We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.
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References
X. Liu, “Progress in stability of impulsive systems with applications to population growth models,” in: A. A. Martynyuk (editor), Advances in Stability Theory at the End of the 20 th Century, Taylor and Francis, London (2003), pp. 321–340.
V. I. Slyn’ko, Stability of Motion of Mechanical Systems: Hybrid Models [in Russian], Author’s Abstract of the Doctoral-Degree Thesis (Physics and Mathematics), Kiev (2009).
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989).
M. O. Perestyuk and O. S. Chernikova, “Some modern aspects of the theory of impulsive differential equations,” Ukr. Mat. Zh., 60, No. 1, 81–94 (2008) 60, No. 1, 91–107 (2008).
N. A. Perestyuk, “On the problem of stability of the equilibrium position of impulsive systems,” God. VUZ, Prilozh. Mat., Sofia, 11, Kn. 1, 145–150 (1976).
N. A. Perestyuk, “Stability of solutions of linear impulsive systems,” Vestn. Kiev Univ., Ser. Mat. Mekh., No. 19, 71–76 (1977).
A. O. Ignat’ev, O. A. Ignat’ev, and A. A. Soliman, “Asymptotic stability and instability of the solutions of systems with impulse action,” Math. Notes, 80, No. 4, 491–499 (2006).
A. O. Ignat’ev, “On the stability of invariant sets of systems with impulse effect,” Nonlinear Anal., 69, 53–72 (2008).
A. O. Ignat’ev and O. A. Ignat’ev, “Stability of solutions of systems with impulse effect,” in: Progr. Nonlinear Analysis Research, Nova Science (2009), pp. 363–389.
V. I. Slyn’ko, “Construction of Poincaré maps for a holonomic mechanical system with two degrees of freedom in the presence of impacts,” Prikl. Mekh., 44, No. 5, 115–122 (2008).
S. V. Babenko and V. I. Slyn’ko, “Stability of motion of nonlinear impulsive systems in the critical cases,” Dopov. Nats. Akad. Nauk Ukr., No. 6, 46–52 (2008).
A. A. Martynyuk and A. Yu. Obolenskii, Investigation of the Stability of Autonomous Systems of Comparison [in Russian], Preprint No. 78.28, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1978).
A. Yu. Obolenskii, “On the stability of the systems of comparison,” Dopov. Akad. Nauk Ukr. RSR, No. 8, 607–611 (1979).
A. Yu. Obolenskii, “On the stability of the linear systems of comparison,” Mat. Fiz. Nelin. Mekh., Issue 1, 51–55 (1984).
T. K. Sirazetdinov, Stability of Systems with Distributed Parameters [in Russian], Nauka, Novosibirsk (1987).
A. A. Movchan, “On the direct Lyapunov method in the problems of stability of elastic systems,” Prikl. Mat. Mekh., 23, No. 3, 483–493 (1959).
A. A. Martynyuk, V. Lakshmikantham, and S. Leela, Stability of Motion: The Method of Comparison [in Russian], Naukova Dumka, Kiev (1991).
M. W. Hirsch, “Stability and convergence in strongly monotone dynamical systems,” J. Reine Angew. Math., 383, 1–53 (1988).
T. V. Kedyk and A. Yu. Obolenskii, “On the stability of hybrid quasimonotone extensions with respect to two measures,” Dokl. Akad. Nauk. Ukr. SSR, No. 8, 80–82 (1991).
T. V. Kedyk, “On the invariant manifolds of hybrid quasimonotone extensions,” Dokl. Akad. Nauk. Ukr. SSR, No. 10, 8–11 (1991).
A. Yu. Obolenskii, “Stability of solutions of autonomous Wazéwski systems with delayed action,” Ukr. Mat. Zh., 35, No. 5, 574–579 (1983); English translation: Ukr. Math. J., 35, No. 5, 486–492 (1983).
S. G. Mikhlin, Linear Partial Differential Equations [in Russian], Vysshaya Shkola, Moscow (1977).
V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1967).
M. O. Perestyuk and V. V. Marynets’, Theory of Equations of Mathematical Physics. A Course of Lectures [in Ukrainian], Lybid’, Kyiv (1993).
A. I. Dvirnyi and V. I. Slyn’ko, “On the stability of linear impulsive systems with respect to a cone,” Dopov. Nats. Akad. Nauk Ukr., No. 4, 37–43 (2004).
M. A. Krasnosel’skii, E. A. Lifshits, and A. V. Sobolev, Positive Linear Systems [in Russian], Nauka, Moscow (1985).
P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).
A. I. Dvirnyi, “On the estimation of the boundary of robustness for a linear impulsive system,” Dopov. Nats. Akad. Nauk Ukr., No. 9, 34–39 (2003).
V. I. Slyn’ko, “On the exponential stability of a linear impulsive system in a Hilbert space,” Dopov. Nats. Akad. Nauk Ukr., No. 12, 44–47 (2002).
Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel’, Functional Analysis [in Russian], Vyshcha Shkola, Kiev (1990).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 7, pp. 904–923, July, 2011.
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Dvirnyi, A.I., Slyn’ko, V.I. On the stability of abstract monotone impulsive differential equations in terms of two measures. Ukr Math J 63, 1042–1064 (2011). https://doi.org/10.1007/s11253-011-0563-3
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DOI: https://doi.org/10.1007/s11253-011-0563-3