Systems of fractional Langevin equations of Riemann-Liouville and Hadamard types

Advances in Difference Equations

Volumn 2015, Issue 1, 2015, Pages

  Sudsutad, Weerawat ^a,b   Ntouyas, Sotiris K ^c,d   Tariboon, Jessada ^a,b  

^a KING MONGKUT'S UNIVERSITY OF TECHNOLOGY NORTH BANGKOK   (Thailand)

^b Thailand Ministry of Science and Technology   (Thailand)

^c UNIVERSITY OF IOANNINA   (Greece)

^d KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

Author keywords

coupled system; existence; fixed point theorems; fractional integral boundary conditions; Hadamard fractional derivative; Langevin equation; Riemann Liouville fractional derivative; uniqueness

Indexed keywords

EID: 84938316934     PISSN: 16871839     EISSN: 16871847     Source Type: Journal    
DOI: 10.1186/s13662-015-0566-8     Document Type: Article

References (37)

1. Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993)
2. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)
3. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
4. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Boston (2012)
5. Ahmad, B, Nieto, JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011, 36 (2011)
6. Ahmad, B, Ntouyas, SK, Alsaedi, A: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011, Article ID 107384 (2011)
7. Ahmad, B, Ntouyas, SK, Alsaedi, A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013, Article ID 320415 (2013)
8. Ahmad, B, Nieto, JJ: Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013, Article ID 149659 (2013)
9. Agarwal, RP, Zhou, Y, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 1095-1100 (2010)
10. Baleanu, D, Mustafa, OG, Agarwal, RP: On Lp$L^{p}$-solutions for a class of sequential fractional differential equations. Appl. Math. Comput. 218, 2074-2081 (2011)
11. O’Regan, D, Stanek, S: Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 71, 641-652 (2013)
12. Zhang, L, Ahmad, B, Wang, G, Agarwal, RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249, 51-56 (2013)
13. Liu, X, Jia, M, Ge, W: Multiple solutions of a p-Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013, 126 (2013)
14. Hadamard, J: Essai sur l’etude des fonctions donnees par leur developpment de Taylor. J. Math. Pures Appl. 8, 101-186 (1892)
15. Butzer, PL, Kilbas, AA, Trujillo, JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 269, 387-400 (2002)
16. Butzer, PL, Kilbas, AA, Trujillo, JJ: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1-27 (2002)
17. Butzer, PL, Kilbas, AA, Trujillo, JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1-15 (2002)
18. Kilbas, AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 38, 1191-1204 (2001)
19. Kilbas, AA, Trujillo, JJ: Hadamard-type integrals as G-transforms. Integral Transforms Spec. Funct. 14, 413-427 (2003)
20. Glushak, AV, Manaenkova, TA: Direct and inverse problems for an abstract differential equation containing Hadamard fractional derivatives. Differ. Equ. 47, 1307-1317 (2011)
21. Kassim, MDA: Well-posedness for a Cauchy fractional differential problem with Hilfer type fractional derivative. PhD thesis, King Fahd University of Petroleum and Minerals, Saudi Arabia (2011)
22. Bazhlekova, E: Fractional evolution equations in Banach spaces. PhD thesis, Eindhoven University of Technology, Eindhoven (2001)
23. Kostic, M: Abstract Volterra Integro-Differential Equations. CRC Press, Boca Raton (2015)
24. Prüss, J: Evolutionary Integral Equations and Applications. Birkhäuser, Basel (1993)
25. Coffey, WT, Kalmykov, YP, Waldron, JT: The Langevin Equation, 2nd edn. World Scientific, Singapore (2004)
26. Lim, SC, Li, M, Teo, LP: Langevin equation with two fractional orders. Phys. Lett. A 372, 6309-6320 (2008)
27. Lim, SC, Teo, LP: The fractional oscillator process with two indices. J. Phys. A, Math. Theor. 42, Article ID 065208 (2009)
28. Uranagase, M, Munakata, T: Generalized Langevin equation revisited: mechanical random force and self-consistent structure. J. Phys. A, Math. Theor. 43, Article ID 455003 (2010)
29. Denisov, SI, Kantz, H, Hänggi, P: Langevin equation with super-heavy-tailed noise. J. Phys. A, Math. Theor. 43, Article ID 285004 (2010)
30. Lozinski, A, Owens, RG, Phillips, TN: The Langevin and Fokker-Planck equations in polymer rheology. In: Handbook of Numerical Analysis, vol. 16 pp. 211-303 (2011)
31. Lizana, L, Ambjörnsson, T, Taloni, A, Barkai, E, Lomholt, MA: Foundation of fractional Langevin equation: harmonization of a many-body problem. Phys. Rev. E 81, Article ID 051118 (2010)
32. Gambo, YY, Jarad, F, Baleanu, D, Abdeljawad, T: On Caputo modification of the Hadamard fractional derivative. Adv. Differ. Equ. 2014, 10 (2014)
33. Ahmad, B, Eloe, PW: A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices. Commun. Appl. Nonlinear Anal. 17, 69-80 (2010)
34. Ahmad, B, Nieto, JJ, Alsaedi, A, El-Shahed, M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 13, 599-606 (2012)
35. Tariboon, J, Ntouyas, SK, Thaiprayoon, C: Nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions. Adv. Math. Phys. 2014, Article ID 372749 (2014)
36. Diethelm, K: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
37. Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003)