Existence and Stability of Solutions of Katugampola-Caputo Type Implicit Fractional Differential Equations with Impulses

Year 2018, , 162 - 174, 25.12.2018

M. Janaki , K. Kanagarajan , E. M. Elsayed

https://doi.org/10.33401/fujma.437210

Cited By: 1

Abstract

This paper investigates the existence and Ulam stability of solutions for impulsive nonlinear fractional implicit differential equations with finite delay via Katugampola fractional derivative in Caputo sense. Our results are based on some standard fixed point theorems. Some examples are presented to illustrate the main results.

Keywords

Existence, Fixed point, Implicit fractional differential equations, Katugampola fractional derivative, Stability

References

• [1] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012.
• [2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [3] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.
• [4] S. Abbas, M. Benchohra, G. M. N’Gu´er´ekata, Topics in Fractional Differential Equations, Springer Verlag, New York, 2012.
• [5] A. Allaberen, D. Fadime, P. Zehra, A note on the fractional hyperbolic differential and difference equations, Appl. Math. Comput., 217(9) (2011), 4654–4664.
• [6] A. Ashyralyev, F. Dal, Z. Pinar, On the numerical solution of fractional hyperbolic partial differential equations, Math. Probl. Eng., (2009), 1–11.
• [7] D. Baleanu, Z. B. G¨ uvenc, J. A. T. Machado, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.
• [8] R. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1993.
• [9] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
• [10] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [11] M. Benchohra, J. Henderson, S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, 2006.
• [12] J. R. Graef, J. Henderson, A. Ouahab, Impulsive Differential Inclusions. A Fixed Point Approach, De Gruyter, Berlin/Boston, 2013.
• [13] J. Henderson, J. Ouahab, Impulsive differential inclusions with fractional order, Comput. Math. Appl., 59 (2010), 1191–1226.
• [14] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
• [15] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
• [16] W. Albarakati, M. Benchohra, S. Bouriah, Existence and stability results for nonlinear implicit fractional differential equations with delay and impulses, Differ. Equ. Appl., 8(2) (2016), 273–293.
• [17] E. E. Ndiyo, Existence result for solution of second order impulsive differential inclusion to dynamic evolutionary process, Amer. J. Math. Stat., 7(2) (2017), 89–92.
• [18] J. Wang, M. Feckan, Y. Zhou, Ulam’s type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395 (2012), 258–264.
• [19] L. Zhang, G. Wang, Existence of solutions for nonlinear fractional differential equations with impulses and anti-Periodic boundary conditions, Electron. J. Qual. Theory Differ. Equ., 7 (2011), 1–11.
• [20] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. App., 6(4) (2014), 1–15.
• [21] U. N. Katugampola, New approach to a generalized fractional integral, App. Math. Comput., 218(3) (2011), 860–865.
• [22] U. N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, (2016), arXiv:1411.5229v2[math.CA].
• [23] G. Wang, B. Ahmad, L. Zhang, Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput. Math. Appl., 62 (2011), 1389-1397.
• [24] D. Vivek, K. Kanagarajan, S. Harikrishnan, Theory and analysis of impulsive type pantograph equations with Katugampola fractional derivative, J. Vib. Test. Sys. Dynm., 2(1) (2018), 9–20.
• [25] D. D. Bainov, S. G. Hristova, Integral inequalities of Gronwall type for piecewise continuous functions, J. Appl. Math. Stoc. Anal., 10 (1997), 89–94.
• [26] A. Ali, F. Rabiei, K. Shah, On Ulam’s type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions, J. Nonlinear Sci. Appl., 10 (2017), 4760–4775.
• [27] F. Haq, K. Shah, G. U. Rahman, Hyers-Ulam stability to a class of fractional differential equations with boundary conditions, Int. J. Appl. Comput. Math, Springer (India) Private Ltd., (2017).
• [28] A. Khan, K. Shah, Y. Li, T. S. Khan, Ulam type stability for a coupled system of boundary value problems of nonlinear fractional differential equations, J. Func. Spaces, 2017 (2017), Article ID 3046013, 8 pages.
• [29] K. Shah, C. Tunc, Existence theory and stability analysis to a system of boundary value problem, J. Taib. Uni. Sci., 11 (2017), 1330–1342.
• [30] A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.