Nonlinear difference equations with asymptotically stable solutions
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We establish conditions of asymptotic stability for all solutions of the equation X n+1=F(X n ), n≥0, in the Banach space E in the case where r(F′(x))<1 ∀ x ∈ E, r′(x) is the spectral radius of F′(x). An example of an equation with an unstable solution is given.
KeywordsBanach Space Difference Equation Asymptotic Stability Trivial Solution Spectral Radius
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- 1.A. Halanay and D. Wexler, Qualitative Theory of Pulse Systems, Editura Academiei Republicii Socialiste România, Bucharest (1968).Google Scholar
- 2.D. I. Martynyuk, Lectures on the Qualitative Theory of Difference Equations [in Russian], Naukova Dumka, Kiev (1972).Google Scholar
- 3.A. N. Sharkovskii, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1986).Google Scholar
- 4.A. N. Sharkovskii, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of One-Dimensional Maps [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
- 5.V. E. Slyusarchuk, Stability of Solutions of Difference Equations in Banach Spaces [in Russian], Naukova Dumka, Kiev (1989).Google Scholar
- 5.V. E. Slyusarchuk, Stability of Solutions of Difference Equations in Banach Spaces [in Russian], Candidate Degree Thesis (Physics and Mathematics), Chernovtsy (1972).Google Scholar
- 9.V. E. Slyusarchuk, “On the instability of autonomic systems with respect to a linear approximation,” in: Asymptotic Methods and Their Applications to Problems of Mathematical Physics [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1990), pp. 112–114.Google Scholar
- 10.M. L. Sverdan and E. F. Tsar'kov, Stability of Stochastic Pulse Systems [in Russian], Riga Technical University, Riga (1994).Google Scholar
- 14.A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1968).Google Scholar
- 15.L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).Google Scholar
- 16.L. Collatz, Functional Analysis and Numerical Mathematics, Academic Press, New York (1966).Google Scholar