HYERS-ULAM STABILITY FOR AN <i>N<sup>TH</sup></i> ORDER DIFFERENTIAL EQUATION USING FIXED POINT APPROACH

Ramdoss Murali; Choonkil Park; Arumugam Ponmana Selvan

doi:10.11948/20190093

2021 Volume 11 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(2): 614-631

Next Article Previous Article

Ramdoss Murali, Choonkil Park, Arumugam Ponmana Selvan. HYERS-ULAM STABILITY FOR AN NTH ORDER DIFFERENTIAL EQUATION USING FIXED POINT APPROACH[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 614-631. doi: 10.11948/20190093

Citation:

Ramdoss Murali, Choonkil Park, Arumugam Ponmana Selvan. HYERS-ULAM STABILITY FOR AN N^TH ORDER DIFFERENTIAL EQUATION USING FIXED POINT APPROACH[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 614-631. doi: 10.11948/20190093

HYERS-ULAM STABILITY FOR AN N^TH ORDER DIFFERENTIAL EQUATION USING FIXED POINT APPROACH

1.
PG and Research Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur-635601, Vellore Dist., Tamil Nadu, India
2.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Corresponding author: Email address: baak@hanyang.ac.kr (C. Park)

Abstract

Abstract

In this paper, we prove the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of the $ n^{th} $ order differential equation of the form

$ x^{(n)}(t) = f(t, x(t)) $

and

$ x^{(n)}(t) = f \left( t, x(t), x'(t), x''(t),\cdots , x^{(n-1)}(t) \right) $

with initial conditions

$ x(a) = x_0 , x'(a) = x_1 , x''(a) = x_2 , \cdots , x^{(n-1)}(a) = x_{n-1} $

for all $ t \in I = [a, b] \subset \mathbb{R} $ and $ x \in C^{(n)}(I) $ by using fixed point method in the sense of Cadariu and Radu.
- Hyers-Ulam stability /
- Hyers-Ulam-Rassias stability /
- fixed point method /
- differential equation
MSC: 65Mxx, 49J20, 39B72, 47H10, 39B82

FullText(HTML)

References

[1]	R. P. Agarwal, B. Xu and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 2003, 288, 852–869. doi: 10.1016/j.jmaa.2003.09.032 CrossRef Google Scholar
[2]	Q. H. Alqifiary and J. K. Miljanovic, Note on the stability of system of differential equations $\dot x$(t) = f(t, x(t)), Gen. Math. Notes, 2014, 20, 27–33. $\dot x$(t) = f(t, x(t))" target="_blank">Google Scholar
[3]	C. Alsina and R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 1998, 2, 373–380. Google Scholar
[4]	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 1950, 2, 64–66. Google Scholar
[5]	D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Am. Math. Soc., 1951, 57, 223–237. doi: 10.1090/S0002-9904-1951-09511-7 CrossRef Google Scholar
[6]	M. Burger, N. Ozawa and A. Thom, On Ulam stability, Israel J. Math., 2013, 193, 109–129. doi: 10.1007/s11856-012-0050-z CrossRef Google Scholar
[7]	L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 2003, 4(1), Article 4, 7 pp. Google Scholar
[8]	L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber., 2004, 346, 43–52. Google Scholar
[9]	J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Am. Math. Soc., 1968, 74, 305–309. doi: 10.1090/S0002-9904-1968-11933-0 CrossRef Google Scholar
[10]	Z. Došlá, P. Liška and M. Marini, Asymptotic problems for functional differential equations via linearization method, J. Fixed Point Theory Appl., 2019, 21, Art. No. 4. doi: 10.1007/s11784-018-0642-2 CrossRef Google Scholar
[11]	P. Gǎvruta and L. Gǎvruta, A new method for the generalized Hyers-UlamRassias stability, Int. J. Nonlinear Anal., 2010, 2, 11–18. Google Scholar
[12]	P. Gǎvruta, S. Jung and Y. Li, Hyers-Ulam stability for the second order linear differential equations with boundary conditions, Elec. J. Diff. Equ., 2011, 2011, Art. No. 80. Google Scholar
[13]	D. H. Hyers, On the stability of a linear functional equation, Proc. Nat. Acad. Sci. USA, 1941, 27, 222–224. doi: 10.1073/pnas.27.4.222 CrossRef Google Scholar
[14]	S. Jung, A fixed point approach to the stability of isometries, J. Math. Anal. Appl., 2007, 329(2), 879–890. doi: 10.1016/j.jmaa.2006.06.098 CrossRef Google Scholar
[15]	S. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., 2006, 320, 549–561. doi: 10.1016/j.jmaa.2005.07.032 CrossRef Google Scholar
[16]	S. Jung, A fixed point approach to the stability of differential equations y'(x) = F(x, y), Bull. Malays. Math. Sci. Soc. (2), 2010, 33, 47–56. Google Scholar
[17]	T. Li, A. Zada and S. Faisal, Hyers-Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl., 2016, 9, 2070–2075. doi: 10.22436/jnsa.009.05.12 CrossRef Google Scholar
[18]	R. Murali and A. Ponmana Selvan, On the generalized Hyers-Ulam stability of linear ordinary differential equations of higher order, Int. J. Pure Appl. Math., 2017, 117(12), 317–326. Google Scholar
[19]	R. Murali and A. Ponmana Selvan, Hyers-Ulam stability of nth order differential equation, Contemp. Stud. Discrete Math., 2018, 2, 45–50. Google Scholar
[20]	R. Murali and A. Ponmana Selvan, Hyers-Ulam-Rassias stability for the linear ordinary differential equation of third order, Kragujevac J. Math., 2018, 42, 579–590. doi: 10.5937/KgJMath1804579M CrossRef Google Scholar
[21]	R. Murali and A. Ponmana Selvan, Ulam stability of a differential equation of second order: A fixed point approach, J. Phys. : IOP Conf. Series, 2018, 1139, Art. ID 012051. doi: 10.1088/1742-6596/1139/1/012051 CrossRef Google Scholar
[22]	A. Najati, J. Lee, C. Park and Th. M. Rassias, On the stability of a Cauchy type functional equation, Demonstr. Math., 2018, 51, 323–331. doi: 10.1515/dema-2018-0026 CrossRef Google Scholar
[23]	C. Park, J. M. Rassias, A. Bodaghi and S. Kim, Approximate homomorphisms from ternary semigroups to modular spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 2019, 113, 2175–2188. doi: 10.1007/s13398-018-0608-7 CrossRef Google Scholar
[24]	S. Pinelas, V. Govindan and K. Tamilvanan, Stability of a quartic functional equation, J. Fixed Point Theory Appl., 2018, 20, Art. No. 148. doi: 10.1007/s11784-018-0629-z CrossRef Google Scholar
[25]	V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 2003, 4(1), 91–96. Google Scholar
[26]	T. M. Rassias, On the stability of the linear mappings in Banach spaces, Proc. Am. Math. Soc., 1978, 72, 297–300. doi: 10.1090/S0002-9939-1978-0507327-1 CrossRef Google Scholar
[27]	S. E. Takahasi, T. Miura and S. Miyajima, On the Hyers-Ulam stability of the Banach space-valued differential equation y' = αy, Bull. Korean Math. Soc., 2002, 39, 309–315. doi: 10.4134/BKMS.2002.39.2.309 CrossRef Google Scholar
[28]	S. M. Ulam, Problem in Modern Mathematics, Willey, New York, 1960. Google Scholar
[29]	T. Xu, On the stability of multi-Jensens mappings in β-normed spaces, Appl. Math. Lett., 2012, 25, 1866–1870. doi: 10.1016/j.aml.2012.02.049 CrossRef Google Scholar