A note on impulsive fractional evolution equations with nondense domain. (English) Zbl 1246.34015

Summary: We are concerned with the existence of integral solutions for nondensely defined fractional functional differential equations with impulse effects. Some errors in the existing paper concerned with nondensely defined fractional differential equations are pointed out, and correct formula of integral solutions is established by using integrated semigroup and some probability densities. Sufficient conditions for the existence are obtained by applying the Banach contraction mapping principle. An example is also given to illustrate our results.


34A08 Fractional ordinary differential equations
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[1] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, London, UK, 1974. · Zbl 0292.26011
[2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0943.82582
[3] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542
[4] D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609-625, 1996. · Zbl 0881.34005
[5] S. Aizicovici and M. McKibben, “Existence results for a class of abstract nonlocal Cauchy problems,” Nonlinear Analysis A, vol. 39, no. 5, pp. 649-668, 2000. · Zbl 0954.34055
[6] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229-248, 2002. · Zbl 1014.34003
[7] S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211-255, 2004. · Zbl 1068.35037
[8] M. Benchohra and B. A. Slimani, “Existence and uniqueness of solutions to impulsive fractional differential equations,” Electronic Journal of Differential Equations, vol. 10, pp. 1-11, 2009. · Zbl 1178.34004
[9] G. M. Mophou and G. M. N’Guérékata, “On integral solutions of some nonlocal fractional differential equations with nondense domain,” Nonlinear Analysis A, vol. 71, no. 10, pp. 4668-4675, 2009. · Zbl 1178.34005
[10] V. Daftardar-Gejji and H. Jafari, “Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1026-1033, 2007. · Zbl 1115.34006
[11] M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433-440, 2002. · Zbl 1005.34051
[12] Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 4465-4475, 2010. · Zbl 1260.34017
[13] J. Cao, Q. Yang, and Z. Huang, “Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations,” Nonlinear Analysis A, vol. 74, no. 1, pp. 224-234, 2011. · Zbl 1213.34089
[14] X.-B. Shu, Y. Lai, and Y. Chen, “The existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis A, vol. 74, no. 5, pp. 2003-2011, 2011. · Zbl 1227.34009
[15] Z. Tai and S. Lun, “On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 25, no. 2, pp. 104-110, 2012. · Zbl 1236.93024
[16] J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis, vol. 12, no. 1, pp. 262-272, 2011. · Zbl 1214.34010
[17] G. Da Prato and E. Sinestrari, “Differential operators with nondense domain,” Annali della Scuola Normale Superiore di Pisa, vol. 14, no. 2, pp. 285-344, 1987. · Zbl 0652.34069
[18] H. R. Thieme, ““Integrated semigroups” and integrated solutions to abstract Cauchy problems,” Journal of Mathematical Analysis and Applications, vol. 152, no. 2, pp. 416-447, 1990. · Zbl 0738.47037
[19] H. R. Thieme, “Semiflows generated by Lipschitz perturbations of non-densely defined operators,” Differential and Integral Equations, vol. 3, no. 6, pp. 1035-1066, 1990. · Zbl 0734.34059
[20] M. Adimy, H. Bouzahir, and K. Ezzinbi, “Existence for a class of partial functional differential equations with infinite delay,” Nonlinear Analysis A, vol. 46, no. 1, pp. 91-112, 2001. · Zbl 1003.34068
[21] K. Ezzinbi and J. H. Liu, “Nondensely defined evolution equations with nonlocal conditions,” Mathematical and Computer Modelling, vol. 36, no. 9-10, pp. 1027-1038, 2002. · Zbl 1035.34063
[22] M. Benchohra, E. P. Gatsori, J. Henderson, and S. K. Ntouyas, “Nondensely defined evolution impulsive differential inclusions with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 307-325, 2003. · Zbl 1039.34056
[23] E. P. Gatsori, “Controllability results for nondensely defined evolution differential inclusions with nonlocal conditions,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 194-211, 2004. · Zbl 1059.34037
[24] N. Abada, M. Benchohra, and H. Hammouche, “Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions,” Journal of Differential Equations, vol. 246, no. 10, pp. 3834-3863, 2009. · Zbl 1171.34052
[25] V. Kavitha and M. Mallika Arjunan, “Controllability of non-densely defined impulsive neutral functional differential systems with infinite delay in Banach spaces,” Nonlinear Analysis, vol. 4, no. 3, pp. 441-450, 2010. · Zbl 1200.93020
[26] M. Belmekki and M. Benchohra, “Existence results for fractional order semilinear functional differential equations with nondense domain,” Nonlinear Analysis A, vol. 72, no. 2, pp. 925-932, 2010. · Zbl 1179.26018
[27] W. Arendt, “Vector-valued Laplace transforms and Cauchy problems,” Israel Journal of Mathematics, vol. 59, no. 3, pp. 327-352, 1987. · Zbl 0637.44001
[28] H. Kellerman and M. Hieber, “Integrated semigroups,” Journal of Functional Analysis, vol. 84, no. 1, pp. 160-180, 1989. · Zbl 0689.47014
[29] K. Yosida, Functional Analysis, vol. 123, Springer, Berlin, Germany, 6th edition, 1980. · Zbl 0435.46002
[30] W. Arendt, “Resolvent positive operators,” Proceedings of the London Mathematical Society, vol. 54, no. 2, pp. 321-349, 1987. · Zbl 0617.47029
[31] 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. · Zbl 0986.82037
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