Abstract
In this paper we intend to accomplish two tasks firstly, we address some basic errors in several recent results involving impulsive fractional equations with the Caputo derivative, and, secondly, we study initial value problems for nonlinear differential equations with the Riemann–Liouville derivative of order 0 < α ≤ 1 and the Caputo derivatives of order 1 < δ < 2. In both cases, the corresponding fractional derivative of lower order is involved in the formulation of impulsive conditions.
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Agarwal R.P., Benchohra M., Slimani B.A.: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 44, 1–21 (2008)
Agarwal R.P., Karakoç F.: A survey on oscillation of impulsive delay differential equations. Comp. Math. Appl. 60, 1648–1685 (2010)
Ahmad, B., Nieto, J.J.: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009, 1–11 (2009), ID 708576
Ahmad B., Sivasundaram S.: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 4, 134–141 (2010)
Bai, C.: Existence of positive solutions for a functional fractional boundary value problem. Abstr. Appl. Anal. 2010, 1–13 (2010), Art. ID 127363
Bainov D.D., Simeonov P.S.: Systems with Impulse Effect: Stability, Theory and Applications. In: Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood, Chichester (1989)
Benchohra M., Henderson J., Ntouyas S.K.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)
Benchohra, M., Slimani, B.A.: Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differ. Equ. (no. 10), 1–11 (2009)
Benchohra, M., Hamani, S., Nieto, J.J., Slimani, B.A.: Existence of solutions to differential inclusions with fractional order and impusses. Electron. J. Differ. Equ. (no. 80), 1–18 (2010)
Benchohra, M., Seba, D.: Impulsive fractional differential equations in Banach spaces. Electron. J. Qual. Theory Differ. Equ. Special Edn. I. (no. 8), 1–14 (2009)
Boichuk A.A., Samoilenko A.M.: Generalised inverse operators and Fredholm boundary-value problems. VSP, Utrecht (2004)
Caputo M.: Linear models of dissipation whose Q is almost frequency independent (Part II). Geophys. J. R. Astron. Soc. 13, 529–539 (1967)
Chen J., Tisdell C.C., Yuan R.: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 331, 902–912 (2007)
Goodrich C.: Continuity of solutions to discrete fractional initial value problems. Comp. Math. Appl. 59, 3489–3499 (2010)
Guo D.: Existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal. 68, 2727–2740 (2008)
Guo D.: Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal. 70, 2078–2090 (2009)
Herzallah, M.A.E., Baleanu, D.: Fractional-order variational calculus with generalized boundary conditions. Adv. Differ. Equ. 2011, 1–9 (2011), ID 357580
Kaufmann, E.R.: Impulsive periodic boundary value problems for dynamic equations on time scale. Adv. Differ. Equ. 2009, 1–10 (2009), ID 603271
Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies. Elsevier Science, Amsterdam (2006)
Krasnosel’skiĭ M.A.: Some problems in nonlinear analysis. Amer. Math. Soc. Transl. Ser. 2(10), 345–409 (1958)
Labidi S., Tatar N.: Unboundedness for the Euler–Bernoulli beam equation with a fractional boundary dissipation. Appl. Math. Comput. 161, 697–706 (2005)
Lakshmikantham V.: Theory of fractional functional differential equations. Nonlinear Anal. 69, 3337–3343 (2008)
Lakshmikantham V., Bainov D.D., Simeonov P.S.: Theory of impulsive differential equations. Series in modern applied mathematics. World Scientific, New Jersey (1994)
Lakshmikantham V., Vatsala A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)
Luo Z., Nieto J.J.: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. 70, 2248–2260 (2009)
Luo Z., Jing Z.: Periodic boundary value problem for first-order impulsive functional differential equations. Comp. Math. Appl. 55, 2094–2107 (2008)
Podlubny I.: Fractional Differential Equations, Mathematics in Sciences and Applications. Academic Press, New York (1999)
Sabatier J., Agrawal O.P., Tenreiro-Machado J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, The Netherlands (2007)
Samko S.G., Kilbas A.A., Mirichev O.I.: Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland (1993)
Samoĭlenko A.M., Perestyuk N.A.: Impulsive Differential Equations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises. World Scientific, New Jersey (1995)
Shu, X.B., Lai, Y., Chen, Y.: The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. (2012) (in press)
Tian Y., Bai Z.: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comp. Math. Appl. 59, 2601–2609 (2010)
Tian Y., Ge W.: Variational methods to Sturm–Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 72, 277–287 (2010)
Wang, J., Zhou, Y., Fecčkan, M.: On recent developments in the theory of boundary value problem for impulsive fractional differential equations. Comp. Math. Appl. doi:10.1016/j.camwa.2011.12.064
Wei Z., Li Q., Che J.: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 367, 260–272 (2010)
Zhang X., Huang X., Lou Z.: The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay. Nonlinear Anal. Hybrid Syst. 4, 775–781 (2010)
Zeidler E.: Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems. Springer, New York (1986)
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Kosmatov, N. Initial Value Problems of Fractional Order with Fractional Impulsive Conditions. Results. Math. 63, 1289–1310 (2013). https://doi.org/10.1007/s00025-012-0269-3
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DOI: https://doi.org/10.1007/s00025-012-0269-3