System of fractional differential equations with Erdélyi-Kober fractional integral conditions

Natthaphong Thongsalee; Sorasak Laoprasittichok; Sotiris K. Ntouyas; Jessada Tariboon

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 480-497
  • ISSN: 2391-5455

Abstract

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In this paper we study existence and uniqueness of solutions for a system consisting from fractional differential equations of Riemann-Liouville type subject to nonlocal Erdélyi-Kober fractional integral conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.

How to cite

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Natthaphong Thongsalee, et al. "System of fractional differential equations with Erdélyi-Kober fractional integral conditions." Open Mathematics 13.1 (2015): 480-497. <http://eudml.org/doc/275910>.

@article{NatthaphongThongsalee2015,
abstract = {In this paper we study existence and uniqueness of solutions for a system consisting from fractional differential equations of Riemann-Liouville type subject to nonlocal Erdélyi-Kober fractional integral conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.},
author = {Natthaphong Thongsalee, Sorasak Laoprasittichok, Sotiris K. Ntouyas, Jessada Tariboon},
journal = {Open Mathematics},
keywords = {Riemann-Liouville fractional derivative; Erdélyi-Kober fractional integral; System; Existence; Uniqueness; fixed point theorems; fractional differential equations; nonlocal boundary conditions; fixed point theorem},
language = {eng},
number = {1},
pages = {480-497},
title = {System of fractional differential equations with Erdélyi-Kober fractional integral conditions},
url = {http://eudml.org/doc/275910},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Natthaphong Thongsalee
AU - Sorasak Laoprasittichok
AU - Sotiris K. Ntouyas
AU - Jessada Tariboon
TI - System of fractional differential equations with Erdélyi-Kober fractional integral conditions
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 480
EP - 497
AB - In this paper we study existence and uniqueness of solutions for a system consisting from fractional differential equations of Riemann-Liouville type subject to nonlocal Erdélyi-Kober fractional integral conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.
LA - eng
KW - Riemann-Liouville fractional derivative; Erdélyi-Kober fractional integral; System; Existence; Uniqueness; fixed point theorems; fractional differential equations; nonlocal boundary conditions; fixed point theorem
UR - http://eudml.org/doc/275910
ER -

References

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  1. [1] Agarwal R.P., Zhou Y., He Y., Existence of fractional neutral functional differential equations, Comput. Math. Appl., 2010, 59, 1095-1100. Zbl1189.34152
  2. [2] Ahmad B., Nieto J.J., Boundary value problems for a class of sequential integrodifferential equations of fractional order, J. Funct. Spaces Appl., 2013, Art. ID 149659, 8 pp. [WoS] Zbl1267.45018
  3. [3] Ahmad B., Nieto J.J., Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl., 2011, 2011:36. Zbl1275.45004
  4. [4] Ahmad B., Ntouyas S.K., Alsaedi A., New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Difference Equ., 2011, Art. ID 107384, 11 pp. Zbl1204.34005
  5. [5] Ahmad B., Ntouyas S.K., Alsaedi A., A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions, , Math. Probl. Eng., 2013, Art. ID 320415, 9 pp. Zbl1296.34011
  6. [6] Ahmad B., Ntouyas S.K., A fully Hadamard-type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal., 2014, 17 , 348–360. [Crossref] Zbl1312.34005
  7. [7] Ahmad B., Ntouyas S.K., Fractional differential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal., 2012, 15, 362-382. [Crossref] Zbl1279.34003
  8. [8] Ahmad B., Ntouyas S.K., Nonlocal fractional boundary value problems with slit-strips integral boundary conditions, Fract. Calc. Appl. Anal., 2015, 18, 261-280. [Crossref] Zbl06413568
  9. [9] Baleanu D., Diethelm K., Scalas E., Trujillo J.J., Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, 2012. Zbl1248.26011
  10. [10] Baleanu D., Mustafa O.G., Agarwal R.P., On Lp-solutions for a class of sequential fractional differential equations, Appl. Math. Comput., 2011, 218, 2074-2081. Zbl1235.34008
  11. [11] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. 
  12. [12] Liu X., Jia M., Ge W., Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Difference Equ.,2013, 2013:126. 
  13. [13] O’Regan D., Stanek S., Fractional boundary value problems with singularities in space variables, Nonlinear Dynam., 2013, 71, 641-652. [WoS] 
  14. [14] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999. 
  15. [15] Tariboon J., Ntouyas S.K., A. Singubol, Boundary value problems for fractional differential equations with fractional multi-term integral conditions, J. App. Math., 2014, Article ID 806156, 10 pp. 
  16. [16] Thiramanus P., Ntouyas S.K., Tariboon J., Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal., 2014, Article ID 902054, 9 pp. Zbl1305.26054
  17. [17] Zhang L., Ahmad B., Wang G., Agarwal R.P., Nonlinear fractional integro-differential equations on unbounded domains in a Banach space, J. Comput. Appl. Math., 2013, 249, 51–56. Zbl1302.45019
  18. [18] Erdélyi A., Kober H., Some remarks on Hankel transforms, Quart. J. Math., Oxford, Second Ser. 1940, 11, 212-221. [Crossref] Zbl66.0521.01
  19. [19] Kalla S.L., Kiryakova V.S., An H-function generalized fractional calculus based upon compositions of Erdélyi-Kober operators in Lp; Math. Japonica, 1990, 35, 1-21. 
  20. [20] Kiryakova V., Generalized Fractional Calculus and Applications, Pitman Research Notes in Math. 301, Longman, Harlow - J. Wiley, N. York, 1994. Zbl0882.26003
  21. [21] Kober H., On fractional integrals and derivatives, Quart. J. Math. Oxford, 1940, Ser. ll, 193-211. [Crossref] Zbl66.0520.02
  22. [22] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. Zbl0818.26003
  23. [23] Sneddon I.N., The use in mathematical analysis of Erdélyi-Kober operators and some of their applications. In: Fractional Calculus and Its Applications, Proc. Internat. Conf. Held in New Haven, Lecture Notes in Math., 1975, 457, Springer, N. York, 37-79. 
  24. [24] Yakubovich S.B., Luchko Yu.F., The Hypergeometric Approach to Integral Transforms and Convolutions, Mathematics and its Appl. 287, Kluwer Acad. Publ., Dordrecht-Boston-London, 1994. 
  25. [25] Klimek M., On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publ. Office of Czestochowa University of Technology, 2009. 
  26. [26] Malinowska A., Torres D., Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012. Zbl1258.49001
  27. [27] Kaczorek T., Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, Heidelberg, 2011. 
  28. [28] Hristov J., A unified nonlinear fractional equation of the diffusion-controlled surfactant adsorption: Reappraisal and new solution of the Ward–Tordai problem, Journal of King Saud University – Science, 2015, DOI.org/10.1016/j.jksus.2015.03.008. [Crossref] 
  29. [29] Yang X.J., Baleanu D., Srivastava H.M., Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett.,2015, 47, 54-60. Zbl06477315
  30. [30] Yan S.P., Local fractional Laplace series expansion method for diffusion equation arising in fractal heat transfer, Thermal Science, 2015, 19, Suppl. 1, 131-135. 
  31. [31] Yang X.J., Srivastava H.M., Cattani C., Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Romanian Reports in Physics, 2015, 67, 752-761. 
  32. [32] Ahmad B., Nieto J.J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 2009, 58, 1838-1843. Zbl1205.34003
  33. [33] Faieghi M., Kuntanapreeda S., Delavari H., Baleanu D., LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dynam., 2013, 72, 301-309. Zbl1268.93121
  34. [34] Ntouyas S.K., Obaid M., A coupled system of fractional differential equations with nonlocal integral boundary conditions, Adv. Differ. Equ, 2012, 2012:130. [Crossref] 
  35. [35] Senol B., Yeroglu C., Frequency boundary of fractional order systems with nonlinear uncertainties, J. Franklin Inst. 2013, 350, 1908-1925. [WoS] 
  36. [36] Su X., Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 2009, 22, 64-69. Zbl1163.34321
  37. [37] Sun J., Liu Y., Liu G., Existence of solutions for fractional differential systems with antiperiodic boundary conditions, Comput. Math. Appl., 2012, 64, 1557-1566. [WoS] Zbl1268.34157
  38. [38] Wang J., Xiang H., Liu Z., Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, Int. J. Differ. Equ., 2010, Article ID 186928, 12 pp. Zbl1207.34012
  39. [39] Granas A., Dugundji J., Fixed Point Theory, Springer-Verlag, New York, 2003. Zbl1025.47002

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