YERS–ULAM–RASSIAS STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS WITH A GENERALIZED ACTIONS ON THE RIGHT-HAND SIDE

Alexander N. Sesekin     (Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)
Anna D. Kandrina     (Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)

Abstract


The paper considers the Hyers–Ulam–Rassias stability for systems of nonlinear differential equations with a generalized action on the right-hand side, for example, containing impulses — delta functions. The fact that the derivatives in the equation are considered distributions required a correction of the well known Hyers–Ulam–Rassias definition of stability for such equations. Sufficient conditions are obtained that ensure the property under study.


Keywords


Hyers–Ulam–Rassias stability, Differential equations, Generalized actions, Discontinuous trajectories

Full Text:

PDF

References


  1. Hyers D.H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA, 1941. Vol. 27, No. 4. P. 222–224. DOI: 10.1073/pnas.27.4.222
  2. Miller B.M., Rubinovich E.Y. Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations. Autom. Remote Control, 2013. Vol. 74, No. 12. P. 1969–2006. DOI: 10.1134/S0005117913120047
  3. Pavlenko V., Sesekin A.. Ulam–Hyers Stability of First and Second Order Differential Equations with Discontinuous Trajectories. In: Proc. 2022 16th Int. Conf. on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference 2022). V.N. Tkhai (ed.). IEEE Xplore, 2022. P. 1–4. DOI: 10.1109/STAB54858.2022.9807520
  4. Popa D. Hyers–Ulam–Rassias stability of a linear recurrence. J. Math. Anal. Appl., 2005. Vol. 309, No. 2. P. 591–597. DOI: 10.1016/j.jmaa.2004.10.013
  5. Rassias Th.M. On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc., 1978. Vol. 72, No. 2. P. 297–300. DOI: 10.2307/2042795
  6. Rus I.A. Ulam stability of ordinary differential equations. Stud. Univ. Babeş-Bolyai Math., 2009. Vol. 54, No. 4. P. 125–133.
  7. Samoilenko A.M., Perestyuk N.A. Impulsive Differential Equations. World Sci. Ser. Nonlinear Sci. Ser. A, vol. 14. River Edge, NJ: World Sci. Publ. Co. Inc., 1995. 472 p. DOI: 10.1142/2892
  8. Sesekin A.N. Dynamic systems with nonlinear impulse structure. Proc. Steklov Inst. Math., 2000. Suppl. 2. P. S158–S172.
  9. Wang J.R., Fečkan M., Zhou Y. Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl., 2012. Vol. 395, No. 1. P. 258–264. DOI: 10.1016/j.jmaa.2012.05.040
  10.  Zada A., Riaz U., Khan F.U. Hyers–Ulam stability of impulsive integral equations. Boll. Unione. Mat. Ital., 2019. Vol. 12. P. 453–467. DOI: 10.1007/s40574-018-0180-2
  11.  Zavalishchin S.T., Sesekin A.N. Dynamic Impulse Systems: Theory and Applications. Dordrecht: Kluwer Academic Publishers, 1997. 256 p. DOI: 10.1007/978-94-015-8893-5




DOI: http://dx.doi.org/10.15826/umj.2023.1.013

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.