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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Fractional Differential Equations Involving Hadamard Fractional Derivatives with Nonlocal Multi-point Boundary Conditions

Discontinuity, Nonlinearity, and Complexity 9(3) (2020) 421--431 | DOI:10.5890/DNC.2020.09.006

Muthaiah Subramanian, Murugesan Manigandan, Thangaraj Nandha Gopal

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore - 641 020, Tamilnadu, India

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Abstract

In this paper, we investigate the existence and uniqueness of solutions for the Hadamard fractional boundary value problems with nonlocal multipoint boundary conditions. By using Leray-Schauder nonlinear alternative, Leray Schauder degree theory, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, Banach fixed point theorem, Nonlinear Contractions, the existence and uniqueness of solutions are obtained. As an application, two examples are given to demonstrate our results.

References

  1. [1]  Kilbas, A., Saigo, M., and Saxena, R.K. (2004), Generalized mittag-leffler function and generalized fractional calculus operators, Advances in Difference Equations, 15(1), 31-49.
  2. [2]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Amsterdam, Boston, Elsevier.
  3. [3]  Klafter, J., Lim, S.C., and Metzler, R. (2012), Fractional dynamics: Recent advances,World Scientific.
  4. [4]  Podlubny, I. (1999), Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, Academic Press, San Diego-Boston- New York-London-Tokyo-Toronto.
  5. [5]  Sabatier, J., Agrawal, O.P., and Tenreiro Machado, J. A. (2007), Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer Netherlands.
  6. [6]  Ahmad, B. Alsaedi, A., and Alshariff, Alaa. (2015), Existence results for fractional-order differential equations with nonlocal multi-point-strip conditions involving caputo derivative, Advances in Difference Equations, 2015(1), 348.
  7. [7]  Ahmad, B., Ntouyas, S.K., Agarwal, R.P., and Alsaedi, A. (2015), Existence results for sequential fractional integro differential equations with nonlocal multi point and strip conditions, Fractional Calculus and Applied Analysis, 18(1), 261-280.
  8. [8]  Duraisamy, P. and Nandhagopal, T. (2018), Existence and uniqueness of solutions for a coupled system of higher order fractional differential equations with integral boundary conditions, Discontinuity, Nonlinearity, and Complexity, 7(1), 1-14.
  9. [9]  Ntouyas, S.K. and Etemad, S. (2015), On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, Applied Mathematics and Computation, 266, 235-243.
  10. [10]  Rezapour, Sh. and Hedayati, V. (2017), On a caputo fractional differential inclusion with integral boundary condition for convex-compact and nonconvex-compact valued multifunctions, Kragujevac Journal of Mathematics, 41(1), 143- 158.
  11. [11]  Subramanian, M. and Nandhagopal, T. (2019), Solvability of Liouville-Caputo fractional integro-differential equations with non-local generalized fractional integral boundary conditions, Global Journal of Engineering Science and Researches, 6(4), 142-154.
  12. [12]  Subramanian, M., Vidhyakumar, A.R., and Nandhagopal, T. (2019), Analysis of fractional boundary value problem with non-local integral strip boundary conditions, Nonlinear Studies, 26(2), 445-454.
  13. [13]  Subramanian, M., Vidhyakumar, A.R., and Nandhagopal, T. (2019) A fundamental approach on non-integer order differential equation using nonlocal fractional sub-strips boundary conditions, Discontinuity, Nonlinearity, and Complexity, 8(2), 189-199.
  14. [14]  Subramanian,M., Vidhyakumar, A.R., and Nandhagopal, T. (2019), A strategic view on the consequences of classical integral sub-strips and coupled nonlocal multi-point boundary conditions on a combined Caputo fractional differential equation, Proceedings of Jangjeon Mathematical Society, 22(3), 437-453.
  15. [15]  Tabhariti, L. and Dahmani, Z. (2016), High dimensional fractional coupled systems: new existence and uniqueness results, Kragujevac Journal of Mathematics, 40(2), 224-236.
  16. [16]  Vidhyakumar, A.R., Duraisamy, P., Nandhagopal, T., and Subramanian, M. (2018), Analysis of fractional differential equation involving Caputo derivative with nonlocal discrete and multi-strip type boundary conditions, Journal of Physics: Conference Series, 1139(1), 012020.
  17. [17]  Hadamard, J. (1892), Essai sur letude des fonctions donnees par leur developpment de taylor, Journal deMathmatiques Pures et Appliques, 8.
  18. [18]  Ahmad, B., Ntouyas, S.K., Agarwal, R.P., and Alsaedi, A. (2013), New results for boundary value problems of hadamard-type fractional differential inclusions and integral boundary conditions, Boundary Value Problems, 2013(1), 275.
  19. [19]  Bai, Y. and Kong, H. (2017), Existence of solutions for nonlinear caputo-hadamard fractional differential equations via the method of upper and lower solutions, Journal of Nonlinear Sciences and Applications, 10(1), 5744-5752.
  20. [20]  Thiramanus, P., Ntouyas, S.K., and Tariboon, J. (2016), Positive solutions for hadamard fractional differential equations on infinite domain, Advances in Difference Equations, 2016(1), 83.
  21. [21]  Yukunthorn, W., Ahmad, B., Ntouyas, S.K. and Tariboon, J. (2016), On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Analysis: Hybrid Systems, 19, 77-92.
  22. [22]  Qinghua, M., Chao, M., and Wang, J. (2017), A Lyapunov-type inequality for a fractional differential equation with hadamard derivative, Journal of Mathematical Inequalities, 11(1), 135-141.
  23. [23]  Wang, G., Pei, K., Agrawal, R.P., Zhang, L., and Ahmad, B. (2018), Nonlocal hadamard fractional boundary value problemwith hadamard integral and discrete boundary conditions on a half-line, Journal of Computational and Applied Mathematics, 343, 230-239.
  24. [24]  Alsaedi, A., Ntouyas, S.K., Ahmad, B., and Hobiny, A. (2015), Nonlinear hadamard fractional differential equations with hadamard type nonlocal non-conserved conditions, Advances in Difference Equations, 2015(1), 285.