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Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions

  • Natthaphong Thongsalee EMAIL logo , Sotiris K. Ntouyas and Jessada Tariboon
Published/Copyright: May 5, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 2

Abstract

In this paper we study a new class of Riemann-Liouville fractional differential equations subject to nonlocal Erdélyi-Kober fractional integral boundary conditions. Existence and uniqueness results are obtained by using a variety of fixed point theorems, such as Banach fixed point theorem, Nonlinear Contractions, Krasnoselskii fixed point theorem, Leray-Schauder Nonlinear Alternative and Leray-Schauder degree theory. Examples illustrating the obtained results are also presented.

Key Words and Phrases: fractional differential equations; nonlocal boundary conditions; Erdélyi-Kober fractional integral; fixed point theorem
MSC 2010: Primary 34A08; Secondary 34B15

  1. Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 480–497, DOI: 10.1515/fca-2016-0025

References

[1] R.P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59 (2010), 1095–1100.10.1016/j.camwa.2009.05.010Search in Google Scholar

[2] B. Ahmad, J.J. Nieto, Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013 (2013), Art. ID 149659, 8 pp.10.1155/2013/149659Search in Google Scholar

[3] B. Ahmad, J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011 (2011), # 36.10.1186/1687-2770-2011-36Search in Google Scholar

[4] B. Ahmad, S.K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011 (2011) Art. ID 107384, 11 pp.10.1155/2011/107384Search in Google Scholar

[5] B. Ahmad, S.K. Ntouyas, A. Alsaedi, A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013 (2013), Art. ID 320415, 9 pp.10.1155/2013/320415Search in Google Scholar

[6] B. Ahmad, S.K. Ntouyas, Fractional differential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal. 15, No 3 (2012), 362–382; DOI: 10.2478/s13540-012-0027-y; http://www.degruyter.com/view/j/fca.2012.15.issue-3/issue-files/fca.2012.15.issue-3.xml.10.2478/s13540-012-0027-y;Search in Google Scholar

[7] B. Ahmad, S.K. Ntouyas, A fully Hadamard-type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17, No 2 (2014), 348–360; DOI: 10.2478/s13540-014-0173-5; http://www.degruyter.com/view/j/fca.2014.17.issue-2/issue-files/fca.2014.17.issue-2.xml.10.2478/s13540-014-0173-5;Search in Google Scholar

[8] B. Ahmad, S.K. Ntouyas, Nonlocal fractional boundary value problems with slit-strips integral boundary conditions. Fract. Calc. Appl. Anal. 18, No 1 (2015), 261–280; DOI: 10.1515/fca-2015-0017; http://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.10.1515/fca-2015-0017;Search in Google Scholar

[9] D. Baleanu, K. Diethelm, E. Scalas, J.J.Trujillo, Fractional Calculus Models and Numerical Methods. Ser. on Complexity, Nonlinearity and Chaos, World Scientific, Boston, 2012.10.1142/8180Search in Google Scholar

[10] D. Baleanu, O.G. Mustafa, R.P. Agarwal, On Lp-solutions for a class of sequential fractional differential equations. Appl. Math. Comput. 218 (2011), 2074–2081.10.1016/j.amc.2011.07.024Search in Google Scholar

[11] D.W. Boyd and J.S.W. Wong, On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458–464.10.1090/S0002-9939-1969-0239559-9Search in Google Scholar

[12] A. Granas, J. Dugundji, Fixed Point Theory. Springer-Verlag, New York, 2003.10.1007/978-0-387-21593-8Search in Google Scholar

[13] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier Science B.V., Amsterdam, 2006.Search in Google Scholar

[14] V. Kiryakova, Generalized Fractional Calculus and Applications. Pitman Research Notes in Math. 301, Longman, Harlow - J. Wiley, N. York, 1994.Search in Google Scholar

[15] H. Kober, On fractional integrals and derivatives. Quart. J. Math. OxfordSer. ll (1940), 193–211.10.1093/qmath/os-11.1.193Search in Google Scholar

[16] M.A. Krasnoselskii, Two remarks on the method of successive approximations. Uspekhi Mat. Nauk10 (1955), 123–127.Search in Google Scholar

[17] X. Liu, M. Jia, W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative. Adv. Difference Equ. 2013 (2013), # 126.10.1186/1687-1847-2013-126Search in Google Scholar

[18] D. O’Regan, S. Stanek, Fractional boundary value problems with singularities in space variables. Nonlinear Dynam. 71 (2013), 641–652.10.1007/s11071-012-0443-xSearch in Google Scholar

[19] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.Search in Google Scholar

[20] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York, 1993.Search in Google Scholar

[21] I.N. Sneddon, 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. 457, Springer, N. York, 1975, 37–79.10.1007/BFb0067097Search in Google Scholar

[22] S.B. Yakubovich, Yu.F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions. Mathematics and its Appl. 287, Kluwer Acad. Publ., Dordrecht-Boston-London, 1994.10.1007/978-94-011-1196-6_21Search in Google Scholar

[23] L. Zhang, B. Ahmad, G. Wang, R.P. Agarwal, Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 249 (2013), 51–56.10.1016/j.cam.2013.02.010Search in Google Scholar

Received: 2015-4-5
Received: 2016-1-25
Published Online: 2016-5-5
Published in Print: 2016-4-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA-Volume 19-2-2016)
  4. Survey Paper
  5. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
  6. Survey Paper
  7. A free fractional viscous oscillator as a forced standard damped vibration
  8. Research Paper
  9. On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
  10. Research Paper
  11. Nonlinear Dirichlet problem with non local regional diffusion
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  13. Generalized fraction evolution equations with fractional Gross Laplacian
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  15. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
  16. Research Paper
  17. Convolutional approach to fractional calculus for distributions of several variables
  18. Research Paper
  19. Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
  20. Research Paper
  21. Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
  22. Research Paper
  23. Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
  24. Research Paper
  25. Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
  26. Short Paper
  27. Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
  28. Short Paper
  29. Monotonicity of functions and sign changes of their Caputo derivatives
  30. Short Note
  31. On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators
https://doi.org/10.1515/fca-2016-0025
Keywords for this article
fractional differential equations; nonlocal boundary conditions; Erdélyi-Kober fractional integral; fixed point theorem

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA-Volume 19-2-2016)
  4. Survey Paper
  5. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
  6. Survey Paper
  7. A free fractional viscous oscillator as a forced standard damped vibration
  8. Research Paper
  9. On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
  10. Research Paper
  11. Nonlinear Dirichlet problem with non local regional diffusion
  12. Research Paper
  13. Generalized fraction evolution equations with fractional Gross Laplacian
  14. Research Paper
  15. A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
  16. Research Paper
  17. Convolutional approach to fractional calculus for distributions of several variables
  18. Research Paper
  19. Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
  20. Research Paper
  21. Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
  22. Research Paper
  23. Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
  24. Research Paper
  25. Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
  26. Short Paper
  27. Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
  28. Short Paper
  29. Monotonicity of functions and sign changes of their Caputo derivatives
  30. Short Note
  31. On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators
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