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

doi:10.1016/S0252-9602(16)30085-6

摘要/Abstract

摘要：

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

关键词: singular fractional differential equation, impulsive effect, piecewise continuous solution, fixed point theorem

Abstract:

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

Key words: singular fractional differential equation, impulsive effect, piecewise continuous solution, fixed point theorem

中图分类号:

92D25

刘玉记. PIECEWISE CONTINUOUS SOLUTIONS OF INITIAL VALUE PROBLEMS OF SINGULAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH IMPULSE EFFECTS[J]. 数学物理学报(英文版), 2016, 36(5): 1492-1508.

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.

[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