Mittag-Leffler-Hyers-Ulam-Rassias stability of cubic functional equation

Document Type : Original Article

Authors

1 Department of Mathematics, Razi University, Kermanshah, Iran

2 Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, Iran

3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Abstract

IIn this paper, we prove the Mittag-Leffler-Hyers-Ulam-Rassias stability for cubic functional equation by using the fixed point alternative theorem. As a consequence, we show that the cubic multipliers are superstable under some conditions.

Keywords


[1] J. Aczel, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
[2] M. Akkouchi, Hyers-Ulam-Rassias Stability of nonlinear Volterra integral equations via a  xed point approach, Acta. Univ.
Apulen. Math. Inform, 26 2011, 257-266.
[3] Q.H. Alqi ary, J. K. Miljanovic, Note on the stability of system of differential equations x_ (t) = f(t; x(t)), Gen, 20(1) 2014,
27-33.
[4] D. Amir, Characterizations of Inner Product Spaces, Birkhuser, Basel, 1986.
[5] T. Aoki, On the stability of the linear transformation in Banach spaces, Journal of the mathematical society of Japan, 2(1-2)
(1950), 64-66.
[6] A. Bodaghi, I.A. Alias, M. Eshaghi Gordji, On the stability of quadratic double centralizers and quadratic multipliers: a
 xed point approach, Journal of Inequalities and Applications, 2011(1) 2011, 1-9.
[7] A. Bodaghi, I.A. Alias, M.H. Ghahramani1, Approximately cubic functional equations and cubic multipliers, Journal of
Inequalities and Applications, 2011(1) 2011, 1-8.
[8] A. Bodaghi, C. Park, O.T Mewomo, Multiquartic functional equations, Advances in Difference Equations, 2019(1) 2019,
1-10.
[9] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Mathematical
Journal ,16(2) 1949, 385-397.
[10] J.B. Diaz, B. Margolis, A  xed point theorem of the alternative for contractions on a generalized complete metric space,
Bulletin of the American Mathematical Society, 74(2) 1968, 305-309
[11] M. Eshdaghi Gordji, S. Shams, M. Ramezani, A. Ebadian, Approximately cubic double centralizers, Nonlinear Func Anal
Appl, 15(3) 2010, 503-512.
[12] D.H. Hyers, On the stability of the linear functional equation, Proceedings of the national academy of sciences of the United
States of America, 27(4) 1941, 222-224.
[13] D.H. Hyers, G. Isac, T.M. Rassias, Stability of Functional Equation in Several Variables, Rirkhauser, Basel, 1998.
[14] P. Jordan, J. Von Neumann, On inner products in linear metric spaces, Annals of Mathematics, 36(3) 1935, 719-723.
[15] K.W. Jun, H.M. Kim, The generalized Hyers - Ulam - Rassias stability of a cubic functional equation, Journal ofMathematical
Analysis and Applications, 274(2) 2002, 867-878.
[16] K.W. Jun, H.M. Kim, On the Hyers - Ulam - Rassias stability of a general cubic functional equation, Mathematical Inequal-
ities and Applications, 6(2) 2003, 289-302.
[17] V. Kalvandi, M.E. Samei, New stability results for a sum-type fractional q-integro-differential equation, J. Adv. Math. Stud,
12(2) 2019, 201-209.
[18] R. Murali, A. Ponmana Selvan, Mittag-LefflerHyersUlam stability of a linear differential equations of  rst order using Laplace
transforms, Canad. J. Appl. Math, 2(2) 2020, 47-59.
[19] C. Park, A. Bodaghi, Two multi-cubic functional equations and some results on the stability in modular spaces, Journal of
Inequalities and Applications 2020(1) 2020, 1-16.

[20] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American mathematical society
72(2) 1978, 297-300
[21] T.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, Journal of Mathematical Analysis and
Applications, 246(2) 2000, 352-378.
[22] T.M. Rassias, On the stability of functional equations in Banach spaces, Journal of Mathematical Analysis and Applications
251(1) 2000, 264-284.
[23] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematica, 62(1) 2000,
23-130.
[24] T.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic, Dordrecht, Boston and London, 2003.
[25] M.E. Samei, V. Hedayati, Sh. Rezapour, Existence results for a fraction hybrid differential inclusion with caputo-hadamard
type fractional derivative, Advances in Difference Equations, 2019(1) 2019, 1-15.
[26] F. Skof, Propriet locall e approssimazione di operatori. Rend sem Math Fis Milano, Rendiconti del Seminario Matematico e
Fisico di Milano, 53(1) 1983, 113-129.
[27] S.M. Ulam, Problem in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1960.
[28] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
[29] J. Xue, Hyers-Ulam stability of linear differential equations of second order with constant coefficient, talian Journal of Pure
and Applied Mathematics, 32 2014, 419-424.

[1] J. Aczel, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
[2] M. Akkouchi, Hyers-Ulam-Rassias Stability of nonlinear Volterra integral equations via a  xed point approach, Acta. Univ.
Apulen. Math. Inform, 26 2011, 257-266.
[3] Q.H. Alqi ary, J. K. Miljanovic, Note on the stability of system of differential equations x_ (t) = f(t; x(t)), Gen, 20(1) 2014,
27-33.
[4] D. Amir, Characterizations of Inner Product Spaces, Birkhuser, Basel, 1986.
[5] T. Aoki, On the stability of the linear transformation in Banach spaces, Journal of the mathematical society of Japan, 2(1-2)
(1950), 64-66.
[6] A. Bodaghi, I.A. Alias, M. Eshaghi Gordji, On the stability of quadratic double centralizers and quadratic multipliers: a
 xed point approach, Journal of Inequalities and Applications, 2011(1) 2011, 1-9.
[7] A. Bodaghi, I.A. Alias, M.H. Ghahramani1, Approximately cubic functional equations and cubic multipliers, Journal of
Inequalities and Applications, 2011(1) 2011, 1-8.
[8] A. Bodaghi, C. Park, O.T Mewomo, Multiquartic functional equations, Advances in Difference Equations, 2019(1) 2019,
1-10.
[9] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Mathematical
Journal ,16(2) 1949, 385-397.
[10] J.B. Diaz, B. Margolis, A  xed point theorem of the alternative for contractions on a generalized complete metric space,
Bulletin of the American Mathematical Society, 74(2) 1968, 305-309
[11] M. Eshdaghi Gordji, S. Shams, M. Ramezani, A. Ebadian, Approximately cubic double centralizers, Nonlinear Func Anal
Appl, 15(3) 2010, 503-512.
[12] D.H. Hyers, On the stability of the linear functional equation, Proceedings of the national academy of sciences of the United
States of America, 27(4) 1941, 222-224.
[13] D.H. Hyers, G. Isac, T.M. Rassias, Stability of Functional Equation in Several Variables, Rirkhauser, Basel, 1998.
[14] P. Jordan, J. Von Neumann, On inner products in linear metric spaces, Annals of Mathematics, 36(3) 1935, 719-723.
[15] K.W. Jun, H.M. Kim, The generalized Hyers - Ulam - Rassias stability of a cubic functional equation, Journal ofMathematical
Analysis and Applications, 274(2) 2002, 867-878.
[16] K.W. Jun, H.M. Kim, On the Hyers - Ulam - Rassias stability of a general cubic functional equation, Mathematical Inequal-
ities and Applications, 6(2) 2003, 289-302.
[17] V. Kalvandi, M.E. Samei, New stability results for a sum-type fractional q-integro-differential equation, J. Adv. Math. Stud,
12(2) 2019, 201-209.
[18] R. Murali, A. Ponmana Selvan, Mittag-LefflerHyersUlam stability of a linear differential equations of  rst order using Laplace
transforms, Canad. J. Appl. Math, 2(2) 2020, 47-59.
[19] C. Park, A. Bodaghi, Two multi-cubic functional equations and some results on the stability in modular spaces, Journal of
Inequalities and Applications 2020(1) 2020, 1-16.
[20] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American mathematical society
72(2) 1978, 297-300
[21] T.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, Journal of Mathematical Analysis and
Applications, 246(2) 2000, 352-378.
[22] T.M. Rassias, On the stability of functional equations in Banach spaces, Journal of Mathematical Analysis and Applications
251(1) 2000, 264-284.
[23] T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematica, 62(1) 2000,
23-130.
[24] T.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic, Dordrecht, Boston and London, 2003.
[25] M.E. Samei, V. Hedayati, Sh. Rezapour, Existence results for a fraction hybrid differential inclusion with caputo-hadamard
type fractional derivative, Advances in Difference Equations, 2019(1) 2019, 1-15.
[26] F. Skof, Propriet locall e approssimazione di operatori. Rend sem Math Fis Milano, Rendiconti del Seminario Matematico e
Fisico di Milano, 53(1) 1983, 113-129.
[27] S.M. Ulam, Problem in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1960.
[28] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.
[29] J. Xue, Hyers-Ulam stability of linear differential equations of second order with constant coefficient, talian Journal of Pure
and Applied Mathematics, 32 2014, 419-424.
Volume 2, Issue 3
September 2021
Pages 14-21
  • Receive Date: 26 June 2021
  • Revise Date: 26 July 2021
  • Accept Date: 06 August 2021
  • First Publish Date: 07 August 2021