Li, Xuezhu, Medveď, Milan, and Wang, Jin Rong. "Generalized Boundary Value Problems for Nonlinear Fractional Langevin Equations." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 53.2 (2014): 85-100. <http://eudml.org/doc/262208>.

@article{Li2014,
abstract = {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.},
author = {Li, Xuezhu, Medveď, Milan, Wang, Jin Rong},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Nonlinear fractional Langevin equations; boundary value problems; existence; fixed point theorem; nonlinear fractional Langevin equations; boundary value problems; existence; fixed point theorem},
language = {eng},
number = {2},
pages = {85-100},
publisher = {Palacký University Olomouc},
title = {Generalized Boundary Value Problems for Nonlinear Fractional Langevin Equations},
url = {http://eudml.org/doc/262208},
volume = {53},
year = {2014},
}

TY - JOUR
AU - Li, Xuezhu
AU - Medveď, Milan
AU - Wang, Jin Rong
TI - Generalized Boundary Value Problems for Nonlinear Fractional Langevin Equations
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2014
PB - Palacký University Olomouc
VL - 53
IS - 2
SP - 85
EP - 100
AB - 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.
LA - eng
KW - Nonlinear fractional Langevin equations; boundary value problems; existence; fixed point theorem; nonlinear fractional Langevin equations; boundary value problems; existence; fixed point theorem
UR - http://eudml.org/doc/262208
ER -

References

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  1. Baleanu, D., Machado, J. A. T., Luo, A. C.-J., Fractional Dynamics and Control, Springer, Berlin, 2012. (2012) Zbl1231.93003MR2905887
  2. Diethelm, K., The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics Vol. 2014, dmlbpublisherSpringer, Berlin, 2010. (2010) Zbl1215.34001MR2680847
  3. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Vol. 204, Elsevier Science, 2006. (2006) Zbl1092.45003MR2218073
  4. Lakshmikantham, V., Leela, S., Vasundhara Devi, J., Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009. (2009) Zbl1188.37002
  5. Miller, K. S., Ross, B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, 1993. (1993) MR1219954
  6. Michalski, M. W., Derivatives of Noninteger Order and Their Applications, Dissertationes Mathematicae 328, Inst. Math., Polish Acad. Sci., 1993. (1993) Zbl0880.26007MR1247113
  7. Podlubny, I., Fractional Differential Equations, Academic Press, 1999. (1999) Zbl0924.34008MR1658022
  8. Tarasov, V. E., Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, dmlbpublisherSpringer, 2011. (2011) MR2796453
  9. Staněk, S., 10.1016/j.amc.2012.09.008, Appl. Math. Comput. 219 (2012), 2361–2370. (2012) Zbl1308.34104MR2988118DOI10.1016/j.amc.2012.09.008
  10. Staněk, S., 10.2478/s11533-012-0141-4, Cent. Eur. J. Math. 11 (2013), 574–593. (2013) Zbl1262.34008MR3016324DOI10.2478/s11533-012-0141-4
  11. O’Regan, D., Staněk, S., 10.1007/s11071-012-0443-x, Nonlinear Dyn. 71 (2013), 641–652. (2013) Zbl1268.34023MR3030127DOI10.1007/s11071-012-0443-x
  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. (2011) 
  13. Agarwal, R. P., O’Regan, D., Staněk, S., 10.1016/j.jmaa.2010.04.034, J. Math. Anal. Appl. 37 (2010), 57–68. (2010) DOI10.1016/j.jmaa.2010.04.034
  14. Agarwal, R. P., Benchohra, M., Hamani, S., 10.1007/s10440-008-9356-6, Acta. Appl. Math. 109 (2010), 973–1033. (2010) Zbl1198.26004MR2596185DOI10.1007/s10440-008-9356-6
  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. (2010) Zbl1245.34008MR2676818
  16. Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., TMA 72 (2010), 916–924. (2010) Zbl1187.34026MR2579357
  17. Benchohra, M., Henderson, J., Ntouyas, S. K., Ouahab, A., 10.1016/j.jmaa.2007.06.021, J. Math. Anal. Appl. 338 (2008), 1340–1350. (2008) Zbl1209.34096MR2386501DOI10.1016/j.jmaa.2007.06.021
  18. Mophou, G. M., N’Guérékata, G. M., 10.1016/j.amc.2009.12.062, Appl. Math. Comput. 216 (2010), 61–69. (2010) Zbl1191.34098MR2596132DOI10.1016/j.amc.2009.12.062
  19. Wang, J., Fec̆kan, M., Zhou, Y., 10.4310/DPDE.2011.v8.n4.a3, Dynam. Part. Differ. Eq. 8 (2011), 345–361. (2011) Zbl1264.34014MR2901608DOI10.4310/DPDE.2011.v8.n4.a3
  20. Wang, J., Zhou, Y., Analysis of nonlinear fractional control systems in Banach spaces, Nonlinear Anal., TMA 74 (2011), 5929–5942. (2011) Zbl1223.93059MR2833364
  21. Zhang, S., 10.1016/S0022-247X(02)00583-8, J. Math. Anal. Appl. 278 (2003), 136–148. (2003) Zbl1026.34008MR1963470DOI10.1016/S0022-247X(02)00583-8
  22. Zhou, Y., Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal., RWA 11 (2010), 4465–4475. (2010) Zbl1260.34017MR2683890
  23. Zhou, Y., Jiao, F., Li, J., Existence and uniqueness for p -type fractional neutral differential equations, Nonlinear Anal., TMA 71 (2009), 2724–2733. (2009) Zbl1175.34082MR2532797
  24. Lutz, E., 10.1103/PhysRevE.64.051106, Phys. Rev. E 64, 051106 (2001), 1–4. (2001) DOI10.1103/PhysRevE.64.051106
  25. Fa, K. S., 10.1103/PhysRevE.73.061104, Phys. Rev. E 73, 061104 (2006), 1–4. (2006) DOI10.1103/PhysRevE.73.061104
  26. Fa, K. S., 10.1140/epje/i2007-10224-2, Eur. Phys. J. E 24 (2007), 139–143. (2007) DOI10.1140/epje/i2007-10224-2
  27. Kobolev, V., Romanov, E., 10.1143/PTPS.139.470, Prog. Theor. Phys. Suppl. 139 (2000), 470–476. (2000) DOI10.1143/PTPS.139.470
  28. Picozzi, S., West, B., 10.1103/PhysRevE.66.046118, Phys. Rev. E 66, 046118 (2002), 1–12. (2002) MR1935186DOI10.1103/PhysRevE.66.046118
  29. Bazzani, A., Bassi, G., Turchetti, G., 10.1016/S0378-4371(03)00073-6, Physica A 324 (2003), 530–550. (2003) Zbl1050.82029MR1982904DOI10.1016/S0378-4371(03)00073-6
  30. Lim, S. C., Li, M., Teo, L. P., 10.1016/j.physleta.2008.08.045, Phys. Lett. A 372 (2008), 6309–6320. (2008) Zbl1225.82049MR2462401DOI10.1016/j.physleta.2008.08.045
  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. (2010) Zbl1207.34007MR2575288
  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. (2010) Zbl1275.34005MR2721923
  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. (2012) Zbl1238.34008MR2846866
  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. (2011) Zbl1228.34016MR2783108
  35. Ibrahim, R. W., Existence of nonlinear Lane–Emden equation of fractional order, Math. Notes, Miskolc 13 (2012), 39–52. (2012) Zbl1265.34216MR2954543
  36. Sandev, T., Tomovski, Ž., Dubbeldam, J. L. A., 10.1016/j.physa.2011.05.039, Physica A 390 (2011), 3627–3636. (2011) DOI10.1016/j.physa.2011.05.039
  37. Sandev, T., Metzler, R., Tomovski, Ž., 10.2478/s13540-012-0031-2, Fract. Calc. Appl. Anal. 15 (2012), 426–450. (2012) Zbl1274.82045MR2944109DOI10.2478/s13540-012-0031-2
  38. Smart, D. R., Fixed Point Theorems, Cambridge University Press, Cambridge, 1980. (1980) Zbl0427.47036MR0467717
  39. Wang, J., Dong, X., Zhou, Y., 10.1016/j.cnsns.2011.12.002, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 3129–3139. (2012) Zbl1298.45011MR2904208DOI10.1016/j.cnsns.2011.12.002

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