Existence, Uniqueness and Stability of Nonlinear Implicit Fractional Dynamical Equation with Impulsive condition on Time Scales

References

[1] Diethelm, K., Freed, A.D., On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. Scientific Computing in Chemical Engineering II. Springer, Berlin, Heidelberg, (1999), 217-224.10.1007/978-3-642-60185-9_24Search in Google Scholar

[2] Glöckle, W.G., Nonnenmacher, T.F., A fractional calculus approach to self-similar protein dynamics. Biophysical Journal, 68(1) (1995), 46-53.10.1016/S0006-3495(95)80157-8Search in Google Scholar

[3] Zhang, S., Existence of solution for a boundary value problem of fractional order. Acta Mathematica Scientia, 26(2) (2006), 220-228.10.1016/S0252-9602(06)60044-1Search in Google Scholar

[4] Benchohra, M., Graef, J. R., Hamani, S., Existence results for boundary value problems with non-linear fractional differential equations. Applicable Analysis, 87(7) (2008), 851-863.10.1080/00036810802307579Search in Google Scholar

[5] Wang, T., Feng, X., Existence and uniqueness of fractional differential equations with integral boundary conditions. The Journal of Nonlinear Sciences and its Applications 1(4) (2008), 206-212.10.22436/jnsa.001.04.02Search in Google Scholar

[6] Ahmad, B., Nieto, J.J., Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems, 2009(1) (2009), 708576.10.1155/2009/708576Search in Google Scholar

[7] Ahmad, B., Sivasundaram, S., Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Analysis: Hybrid Systems, 3(3) (2009), 251-258.10.1016/j.nahs.2009.01.008Search in Google Scholar

[8] Liu, X., Jia, M., Baofeng, Wu., Existence and uniqueness of solution for fractional differential equations with integral boundary conditions. Electronic Journal of Qualitative Theory of Differential Equations, 2009(69) (2009), 1-10.10.14232/ejqtde.2009.1.69Search in Google Scholar

[9] Agarwal, R. P., Benchohra, M., Hamani,S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Applicandae Mathematicae, 109(3) (2010), 973-1033.10.1007/s10440-008-9356-6Search in Google Scholar

[10] Benchohra, M., Ouaar, F., Existence results for nonlinear fractional differential equations with integral boundary conditions. Bulletin of Mathematical Analysis and Applications, 2(2) (2010), 7-15.Search in Google Scholar

[11] Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., Theory of impulsive differential equations. Volume 6, World scientific, 1989.10.1142/0906Search in Google Scholar

[12] Benchohra, M., Henderson, J., Ntouyas, S.K., Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Volume 2, New York, 2006.10.1155/9789775945501Search in Google Scholar

[13] Hilger,S., EinMakettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten (Ph.D thesis), Universitt Wurzburg, 1988.Search in Google Scholar

[14] Naidu, D., Singular perturbations and time scales in control theory and applications: an overview. Dynamics of Continuous Discrete and Impulsive Systems Series B, 9 (2002), 233-278.Search in Google Scholar

[15] Zhuang, K., Periodic solutions for a stage-structure ecological model on time scales. Electronic Journal of Differential Equations, 2007 (2007), 1-7.Search in Google Scholar

[16] Ferhan, A. M., Biles, D. C., Lebedinsky, A., An application of time scales to economics. Mathematical and Computer Modelling, 43(7) (2006), 718-726.10.1016/j.mcm.2005.08.014Search in Google Scholar

[17] Ferhan, A. M., Uysal, F., A production–inventory model of HMMS on time scales. Applied Mathematics Letters, 21(3) (2008), 236-243.10.1016/j.aml.2007.03.013Search in Google Scholar

[18] Bohner, M., Peterson, A., Dynamic Equations on Time Scales. Birkhäuser, Basel, 2001.10.1007/978-1-4612-0201-1Search in Google Scholar

[19] Bohner, M., Peterson, A., eds.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, MA, 2003.10.1007/978-0-8176-8230-9Search in Google Scholar

[20] Agarwal, R. P., Bohner, M., Basic calculus on time scales and some of its applications, Results in Mathematics, 35 (1999), 3-22.10.1007/BF03322019Search in Google Scholar

[21] Liu, H., Xiang, X., A class of the first order impulsive dynamic equations on time scales, Nonlinear Analysis: Theory, Methods and Applications, 69(9) (2008), 2803-2811.10.1016/j.na.2007.08.052Search in Google Scholar

[22] Kumar, V., Malik, M., Existence and stability of fractional integro differential equation with non-instantaneous integrable impulses and periodic boundary condition on time scales, Journal of King Saud University - Science (2018), doi: https://doi.org/10.1016/j.jksus.2018.10.01110.1016/j.jksus.2018.10.011Search in Google Scholar

[23] Ahmadkhanlu, A., Jahanshahi,M., On the existence and uniqueness of solution of initial value problem for fractional order differential equations on time scales. Bulletin of the Iranian Mathematical Society, 38(1) (2012), 241-252.Search in Google Scholar

[24] Benkhettou, N., Hammoudi, A., Torres, D.F.M., Existence and uniqueness of solution for a fractional Riemann-Liouville initial value problem on time scales. Journal of King Saud University-Science, 28 (2016), 87-92.Search in Google Scholar

[25] Yan, R.A., Sun, S.R., Han, Z.L., Existence of solutions of boundary value problems for Caputo fractional differential equations on time scales. Bulletin of the Iranian Mathematical Society, 42(2) (2016), 247-262.Search in Google Scholar

[26] Agarwal, R.P., Benchohra, M., O’Regan, D., Ouahab, A., Second order impulsive dynamic equations on time scales. Functional Differential Equations, 11(3-4) (2004), 223-234.Search in Google Scholar

[27] Wang, J.R., Feĉkan, M., Zhou, Y., Ulam’s type stability of impulsive ordinary differential equations. Journal of Mathematical Analysis and Applications, 395(1) (2012), 258-264.10.1016/j.jmaa.2012.05.040Search in Google Scholar

[28] Zada, A., Faisal, S., Li, Y., On the Hyers-Ulam stability of first-order impulsive delay differential equations. Journal of Function Spaces, 2016 (2016).10.1155/2016/8164978Search in Google Scholar

[29] Haq, F., Shah, K., Rahman, G., Shahzad, M., Hyers-Ulam stability to a class of fractional differential equations with boundary conditions. International Journal of Applied and Computational Mathematics, 3(1) (2017), 1135-1147.10.1007/s40819-017-0406-5Search in Google Scholar

[30] Shen, Y., The Ulam stability of first order linear dynamic equations on time scales. Results in Mathematics, 72(4) (2017), 1881-1895.10.1007/s00025-017-0725-1Search in Google Scholar

[31] András, S., Mészáros, A.R., Ulam-Hyers stability of dynamic equations on time scales via Picard operators. Applied Mathematics and Computation, 219(9) (2013), 4853-4864.10.1016/j.amc.2012.10.115Search in Google Scholar