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Arabian Journal of Mathematics

, Volume 8, Issue 2, pp 79–94 | Cite as

Boundedness and power-type decay of solutions for a class of generalized fractional Langevin equations

  • Ahmad M. AhmadEmail author
  • Khaled M. Furati
  • Nasser-Eddine Tatar
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Abstract

In this paper we study the long-time behavior of solutions for a general class of Langevin-type fractional integro-differential equations. The involved fractional derivatives are either of Riemann–Liouville or Caputo type. Reasonable sufficient conditions under which the solutions are bounded or decay like power functions are established. For this purpose, we combine and generalize some well-known integral inequalities with some crucial estimates. Our findings are supported by examples and special cases.

Mathematics Subject Classification

34A08 34A12 34C11 

Notes

Acknowledgements

The authors would like to acknowledge the support provided by King Fahd University of Petroleum and Minerals (KFUPM) through project No. IN161010.

References

  1. 1.
    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(3), 973–1033 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Agarwal, R.P.; Ntouyas, S.K.; Ahmad, B.; Alhothuali, M.S.: Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions. Adv. Differ. Equations 2013(1), 1–9 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aghajani, A.; Jalilian, Y.; Trujillo, J.: On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 15(1), 44–69 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Anguraj, A.; Karthikeyan, P.; Trujillo, J.: Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition. Adv. Differ. Equations 2011(1), 690653 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Baghani, O.: On fractional Langevin equation involving two fractional orders. Commun. Nonlinear Sci. Numer. Simul. 42, 675–681 (2017)MathSciNetGoogle Scholar
  6. 6.
    Băleanu, D.; Agarwal, R.P.; Mustafa, O.G.; Coşulschi, M.: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A Math. Theor. 44(5), 055203 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Băleanu, D.; Mustafa, O.G.; Agarwal, R.P.: On the solution set for a class of sequential fractional differential equations. J. Phys. A Math. Theor. 43(38), 385209 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Baleanu, D.; Nazemi, S.Z.; Rezapour, S.: The existence of positive solutions for a new coupled system of multiterm singular fractional integrodifferential boundary value problems. Abstr. Appl. Anal. 2013, Article ID 368659 (2013)Google Scholar
  9. 9.
    Camargo, R.F.; Chiacchio, A.O.; Charnet, R.; de Oliveira, E.C.: Solution of the fractional Langevin equation and the Mittag–Leffler functions. J. Math. Phys. 50(6), 063507 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Coffey, W.; Kalmykov, Y.P.; Waldron, J.: The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering. World Scientific, Singapore (2004)zbMATHGoogle Scholar
  11. 11.
    Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204(2), 609–625 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    El-Sayed, A.; El-Salam, S.A.: Lp-solution of weighted Cauchy-type problem of a diffreintegral functional equation. Int. J. Nonlinear Sci. 5(3), 281–288 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Furati, K.M.; Tatar, N.E.: An existence result for a nonlocal fractional differential problem. J. Fract. Calculus 26, 43–51 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Furati, K.M.; Tatar, Ne: Behavior of solutions for a weighted Cauchy-type fractional differential problem. J. Fract. Calculus 28, 23–42 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Furati, K.M.; Tatar, Ne: Power-type estimates for a nonlinear fractional differential equation. Nonlinear Anal. Theory Methods Appl. 62(6), 1025–1036 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Furati, K.M.; Tatar, Ne: Long time behavior for a nonlinear fractional model. J. Math. Anal. Appl. 332(1), 441–454 (2007)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gayathri, B.; Murugesu, R.; Rajasingh, J.: Existence of solutions of some impulsive fractional integrodifferential equations. Int. J. Math. Anal. 6(17), 825–836 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hernández, E.; O’Regan, D.; Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. Theory Methods Appl. 73(10), 3462–3471 (2010)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kassim, M.D.; Furati, K.M.; Tatar, Ne: Asymptotic behavior of solutions to nonlinear initial-value fractional differential problems. Electron. J. Differ. Equations 2016(291), 1–14 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kendre, S.; Jagtap, T.; Kharat, V.: On nonlinear fractional integrodifferential equations with non local condition in Banach spaces. Nonlinear Anal. Differ. Equations 1, 129–141 (2013)Google Scholar
  21. 21.
    Kilbas, A.; Bonilla, B.; Trujillo, J.; et al.: Fractional integrals and derivatives and differential equations of fractional order in weighted spaces of continuous functions. Doklady Natsionalnoi Akademii Nauk Belarusi 44(6), 18–22 (2000)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier Science Limited, Amsterdam (2006)zbMATHGoogle Scholar
  23. 23.
    Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy’s Equation and Jensen’s Inequality. Springer, New York (2009)zbMATHGoogle Scholar
  24. 24.
    Li, X.; Medveď, M.; Wang, J.R.: Generalized boundary value problems for nonlinear fractional Langevin equations. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 53(2), 85–100 (2014)MathSciNetGoogle Scholar
  25. 25.
    Mainardi, F.; Pironi, P.: The fractional Langevin equation: Brownian motion revisited (2008). arXiv:0806.1010 (preprint)
  26. 26.
    Medveď, M.: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl. 214(2), 349–366 (1997)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Medved, M.; Pospíšsil, M.: Asymptotic integration of fractional differential equations with integrodifferential right-hand side. Math. Model. Anal. 20(4), 471–489 (2015)MathSciNetGoogle Scholar
  28. 28.
    Michalski, M.W.: Derivatives of Noninteger Order and Their Applications, vol. 328. Polska Akademia Nauk, Institut Matematyczny, Warszawa (1993)zbMATHGoogle Scholar
  29. 29.
    Miller, K.S.; Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  30. 30.
    Momani, S.; Hadid, S.: Lyapunov stability solutions of fractional integrodifferential equations. Int. J. Math Math. Sci. 2004(47), 2503–2507 (2004)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Mustafa, O.G.; Baleanu, D.: On the asymptotic integration of a class of sublinear fractional differential equations (2009). arXiv:0904.1495 (preprint)
  32. 32.
    Pinto, M.: Integral inequalities of Bihari-type and applications. Funkcialaj Ekvacioj 33(3), 387–403 (1990)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Podlubny, I.; Petraš, I.; Vinagre, B.M.; O’leary, P.; Dorčák, L.: Analogue realizations of fractional-order controllers. Nonlinear Dyn. 29(1–4), 281–296 (2002)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)zbMATHGoogle Scholar
  35. 35.
    Sitho, S.; Ntouyas, S.K.; Tariboon, J.: Existence results for hybrid fractional integro-differential equations. Bound. Value Probl. 2015(1), 1–13 (2015)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Sudsutad, W.; Tariboon, J.: Nonlinear fractional integro-differential Langevin equation involving two fractional orders with three-point multi-term fractional integral boundary conditions. J. Appl. Math. Comput. 43(1–2), 507–522 (2013)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Tatar, Ne: Existence results for an evolution problem with fractional nonlocal conditions. Comput. Math. Appl. 60(11), 2971–2982 (2010)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Thaiprayoon, C.; Ntouyas, S.K.; Tariboon, J.: On the nonlocal katugampola fractional integral conditions for fractional Langevin equation. Adv. Differ. Equations 2015(1), 1–16 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Torres, C.: Existence of solution for fractional Langevin equation: Variational approach. Electron. J. Qual. Theory Differ. Equations 54, 2014 (2014)MathSciNetGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Ahmad M. Ahmad
    • 1
    Email author
  • Khaled M. Furati
    • 1
  • Nasser-Eddine Tatar
    • 1
  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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