Best Ulam constants for two-dimensional nonautonomous linear differential systems
Douglas R. Anderson
Department of Mathematics, Concordia College, Moorhead, Minnesota, USA
Search for more papers by this authorCorresponding Author
Masakazu Onitsuka
Department of Applied Mathematics, Okayama University of Science, Okayama, Japan
Correspondence
Masakazu Onitsuka, Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan.
Email: [email protected]
Search for more papers by this authorDonal O'Regan
School of Mathematical and Statistical Sciences, University of Galway, Galway, Ireland
Search for more papers by this authorDouglas R. Anderson
Department of Mathematics, Concordia College, Moorhead, Minnesota, USA
Search for more papers by this authorCorresponding Author
Masakazu Onitsuka
Department of Applied Mathematics, Okayama University of Science, Okayama, Japan
Correspondence
Masakazu Onitsuka, Department of Applied Mathematics, Okayama University of Science, Okayama 700-0005, Japan.
Email: [email protected]
Search for more papers by this authorDonal O'Regan
School of Mathematical and Statistical Sciences, University of Galway, Galway, Ireland
Search for more papers by this authorAbstract
This study deals with the Ulam stability of nonautonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two-dimensional nonautonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on , but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for nonautonomous systems other than periodic systems.
CONFLICT OF INTEREST STATEMENT
The authors declare no conflicts of interest.
REFERENCES
- 1R. P. Agarwal, B. Xu, and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl. 288 (2003), no. 2, 852–869.
- 2D. R. Anderson and M. Onitsuka, Best constant for Hyers–Ulam stability of two step sizes linear difference equations, J. Math. Anal. Appl. 496 (2021), no. 2, 124807.
- 3D. R. Anderson and M. Onitsuka, Hyers–Ulam stability for differential systems with constant coefficient matrix, Results Math. 77 (2022), no. 3, 136.
- 4D. R. Anderson, M. Onitsuka, and J. M. Rassias, Best constant for Ulam stability of first-order -difference equations with periodic coefficient, J. Math. Anal. Appl. 491 (2020), no. 2, 124363.
- 5L. Backes and D. Dragičević, Shadowing for nonautonomous dynamics, Adv. Nonlinear Stud. 19 (2019), no. 2, 425–436.
- 6L. Backes and D. Dragičević, Shadowing for infinite -dimensional dynamics and exponential trichotomies, Proc. R. Soc. Edin. A 151 (2021), no. 3, 863–884.
- 7L. Backes and D. Dragičević, A general approach to nonautonomous shadowing for nonlinear dynamics, Bull. Sci. Math. 170 (2021), 102996.
- 8L. Backes, D. Dragičević, M. Onitsuka, and M. Pituk, Conditional Lipschitz shadowing for ordinary differential equations, J. Dyn. Diff. Equ. (2023). https://doi.org/10.1007/s10884-023-10246-6.
- 9L. Backes, D. Dragičević, M. Pituk, and L. Singh, Weighted shadowing for delay differential equations, Arch. Math. 119 (2022), no. 5, 539–552.
- 10A.-R. Baias, F. Blaga, and D. Popa, On the best Ulam constant of a first -order linear difference equation in Banach spaces, Acta Math. Hungar. 163 (2021), no. 2, 563–575.
- 11A.-R. Baias and D. Popa, On the best Ulam constant of the second order linear differential operator, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114 (2020), no. 1, 23.
- 12A.-R. Baias and D. Popa, On the best Ulam constant of a higher order linear difference equation, Bull. Sci. Math. 166 (2021), 102928.
- 13C. Buşe, V. Lupulescu, and D. O'Regan, Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients, Proc. R. Soc. Edin. A 150 (2020), no. 5, 2175–2188.
- 14J. Brzdęk, K. Ciepliński, and Z. Leśniak, On Ulam's type stability of the linear equation and related issues, Discrete Dyn. Nat. Soc. 2014 (2014), 536791.
- 15J. Brzdęk, D. Popa, I. Raşa, and B. Xu, Ulam stability of operators, Mathematical Analysis and its Applications, Academic Press, London, 2018.
- 16C. Buşe, D. O'Regan, O. Saierli, and A. Tabassum, Hyers–Ulam stability and discrete dichotomy for difference periodic systems, Bull. Sci. Math. 140 (2016), no. 8, 908–934.
- 17W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Math, vol. 629, Springer-Verlag, Berlin, 1978.
10.1007/BFb0067780 Google Scholar
- 18D. Dragičević, Hyers–Ulam stability for a class of perturbed Hill's equations, Results Math. 76 (2021), no. 3, 129.
- 19D. Dragičević, Hyers–Ulam stability for nonautonomous semilinear dynamics on bounded intervals, Mediterr. J. Math. 18 (2021), no. 2, 71.
- 20D. Dragičević and M. Pituk, Shadowing for nonautonomous difference equations with infinite delay, Appl. Math. Lett. 120 (2021), 107284.
- 21R. Fukutaka and M. Onitsuka, Best constant in Hyers–Ulam stability of first-order homogeneous linear differential equations with a periodic coefficient, J. Math. Anal. Appl. 473 (2019), no. 2, 1432–1446.
- 22R. Fukutaka and M. Onitsuka, Best constant for Ulam stability of Hill's equations, Bull. Sci. Math. 163 (2020), 102888.
- 23J. K. Hale, Ordinary differential equations, Pure and Applied Mathematics, vol. 11, Wiley-Interscience, New York, 1969.
- 24W. Kelley and A. Peterson, The theory of differential equations: classical and qualitative, Pearson Prentice Hall, Upper Saddle River, NJ, 2004.
- 25A. N. Michel, L. Hou and D. Liu, Stability of dynamical systems: on the role of monotonic and non-monotonic Lyapunov functions, 2nd ed., Systems and Control: Foundations and Applications, Birkhäuser, Cham, 2015.
10.1007/978-3-319-15275-2 Google Scholar
- 26D. Popa and I. Raşa, On the best constant in Hyers–Ulam stability of some positive linear operators, J. Math. Anal. Appl. 412 (2014), no. 1, 103–108.
- 27D. Popa and I. Raşa, Best constant in Hyers–Ulam stability of some functional equations, Carpathian J. Math. 30 (2014), no. 3, 383–386.
- 28D. Popa and I. Raşa, Best constant in stability of some positive linear operators, Aequationes Math. 90 (2016), no. 4, 719–726.
