Abstract
This paper deals with impulsive differential-difference equations for which the impulses are realized at moments when the integral curve of the initial value problem considered meets fixed curves in the extended phase space of the equation. The corresponding impulsive differential-difference inequalities are considered. Sufficient conditions are given for the absence of the phenomenon “beating” and for the continuation of the solutions of the initial value problems for impulsive differential-difference equations and inequalities. Monotone sequences of lower and upper solutions of the initial value problem for the impulsive differential-difference equations are constructed. Under some natural assumptions it is proved that these sequences are convergent to the solutions of the same problem.
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References
Bainov D. and Dishliev A., Quasiuniqueness, uniqueness and continuability of the solutions of impulsive functional differential equations, Rendiconti di Matematica, Serie VII, 15 (1995) Roma, 391–404.
Dishliev A. and Bainov D., Sufficient conditions for absence of “beating” in systems of differential equations with impulses, Applicable Analysis, 18 (1984), 67–73.
Dishliev A. and Bainov D., Conditions for the absence of the phenomenon “beating” for systems of impulsive differential equations, Bulletin of the Institute of Mathematics Academia Sinica, 13 (1985), no. 3, 237–256.
Hristova S. and Bainov D., A projection-iterative method for finding periodic solutions of nonlinear systems of difference-differential equations with impulses, J. Approx. Theory, 49 (1987), no. 4, 311–320.
Hristova S. and Bainov D., Numeric-analytic method for finding the periodic solutions of nonlinear differential-difference equations with impulses, Computing, 38 (1987), 363–368.
Hristova S. and Bainov D., Application of the monotone-iterative techniques of V. Lakshmikantham for solving the initial value problem for impulsive differential-difference equations, Rocky Mount. J. of Mathematics, 23 (1993), no. 2, 1–10.
Ladde G., Lakshmikantham V., and Vatsala A., Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Belmonth, 1985.
Lakshmikantham V., Leela S., and Oguztoreli M., Quasi-solutions, vector Lyapunov functions and monotone method, IEEE Trans. Automat. Control, Vol. 26, 1986.
Lakshmikantham V. and Vatsala A., Quasi-solutions and monotone method for systems of nonlinear boundary value problems, J. Math. Anal. Appl., 79 (1981), 38–47.
Lakshmikantham V., Bainov D., and Simeonov P., Theory of Impulsive Differential Equations, World Scientific Publishers, Singapore, 1989.
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© 1997 Springer Basel AG
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Bainov, D., Dishliev, A., Hristova, S. (1997). Monotone Iterative Technique for Impulsive Differential-Difference Equations with Variable Impulsive Perturbations. In: Nürnberger, G., Schmidt, J.W., Walz, G. (eds) Multivariate Approximation and Splines. ISNM International Series of Numerical Mathematics, vol 125. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8871-4_2
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DOI: https://doi.org/10.1007/978-3-0348-8871-4_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9808-9
Online ISBN: 978-3-0348-8871-4
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