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Existence Results for Impulsive Nonlinear Fractional Differential Equations With Nonlocal Boundary Conditions

  • Zhenhai Liu EMAIL logo and Jingyun Lv
Published/Copyright: February 9, 2016
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Abstract

In this paper, we prove the existence and uniqueness of solutions of fractional impulsive differential equations with nonlocal boundary conditions by applying the contraction mapping principle.

References

[1] ANGURAJ, A.-KARTHIKEYAN, P.: Anti-periodic boundary value problem for impulsive fractional integrodifferetial equations, Fract. Calc. Appl. Anal. 13 (2010), 281-294.Search in Google Scholar

[2] AHMAD, B.-NIETO, J. J.: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with antiperiodic boundary conditions, Nonlinear Anal. 69 (2008), 3291-3298.10.1016/j.na.2007.09.018Search in Google Scholar

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

[4] AHMAD, B.-SIVASUNDARAM, S.: Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybrid Syst. 4 (2010), 134-141.10.1016/j.nahs.2009.09.002Search in Google Scholar

[5] BENCHOHRA, M.-HAMANI, S.-NIETO, J. J.: Existence of solutions to differential inclusions with fractional order and impulses, Electron. J. Differential Equations 80 (2010), 1-18.Search in Google Scholar

[6] BENCHOHRA, M.-HAMANI, S.: The method of upper and lower solutions and impulsive fractional differential inclusions, Nonlinear Anal. Hybrid Syst. 3 (2009), 433-440.10.1016/j.nahs.2009.02.009Search in Google Scholar

[7] BALACHANDRAN, K.-KIRUTHIKA, S.-TRUJILLO, J. J.: Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1970-1977.10.1016/j.cnsns.2010.08.005Search in Google Scholar

[8] BALACHANDRAN, K.-KIRUTHIKA, S.-TRUJILLO, J. J.: On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces, Comput. Math. Appl. 62 (2011), 1157-1165.10.1016/j.camwa.2011.03.031Search in Google Scholar

[9] BENCHOHRA,M.-SLIMANI, B. A.: Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential Equations 10 (2009), 1-11.Search in Google Scholar

[10] CHEN, F. L.-CHEN, A. P.-WANG, X. P.: On the solutions for Impulsive Fractional Functional Differential Equations, Differ. Equ. Dyn. Syst. 17 (2009), 379-391.10.1007/s12591-009-0027-5Search in Google Scholar

[11] FEČKAN, M.-ZHOU, Y.-WANG, J. R.: On the concept and existence of solution for impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3050-3060.10.1016/j.cnsns.2011.11.017Search in Google Scholar

[12] KILBAS, A. A.-SRIVASTAVA, H. M.-TRUJILLO, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Math. Stud. 204, Elsevier Science B. V, Amsterdam, 2006.Search in Google Scholar

[13] LAKSHMIKANTHAM, V.-BAINOV, D. D.-SIMEONOV, P. S.: Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.10.1142/0906Search in Google Scholar

[14] LIU, Y.-CHEN, H. B.: Nonlocal boundary value problem for impulsive differntial equations of fractional order, Adv. Difference Equ. 2011, Article ID 404917, 16 pp..10.1155/2011/404917Search in Google Scholar

[15] LIU, Z. H.-LIANG, J. T.: A class of boundary value problems for first-order impulsive integro-differential equations with deviating arguments, J. Comput. Appl. Math. 237 (2013), 477-486.10.1016/j.cam.2012.06.018Search in Google Scholar

[16] LIU, Z. H.-SUN, J. H.: Nonlinear boundary value problems of fractional functional integro-differential equations, Comput. Math. Appl. 64 (2012), 3228-3234.10.1016/j.camwa.2012.02.026Search in Google Scholar

[17] LIU, Z. H.-SUN, J. H.: Nonlinear boundary value problems of fractional differential systems, Comput. Math. Appl. 64 (2012), 463-475.10.1016/j.camwa.2011.12.020Search in Google Scholar

[18] LIU, Z. H.-LI, X. W.: On the controllability of impulsive fractional evolution inclusions in Banach spaces, J. Optim. Theory Appl. 156 (2013), 167-182.10.1007/s10957-012-0236-xSearch in Google Scholar

[19] LIU, Z. H.-LI, X. W.: Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1362-1373.10.1016/j.cnsns.2012.10.010Search in Google Scholar

[20] LIU, Z. H.-LU, L.: A class of BVPs for nonlinear fractional differential equations with p-Laplacian operator, Electron. J. Qual. Theory Differ. Equ. 70 (2012), 1-16.Search in Google Scholar

[21] LIU, Z. H.-HAN, J. F.: Integral boundary value problems for fractional order integrodifferential equations, Dynam. Systems Appl. 21 (2012), 535-548.Search in Google Scholar

[22] LI, F.: Existence and uniqueness of mild solution for fractional integrodifferential equations of neutral type with nonlocal conditions, Math. Slovaca 62 (2012), 921-936.10.2478/s12175-012-0055-4Search in Google Scholar

[23] PODLUBNY, I.: Fractional Differential Equations, Academic Press, San Diego, 1999.Search in Google Scholar

[24] SAMKO, S. G.-KILBAS, A. A.-MARICHEV, O. I.: Fractional Integrals and Derivatives, Theory and Applications. Theory and Applications, Gordon and Breach, Yverdon, 1993.Search in Google Scholar

[25] SHU, X. B.-LAI, Y. Z.-CHEN, Y. M.: The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal. 74 (2011), 2003-2011.10.1016/j.na.2010.11.007Search in Google Scholar

[26] WANG, J. R.-FAN, Z. B.-ZHOU, Y.: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces, J. Optim. Theory Appl. 154 (2012), 292-302.10.1007/s10957-012-9999-3Search in Google Scholar

[27] WANG, J. R.-FEČKAN, M.-ZHOU, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations, Dyn. Partial Differ. Equ. 8 (2011), 345-361.10.4310/DPDE.2011.v8.n4.a3Search in Google Scholar

[28] WANG, J. R.-ZHOU, Y.-WEI, W.: Impulsive problems for fractional evolution equations and optimal controls in infinite dimensional spaces, Topol. Methods Nonlinear Anal. 38 (2011), 17-43.Search in Google Scholar

[29] WANG, J. R.-LI, X. Z.-WEI, W.: On the natural solution of an impulsive fractional differential equation of order q ∈ (1, 2), Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4384-4394.10.1016/j.cnsns.2012.03.011Search in Google Scholar

[30] WANG, J. R.-ZHOU, Y.-FEČKAN, M.: On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl. 64 (2012), 3008-3020.10.1016/j.camwa.2011.12.064Search in Google Scholar

[31] WANG, J. R.-ZHOU, Y.-FEČKAN, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl. 64 (2012), 3389-3405.10.1016/j.camwa.2012.02.021Search in Google Scholar

[32] WANG, H. H.: Existence results for fractional functional differential equations with impulsesl, J. Appl. Math. Comput. (2010), DOI 10. 1007/s12190-010-0465-9.Search in Google Scholar

[33] WANG, G. T.-AHMAD, B.-ZHANG, L. H.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal. 74 (2011), 792-804.10.1016/j.na.2010.09.030Search in Google Scholar

[34] ZHANG, L. H.-WANG, G. T.: 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. Search in Google Scholar

Received: 2012-4-28
Accepted: 2013-3-4
Published Online: 2016-2-9
Published in Print: 2015-12-1

Mathematical Institute Slovak Academy of Sciences

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