Korean Journal of Mathematics

  • 제27권1호
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  • Pages.119-129
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  • 2019
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  • 1976-8605(pISSN)
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  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
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MULTI-ORDER FRACTIONAL OPERATOR IN A TIME-DIFFERENTIAL FORMAL WITH BALANCE FUNCTION

  • Harikrishnan, S. (Department of Mathematics Sri Ramakrishna Mission Vidyalaya College of Arts and Science) ;
  • Ibrahim, Rabha W. (IEEE) ;
  • Kanagarajan, K. (Department of Mathematics Sri Ramakrishna Mission Vidyalaya College of Arts and Science)
  • 투고 : 2018.09.27
  • 심사 : 2019.03.07
  • 발행 : 2019.03.30
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초록

Balance function is one of the joint factors to determine fall in risk theory. It helps to moderate the progression and riskiness of falls for detecting balance and fall risk factors. Nevertheless, the objective measures for balance function require expensive equipment with the assessment of any expertise. We establish the existence and uniqueness of a multi-order fractional differential equations based on ${\psi}$-Hilfer operator on time scales with balance function. This class describes the dynamic of time scales derivative. Our tool is based on the Schauder fixed point theorem. Here, sufficient conditions for Ulam-stability are given.

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참고문헌

  1. R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35, (1999), 3-22. https://doi.org/10.1007/BF03322019
  2. A. Ahmadkhanlu and M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, B. Iran. Math. Soc. 38, (2012), 241-252.
  3. N. Benkhettou, A. Hammoudi, and D. F. M. Torres, Existence and uniqueness of solution for a fractional Riemann-lioville initial value problem on time scales, Journal of King Saud University-Science, 28,(2016),87-92. https://doi.org/10.1016/j.jksus.2015.08.001
  4. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
  5. M. Bohner and A. Peterson, Dtnamica equations on times scale, Birkhauser Boston, Boston, MA.
  6. K.M. Furati, M.D. Kassim, and N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64, (2012), 1616-1626. https://doi.org/10.1016/j.camwa.2012.01.009
  7. H. Gu and J.J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 15 (2015), 344-354.
  8. S. Harikrishnan, Kamal Shah, Dumitru Baleanu, and K. Kanagarajan, Note on the solution of random differential equations via $\psi$-Hilfer fractional derivative, Advances in Difference Equations, 2018, 2018:224.
  9. R. Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.
  10. R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23, (2012), 1-10. https://doi.org/10.1142/S0129167X12500565
  11. R. W. Ibrahim, Ulam-Hyers stability for Cauchy fractional differential equation in the unit disk, Abstract and Applied Analysis. Vol. 2012. Hindawi, 2012.
  12. R. W. Ibrahim, Existence of Ulam Stability for iterative fractional differential equations based on fractional entropy Entropy 17 (5) (2015), 3172-3181. https://doi.org/10.3390/e17053172
  13. P. Muniyappan and S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math., 102, (2015), 631-642.
  14. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  15. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Sci. Publishers, Yverdon, 1993.
  16. D. Vivek, K. Kanagarajan, and E.M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15, (2018), 1-15. https://doi.org/10.1007/s00009-017-1047-y
  17. J. Wang, L. Lv, and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron J. Qual. Theory Differ. Equ., 63, (2011), 1-10.