The paper firstly studies existence condition of solution via the corresponding characteristic eigenvalues condition. Secondly, we derive the representation for the general solution by using the method of constant variation. Finally, we use the representation of the general solution to investigate Ulam-Hyers stability (UHS) and Ulam-Hyers-Rassias stability (UHRS).
References
[1]
Hahn, W. (1949) Über Orthogonalpolynome, die-Differenzengleichungen genügen. MathematischeNachrichten, 2, 4-34. https://doi.org/10.1002/mana.19490020103
[2]
Annaby, M.H., Hamza, A.E. and Aldwoah, K.A. (2012) Hahn Difference Operator and Associated Jackson-Nörlund Integrals. JournalofOptimizationTheoryandApplications, 154, 133-153. https://doi.org/10.1007/s10957-012-9987-7
[3]
Hamza, A.E., Zaghrout, A.S. and Ahmed, S.M. (2013) Characterization of Stability of First Order Hahn Difference Equations. Journal of Advances in Mathematics, 5, 678-687.
[4]
Hamza, A.E. and Ahmed, S.M. (2013) Existence and Uniqueness of Solutions of Hahn Difference Equations. AdvancesinDifferenceEquations, 2013, Article No. 316. https://doi.org/10.1186/1687-1847-2013-316
[5]
Abdelkhaliq, M.M. andHamza, A.E. (2018) Stability of Hahn Difference Equations in Banach Algebras. Communications of the Korean Mathematical Society, 33, 1141-1158.
[6]
Hıra, F. (2023) Hahn Laplace Transform and Its Applications. DemonstratioMathematica, 56, Article ID: 20230259. https://doi.org/10.1515/dema-2023-0259
[7]
Hıra, F. (2023) On-Differential Transform Method. Journal of Physics A: MathematicalandTheoretical, 56, Article ID: 325202. https://doi.org/10.1088/1751-8121/ace503
[8]
Oraby, K. and Hamza, A. (2020) Taylor Theory Associated with Hahn Difference Operator. JournalofInequalitiesandApplications, 2020, Article No. 124. https://doi.org/10.1186/s13660-020-02392-y
[9]
Hamza, A.E. and Shehata, E.M. (2016) Existence and Uniqueness of Solutions of General Quantum Difference Equations. Advances in Dynamical Systems and Applications, 11, 45-58.
[10]
Makarash, S.D. (2016) Leibnizs Rule and Fubinis Theorem Associated with Hahn Difference Operators. JournalofAdvancesinMathematics, 12, 6335-6346. https://doi.org/10.24297/jam.v12i6.3836
[11]
Sitthiwirattham, T. (2016) On a Nonlocal Boundary Value Problem for Nonlinear Second-Order Hahn Difference Equation with Two Different-derivatives. AdvancesinDifferenceEquations, 2016, Article No. 116. https://doi.org/10.1186/s13662-016-0842-2
[12]
Hamza, A.E. and Ahmed, S.M. (2013) Theory of Linear Hahn Difference Equations. Journal of Advances in Mathematics, 4, 441-461.
[13]
MacQuarrie, L., Saad, N. and Islam, M.S. (2021) Asymptotic Iteration Method for Solving Hahn Difference Equations. AdvancesinDifferenceEquations, 2021, Article No. 354. https://doi.org/10.1186/s13662-021-03511-9
[14]
Chen, K. and Wang, J. (2024) Ulam Type Stability for Nonlinear Hahn Difference Equations with Delay. ElectronicJournalofDifferentialEquations, No. 77, 1-15. https://doi.org/10.58997/ejde.2024.77
[15]
Ulam, S.M. (1960) A Collection of Mathematical Problems. Interscience Publishers.
[16]
Hyers, D.H. (1941) On the Stability of the Linear Functional Equation. Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224. https://doi.org/10.1073/pnas.27.4.222
[17]
Rassias, T.M. (1978) On the Stability of the Linear Mapping in Banach Spaces. ProceedingsoftheAmericanMathematicalSociety, 72, 297-300. https://doi.org/10.1090/s0002-9939-1978-0507327-1
[18]
Gavruta, P. (1994) A Generalization of the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings. JournalofMathematicalAnalysisandApplications, 184, 431-436. https://doi.org/10.1006/jmaa.1994.1211
[19]
Ger, R. and Semrl, P. (1996) The Stability of the Exponential Equation. ProceedingsoftheAmericanMathematicalSociety, 124, 779-787. https://doi.org/10.1090/s0002-9939-96-03031-6
Jung, S.M. (2001) Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press.
[22]
Rassias, T.M. (2000) On the Stability of Functional Equations and a Problem of Ulam. ActaApplicandaeMathematicae, 62, 23-130. https://doi.org/10.1023/a:1006499223572
[23]
Rus, I.A. (2009) Ulam Stability of Ordinary Differential Equations. Studia Universitatis Babes-Bolyai Mathematica, 54, 125-133.
[24]
Jung, S. (2005) Hyers-Ulam Stability of Linear Differential Equations of First Order, Iii. JournalofMathematicalAnalysisandApplications, 311, 139-146. https://doi.org/10.1016/j.jmaa.2005.02.025
[25]
Gordji, M.E., Cho, Y.J., Ghaemi, M.B., et al. (2011) Stability of the Second Order Partial Differential Equations. Journal of Inequalities and Applications, 2011, Article No. 81.
[26]
Xue, J. (2014) Hyers-Ulam Stability of Linear Differential Equations of Second Order with Constant Coefficient. Italian Journal of Pure and Applied Mathematics, 32, 419-424.
[27]
Baias, A.R. and Popa, D. (2019) On Ulam Stability of a Linear Difference Equation in Banach Spaces. BulletinoftheMalaysianMathematicalSciencesSociety, 43, 1357-1371. https://doi.org/10.1007/s40840-019-00744-6
[28]
Alghamdi, M., Aljehani, A. and E. Hamza, A. (2021) Hyers-Ulam-Rassias Stability of Abstract Second-Order Linear Dynamic Equations on Time Scales. JournalofMathematicsandComputerScience, 24, 110-118. https://doi.org/10.22436/jmcs.024.02.02
[29]
Chen, K. and Si, Y. (2025) Ulam Type Stability for the Second-Order Linear Hahn Difference Equations. AppliedMathematicsLetters, 160, Article ID: 109355. https://doi.org/10.1016/j.aml.2024.109355