ULAM TYPE STABILITY FOR A CLASS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

Serkan Aslıyüce, Süleyman Öğrekçi

DOI Number
https://doi.org/10.22190/FUMI220606028A
First page
429
Last page
435

Abstract


In this paper, we are concerned with the stability problem of a general class of second order nonlinear differential equations in the sense of Hyers-Ulam-Rassias and Hyers-Ulam. In our proofs, we show that some of the common restrictions widely used in well-known papers concerned with similar problems on bounded intervals are unnecessary. Therefore, we obtain stability results for second order differential equations with few assumptions on bounded intervals.


Keywords

Hyers-Ulam-Rassias stability, differential equations, generalized metric.

Full Text:

PDF

References


C. Alsine and R. Ger: On Some Inequalities and Stability Results Related to the Exponential Function. J. Inequal. Appl. 2 (1998), 373-380.

Y. Basci, S. Ogrekci and A. Misir: Hyers-Ulam-Rassias Stability for Abel-Riccati Type First-Order Differential Equations. In: GU. J. Sci. 32(4) (2019), pp. 1238-1252.

Y. Basci, A. Misir and S. Ogrekci: On the Stability Problem of Differential Equations in the Sense of Ulam. Results Math.75(6) (2020), doi:10.1007/s00025-019-1132-6.

J. B. Diaz and B. Margolis: A xed point theorem of alternative, for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74 (1968), 305-309.

D. H. Hyers: On the Stability of the Linear Functional Equation. Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.

E. B. Saff and R. S. Varga: Jung, S. M. A xed point approach to the stability of differential equations y0 = F(x; y). Bull. Malays. Math. Sci. Soc. 33(1) (2010), 47-56.

S. M. Jung: Hyers-Ulam stability of a system of rst order linear differential equations with constant coefficients. Journal of Mathematical Analysis and Applications 320 (2006), 549-561.

T. Miura, S. Miyajima and S. H. Takahasi: A Characterization of Hyers-Ulam Stability of First Order Linear Differential Operators. J. Math. Anal. Appl. 286 (2003), pp. 136-146.

T. Miura, S. Miyajima and S. H. Takahasi: Hyers-Ulam Stability of Linear Differential Operator with Constant Coeffcients. Math. Nacr. 258 (2003), 90-96.

M. Obloza: Connections Between Hyers and Lyapunov Stability of the Ordinary Differential Equations. Rocznik. Nauk. Dydakt. Prace. Mat. 14 (1997), 141-146.

M. Obloza: Hyers-Ulam Stability of the Linear Differential Equations. Rocznik. Nauk. Dydakt. Prace. Mat. 13 (1993), 259-270.

S. Ogrekci: Stability of delay differential equations in the sense of Ulam on unbounded intervals. An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 9(2) (2019), 125-131.

D. Popa and G. Pugna: Hyers-Ulam Stability of Euler's Differential Equation. Results in Mathematics 69 (2016), 317-325.

D. Popa and I. Rasa: On the Hyers-Ulam stability of the linear differential equation. Journal of Mathematical Analysis and Applications 381 (2011), 530-537.

T. Rasham and et. al.: Common fixed point results for new Ciric-type rational multivalued F-contraction with an application. Journal of Fixed Point Theory and Applications 20 (2018), 45.

T. Rasham and et. al.: Fixed point results for a pair of fuzzy mappings and related applications in b-metric like spaces. Advances in Difference Equations 2021.1 (2021), p. 259.

T. Rasham and et. al.: Multivalued Fixed Point Results in Dislocated b-Metric Spaces with Application to the System of Nonlinear Integral Equations"". Symmetry 11 (2019), 40. (2019).

T. Rasham and et. al.: On a pair of fuzzy dominated mappings on closed ball in the multiplicative metric space with applications. Fuzzy Sets and Systems 437 (2022), 81-96.

T. Rasham and et. al.: On pairs of fuzzy dominated mappings and applications. Advances in Difference Equations 2021 (2021), 417.

T. Rasham and et. al.: Sufficient conditions to solve two systems of integral equations via fixed point results. Journal of Inequalities and Applications 2019.1 (2019), 182.

T. M. Rassias: On the Stability of Linear Mappings in Banach Spaces. Proc. Amer. Math. Soc. 72 (1978), 297-300.

H. Rezai, S. M. Jung and T. M. Rassias: Laplace transform and Hyers-Ulam stability of linear differential equations. Journal of Mathematical Analysis and Applications 403(1) (2013), 244-251.

Y. Shen: The Ulam Stability of First Order Linear Dynamic Equations on Time Scales. Results in Mathematics 72(4) (2017), 1881-1895.

S. H. Takahasi, T. Miura, and S. Miyajima: The Hyers-Ulam Stability Constants of First Order Linear Differential Operators. Bull. Korean Math. Soc. 39 (2002), 309-315.

C. Tunc and E. Bicer: Hyers-Ulam-Rassias Stability for a First Order Functional Differential Equation. J. Math. Fund. Sci. 47(2) (2015), 143-153.

S. M. Ulam: A Collection of Mathematical Problems. Interscience, New York, 1960.




DOI: https://doi.org/10.22190/FUMI220606028A

Refbacks

  • There are currently no refbacks.




© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)