Articles

PIECEWISE CONTINUOUS SOLUTIONS OF INITIAL VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS

  • Yuji LIU
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  • Department of Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China

Received date: 2014-07-25

  Online published: 2016-10-25

Supported by

Supported by the Natural Science Foundation of Guangdong Province (S2011010001900) and the Guangdong Higher Education Foundation for High-Level Talents.

Abstract

Results on the existence of piecewise continuous solutions for two classes of initial value problems of impulsive singular fractional differential equations are obtained.

Cite this article

Yuji LIU . PIECEWISE CONTINUOUS SOLUTIONS OF INITIAL VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS[J]. Acta mathematica scientia, Series B, 2016 , 36(5) : 1492 -1508 . DOI: 10.1016/S0252-9602(16)30085-6

References

[1] Agarwal R P, Benchhra M, Slimani B A. Existence results for differential equations with fractional order and impulses. Memoirs Differ Equ Math Phys, 2008, 44:1-21
[2] Anguraj A, Karthikeyan P, Rivero M, Trujillo J J. On new existence results for fractional integro-differential equations with impulsive and integral conditions. Comput Math Appl, 2014, 66:2587-2594
[3] Ahmad B, Nieto J J. Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan J Math, 2011, 15:981-993
[4] Ahmad B, Sivasundaram S. Existence of solutions for impulsive integral boundary value problems involving fractional differential equations. Nonlinear Anal, Hybrid Syst, 2009, 3:251-258
[5] Ahmad B, Sivasundaram S. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal, Hybrid Syst, 2010, 4:134-141
[6] Ahmad B, Wang G. A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations. Comput Math Appl, 2011, 62:1341-1349
[7] Bai C. Impulsive periodic boundary value problems for fractional differential equation involving RiemannLiouville sequential fractional derivative. J Math Anal Appl, 2011, 384:211-231
[8] Babakhani A. Existence and uniqueness of solution for class of fractional order differential equations on an unbounded domain. Adv Differ Equ, 2012, 2012:41
[9] Feckan M, Wang J, Zhou Y. On the concept and existence of solution for impulsive fractional differential equations. Commun Nonlinear Sci Numer Simul, 2012, 17:3050-3060
[10] Feckan M, Zhou Y, Wang J. Response to "Comments on the concept of existence of solution for impulsive fractional differential equations[Commun Nonlinear Sci Numer Simul 2014;19:401-3.]". Commun Nonlinear Sci Numer Simul, 2014, DOI:http://dx.doi.org/10.1016/j.cnsns.2014.04.014
[11] Guo T, Jiang W. Impulsive problems for fractional differential equations with boundary value conditions. Comput Math Appl, 2012, 64:3281-3291
[12] Hilfer R. Applications of Fractional Calculus in Physics. River Edge, NJ:World Scientific Publishing Co Inc, 2000
[13] Henderson J, Ouahab A. Impulsive differential inclusions with fractional order. Comput Math Appl, 2010, 59:1191-1226
[14] Ke T, Luo M. Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders. Commun Nonlinear Sci Numer Simulat, 2014, 19:1661-1668
[15] Li X, Chen F, Li X. Generalized anti-periodic boundary value problems of impulsive fractional differential equations. Commun Nonlinear Sci Numer Simul, 2013, 18:28-41
[16] Liu Z, Li X. Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations. Commun Nonlinear Sci Numer Simulat, 2013, 18:1362-1373
[17] Lakshmikantham V V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations. Singapore:World Scientific, 1989
[18] Podlubny I. Geometric and physical interpretation of fractional integration and fractional differentiation. Fract Calc Appl Anal, 2002, 5(4):367-386
[19] Podlubny I. Fractional Differential Equations. London:Academic Press, 1999
[20] Rashid M H M, Al-Omari A. Local and global existence of mild solutions for impulsive fractional semilinear integro-differential equation. Commun Nonlinear Sci Numer Simul, 2011, 16:3493-3503
[21] Rehman a M, Eloe P W. Existence and uniqueness of solutions for impulsive fractional differential equations. Appl Math Comput, 2013, 224:422-431
[22] Stamova I. Global stability of impulsive fractional differential equations. Appl Math Comput, 2014, 237:605-612
[23] Su X, Chen Y, Lai Y. The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal, 2011, 74:2003-2011
[38] Stamova I, Stamov G. Stability analysis of impulsive functional systems of fractional order. Commun Nonlinear Sci Numer Simul, 2014, 19:702-709
[25] Tian Y, Bai Z. Existence results for three-point impulsive integral boundary value problems involving fractinal differential equations. Comput Math Appl, 2010, 59:2601-2609
[26] Wang G, Ahmad B, Zhang L. Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal, 2011, 74:792-804
[27] Wang G, Ahmad B, Zhang L. Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Comput Math Appl, 2011, 62:1389-1397
[28] Wang G, Ahmad B, Zhang L, Nieto J J. Comments on the concept of existence of solution for impulsive fractional differential equations. Commun Nonlinear Sci Numer Simul, 2014, 19:401-403
[29] Wang J, Zhou Y, Feckan M. Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput Math Appl, 2012, 64:3389-3405
[30] Wang J, Zhou Y, Feckan M. On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput Math Appl, 2012, 64:3008-3020
[31] Wang J, Zhou Y. A class of nonlinear differential equations with fractional integrable impulses. Commun Nonlinear Sci Numer Simulat, 2014, 19:3001-3010
[32] Wang X. Impulsive boundary value problem for nonlinear differential equations of fractional order. Comput Math Appl, 2011, 62:2383-2391
[33] Zhou J, Feng M. Green's function for Sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application. Boundary Value Problems, 2014, 2014:69
[34] Zhang X, Zhu C, Wu Z. Solvability for a coupled system of fractional differential equations with impulses at resonance. Boundary Value Problems, 2013, 2013:80
[35] Liu Y, Ahmad B. A study of impulsive multiterm fractional differential equations with single and multiple base points and applications. Scientific World J, 2014, 2014:Article ID 194346
[36] Liu Y, Nieto J J, Otero-Zarraquinos O. Existence results for a coupled system of nonlinear singular fractional differential equations with impulse effects. Math Problems Engin, 2013, 2013:Article ID 498781
[37] Ahmad B, Nieto J J. Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan J Math, 2011, 15:981-993
[38] Stamova I, Stamov G. Stability analysis of impulsive functional systems of fractional order. Commun Nonlinear Sci Numer Simul, 2014, 19:702-709

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