Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃

• Autorzy: Inoan Daniela , Marian Daniela

Identyfikatory

DOI

10.1515/dema-2022-0198

• Języki publikacji: EN

Abstrakty

EN

The Laplace transform method is applied in this article to study the semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order n, with convolution-type kernel. This kind of stability extends the original Hyers-Ulam stability whose study originated in 1940. A general integral equation is formulated first, and then some particular cases (polynomial function and exponential function) for the function from the kernel are considered.

Słowa kluczowe

EN

Volterra integro-differential equation; Laplace transform; semi-Hyers-Ulam-Rassias stability

PL

równanie różniczkowe Volterry; transformata Laplace'a; stabilność Hyersa-Ulama-Rassiasa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20220198
• Opis fizyczny: Bibliogr. 36 poz.

Twórcy

autor

Inoan Daniela

• Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114, Cluj-Napoca, Romania

autor

Marian Daniela

• Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114, Cluj-Napoca, Romania

Bibliografia

• [1] S. M. Ulam, A Collection of Mathematical Problems, Interscience, New York, 1960.
• [2] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224, DOI: https://doi.org/10.1073/pnas.27.4.222.
• [3] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Semin. Univ. Hambg. 62 (1992), 59–64, DOI: https://doi.org/10.1007/BF02941618.
• [4] T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002), 604–616, DOI: https://doi.org/10.1016/S0022-247X(02)00181-6.
• [5] Y.-H. Lee, S. Jung and M. Th. Rassias. Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal. 12 (2018), no. 1, 43–61, DOI: https://doi.org/10.7153/jmi-2018-12-04.
• [6] E. Elqorachi and M. Th. Rassias, Generalized Hyers-Ulam stability of trigonometric functional equations, Mathematics 6 (2018), no. 5, 83, DOI: https://doi.org/10.3390/math6050083.
• [7] S.-M. Jung, K. S. Lee, M. Th. Rassias, and S. M. Yang, Approximation properties of solutions of a mean value-type functional inequality, II, Mathematics 8 (2020), 1299, DOI: https://doi.org/10.3390/math8081299.
• [8] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011.
• [9] J. Brzdek, D. Popa, I. Rasa, and B. Xu, Ulam Stability of Operators, Elsevier, Amsterdam, The Netherlands, 2018.
• [10] M. Obloza, Hyers stability of the linear differential equation, Rocznik Nauk-Dydakt. Prace Mat. 13 (2013), 259–270.
• [11] C. Alsina and R. Ger, On some inequalities and stability results related to exponential function, J. Inequal. Appl. 2 (1998), 373–380.
• [12] D. S. Cimpean and D. Popa, On the stability of the linear differential equation of higher order with constant coefficients, Appl. Math. Comput. 217 (2010), 4141–4146, DOI: https://doi.org/10.1016/j.amc.2010.09.062.
• [13] S.-M. Jung, Hyers-Ulam stability of linear differential equations of first order, III, J. Math. Anal. Appl. 311 (2005), 139–146, DOI: https://doi.org/10.1016/j.jmaa.2005.02.025.
• [14] D. Marian, S. A. Ciplea, and N. Lungu, On Ulam-Hyers stability for a system of partial differential equations of first order, Symmetry 12 (2020), no. 7, 1060, DOI: https://doi.org/10.3390/sym12071060.
• [15] D. Otrocol, Ulam stabilities of differential equation with abstract Volterra operator in a Banach space, Nonlinear Funct. Anal. Appl. 15 (2010), no. 4, 613–619.
• [16] D. Popa and I. Rasa, Hyers-Ulam stability of the linear differential operator with non-constant coefficients, Appl. Math. Comput. 219 (2012), 1562–1568, DOI: https://doi.org/10.1016/j.amc.2012.07.056.
• [17] S. E. Takahasi, H. Takagi, T. Miura and S. Miyajima, The Hyers-Ulam stability constant of first order linear differential operators, J. Math. Anal. Appl. 296 (2004), 403–409, DOI: https://doi.org/10.1016/j.jmaa.2003.12.044.
• [18] M. R. Abdollahpour and M. Th Rassias, Hyers-Ulam stability of hypergeometric differential equations, Aequationes Math. 93 (2019), no. 4, 691–698, DOI: https://doi.org/10.1007/s00010-018-0602-3.
• [19] M. R. Abdollahpour, R. Aghayari, and M. Th Rassias, Hyers-Ulam stability of associated Laguerre differential equations in a subclass of analytic functions, J. Math. Anal. Appl. 437 (2016), 605–612, DOI: https://doi.org/10.1016/j.jmaa.2016.01.024.
• [20] A. Prastaro and Th. M. Rassias, Ulam stability in geometry of PDE’s, Nonlinear Funct. Anal. Appl. 8 (2003), no. 2, 259–278.
• [21] S.-M. Jung, Hyers-Ulam stability of linear partial differential equations of first order, Appl. Math. Lett. 22 (2009), 70–74, DOI: https://doi.org/10.1016/j.aml.2008.02.006.
• [22] N. Lungu and S. Ciplea, Ulam-Hyers-Rassias stability of pseudoparabolic partial differential equations, Carpatian J. Math. 31 (2015), no. 2, 233–240.
• [23] N. Lungu and D. Popa, Hyers-Ulam stability of a first order partial differential equation, J. Math. Anal. Appl. 385 (2012), 86–91, DOI: https://doi.org/10.1016/j.jmaa.2011.06.025.
• [24] D. Marian, Semi-Hyers-Ulam-Rassias stability of the convection partial differential equation via Laplace transform, Mathematics 9 (2021), 2980, DOI: https://doi.org/10.3390/math9222980.
• [25] A. K. Tripathy, Hyers-Ulam Stability of Ordinary Differential Equations, Taylor and Francis, Boca Raton, 2021.
• [26] L. Cadariu, The generalized Hyers-Ulam stability for a class of the Volterra nonlinear integral equations, Sci. Bull. Politehnica Univ. Timis. Trans. Math. Phys. 56 (2011), 30–38.
• [27] L. P. Castro and A. M. Simões, Different types of Hyers-Ulam-Rassias stabilities for a class of integro-differential equations, Filomat 31 (2017), 5379–5390, DOI: https://doi.org/10.2298/FIL1717379C.
• [28] L. P. Castro and A. M. Simões, Hyers-Ulam-Rassias stability of nonlinear integral equations through the Bielecki metric, Math. Meth. Appl. Sci. 41 (2018), no. 17, 7367–7383, DOI: https://doi.org/10.1002/mma.4857.
• [29] V. Ilea and D. Otrocol, Existence and uniqueness of the solution for an integral equation with supremum, via w-distances, Symmetry 12 (2020), no. 9, 1554, DOI: https://doi.org/10.3390/sym12091554.
• [30] D. Marian, S. A. Ciplea, and N. Lungu, On a functional integral equation, Symmetry 13 (2021), no. 8, 1321, DOI: https://doi.org/10.3390/sym13081321.
• [31] H. Rezaei, S.-M. Jung, and Th. M. Rassias, Laplace transform and Hyers-Ulam stability of linear differential equations, J. Math. Anal. Appl. 403 (2013), 244–251, DOI: https://doi.org/10.1016/j.jmaa.2013.02.034.
• [32] Q. Alqifiary and S-M. Jung, Laplace transform and generalized Hyers-Ulam stability of linear differential equations, Electron. J. Differ. Equ. 80 (2014), 1–11.
• [33] E. Bicer and C. Tunc, On the Hyers-Ulam stability of Laguerre and Bessel equations by Laplace transform method, Nonlinear. Dyn. Syst. 17 (2017), no. 4, 340–346.
• [34] R. Murali and A. Ponmana Selvan, Mittag-Leffler-Hyers-Ulam stability of a linear differential equation of first order using Laplace transforms, Canad. J. Appl. Math. 2 (2020), 47–59.
• [35] Y. Shen and W. Chen, Laplace transform method for the Ulam stability of linear fractional differential equations with constant coefficients, Mediterr. J. Math. 14 (2017), 25, DOI: https://doi.org/10.1007/s00009-016-0835-0.
• [36] D. Inoan and D. Marian, Semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel via Laplace transform, Symmetry 13 (2021), no. 11, 2181, DOI: https://doi.org/10.3390/sym13112181.