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The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients

Na Song, Zheng-De Xia and Qiang Hou
From the journal Mathematica Slovaca

Abstract

In this paper, we study the existence and global exponential stability of positive piecewise pseudo almost periodic solutions for the impulsive Lasota-Wazewska model with multiply time-varying delays when coefficients are piecewise pseudo almost periodic. Under proper conditions, by using the Gronwall’s inequation, we establish some criteria to ensure that the solution of this model stability exponentially to a positive piecewise pseudo almost periodic solution. Moreover, an example and its numerical simulation are given to illustrate the theoretical results.

MSC 2010: Primary 34C27; 34A37

  1. Communicated by Marcus Waurick

References

[1] Ahmad, S.—Stamov, G. T.: Almost periodic solutions of N-dimensional impulsive competitive systems, Nonlinear Anal. Real World Appl. 10 (2009), 1846–1853.Search in Google Scholar

[2] Bainov, D. D.—Simeonov, P. S.: Impulsive Differential Equation: Periodic Solutions and Applications, London, Longman Scientific and Technical, 1993.Search in Google Scholar

[3] Bainov, D. D.—Simeonov, P. S.: System with Impulsive Effect: Stability, Theory and Applications, New York, John Wiley and Sons, 1986.Search in Google Scholar

[4] Berezansky, L.—Braverman, E.—Idels, L.: Nicholson blowflies differential equations revisited: main results and open problems, Appl. Math. Model. 34 (2010), 1405–1410.Search in Google Scholar

[5] Chen, F. D.: Periodic solutions and almost periodic solutions for a delay multispecies Logarithmic population model, Appl. Math. Comput. 171 (2005), 760–770.Search in Google Scholar

[6] Chen, L. J.—Chen, F. D.: Positive periodic solution of the discrete Lasota-Wazewska model with impulse, J. Difference Equ. Appl. 20 (2014), 406–412.Search in Google Scholar

[7] Cherif, F.—Miraoui, M.: New results for a Lasota-Wazewska model, Int. J. Biomath. 12 (2019), 1950019.Search in Google Scholar

[8] Duan, L.—Lihong, H.—Chen, Y.: Global exponential stability of periodic suolutions to a delay Lasota-Wazewska model with discontinuous harvesting, Proc. Amer. Math. Soc. 144(2) (2016), 561–573.Search in Google Scholar

[9] Duan, L.—Fang, X.—Huang, C.-X.: Global exponential convergence in a delayed almost periodic Nicholson’s blowflies model with discontinuous harvesting, Math. Methods Appl. Sci. 39 (2016), 2821–2839.Search in Google Scholar

[10] Faria, T.—Oliveira, J. J. On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model, AIMS Journals, Search in Google Scholar

[11] Gopalsamy, K.—Trofimchuk, S.: Almost periodic solution of Lasota-Wazewska-type delay differential equation, J. Math. Anal. Appl. 237 (1999), 106–127.Search in Google Scholar

[12] Graef, J. R.—Qian, C.—Spikes, P. W.: Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull. 38 (1996), 275–283.Search in Google Scholar

[13] Gyori, I.—Trofimchuk, S.: Global attractivity in(t) = –δx(t) + pf(x(tτ)), Dynam. Systems Appl. 8 (1999), 197–210.Search in Google Scholar

[14] Henríquez, H. R.—Andrade, B. D.—Rabelo, M.: Existence of almost periodic solutions for a class of abstract impulsive differential equations, ISRN Math. (2011), Search in Google Scholar

[15] Hernández, E.—Rabello, M.—Henríquez, H. R.: Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl. 331 (2007), 1135–1158.Search in Google Scholar

[16] Huang, Z. D.—Gong, S. H.—Wang, L. J.: Positive almost periodic solution for a class of Lasota-Wazewska model with multiple timing-varing delays, Comput. Math. Appl. 61 (2011), 755–760.Search in Google Scholar

[17] Kulenovic, M. R. S.—Ladas, G.—Sficas, Y. G.: Global attractivity in population dynamics, Comput. Math. Appl. 18 (1989), 925–928.Search in Google Scholar

[18] Lakshmikantham V.—Bainov D. D.—Simeonov, P. S.: Theory of Impulsive Differential Equations, Singapore, New Jersey, London, World Scientific, 1989.Search in Google Scholar

[19] Liu, J. W.—Zhang, C. Y.: Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations, Adv. Differ. Equ. 11 (2013), 1–21.Search in Google Scholar

[20] Rihani, S.—Kessab, A.—Chérif, F.: Pseudo almost S. periodic solutions for a Lasota-Wazewska model, Electron. J. Differential Equations 62 (2016), 1–17.Search in Google Scholar

[21] Samoilenko, A. M.—Perestyuk, N. A.: Impulsive Differential Equations, Singapore, World Scientific, 1995.Search in Google Scholar

[22] Song, N.—Li, H.-X.—Chen, C.-H.: Piecewise weighted pseudo almost periodic functions and applications to impulsive differential equations, Math. Slovaca 66(5) (2016), 1–18.Search in Google Scholar

[23] Song, N.—Xia, Z.: Almost periodic solutions for implusive Lasota-Wazewska Model with discontinuous coefficients, Int. Math. Forum 17 (2017), 841–852.Search in Google Scholar

[24] Stamov, G. T.: Almost Periodic Solutions of Impulsive Differential Equations, Berlin, Heidelberg, Springer-Verlag, 2012.Search in Google Scholar

[25] Stamov, G. T.: On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model, Appl. Math. Lett. 22 (2009), 516–520.Search in Google Scholar

[26] Stamov, G. T.—Alzabut, J. O.: Almost periodic solutions for abstract impulsive differential equations, Nonlinear Anal. 72 (2010), 2457–2464.Search in Google Scholar

[27] Stamov, G. T.—Stamova, I. M.—Cao, J.: Uncertain impulsive functional differential systems of fractional order and almost periodicity, J. Franklin Inst. 355 (2018), 5310–5323.Search in Google Scholar

[28] Tan, Y.—Jingb, K.. Existence and global exponential stability of almost periodic solution for delayed competitive neural networks with discontinuous activations, Math. Methods Appl. Sci. 41(5) (2018), 1954–1965.Search in Google Scholar

[29] Xia, Z. N.—Fan, M.: Weighted Stepanov-like pseudo almost automorphy and applications, Nolinear Anal. 75 (2012), 2378–2397.Search in Google Scholar

[30] Wazewska-Czyzewska, M.—Lasota, A.: Mathematical problems of the dynamics of a system of red blood cells, Mat. Stosow 6 (1976), 23–40.Search in Google Scholar

[31] Wang, L.—Yu, M.—Niu, P.: Periodic solution and almost periodic solution of impulsive Lasota-Wazewska model with multiple time-varying delays, Comp. Math. Appl. 64(8) (2012), 2383–2394.Search in Google Scholar

[32] Zhou, H.—Jiang, W.: Existence and stability of positive almost periodic solution for stochastic Lasota-Wazewska model, J. Appl. Math. Comput. 47 (2015), 61–71.Search in Google Scholar

[33] Zhou, H.—Zhou, Z.—Wang, Q.: Positive almost periodic solution for a class of Lasota-Wazewska model with infinite delays, Appl. Math. Comput. 218(8) (2011), 4501–4506.Search in Google Scholar

Received: 2019-04-04
Accepted: 2019-08-17
Published Online: 2020-03-10
Published in Print: 2020-04-28

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