On Fractional Integro-differential Equations with State-Dependent Delay and Non-Instantaneous Impulses





In this paper, we prove the existence of mild solution of the fractional integro-differential equations with state-dependent delay with not instantaneous impulses. The existence results are obtained under the conditions in respect of Kuratowski‘s measure of non- compactness. An example is also given to illustrate the results.


Non-instantaneous impulsive conditions , fractional integro-differential equations , Caputo fractional derivative , mild solution , fixed point , state-dependent delay
  • Khalida Aissani Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abb`es, PO Box 89, 22000, Sidi Bel-Abb`es, Algeria.
  • Mouffak Benchohra Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abb`es, PO Box 89, 22000, Sidi Bel-Abb`es, Algeria. - University of Bechar PO Box 417, 08000, Bechar, Algeria
  • Nadia Benkhettou University of Bechar PO Box 417, 08000, Bechar, Algeria.
  • Pages: 61–75
  • Date Published: 2019-04-01
  • Vol. 21 No. 1 (2019)
[1] S. Abbas, M. Benchohra, J. Graef and J. Henderson, Implicit Fractional Differential and Integral Equations; Existence and Stability, De Gruyter, Berlin, 2018.
[2] S. Abbas, M. Benchohra and G.M. N‘Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012.
[3] S. Abbas, M. Benchohra and G.M. N‘Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
[4] R. P. Agarwal, S. Hristova, and D. O‘Regan, Non-instantaneous Impulses in Differential Equations. Springer, Cham, 2017.
[5] R. P. Agarwal, M. Meehan, and D. O‘Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.
[6] R.P. Agarwal, M. Benchohra and B.A. Slimani, Existence results for differential equations with fractional order impulses, Mem. Differential Equations. Math. Phys., 44 (2008), 1 21.
[7] A. Anguraj and P. Karthikeyan, Anti-periodic boundary value problem for impulsive fractional integro differential equations, Fract. Calc. Appl. Anal. 13 (2010), 1-13.
[8] A. Anguraj and S. Kanjanadevi, Existence results for fractional non-instantaneous impulsive integro-differential equations with nonlocal conditions, Dynam. Cont. Disc. Ser. A 23 (2016), 429-445.
[9] A. Anguraj and S. Kanjanadevi, Non-instantaneous impulsive fractional neutral differential equations with state-dependent delay, Progr. Fract. Differ. Appl. 3(3) (2017), 207-218
[10] K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional impulsive semi-linear evolution equations, Electron. J. Qual. Theor. Differ. Equat., 2010(4)(2010), 112.
[11] K. Balachandran, S.Kiruthika and J.J. Trujillo, Existence results for fractional impulsive integro-differential equations in Banach spaces, Commun. Nonlinear Sci. Num. Simul. 16 (2011), 1970-1977.
[12] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012.
[13] J. Bana ́s and K. Goebel, Measures of Noncompactness in Banach Spaces, of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1980.
[14] M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, Vol 2, New York, 2006.
[15] M. Benchohra and S. Litimein, Existence results for a new class of fractional integro-differential equations with state dependent delay, Mem. Differ. Equa. Math. Phys. 74 (2018), 27-38.
[16] D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998), 109-138.
[17] L. Debnath and D. Bhatta, Integral Transforms and Their Applications (Second Edition), CRC Press, 2007.
[18] K. Diethelm, The Analysis of Fractional Differential Equations. Springer, Berlin, 2010.
[19] G. R. Gautam and J. Dabas, Existence result of fractional functional integro-differential equation with not instantaneous impulse, Int. J. Adv. Appl. Math. Mech. 1(3) (2014), 11-21.
[20] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funk. Ekvacioj, 21 (1) (1978), 11-41.
[21] H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7 (12) (1983), 1351-1371.
[22] E. Hernández, A. Prokopczyk, and L. Ladeira, A note on partial functional differential equations with state-dependent delay, Nonlinear Anal. RWA, 7 (2006), 510-519.
[23] E. Hernández and D. O‘Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013), 1641-1649
[24] R. Hilfer, Applications of Fractional Calculus in Physics. Singapore, World Scientific, 2000.
[25] Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Unbounded Delay, Springer-Verlag, Berlin, 1991.
[26] A. A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam, 2006.
[27] P. Kumar, R. Haloi, D. Bahuguna and D. N. Pandey, Existence of solutions to a new class of abstract non-instantaneous impulsive fractional integro-differential equations, Nonlin. Dynam. Syst. Theor. 16 (1) (2016), 73-85.
[28] V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equa- tions, World Scientific, NJ, 1989.
[29] P. Li and C. J. Xu , Mild solution of fractional order differential equations with not instanta- neous impulses, Open Math, 13 (2015), 436-443.
[30] F. Mainardi, P. Paradisi and R. Gorenflo, Probability distributions generated by fractional diffusion equations, in Econophysics: An Emerging Science, J. Kertesz and I. Kondor, Eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.
[31] M. Meghnafi, M. Benchohra and K. Aissani, Impulsive fractional evolution equations with state-dependent delay, Nonlinear Stud. 22 (4)(2015), 659-671.
[32] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equa- tions, John Wiley, New York, 1993.
[33] H. M´onch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4 (1980), 985–999.
[34] D. N. Pandey, S. Das and N. Sukavanam, Existence of solution for a second-order neutral differential equation with state dependent delay and non-instantaneous impulses, Int. J. Nonlin. Sci. 18(2)(2014), 145-155.
[35] M. Pierri, D. O‘Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput. 219 (2013), 6743- 6749.
[36] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[37] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
[38] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010
[39] Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press Elsevier, 2016.

Most read articles by the same author(s)


Download data is not yet available.



How to Cite

K. . Aissani, M. . Benchohra, and N. . Benkhettou, “On Fractional Integro-differential Equations with State-Dependent Delay and Non-Instantaneous Impulses”, CUBO, vol. 21, no. 1, pp. 61–75, Apr. 2019.