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On General Solution for Fractional Differential Equations with Not Instantaneous Impulses

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Języki publikacji
EN
Abstrakty
EN
In this paper we mainly study a kind of fractional differential equations with not instantaneous impulses, and find the equivalent equations of the impulsive system. The obtained result discovers that there exist general solution for the impulsive system. Next, an example is given to illustrate the obtained result.
Wydawca
Rocznik
Strony
355--369
Opis fizyczny
Bibliogr. 28 poz.
Twórcy
autor
  • School of Electronic Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
autor
  • School of Chemical and Environmental Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
autor
  • School of Electronic Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
Bibliografia
  • [1] Ahmad B, Sivasundaram S. Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Analysis: Hybrid Systems. 2009; 3 (3): 251-258. doi: 10.1016/j.nahs.2009.01.008.
  • [2] Ahmad B, Sivasundaram S. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems. 2010; 4(1): 134-141. doi: 10.1016/j.nahs.2009.09.002.
  • [3] Ahmad B, Wang G. A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations. Computers and Mathematics with Applications. 2011; 62 (3): 1341-1349. doi: 10.1016/j.camwa.2011.04.033.
  • [4] Tian Y, Bai Z. Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Computers and Mathematics with Applications. 2010; 59 (8): 2601-2609. doi: 10.1016//j.camwa.2010.01.028.
  • [5] Cao J, Chen H. Some results on impulsive boundary value problem for fractional differential inclusions. Electronic Journal of Qualitative Theory of Differential Equations. 2011; 2011 (11): 1-24. doi: 10.14232/ejqtde.2011.1.11.
  • [6] Wang G, Ahmad B, Zhang L. Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Analysis: Theory, Methods, Applications. 2011; 74 (3): 792-804. doi: 10.1016/j.na.2010.09.030.
  • [7] Zhang X, Huang X, Liu Z. The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay. Nonlinear Analysis: Hybrid Systems. 2010; 4 (4): 775-781. doi: 10.1016/j.nahs.2010.05.007.
  • [8] Wang G, Ahmad B, Zhang L. Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Computers and Mathematics with Applications. 2011; 62 (3): 1389-1397. doi: 10.1016/j.camwa.2011.04.004.
  • [9] Wang X. Impulsive boundary value problem for nonlinear differential equations of fractional order. Computers and Mathematics with Applications. 2011; 62 (5): 2383-2391. doi: 10.1016/j.camwa.2011.07.026.
  • [10] Feckan M, Zhou Y, Wang JR. On the concept and existence of solution for impulsive fractional differential equations. Communications in Nonlinear Science and Numerical Simulation. 2012; 17 (7): 3050-3060. doi: 10.1016/j.cnsns.2011.11.017.
  • [11] Stamova I, Stamov G. Stability analysis of impulsive functional systems of fractional order. Communications in Nonlinear Science and Numerical Simulation. 2014; 19 (3): 702-709. doi: 10.1016/j.cnsns.2013.07.005.
  • [12] Abbas S, Benchohra M. Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order. Nonlinear Analysis: Hybrid Systems. 2010; 4 (3): 406-413. doi: 10.1016/j.nahs.2009.10.004.
  • [13] Abbas S, Benchohra M. Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay. Fractional Calculus and Applied Analysis. 2010; 13 (3): 225-242.
  • [14] Abbas S, Agarwal RP, Benchohra M. Darboux problem for impulsive partial hyperbolic differential equations of fractional order with variable times and infinite delay, Nonlinear Analysis: Hybrid Systems. 2010; 4 (4): 818-829. doi: 10.1016/j.nahs.2010.06.001.
  • [15] Abbas S, Benchohra M, Gomiewicz L. Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Scientiae Mathematicae Japonicae. 2010; 72 (l): 49-60.
  • [16] Benchohra M, Seba D. Impulsive partial hyperbolic fractional order differential equations in Banach spaces. Journal of Fractional Calculus and Applications. 2011; 1 (4): 1-12.
  • [17] Guo TL, Zhang KJ. Impulsive fractional partial differential equations. Applied Mathematics and Computation. 2015; 257: 581-590. doi: 10.1016/j.amc.2014.05.101.
  • [18] Hernandez E, O’Regan D. On a new class of abstract impulsive differential equations. Proceedings of the American Mathematical Society. 2013; 141 (5): 1641-1649. doi: 10.1090/S0002-9939-2012-11613-2.
  • [19] Pierri M, O’Regan D, Rolnik V. Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Applied Mathematics and Computation. 2013; 219 (12): 6743-6749. doi: 10.1016/j.amc.2012.12.084.
  • [20] Li PL, Xu CJ. Mild solution of fractional order differential equations with not instantaneous impulses. Open Mathematics. 2015; 13: 436-443. doi: 10.1515/math-2015-0042.
  • [21] Zhang X, Zhang X, Zhang M. On the concept of general solution for impulsive differential equations of fractional order q ϵ (0,1). Applied Mathematics and Computation. 2014; 247: 72-89. doi: 10.1016/j.amc.2014.08.069.
  • [22] Zhang X. On the concept of general solutions for impulsive differential equations of fractional order q ϵ (1, 2). Applied Mathematics and Computation. 2015; 268: 103-120. doi: 10.1016/j.amc.2015.05.123.
  • [23] Zhang X. The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect. Advances in Difference Equations. 2015; 2015, Article ID 215 (16 p.). doi: 10.1186/s13662-015-0552-1.
  • [24] Zhang X, Agarwal P, Liu Z, Peng H. The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ϵ (1, 2). Open Mathematics. 2015; 13 (l): 908-923. doi: 10.1515/math-2015-0073.
  • [25] Zhang X, Shu T, Cao H, Liu Z, Ding W. The general solution for impulsive differential equations with Hadamard fractional derivative of order q ϵ (1, 2). Advances in Difference Equations. 2016; 2016, Article ID 14 (36 p.). doi: 10.1186/s13662-016-0744-3.
  • [26] Zhang X, Zhang X, Liu Z, Peng H, Shu T, Yang S. The general solution of impulsive systems with Caputo-Hadamard fractional derivative of order q € C(ℜ(q) € (1, 2)). Mathematical Problems in Engineering. 2016; 2016, Article ID 8101802 (20 p.). doi: 10.1155/2016/8101802.
  • [27] Kilbas AA, Srivastava HH, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
  • [28] Diethelm K, Ford NJ. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications. 2002 ;265 (2): 229-248. doi: 10.1006/jmaa.2000.7194.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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