Hyers-Ulam-Rassias Stability for Abel-Riccati Type First-Order Differential Equations

Year 2019, , 1238 - 1252, 01.12.2019

Yasemin Bascı , Suleyman Ögrekcı , Adil Mısır

https://doi.org/10.35378/gujs.493396

Cited By: 1

Abstract

This paper examines Hyers-Ulam (HU), Hyers-Ulam-Rassias (HUR)
and Hyers-Ulam-Rassias-Gavruta (HURG) stability of the first-order differential
equation including Bernoulli’s, Riccati and Abel with given initial condition.

Keywords

Abel differential equation, Hyers-Ulam stability, Hyers-Ulam-Rassias stability, Hyers-Ulam-Rassias-Gavruta stability, Riccati differential equation

References

• Referans1 Alqifiary, Q.H.: Note on the stability for linear systems of differential equations. International Journal of Applied Mathematical Research. 3, no. 1, 15-22 (2014).
• Referans2 Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, no.4, 373-380 (1998).
• Referans3 András, S., Mészáros, A.R.: Ulam-Hyers stability of dynamic equations on time scales via Picard operators. Appl. Math. Comput. 219, 4853-4864 (2013).
• Referans4 Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan. 2, 64-66 (1950).
• Referans5 Bourgin, D.G.:Classes of transformations and bordering transformations. Bull. Amer. Math. Soc. 57, 223-237 (1951).
• Referans6 Cãdariu, L., Radu, V.: On the Stability of the Cauchy Functional Equation: A Fixed Point Approach, Iteration theory (ECIT ’02), Grazer Math. Ber. 346, 43-52 (2004).
• Referans7 Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27, 222-224 (1941).
• Referans8 Jung, S.M.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17, no. 10, 1135-1140 (2004).
• Referans9 Jung, S.M.: Hyers-Ulam stability of linear differential equations of first order III. J. Math. Anal. Appl. 311, no. 1, 139-146 (2005).
• Referans10 Jung, S.M.: Hyers-Ulam stability of linear differential equations of first order II. Appl. Math. Lett. 19, no. 9, 854-858 (2006).
• Referans11 Jung, S.M.: A fixed point approach to the stability of differential equations . Bull. Malays. Math. Sci. Soc. (2) 33, no. 1, 47-56 (2010).
• Referans12 Jung, S.M., Brzdek, J.: Hyers-Ulam stability of the delay Equation . Abstract and Applied Analysis 2010, Article ID 372176, 10 pages (2010).
• Referans13 Jung, S.M., Rassias, T.M.: Generalized Hyers-Ulam stability of Riccati differential equation. Math. Inequal. Appl. 11, no. 4, 777-782 (2008).
• Referans14 Li, X., Wang, J.: Ulam-Hyers-Rassias stability of semilinear differential equations with impulses. Electronic J. Differential Equations. 2013, no. 172, 1-8 (2013).
• Referans15 Miura, T.: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Japan. 55, 17-24 (2002).
• Referans16 Miura, T., Jung, S.M., Takahasi, S.E.: Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations . J. Korean Math. Soc. 41, 995-1005 (2004).
• Referans17 Miura, T., Miyajima, S., Takahasi, S.E.: A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. Appl. 286, 136-146 (2003).
• Referans18 Miura, T., Takahasi, S.E., Choda, H.:On the Hyers-Ulam stability of real continuous function valued differentiable map. Tokyo J. Math. 24, 467-476 (2001).
• Referans19 Obloza, M.: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 13,no. 1, 259-270 (1993).
• Referans20 Obloza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 14,no. 1, 141-146 (1997).
• Referans21 Onitsuka, M., Shoji, T.: Hyers-Ulam stability of first-order homogeneous linear differential equations with a real-valued coefficient. Appl. Math. Lett. 63, 102-108 (2017).
• Referans22 Qarawani, M.N.: On Hyers-Ulam-Rassias stability for Bernoulli’s and first order linear and nonlinear differential equations. British J. Math. and Computer Science. 4, no. 11, 1615-1628 (2014).
• Referans23 Otrocol, D., Ilea, V.: Ulam stability for a delay differential equation. Central European Journal of Mathematics. 7, 1296-1303 (2013).
• Referans24 Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, no. 2, 297-300 (1978).
• Referans25 Rus, I.A.: Ulam stability of ordinary differential equations. Studia Univ. ”Babeş-Bolyai”, Mathematica. 54, no. 4, 125-134 (2009).
• Referans26 Rus, I.A.: Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 26, no. 1, 103-107 (2010).
• Referans27 Shen, Y.: The Ulam stability of first order linear dynamic equations on time scales. Results Math. Online first. (2017), 2017 Springer International Publishing AG DOI 10.1007/s00025-017-0725-1.
• Referans28 Takahasi, S.E., Miura, T., Miyajima, S.: On the Hyers-Ulam stability of the Banach space-valued differential equation . Bull. Korean Math. Soc. 39, 309-315 (2002).
• Referans29 Takahasi, S.E., Takagi, H., Miura, T., Miyajima, S.: The Hyers-Ulam stability constants of first order linear differential operators. J. Math. Anal. Appl. 296, 403-409 (2004).
• Referans30 Tunç, C., Biçer, E.:Hyers-Ulam-Rassias stability for a first-order functional differential equation. Journal of Mathematical and Fundamental Sciences. 47, no. 2, 143-153 (2015).
• Referans31 Wang, G., Zhou, M., Sun, L.: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 21, 1024-1028 (2008).
• Referans32 Zada, A., Shah, S.O., Ismail, S., Li, T.: Hyers-Ulam stability in terms of dichotomy of first order linear dynamic systems. Penjab Uni. J. Math. 49, no. 3, 37-47 (2017).