Existence, Uniqueness and UHR Stability of Solutions to Nonlinear Ordinary Differential Equations with Noninstantaneous Impulses

Xuping Zhang 1  and Zhen Xin 2
  • 1 Department of Mathematics, Northwest Normal University, #967 anning east road, Lanzhou, China
  • 2 Department of Mathematics, Northwest Normal University, Gansu 730070, Lanzhou, China
Xuping Zhang
  • Corresponding author
  • Department of Mathematics, Northwest Normal University, #967 anning east road, Lanzhou, 730070, China
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Zhen Xin
  • Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China
  • Search for other articles:
  • degruyter.comGoogle Scholar


We consider the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions to the initial value problem with noninstantaneous impulses on ordered Banach spaces. The existence and uniqueness of solutions for nonlinear ordinary differential equation with noninstantaneous impulses is obtained by using perturbation technique, monotone iterative method and a new estimation technique of the measure of noncompactness under the situation that the corresponding noninstantaneous impulsive functions gi are compact and not compact, respectively. Furthermore, the UHR stability of solutions is also obtained, which provides an approach to find approximate solution to noninstantaneous impulsive equations in the sense of UHR stability.

  • [1]

    A. D. Myshkis and A. M. Samoilenko, Sytems with impulsive at fixed moments of time. Mat. Sb. 74 (1967), 202–208.

  • [2]

    D. D. Bainov and P. S. Simeonov, Impulsive differential equations, in: Series on Advances in Mathematics for Applied Sciences, vol. 28. World Scientific, Singapore, 1995.

  • [3]

    D. D. Bainov, Y. I. Domshlak and P. S. Simeonov, Sturmian comparison theory for impulsive differential inequalities and equations. Arch. Math. (Basel) 67(1) (1996), 35–49.

  • [4]

    D. D. Bainov and P. S. Simeonov, Oscillation theory of impulsive differential equations. International Publications, Orlando, FL, 1998.

  • [5]

    V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive differential equations. Series in Modern Applied Mathematics, vol. 6. World Scientific, Singapore, 1989.

  • [6]

    J. Liu and Z. Zhao, Multiple solutions for impulsive problems with non-autonomous perturbations, Appl. Math. Lett. 64 (2017), 143–149.

  • [7]

    X. Hao and L. Liu, Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces, Math. Methods Appl. Sci. 40(13) (2017), 4832-4841.

  • [8]

    M. Zuo, X. Hao, L. Liu and Y. Cui, Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl. 161 (2017), 15.

  • [9]

    J. Jiang, L. Liu and Y. Wu, Positive solutions for second order impulsive differential equations with Stieltjes integral boundary conditions. Adv. Difference Equ. 124 18 (2012).

  • [10]

    X. Hao, L. Liu and Y. Wu, Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun., Nonlinear Sci., Numer., Simulat. 16(1) (2011), 101–111.

  • [11]

    A. M. Samoilenko, N. A. Perestyuk and Y. Chapovsky, Impulsive differential equations. World Scientific, Singapore, 1995.

  • [12]

    M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive differential equations and inclusions. Hindawi Publishing Corporation, Cairo, 2006.

  • [13]

    X. Hao, M. Zuo and L. Liu, Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities, Appl. Math. Lett. 82 (2018), 24–31.

  • [14]

    X. Hao, L. Liu and Y. Wu, Iterative solution for nonlinear impulsive advection reaction diffusion equations, J. Nonlinear Sci. Appl. 9(6) (2016), 4070–4077.

  • [15]

    X. Hao and L. Liu, Mild solutions of impulsive semilinear neutral evolution equations in Banach spaces, J. Nonlinear Sci. Appl. 9(12) (2016), 6183–6194.

  • [16]

    X. Hao, L. Zhang and L. Liu, Positive solutions of higher order fractional integral boundary value problem with a parameter, Nonlinear Anal. Model. Control 24(2) (2019), 210–223.

  • [17]

    E. Hernández and D. O’Regan, On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141 (2013), 1641–1649.

  • [18]

    M. Pierri, D. O’Regan and V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses. Appl. Math. Comput. 219 (2013), 6743–6749.

  • [19]

    M. Fečkan, J. Wang and Y. Zhou, Existence of periodic solutions for nonlinear evolution equations with non-instantaneous impulses. Nonauton. Dyn. Syst. 1 (2014), 93–101.

  • [20]

    A. Zada, O. Shah and R. Shah, Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput. 271 (2015), 512–518.

  • [21]

    D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. (1941), 222–224.

  • [22]

    S. W. Du and V. Lakshmikantham, Monotone iterative technique for differential equations in a Banach space, J. Math. Anal. Appl. 87(2) (1982), 454–459.

  • [23]

    P. Chen, Y. Li and X. Zhang, Double perturbations for impulsive differential equations in Banach spaces, Taiwanese J. Math. 20(5) (2016), 1065–1077.

  • [24]

    P. Chen, Y. Li and X. Zhang, Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces, Appl. Math. 59(1) (2014), 99–120.

  • [25]

    S. M. Ulam, A collection of the mathematical problems, Interscience Publisher, New York, London, 1960.

  • [26]

    S. M. Ulam, Problem in modern mathematics, Science Editions., John Wiley and Sons, Inc., New York, 1964.

  • [27]

    M. Obłza, Hyers stability of the linear differential equation, Rocz. Nauk.-Dydakt. Pr. Mat. 13 (1993), 259–270.

  • [28]

    C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl. 2(4) (1998), 373–380.

  • [29]

    S. E. Takahasi, H. Takagi, T. Miura and S. Miyajima, The Hyers-Ulam stability constants of first order linear differential operators, J. Math. Anal. Appl. 296 (2004), 403–409.

  • [30]

    T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Am. Math. Soc. 72 (1978), 297–300.

  • [31]

    D. Barbu, Buşe, C. and A. Tabassum, Hyers-Ulam stability and discrete dichotomy. J. Math. Anal. Appl. 42 (2015), 1738–1752.

  • [32]

    C. Buşe, D. O’Regan, Saierli O and A. Tabassum, Hyers-Ulam stability and discrete dichotomy for difference periodic systems. Bull. Sci. Math. 140 (2016), 908–934.

  • [33]

    J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Pekker, New York, 1980.

  • [34]

    K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985.

  • [35]

    D. Guo and J. Sun, Ordinary differential equations in abstract spaces, Shandong Science and Technology, Jinan, 1989.

  • [36]

    P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math. 63(3–4) (2013), 731–744.

  • [37]

    H. P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal. 7(12) (1983), 1351–1371.

  • [38]

    W. V. Petryshyn, Structure of the fixed points sets of k-set-contractions, Arch. Rational. Mech. Anal. 40 (1971), 312–328.

  • [39]

    D. D. Bainov and P. S. Simeonov, Integral inequalities and applications. Mathematics and its Applications (East European Series), 57. Kluwer Academic Publishers Group, Dordrecht, 1992.

Purchase article
Get instant unlimited access to the article.
Log in
Already have access? Please log in.

Log in with your institution

Journal + Issues

The International Journal of Nonlinear Sciences and Numerical Simulation publishes original papers on all subjects relevant to nonlinear sciences and numerical simulation. The journal is directed at researchers in nonlinear sciences, engineers, and computational scientists, economists, and others, who either study the nature of nonlinear problems or conduct numerical simulations of nonlinear problems.