Kamal transform and Ulam stability of differential equations. (English) Zbl 07905192
Summary: In the growth of the field of functional-differential equations and their Ulam stability, many researchers have utilized various methods to prove the Ulam stability of functional and differential equations. Hyers method and the fixed-point method are remarkably applied by many researchers to investigate the Ulam stability of functional and differential equations. In this research work, we propose a new method for investigating the Ulam stability of linear differential equations by using Kamal transform.
MSC:
| 44A10 | Laplace transform |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
| 34A40 | Differential inequalities involving functions of a single real variable |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
References:
| [1] | L. Aiemsomboon, W. Sintunavarat, Stability of the generalized logarithmic functional equations arising from fixed point theory, Rev. R. Acad. Cienc. Ex-actas Fís. Nat. Ser. A Math., 2018, 112, 229-238. · Zbl 06836246 |
| [2] | T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 1951, 2, 64-66. · Zbl 0040.35501 |
| [3] | C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 1998, 2, 373-380. · Zbl 0918.39009 |
| [4] | Q. H. Alqifiary, S. Jung, Laplace transform and generalized Hyers-Ulam stability of differential equations, Elec. J. Differential Equ., 2014, 80, 11 pages. · Zbl 1290.34059 |
| [5] | Q. H. Alqifiary, J. K. Miljanovic, Note on the stability of system of differential equations ẋ(t) = f (t, x(t)), Gen. Math. Notes, 2014, 20, 27-33. |
| [6] | N. Eghbali, V. Kalvandi, J. M. Rassias, A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation, Open Math., 2016, 14 237-246. · Zbl 1351.45010 |
| [7] | J. Huang, S. Jung, Y. Li, On Hyers-Ulam stability of nonlinear differential equations, Bull. Korean Math. Soc., 2015, 52, 685-697. · Zbl 1370.34098 |
| [8] | D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 1941, 27, 222-224. · Zbl 0061.26403 |
| [9] | S. Jung, Hyers-Ulam stability of linear differential equation of first order, Appl. Math. Lett., 2004, 17, 1135-1140. · Zbl 1061.34039 |
| [10] | S. Jung, Hyers-Ulam stability of linear differential equations of first order (III), J. Math. Anal. Appl., 2005, 311, 139-146. · Zbl 1087.34534 |
| [11] | S. Jung, Hyers-Ulam stability of linear differential equations of first order (II), Appl. Math. Lett., 2006, 19, 854-858. · Zbl 1125.34328 |
| [12] | S. Jung, Hyers-Ulam stability of a system of first order linear differential equa-tions with constant coefficients, J. Math. Anal. Appl., 2006, 320, 549-561. · Zbl 1106.34032 |
| [13] | S. Jung, Approximate solution of a linear differential equation of third order, Bull. Malay. Math. Sci. Soc., 2012, 35, 1063-1073. · Zbl 1253.34023 |
| [14] | S. Jung, D. Popa, M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric spaces, J. Global Optim., 2014, 59, 13-16. |
| [15] | H. Kim, H. Shin, Approximate Lie * -derivations on ρ-complete convex modular algebras, J. Appl. Anal. Comput., 2019, 9, 765-776. · Zbl 1470.39053 |
| [16] | Y. Lee, S. Jung, M. Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal., 2018, 12, 43-61. · Zbl 1412.39028 |
| [17] | T. Li, A. Zada, S. Faisal, Hyers-Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl., 2016, 9, 2070-2075. · Zbl 1337.34052 |
| [18] | Y. Li, Y. Shen, Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett., 2010, 23, 306-309. · Zbl 1188.34069 |
| [19] | Z. Lu, G. Lu, Y. Jin, C. Park, The stability of additive (α, β)-functional equa-tions, J. Appl. Anal. Comput., 2019, 9, 2295-2307. · Zbl 1462.39022 |
| [20] | T. Miura, S. Jung, S. E. Takahasi, Hyers-Ulam-Rassias stability of the Banach space valued linear differential equation y ′ = λy, J. Korean Math. Soc., 2004, 41, 995-1005. · Zbl 1069.34079 |
| [21] | R. Murali, 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. |
| [22] | R. Murali, A. Ponmana Selvan, Hyers-Ulam-Rassias stability for the linear ordinary differential equation of third order, Kragujevac J. Math., 2018, 42(4), 579-590. · Zbl 1488.34081 |
| [23] | M. Obloza, Hyers stability of the linear differential equation, Rockznik Nauk-Dydakt. Prace Math.,1993, 13, 259-270. · Zbl 0964.34514 |
| [24] | M. Obloza, Connection between Hyers and Lyapunov stability of the ordinary differential equations, Rockznik Nauk-Dydakt. Prace Math., 1997, 14, 141-146. · Zbl 1159.34332 |
| [25] | M. Onitsuka, T. Shoji, Hyers-Ulam stability of first order homogeneous linear differential equations with a real valued co-efficients, Appl. Math. Lett., 2017, 63, 102-108. · Zbl 1351.34066 |
| [26] | I. A. Rus, Ulam stabilities of ordinary differential equations in Banach space, Carpathian J. Math., 2010, 26, 103-107. · Zbl 1224.34164 |
| [27] | S. E. Takahasi, T. Miura, S. Miyajima, On the Hyers-Ulam stability of the Banach space-valued differential equation y ′ = αy, Bull. Korean Math. Soc., 2002, 39, 309-315. · Zbl 1011.34046 |
| [28] | G. Wang, M. Zhou, L. Sun, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 2008, 21, 1024-1028. · Zbl 1159.34041 |
| [29] | J. Xue, Hyers-Ulam stability of linear differential equations of second order with constant coefficient, Italian J. Pure Appl. Math., 2014, 32, 419-424. · Zbl 1333.34095 |
| [30] | S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York, 1960. · Zbl 0086.24101 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
