Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 123-141, 2017

Pseudo almost periodicity of fractional integro-differential equations with impulsive effects in Banach spaces

Zhinan Xia

Received July 22, 2015.   First published February 24, 2017.

Zhinan Xia, Department of Applied Mathematics, Zhejiang University of Technology, Liuhe Road 288, Hangzhou 310023, Zhejiang, China, e-mail: xiazn299@zjut.edu.cn

Abstract: In this paper, for the impulsive fractional integro-differential equations involving Caputo fractional derivative in Banach space, we investigate the existence and uniqueness of a pseudo almost periodic $PC$-mild solution. The working tools are based on the fixed point theorems, the fractional powers of operators and fractional calculus. Some known results are improved and generalized. Finally, existence and uniqueness of a pseudo almost periodic $PC$-mild solution of a two-dimensional impulsive fractional predator-prey system with diffusion are investigated.

Keywords: impulsive fractional integro-differential equation; pseudo almost periodicity; probability density; fractional powers of operator

Classification (MSC 2010): 34A37, 26A33, 34C27

DOI: 10.21136/CMJ.2017.0398-15

Full text available as PDF.


References:
  [1] K. Adolfsson, J. Enelund, P. Olsson: On the fractional order model of viscoelasticity. Mech. Time-Depend. Mat. 9 (2005), 15-34. DOI 10.1007/s11043-005-3442-1
  [2] E. Ait Dads, O. Arino: Exponential dichotomy and existence of pseudo almost-periodic solutions of some differential equations. Nonlinear Anal., Theory Methods Appl. 27 (1996), 369-386. DOI 10.1016/0362-546X(95)00027-S | MR 1393143 | Zbl 0855.34055
  [3] M. U. Akhmet, M. Beklioglu, T. Ergenc, V. I. Tkachenko: An impulsive ratio-dependent predator-prey system with diffusion. Nonlinear Anal., Real World Appl. 7 (2006), 1255-1267. DOI 10.1016/j.nonrwa.2005.11.007 | MR 2260913 | Zbl 1114.35097
  [4] J. Cao, Q. Yang, Z. Huang: Optimal mild solutions and weighted pseudo-almost periodic classical solutions of fractional integro-differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 224-234. DOI 10.1016/j.na.2010.08.036 | MR 2734991 | Zbl 1213.34089
  [5] Y. K. Chang, R. Zhang, G. M. N'Guérékata: Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations. Comput. Math. Appl. 64 (2012), 3160-3170. DOI 10.1016/j.camwa.2012.02.039 | MR 2989344 | Zbl 1268.34010
  [6] F. Chérif: Pseudo almost periodic solutions of impulsive differential equations with delay. Differ. Equ. Dyn. Syst. 22 (2014), 73-91. DOI 10.1007/s12591-012-0156-0 | MR 3149175 | Zbl 1298.34132
  [7] A. Debbouche, M. M. El-borai: Weak almost periodic and optimal mild solutions of fractional evolution equations. Electron. J. Differ. Equ. (electronic only) 2009 (2009), No. 46, 8 pages. MR 2495851 | Zbl 1171.34331
  [8] T. Diagana, G. M. N'Guérékata: Pseudo almost periodic mild solutions to hyperbolic evolution equations in intermediate Banach spaces. Appl. Anal. 85 (2006), 769-780. DOI 10.1080/00036810600708499 | MR 2232421 | Zbl 1103.34051
  [9] H.-S. Ding, J. Liang, G. M. N'Guérékata, T. J. Xiao: Mild pseudo-almost periodic solutions of nonautonomous semilinear evolution equations. Math. Comput. Modelling 45 (2007), 579-584. DOI 10.1016/j.mcm.2006.07.006 | MR 2286345 | Zbl 1165.34387
  [10] M. Enelund, P. Olsson: Damping described by fading memory-analysis and application to fractional derivative models. Int. J. Solids Struct. 36 (1999), 939-970. DOI 10.1016/S0020-7683(97)00339-9 | MR 1666097 | Zbl 0936.74023
  [11] A. M. Fink: Almost Periodic Differential Equations. Lecture Notes in Mathematics 377, Springer, New York (1974). DOI 10.1007/BFb0070324 | MR 0460799 | Zbl 0325.34039
  [12] H. R. Henríquez, B. de Andrade, M. Rabelo: Existence of almost periodic solutions for a class of abstract impulsive differential equations. ISRN Math. Anal. 2011 (2011), Article ID 632687, 21 pages. DOI 10.5402/2011/632687 | MR 2784886 | Zbl 1242.34110
  [13] J. Hong, R. Obaya, A. Sanz: Almost-periodic-type solutions of some differential equations with piecewise constant argument. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 45 (2001), 661-688. DOI 10.1016/S0362-546X(98)00296-X | MR 1841201 | Zbl 0996.34062
  [14] Y. Li, C. Wang: Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales. Adv. Difference Equ. 2012 (2012), Article ID 77, 24 pages. DOI 10.1186/1687-1847-2012-77 | MR 2946504 | Zbl 1294.34085
  [15] J. Liu, C. Zhang: Existence and stability of almost periodic solutions for impulsive differential equations. Adv. Difference Equ. 2012 (2012), Article ID 34, 14 pages. DOI 10.1186/1687-1847-2012-34 | MR 2935667 | Zbl 1291.34076
  [16] J. Liu, C. Zhang: Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations. Adv. Difference Equ. 2013 (2013), 2013:11, 21 pages. DOI 10.1186/1687-1847-2013-11 | MR 3019356
  [17] J. Liu, C. Zhang: Existence and stability of almost periodic solutions to impulsive stochastic differential equations. Cubo 15 (2013), 77-96. DOI 10.4067/s0719-06462013000100005 | MR 3087596 | Zbl 1292.34054
  [18] J. Liu, C. Zhang: Existence of almost periodic solutions for impulsive neutral functional differential equations. Abstr. Appl. Anal. 2014 (2014), Article ID 782018, 11 pages. DOI doi.org/10.1155/2014/782018 | MR 3251537
  [19] A. Pazy: Semigroup of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer, New York (1983). DOI 10.1007/978-1-4612-5561-1 | MR 0710486 | Zbl 0516.47023
  [20] I. Podlubny: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering 198, Academic Press, San Diego (1999). MR 1658022 | Zbl 0924.34008
  [21] A. M. Samoilenko, N. A. Perestyuk: Impulsive Differential Equations. World Scientific Series on Nonlinear Science. Series A. 14, World Scientific, Singapore (1995). DOI 10.1142/9789812798664 | MR 1355787 | Zbl 0837.34003
  [22] G. T. Stamov: Almost Periodic Solutions of Impulsive Differential Equations. Lecture Notes in Mathematics 2047, Springer, Berlin (2012). DOI 10.1007/978-3-642-27546-3 | MR 2934087 | Zbl 1255.34001
  [23] G. T. Stamov, J. O. Alzabut: Almost periodic solutions for abstract impulsive differential equations. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72 (2010), 2457-2464. DOI 10.1016/j.na.2009.10.042 | MR 2577811 | Zbl 1190.34067
  [24] G. T. Stamov, I. M. Stamova: Almost periodic solutions for impulsive fractional differential equations. Dyn. Syst. 29 (2014), 119-132. DOI 10.1080/14689367.2013.854737 | MR 3170642 | Zbl 1320.34012
  [25] J. R. Wang, M. Fečkan, Y. Zhou: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8 (2011), 345-361. DOI 10.4310/DPDE.2011.v8.n4.a3 | MR 2901608 | Zbl 1264.34014
  [26] C. Zhang: Pseudo almost perioidc solutions of some differential equations. J. Math. Anal. Appl. 181 (1994), 62-76. DOI 10.1006/jmaa.1994.1005 | MR 1257954 | Zbl 0796.34029
  [27] C. Zhang: Pseudo almost perioidc solutions of some differential equations. II. J. Math. Anal. Appl. 192 (1995), 543-561. DOI 10.1006/jmaa.1995.1189 | MR 1332227 | Zbl 0826.34040


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://link.springer.com/journal/10587.

[Previous Article] [Next Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]