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

• Autorzy: Natthaphong Thongsalee , Sorasak Laoprasittichok , Sotiris K. Ntouyas , Jessada Tariboon
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Riemann-Liouville fractional derivative; Erdélyi-Kober fractional integral; System; Existence; Uniqueness; fixed point theorems

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2015-08-28

zaakceptowano

2015-11-19

online

2015-11-30

Twórcy

autor

Natthaphong Thongsalee

• Nonlinear Dynamic Analysis Research Center, Department of Mathematics,
  Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

autor

Sorasak Laoprasittichok

• Nonlinear Dynamic Analysis Research Center, Department of Mathematics,
  Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

autor

Sotiris K. Ntouyas

• Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece and Nonlinear Analysis and Applied
  Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203,
  Jeddah 21589, Saudi Arabia

autor

Jessada Tariboon

• Nonlinear Dynamic Analysis Research Center, Department of Mathematics,
  Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

Bibliografia

• [1] Agarwal R.P., Zhou Y., He Y., Existence of fractional neutral functional differential equations, Comput. Math. Appl., 2010, 59, 1095-1100.
• [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]
• [3] Ahmad B., Nieto J.J., Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl., 2011, 2011:36.
• [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.
• [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.
• [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]
• [7] Ahmad B., Ntouyas S.K., Fractional differential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal., 2012, 15, 362-382. [Crossref]
• [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]
• [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.
• [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.
• [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] Liu X., Jia M., Ge W., Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Difference Equ.,2013, 2013:126.
• [13] O’Regan D., Stanek S., Fractional boundary value problems with singularities in space variables, Nonlinear Dynam., 2013, 71, 641-652. [WoS]
• [14] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999.
• [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] 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.
• [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.
• [18] Erdélyi A., Kober H., Some remarks on Hankel transforms, Quart. J. Math., Oxford, Second Ser. 1940, 11, 212-221. [Crossref]
• [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] Kiryakova V., Generalized Fractional Calculus and Applications, Pitman Research Notes in Math. 301, Longman, Harlow - J. Wiley, N. York, 1994.
• [21] Kober H., On fractional integrals and derivatives, Quart. J. Math. Oxford, 1940, Ser. ll, 193-211. [Crossref]
• [22] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.
• [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] 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] Klimek M., On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publ. Office of Czestochowa University of Technology, 2009.
• [26] Malinowska A., Torres D., Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.
• [27] Kaczorek T., Selected Problems of Fractional Systems Theory, Springer-Verlag, Berlin, Heidelberg, 2011.
• [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] 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.
• [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] 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] 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.
• [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.
• [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] Senol B., Yeroglu C., Frequency boundary of fractional order systems with nonlinear uncertainties, J. Franklin Inst. 2013, 350, 1908-1925. [WoS]
• [36] Su X., Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 2009, 22, 64-69.
• [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]
• [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.
• [39] Granas A., Dugundji J., Fixed Point Theory, Springer-Verlag, New York, 2003.

Identyfikatory

DOI

10.1515/math-2015-0079