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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
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

On Caputo-Hadamard Type Fractional Differential Equations with Nonlocal Discrete Boundary Conditions

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 185--194 | DOI:10.5890/DNC.2021.06.002

Murugesan Manigandan$^{1}$, Muthaiah Subramanian$^{2}$, Palanisamy Duraisamy$^{3}$, Thangaraj Nandha Gopal$^{1}$

$^{1}$ Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore - 641 020, Tamilnadu, India

$^{2}$ Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore, India

$^3$ Department of Mathematics, Gobi Arts and Science College, Gobichettipalayam, Tamilnadu, India

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Abstract

This paper studies a new class of boundary value problems of Caputo-Hadamard fractional differential equations of order $\varrho\in (2, 3]$ supplemented with nonlocal multi-point (discrete) boundary conditions. Existence and uniqueness results for the given problem have obtained by applying standard fixed-point theorems. Finally, two examples are given to illustrate the validity of our main results.

Acknowledgments

The corresponding author was supported by the minor research project funded by University Grants Commissions (F.No.4-4/2015-16 (MRP/UGC-SERO)).

References

  1. [1]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Amsterdam, Boston, Elsevier.
  2. [2]  Kilbas, A., Saigo, M., and Saxena, R.K. (2004), Generalized Mittag-Leffler function and generalized fractional calculus operators, Adv. Difference Equ., 15(1), 31-49.
  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, 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 Alsharif, A. (2015), Existence results for fractional-order differential equations with nonlocal multi-point-strip conditions involving Caputo derivative, Adv. Difference Equ., 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, Fract. Calc. Appl. Anal., 18(1), 261-280.
  8. [8]  Duraisamy, P. and Nandha Gopal, 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]  Duraisamy, P., Vidhya Kumar, A.R., Nandha Gopal, T., and Subramanian, M. (2018), Influence of nonlocal discrete and integral boundary conditions involving Caputo derivative in boundary value problem, J. Phys. Conf. Ser., 1139(1), 012014.
  10. [10]  Ntouyas, S.K. and Etemad, S. (2016), On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, Appl. Math. Comput., 266(1), 235-246.
  11. [11]  Subramanian, M., Manigandan, M., and Nandha Gopal, T. (2019), Existence of Solutions for Nonlocal Boundary Value Problem of Hadamard Fractional Differential Equations, Advances in the Theory of Nonlinear Analysis and its Application, 3(3), 162-173.
  12. [12]  Subramanian, M., Vidhya Kumar, A.R., and Nandha Gopal, T. (2019), Analysis of fractional boundary value problem with non-local integral strip boundary conditions, Nonlinear Stud., 26(2), 445-454.
  13. [13]  Subramanian, M., Vidhya Kumar, A.R., and Nandha Gopal, 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, Proc. Jangjeon Math. Soc., 22(3), 437-453.
  14. [14]  Subramanian, M., Vidhya Kumar, A.R., and Nandha Gopal, 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.
  15. [15]  Subramanian, M., Vidhya Kumar, A.R., and Nandha Gopal, T. (2019), Influence of coupled nonlocal slit-strip conditions involving Caputo derivative in fractional boundary value problem, Discontinuity, Nonlinearity, and Complexity, 8(4), 429-445.
  16. [16]  Vidhya Kumar, A.R., Duraisamy, P., Nandha Gopal, T., and Subramanian, M. (2018), Analysis of fractional differential equation involving Caputo derivative with nonlocal discrete and multi-strip type boundary conditions, J. Phys. Conf. Ser., 1139(1), 012020.
  17. [17]  Hadamard, J. (1892), Essai sur letude des fonctions donnees par leur developpement de Taylor, Journal de Mathematiques Pures et Appliquees, 8, 101-186.
  18. [18]  Ahmad, B. and Ntouyas, S.K. 2014, On three-point Hadamard-type fractional boundary value problems, Int. Electron. J. Pure. Appl. Math., 8(4), 31-42.
  19. [19]  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, Bound. Value Probl., 2013(1), 275.
  20. [20]  Alsaedi, A., Ntouyas, S.K., Ahmad, B., and Hobiny, A. (2015), Nonlinear Hadamard fractional differential equations with Hadamard type nonlocal non-conserved conditions, Adv. Difference Equ., 2015(1), 285.
  21. [21]  Bai, Y. and Kong, H. (2017), Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions, J. Nonlinear Sci. Appl., 10(1), 5744-5752.
  22. [22]  Gambo, Y.Y., Jarad, F., Baleanu, D., and Abdeljawad, T. (2014), On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2014(1), 10.
  23. [23]  Jarad, F., Abdeljawad, T., and Baleanu, D. (2012), Caputo-type modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2012(1), 142.
  24. [24]  Thiramanus, P., Ntouyas, S.K., and Tariboon, J. (2016), Positive solutions for Hadamard fractional differential equations on infinite domain, Adv. Difference Equ., 2016(1), 83.
  25. [25]  Yukunthorn, W., Ahmad, B., Ntouyas, S.K., and Tariboon, J. (2019), On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Anal. Hybrid Syst., 19, 77-92.
  26. [26]  Qinghua, M., Chao, M., and Wang, J. (2017), A Lyapunov-type inequality for a fractional differential equation with Hadamard derivative, J. Math. Inequal., 11(1), 135-141.
  27. [27]  Wang, G., Pei, K., Agrawal, R.P., Zhang, L., and Ahmad, B. (2018), Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math., 343, 230-239.

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