AIMS Mathematics, 2020, 5(4): 3714-3730. doi: 10.3934/math.2020240.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type

1 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University Aurangabad, (M.S), 431001, India
2 Department of Mathematics, Maulana Azad College of arts, Science and Commerce, RozaBagh, Aurangabad 431004 (M.S.), India
3 Department of Mathematics, Hodeidah University, Al-Hodeidah, Yemen

This paper deals with a nonlinear implicit fractional differential equation with the anti-periodic boundary condition involving the Caputo-Katugampola type. The existence and uniqueness results are established by applying the fixed point theorems of Krasnoselskii and Banach. Further, by using generalized Gronwall inequality the Ulam-Hyers stability results are proved. To demonstrate the effectiveness of the main results, appropriate examples are granted.
  Figure/Table
  Supplementary
  Article Metrics

Keywords fractional differential equations; Katugampola fractional operator; Ulam-Hyers stability; fixed point theorems; fractional Gronwall inequality

Citation: Saleh S. Redhwan, Sadikali L. Shaikh, Mohammed S. Abdo. Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type. AIMS Mathematics, 2020, 5(4): 3714-3730. doi: 10.3934/math.2020240

References

  • 1. M. Hamid, M. Usman, Z. H. Khan, et al. Dual solutions and stability analysis of flow and heat transfer of Casson fluid over a stretching sheet, Phys. Lett. A, 383 (2019), 2400-2408.    
  • 2. M. Hamid, M. Usman, Z. H. Khan, et al. Numerical study of unsteady MHD flow of Williamson nanofluid in a permeable channel with heat source/sink and thermal radiation, Europ. Phys. J. Plus, 133 (2018), 527.
  • 3. M. Hamid, M. Usman, T. Zubair, et al. Shape effects of MoS2 nanoparticles on rotating flow of nanofluid along a stretching surface with variable thermal conductivity: A Galerkin approach, Int. J. Heat Mass Transfer, 124 (2018), 706-714.    
  • 4. S. T. Mohyud-Din, M. Usman, K. Afaq, et al. Examination of carbon-water nanofluid flow with thermal radiation under the effect of Marangoni convection, Eng. Comput., 34 (2017), 2330-2343.    
  • 5. M. Usman, M. M. Din, T. Zubair, et al. Fluid flow and heat transfer investigation of blood with nanoparticles through porous vessels in the presence of magnetic field, J. Algorithms Comput. Technol., 13 (2018), 1748301818788661.
  • 6. M. Usman, M. Hamid, R. U. Haq, et al. Heat and fluid flow of water and ethylene-glycol based Cu-nanoparticles between two parallel squeezing porous disks: LSGM approach, Int. J. Heat Mass Transfer, 123 (2018), 888-895.    
  • 7. A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 204, 2006.
  • 8. A. B. Malinowska, T. Odzijewicz, D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer, Berlin, 2015.
  • 9. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • 10. R. P. Agarwal, Y. Zhou, Y. He, Existence of fractional neutral functional differential equations, Comput. Math. Appl., 59 (2010), 1095-1100.    
  • 11. D. Baleanu, K. Diethelm, E. Scalas, et al. Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, 3, 2012.
  • 12. 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.
  • 13. U. N. Katugampola, New approach to a generalized fractional integral. Appl. Math. Comput., 218 (2011), 860-865.
  • 14. A. G. Butkovskii, S. S. Postnov, E. A. Postnova, Fractional integro-differential calculus and its control-theoretical applications i-mathematical fundamentals and the problem of interpretation, Autom. Remote Control, 74 (2013), 543-574.    
  • 15. S. Gaboury, R. Tremblay, B. J. Fugere, Some relations involving a generalized fractional derivative operator, J. Inequalities Appl., 167, 2013.
  • 16. R. Herrmann, Fractional Calculus: An Introduction for Physicists, World Scienti c, River Edge, New Jerzey, 2 Eds., 2014.
  • 17. G. Jumarie, On the solution of the stochastic differential equation of exponential growth driven by fractional brownian motion, Appl. Math. Lett., 18 (2005), 817-826.    
  • 18. U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6 (2014), 1-15.
  • 19. U. N. Katugampola, Mellin transforms of the generalized fractional integrals and derivatives, Appl. Math. Comput., 257 (2015), 566-580.
  • 20. U. N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, Preprint, arXiv:1411.5229, 2014.
  • 21. R. Almeida, A Gronwall inequality for a general Caputo fractional operator, arXiv preprint arXiv:1705.10079, 2017.
  • 22. D. S. Oliveira, E. Capelas de Oliveira, Hilfer-Katugampola fractional derivatives, Comput. Appl. Math., 37 (2018), 3672-3690.    
  • 23. S. Abbas, M. Benchohra, J. R. Graef, et al. Implicit Fractional Differential and Integral Equations: Existence and Stability, Walter de Gruyter: London, UK, 2018.
  • 24. M. Benchohra, S. Bouriah, M. A. Darwish, Nonlinear boundary value problem for implicit differential equations of fractional order in Banach spaces, Fixed Point Theory, 18 (2017), 457-470.    
  • 25. M. Benchohra, S. Bouriah, J. R. Graef, Nonlinear implicit differential equations of fractional order at resonance, Electron. J. Differential Equations, 324 (2016), 1-10.
  • 26. M. Benchohra, J. E. Lazreg, Nonlinear fractional implicit differential equations, Commun. Appl. Anal., 17 (2013), 471-482.
  • 27. E. Alvarez, C. Lizama, R. Ponce, Weighted pseudo anti-periodic solutions for fractional integrodifferential equations in Banach spaces, Appl. Math. Comput., 259 (2015), 164-172.
  • 28. F. Chen, A. Chen, X. Wu, Anti-periodic boundary value problems with Riesz-Caputo derivative, Advances Difference Equations, 1 (2019), 119.
  • 29. D. Yang, C. Bai, Existence of solutions for Anti-Periodic fractional differential inclusions involving Riesz-Caputo fractional derivative, Mathematics, 7 (2019), 630.
  • 30. B. Ahmad, J. J. Nieto, Anti-periodic fractional boundary value problems, Comput. Math. Appl., 62 (2011), 1150-1156.    
  • 31. M. Benchohra, J. E. Lazreg, Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative, Studia Universitatis Babes-Bolyai, Mathematica, 62 (2017), 27-38.    
  • 32. L. Palve, M. S. Abdo, S. K. Panchal, Some existence and stability results of HilferHadamard fractional implicit differential fractional equation in a weighted space, preprint: arXiv:1910.08369v1 math. GM, (2019), 20.
  • 33. I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103-107.
  • 34. J. Wang, Y. Zhou, M. Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal., 41 (2013), 113-133.
  • 35. T. A. Burton, C. KirkÙ, A fixed point theorem of Krasnoselskii Schaefer type, Mathematische Nachrichten, 189 (1998), 23-31.    
  • 36. J. V. C. Sousa, E. C. Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, arXiv preprint arXiv:1709.03634, 2017.
  • 37. M. Benchohra, S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure Appl. Anal., 1 (2015), 22-37.
  • 38. A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Math., 5 (2019), 259-272.
  • 39. S. S. Redhwan, S. L. Shaikh, M. S. Abdo, Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations, arXiv preprint arXiv:1911.08884, 2019.

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved