Weighted Hyers-Ulam Stability for Nonlinear Nonautonomous Difference Equations
DOI:
https://doi.org/10.5644/SJM.18.01.07Keywords:
shadowing, exponential dichotomy, nonautonomous dynamicsAbstract
Let $(A_m)_{m\in \mathbb{Z}}$ be a sequence of bounded linear operators acting on an arbitrary Banach space $X$ and admitting an exponential dichotomy. Furthermore, let $f_m \colon X\to X$, $m\in \mathbb{Z}$ be a sequence of Lipschitz maps. Provided that the Lipschitz constants of $f_m$ are uniformly small, we show that a nonlinear difference equation $x_{m+1} = A_mx_m +f_m(x_m), m\in \mathbb{Z}$ exhibits various types of the Hyers-Ulam stability property.
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L. Backes and D. Dragičević, Quasi-shadowing for partially hyperbolic dynamics on Banach spaces, J. Math. Anal. Appl. 492 (2020),124445.
L. Backes and D. Dragičević, Shadowing for infinite dimensional dynamics and exponential trichotomies, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), 863--884.
L. Backes and D. Dragičević, A general approach to nonautonomous shadowing for nonlinear dynamics, Bull. Sci. Math. 170 (2021), 102996.
L. Backes, D. Dragičević and L. Singh, Shadowing for nonautonomous and nonlinear dynamics with impulses, Monatsh. Math (2021). https://doi.org/10.1007/s00605-021-01629-2
D. Barbu, C. Buse and A. Tabassum, Hyers--Ulam stability and discrete dichotomy, J. Math. Anal. Appl. 423 (2015), 1738–1752.
N. Bernardes Jr., P.R. Cirilo, U. B. Darji, A. Messaoudi and E. R. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl. 461 (2018), 796–816.
C. Buse, D. O’Regan, O. Saierli and A. Tabassum, Hyers--Ulam stability and discrete dichotomy for difference periodic systems, Bull. Sci. Math. 140 (2016), 908–934.
C. Buse, V. Lupulescu and D. O' Regan, Hyers--Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients, Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 2175--2188.
W. A. Coppel, Dichotomies in stability theory, Springer Verlag, Berlin, Heidelberg, New-York, 1978.
D. Dragičević, Admissibility, a general type of Lipschitz shadowing and structural stability, Commun. Pure Appl. Anal. 14 (2015), 861–880.
D. Dragičević, Hyers–Ulam stability for a class of perturbed Hill’s equations Results Math 76, 129 (2021). https://doi.org/10.1007/s00025-021-01442-1
D. Dragičević, On the Hyers–Ulam stability of certain nonautonomous and nonlinear difference equations Aequat. Math. 95, 829–840 (2021). https://doi.org/10.1007/s00010-020-00774-7
D. Dragičević, Hyers–Ulam stability for nonautonomous semilinear dynamics on bounded intervals, Mediterr. J. Math. 18, 71 (2021). https://doi.org/10.1007/s00009-021-01729-1
D. Dragičević and M. Pituk, Shadowing for nonautonomous difference equations with infinite delay, Appl. Math. Lett. 120 (2021), 107284.
D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal. 235 (2006), 330–354.
N. T. Huy, Invariant manifolds of admissible classes for semi-linear evolution equations, J. Differential Equations, 246 (2009), 1820–1844.
A. L. Sasu and B. Sasu, Integral equations, dichotomy of evolution families on the half-line and applications, Integral Equations Operator Theory, 66 (2010), 113–140.
A. L. Sasu and B. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line}, Discrete Contin. Dyn. Syst. 33 (2013), 3057–3084.
B. Sasu, Input-output control systems and dichotomy of variational difference equations, J. Difference Equ. Appl. 17 (2011) 889–913.
D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory, Discrete Contin. Dyn. Syst. 33 (2013), 4187 -- 4205.
A. Zada and B. Zada, Hyers-Ulam stability and exponential dichotomy of discrete semigroup, Appl. Math. E-Notes. 19 (2019), 527--534.
