S. HARİKRİSHNANA, Rabha İBRAHİM, K. KANAGARAJAN

An Arbitrary Order Differential Equations on Times Scale

Here existence and stability results of $\psi$-Hilfer fractional differential equations on time scales is obtained. Here sufficient condition for existence and uniqueness of solution by using Schauder's fixed point theorem (FPT) and Banach FPT is produced. In addition, generalized Ulam stability of the proposed problem is also discussed. problem.

Keywords:

Fractional calculus, Existence, Ulam-Hyers-Rassias stability,

PDF

___

[1] A. Ahmadkhanlu, M. Jahanshahi, On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales, Bull. Iranian Math. Soc., 38 (2012), 241-252.
[2] R. P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35 (1999), 3-22.
[3] N. Benkhettou, A. Hammoudi, D. F. M. Torres, Existence and uniqueness of solution for a fractional Riemann-lioville initial value problem on time scales, J. King Saud Univ. Sci., 28 (2016), 87-92.
[4] M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkhauser, Boston, 2003.
[5] M. Bohner, A. Peterson, Dtnamica equations on times scale, Birkhauser, Boston, Boston, MA.
[6] K. M. Furati, M. D. Kassim, N.e-. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616-1626.
[7] H. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 15 (2015), 344-354.
[8] S. Harikrishnan, K. Shah, D. Baleanu, K. Kanagarajan, Note on the solution of random differential equations via y-Hilfer fractional derivative, Adv. Difference Equ., 2018(224) (2018).
[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).
[11] P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math., 102 (2015), 631-642.
[12] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
[13] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, theory and applications, Gordon and Breach Sci. Publishers, Yverdon, 1993.
[14] J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the y-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., (in press).
[15] J. Vanterler da C. Sousa, E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., (in press).
[16] D. Vivek, K. Kanagarajan, E. M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15 (2018), 1-15.
[17] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron J. Qual. Theory Differ. Equ., 63 (2011), 1-10.