Further results on Ulam stability for a system of first-order nonsingular delay differential equations

• Autorzy: Zada, Akbar , Pervaiz, Bakhtawar , Alzabut, Jehad , Shah, Syed Omar
• Języki publikacji: EN

Abstrakty

EN

This paper is concerned with a system governed by nonsingular delay differential equations. We study the β-Ulam-type stability of the mentioned system. The investigations are carried out over compact and unbounded intervals. Before proceeding to the main results, we convert the system into an equivalent integral equation and then establish an existence theorem for the addressed system. To justify the application of the reported results, an example along with graphical representation is illustrated at the end of the paper.

Słowa kluczowe

PL

macierz wykładnicza; stabilność Hyersa-Ulama-Rassiasa

EN

delayed exponential matrix; delay differential system; representation of solutions; β-Hyers-Ulam-Rassias stability

• Czasopismo: Demonstratio Mathematica
• Rocznik: 2020
• Tom: Vol. 53, nr 1
• Strony: 225--235
• Opis fizyczny: Bibliogr. 17 poz., rys.

Twórcy

autor

Zada, Akbar

• Department of Mathematics, University of Peshawar, Peshawar, 25000, Pakistan,

autor

Pervaiz, Bakhtawar

• Department of Mathematics, University of Peshawar, Peshawar, 25000, Pakistan,

autor

Alzabut, Jehad

• Department of Mathematics and General Sciences, Prince Sultan University, 11586, Riyadh, Saudi Arabia,

autor

Shah, Syed Omar

• Department of Mathematics, University of Peshawar, Peshawar, 25000, Pakistan; Department of Physical and Numerical Sciences, Qurtuba University of Science and Information Technology Peshawar, Dera Ismail Khan, Pakistan,

Bibliografia

• [1] D. Ya. Khusainov and G. V. Shuklin, Linear autonomous time-delay system with permutation matrices solving, Stud. Univ. Zilina Math. Ser. 17(2003), 101-108.
• [2] J. Diblik and D. Ya. Khusainov, Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(k−m)+f(k) with commutative matrices, J. Math. Anal. Appl. 318(2006), 63-76, DOI: 10.1016/j.jmaa.2005.05.021.
• [3] Z. You, J. R. Wang, and D. O’Regan, Exponential stability and relative controllability of nonsingular delay systems, Bull. Braz. Math. Soc. (N.S.) 50(2019), 457-479, DOI: 10.1007/s00574-018-0110-z.
• [4] S. M. Ulam, Problems in Modern Mathematics, Rend. Chap. VI, Wiley, New York, 1960.
• [5] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27(1941), no. 4, 222-224, DOI: 10.1073/pnas.27.4.222.
• [6] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), no. 2, 297-300, DOI: 10.2307/2042795.
• [7] T. Li and A. Zada, Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ. 2016(2016), 153, DOI: 10.1186/s13662-016-0881-8.
• [8] S. O. Shah and A. Zada, Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales, Appl. Math. Comput. 359(2019), 202-213, DOI: 10.1016/j.amc.2019.04.044.
• [9] J. Wang, A. Zada, and W. Ali, Ulam’s-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlinear Sci. Numer. Simul. 19(2018), no. 5, 553-560, DOI: 10.1515/ijnsns-2017-0245.
• [10] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2(1950), no. 1-2, 64-66, DOI: 10.2969/jmsj/00210064.
• [11] A. Zada, J. Alzabut, H. Waheed, and I.-L. Popa, Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions, Adv. Difference Equ. 2020(2020), 64, DOI: 10.1186/s13662-020-2534-1.
• [12] M. Ahmad, A. Zada, and J. Alzabut, Hyres-Ulam stability of coupled system of fractional differential equations of Hilfer-Hadamard type, Demonstr. Math. 52(2019), 283-295, DOI: 10.1515/dema-2019-0024.
• [13] G. M. Selvam, D. Baleanu, J. Alzabut, D. Vignesh, and S. Abbas, On Hyers-Ulam Mittag-Leffler stability of discrete fractional Duffing equation with application on inverted pendulum, Adv. Difference Equ. 2020(2020), 456, DOI: 10.1186/s13662-020-02920-6.
• [14] W. Sudsutad, J. Alzabut, S. Nontasawatsri, and C. Thaiprayoon, Stability analysis for a generalized proportional fractional Langevin equation with variable coefficient and mixed integro-differential boundary conditions, J. Nonlinear Funct. Anal. 2020(2020), 23, DOI: 10.23952/jnfa.2020.23.
• [15] X. Wang, M. Arif, and A. Zada, β-Hyers-Ulam-Rassias stability of semilinear nonautonomous impulsive system, Symmetry 11(2019), 231, DOI: 10.3390/sym11020231.
• [16] A. Zada, S. Shaleena, and T. Li, Stability analysis of higher order nonlinear differential equations in β-normed spaces, Math. Meth. App. Sci. 42(2019), no. 4, 1151-1166, DOI: 10.1002/mma.5419.
• [17] R. Bellman, The stability of solutions of linear differential equations, Duke Math. J. 10(1943), no. 4, 643-647, DOI: 10.1215/S0012-7094-43-01059-2.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).

Identyfikatory

DOI

10.1515/dema-2020-0018