Note On Hyers-Ulam Stability Criteria for Third Order Nonlinear Differential Equations with Forcing Term
Keywords:
Forcing Term, Integral Inequality, New Criteria, Hyers-Ulam Stability, Nonlinear Differential Equation
Abstract
The stability of the ordinary differential equations has been investigated and the investigation is ongoing. In this paper we are concerned with note on Hyers-Ulam stability(HUs) criteria for third order nonlinear differential equations with forcing term. The third order nonlinear differential equations invesgated were transformed to integral equation, then, applied Bihari inequality and Gronwall-Bellman-Bihari(GBB) type inequality to arrive at our results. New criteria were established to prove HUs of nonlinear third order differential equations. Finally, examples are given to illustrate correctness our results.
References
[1] Abdollahpour, M. R. Najati,A. and Kim, G .H.(2012). Hyers -Ulam Stability of a differential equation of third order. Int.Journal of Math.Analysis. 6(59), 2943-2948.
[2] Adeyanju, A. A., Adams, D. O., Oni, S. A. and Olutimo, A. L.(2023). On the stability,boundedness and asymptotic behaviour of solutions of differential equations of third-order. International Journal of Mathematics Science and Optimizations: Theory and Applications 9(2), 121-132.
[3] Aligfiary, Q. H. Jung, S. M.(2014). On the Hyers-Ulam Stability of Differential Equations of Second Order. Hindawi Publishing Cooperation Abstract and Appliedd Analysis. 1-8.
[4] Alsina, G. Ger, R.(1988). On Some Inequalities and Stability Result Related to the Exponential Function. J. Inequl. Appl. 2,373-380.
[5] Bihari, I.(1957). Researches of the Boundedness and Stability of the Solutions of nonlinear differential Equations. Acta. Math Acad, Sc. Hung 7,278-291.
[6] Bishop, A.B. and Nnubia, A.C.(2021). Stability of Nonlocal Stochastics Volterra Equations International Journal of Mathematical Sciences and Optimizations:Theory and Applications. 7,(2),48-56.
[7] Dhongade, U D. Deo, S.G. (1973). Some Generalisations of Bellman-Bihari Integral Inequalities., Journal of Mathematical Analysis and Applications. 44,218-226.
[8] Fakunle, I. and Arawomo, P.O. (2023). -Ulam stability Theorems for Second Order Nonlinear Damped Differential Equations with Forcing Term. Journal of the Nigerian Mathematical Society, 42,19-35.
[9] Fakunle, I. and Arawomo, P.O. (2023). Hyers-Ulam-Rassias stability of some Perturbed Nonlinear Second Order Ordinary Differential Equations. Proyecciones Journal of Mathematics. 42(5),1157-1175.
[10] Fakunle,I and Arawomo,P.O.(2022). On Hyers-Ulams stability of a Perturbed Nonlinear Second Differential using Gronwall-Bellman-Bihari Inequality. Nigerian Journal of Mathematics and Applications 32(1),189-201.
[11] Fakunle,I. Arawomo, P.O.(2019). Hyers-Ulam Stability of a Perturbed Generalised Lienard Equation. International Journal of Applied Mathematics. 32(3),479-489.
[12] Fakunle,I. Arawomo, P.O.(2018). Hyers-Ulam Stability of Certain Class of Nonlinear Second Order Differential Equations. International Journal of Pure and Applied Mathematical Sciences. 11(1),55-65.
[13] Fakunle, I. Arawomo, P.O (2018). On Hyers-Ulam Stability of Nonlinear Second Order Ordinary and Functional Differential Equations. International Journal of Differential Equations and Applications 17(1),77-88.
[14] Hyers, D. H (1941). On the Stability of the Linear functional equation. Proceedings of the National Academy of Science of the united States of America, 27,222-224.
[15] Jung, S.-M.(2006). Hyers-Ulam Stability of Linear Differential Equations of First Order(II). J.Math. Anal. Appl. 19,854-858.
[16] Jung, S.-M. (2005). Hyers-Ulam Stability of linear differential equation of first order(III),Math.Lett. Appl.311,139-146.
[17] Jung, S.-M.(2004) Hyers-Ulam Stability of Linear Differential Equations of First Order. Appl.Math.Lett.17(10),11351140.
[18] Li, Y. and Shen, Y.(2010) Hyers-Ulam Stability of Linear Differential Equations of second order. Appl.Math.Lett.23(3), 306-309.
[20] Murray, R. S.(1974). Schum’s Outline of Theory and Problem of Calculus. SI(Metric) Edition, International Edition.
[19] Li, Y.(2010) Hyers-Ulam Stability of Linear Differential Equations u′′ Thai.J.Math.8(2), 215-219.
[21] Murali, R. and PonmanaSelvan, A.(2018). Hyers -Ulam-Rassias Stability for the linear differential equation of third order. Kragujevac Journa of Mathematics 42(4), 579-590.
[22] Murali, R. and PonmanaSelvan, A.(2018) Hyers -Ulam Stability of third order linear Differential equation. Journal of Computer and Mathematical science. 9(10), 1334-1340.
[23] Miura, T. Miyajima, S., Takahasi, S.E.(2003). A characterisation of Hyers -Ulam Stability of first order linear differential operators. J. Math. Anal. Appl.286, 136-146.
[24] Qarawani, M. N.(2012). Hyers-Ulam Stability of Linear and Nonlinear Differential Equations of Second Order. Int. Journal of Applied Mathematical Research. 1,4,422-432.
[25] Qarawani, M. N.(2012). Hyers-Ulam Stability of a Generalised Second Order Nonlinear Differential Equations. Applied Mathematics. 3, 1857-1861.
[26] Rassias, TH.M (1978) : On the Stability of the Linear Mapping in Banach Spaces. Proceedings of the American Mathematical Society, 72,(2),297-300.
[27] Rus,I. A. (2010). Ulam Stability of Ordinary Differential Equation. Studia Universities BabesBolyal Mathematical. 54,4,306-309.
[28] Rus, I. A.(2010) Ulam stability of Ordinary Differential Equations in a Banach Space. Carpathian, J.Math. 126,103-107.
[29] Takahasi,S.E., Miura, T., Miyajima, S. (2002). On the Hyers-UIam Stability of the Banach Space-Valued Differential Equation y′ = λy. Bulletin of the Korean Mathematical Society. 392,309-315.
[30] Tunc,C. and Bicer,E.(2013). Hyers-Ulam Stability of Non-Homogeneous Euler Equations of Third and Fourth Order. Scientific Research and Essays 220-226
[31] Tripathy, A.K. and Satapathy, A.(2014). Hyers-Ulam stability of third order Euler’s Differential equations.Journal of Nonlinear DynamicsID 487257,6pages.
[32] Ulam, S.M.((1960). Problems in Modern Mathematics Science Editions. wily, New York. NY, USA, Chapter 6.
[33] G.Wang, G. Zhou, M., Sun, L.(2008). Hyers-Ulam stability of linear differential equations of first order. Appl.Math.Lett.21,1024-1028.
[2] Adeyanju, A. A., Adams, D. O., Oni, S. A. and Olutimo, A. L.(2023). On the stability,boundedness and asymptotic behaviour of solutions of differential equations of third-order. International Journal of Mathematics Science and Optimizations: Theory and Applications 9(2), 121-132.
[3] Aligfiary, Q. H. Jung, S. M.(2014). On the Hyers-Ulam Stability of Differential Equations of Second Order. Hindawi Publishing Cooperation Abstract and Appliedd Analysis. 1-8.
[4] Alsina, G. Ger, R.(1988). On Some Inequalities and Stability Result Related to the Exponential Function. J. Inequl. Appl. 2,373-380.
[5] Bihari, I.(1957). Researches of the Boundedness and Stability of the Solutions of nonlinear differential Equations. Acta. Math Acad, Sc. Hung 7,278-291.
[6] Bishop, A.B. and Nnubia, A.C.(2021). Stability of Nonlocal Stochastics Volterra Equations International Journal of Mathematical Sciences and Optimizations:Theory and Applications. 7,(2),48-56.
[7] Dhongade, U D. Deo, S.G. (1973). Some Generalisations of Bellman-Bihari Integral Inequalities., Journal of Mathematical Analysis and Applications. 44,218-226.
[8] Fakunle, I. and Arawomo, P.O. (2023). -Ulam stability Theorems for Second Order Nonlinear Damped Differential Equations with Forcing Term. Journal of the Nigerian Mathematical Society, 42,19-35.
[9] Fakunle, I. and Arawomo, P.O. (2023). Hyers-Ulam-Rassias stability of some Perturbed Nonlinear Second Order Ordinary Differential Equations. Proyecciones Journal of Mathematics. 42(5),1157-1175.
[10] Fakunle,I and Arawomo,P.O.(2022). On Hyers-Ulams stability of a Perturbed Nonlinear Second Differential using Gronwall-Bellman-Bihari Inequality. Nigerian Journal of Mathematics and Applications 32(1),189-201.
[11] Fakunle,I. Arawomo, P.O.(2019). Hyers-Ulam Stability of a Perturbed Generalised Lienard Equation. International Journal of Applied Mathematics. 32(3),479-489.
[12] Fakunle,I. Arawomo, P.O.(2018). Hyers-Ulam Stability of Certain Class of Nonlinear Second Order Differential Equations. International Journal of Pure and Applied Mathematical Sciences. 11(1),55-65.
[13] Fakunle, I. Arawomo, P.O (2018). On Hyers-Ulam Stability of Nonlinear Second Order Ordinary and Functional Differential Equations. International Journal of Differential Equations and Applications 17(1),77-88.
[14] Hyers, D. H (1941). On the Stability of the Linear functional equation. Proceedings of the National Academy of Science of the united States of America, 27,222-224.
[15] Jung, S.-M.(2006). Hyers-Ulam Stability of Linear Differential Equations of First Order(II). J.Math. Anal. Appl. 19,854-858.
[16] Jung, S.-M. (2005). Hyers-Ulam Stability of linear differential equation of first order(III),Math.Lett. Appl.311,139-146.
[17] Jung, S.-M.(2004) Hyers-Ulam Stability of Linear Differential Equations of First Order. Appl.Math.Lett.17(10),11351140.
[18] Li, Y. and Shen, Y.(2010) Hyers-Ulam Stability of Linear Differential Equations of second order. Appl.Math.Lett.23(3), 306-309.
[20] Murray, R. S.(1974). Schum’s Outline of Theory and Problem of Calculus. SI(Metric) Edition, International Edition.
[19] Li, Y.(2010) Hyers-Ulam Stability of Linear Differential Equations u′′ Thai.J.Math.8(2), 215-219.
[21] Murali, R. and PonmanaSelvan, A.(2018). Hyers -Ulam-Rassias Stability for the linear differential equation of third order. Kragujevac Journa of Mathematics 42(4), 579-590.
[22] Murali, R. and PonmanaSelvan, A.(2018) Hyers -Ulam Stability of third order linear Differential equation. Journal of Computer and Mathematical science. 9(10), 1334-1340.
[23] Miura, T. Miyajima, S., Takahasi, S.E.(2003). A characterisation of Hyers -Ulam Stability of first order linear differential operators. J. Math. Anal. Appl.286, 136-146.
[24] Qarawani, M. N.(2012). Hyers-Ulam Stability of Linear and Nonlinear Differential Equations of Second Order. Int. Journal of Applied Mathematical Research. 1,4,422-432.
[25] Qarawani, M. N.(2012). Hyers-Ulam Stability of a Generalised Second Order Nonlinear Differential Equations. Applied Mathematics. 3, 1857-1861.
[26] Rassias, TH.M (1978) : On the Stability of the Linear Mapping in Banach Spaces. Proceedings of the American Mathematical Society, 72,(2),297-300.
[27] Rus,I. A. (2010). Ulam Stability of Ordinary Differential Equation. Studia Universities BabesBolyal Mathematical. 54,4,306-309.
[28] Rus, I. A.(2010) Ulam stability of Ordinary Differential Equations in a Banach Space. Carpathian, J.Math. 126,103-107.
[29] Takahasi,S.E., Miura, T., Miyajima, S. (2002). On the Hyers-UIam Stability of the Banach Space-Valued Differential Equation y′ = λy. Bulletin of the Korean Mathematical Society. 392,309-315.
[30] Tunc,C. and Bicer,E.(2013). Hyers-Ulam Stability of Non-Homogeneous Euler Equations of Third and Fourth Order. Scientific Research and Essays 220-226
[31] Tripathy, A.K. and Satapathy, A.(2014). Hyers-Ulam stability of third order Euler’s Differential equations.Journal of Nonlinear DynamicsID 487257,6pages.
[32] Ulam, S.M.((1960). Problems in Modern Mathematics Science Editions. wily, New York. NY, USA, Chapter 6.
[33] G.Wang, G. Zhou, M., Sun, L.(2008). Hyers-Ulam stability of linear differential equations of first order. Appl.Math.Lett.21,1024-1028.
Published
2024-11-30
How to Cite
Fakunle , I., Arawomo , P., Ipinlaye , A. P., & Akintewe , B. (2024). Note On Hyers-Ulam Stability Criteria for Third Order Nonlinear Differential Equations with Forcing Term. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 10(4), 55 - 69. Retrieved from http://ijmso.unilag.edu.ng/article/view/2389
Issue
Section
Articles
Copyright (c) 2024 Authors
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
This is an Open Access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, adaptation, and reproduction in any medium, provided that the original work is properly cited.
