Generalized Hyers-Ulam stability of functional equation deriving from additive and quadratic functions in fuzzy Banach space via two different techniques

A. Bodaghi; M. Arunkumar; S. Karthikeyan; E. Sathya

doi:10.26637/MJM0601/0029

Abstract

In this paper, authors given the generalized Hyers - Ulam stability of the functional equation deriving from additive and quadratic functions
$$
\sum_{j=1}^n f\left(x_i-\frac{1}{n} \sum_{j=1}^n x_j\right)=\sum_{i=1}^n f\left(x_i\right)-n f\left(\frac{1}{n} \sum_{j=1}^n x_j\right)
$$
where $n$ is a positive integer with $n \geq 2$ in Fuzzy Banach space via two different techniques.

Keywords:

Additive, Quadratic, mixed additive-quadratic functional equations, Generalized Ulam - Hyers stability, Fuzzy Banach space, fixed point

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

A. Bodaghi Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran.
M. Arunkumar Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.
S. Karthikeyan Department of Mathematics, R.M.K. Engineering College, Kavarapettai - 601 206, TamilNadu, India.
E. Sathya Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, TamilNadu, India.

Pages: 242-260
Date Published: 01-01-2018
Vol. 6 No. 01 (2018): Malaya Journal of Matematik (MJM)

J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989.

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.

M. Arunkumar, Three Dimensional Quartic Functional Equation In Fuzzy Normed Spaces, Far East Journal of Applied Mathematics, 41(2), (2010), 88-94.

M. Arunkumar, S. Karthikeyan, Solution and stability of n-dimensional mixed Type additive and quadratic functional equation, Far East Journal of Applied Mathematics, Volume 54, Number 1, 2011, 47-64.

M. Arunkumar, John M. Rassias, On the generalized Ulam-Hyers stability of an AQ-mixed type functional equation with counter examples, Far East Journal of Applied Mathematics, Volume 71, No. 2, (2012), 279-305.

M. Arunkumar, Solution and stability of modified additive and quadratic functional equation in generalized 2-normed spaces, International Journal Mathematical Sciences and Engineering Applications, Vol. 7 No. I (January, 2013), 383-391.

M. Arunkumar, P. Agilan, Additive Quadratic functional equation are Stable in Banach space: A Fixed Point Approach, International Journal of pure and Applied Mathematics, Vol. 86, No.6, (2013), 951 - 963.

M. Arunkumar, P. Agilan, Additive Quadratic functional equation are Stable in Banach space: A Direct Method, Far East Journal of Applied Mathematics, Volume 80, No. 1, (2013), $105-121$.

M. Arunkumar, Perturbation of $n$ Dimensional $A Q$ mixed type Functional Equation via Banach Spaces and Banach Algebra: Hyers Direct and Alternative Fixed Point Methods, International Journal of Advanced Mathematical Sciences (IJAMS), Vol. 2 (1), (2014), 34-56.

M. Arunkumar and T. Namachivayam, Stability of $n$ dimensional quadratic functional Equation in Fuzzy normed spaces: Direct And Fixed Point Methods, Proceedings of the International Conference on Mathematical Methods and Computation, Jamal Academic Research Journal an Interdisciplinary, (February 2014), 288-298.

M. Arunkumar, P. Agilan, Stability of A AQC Functional Equation in Fuzzy Normed Spaces: Direct Method, Jamal Academic Research Journal an Interdisciplinary, (2015), $78-86$

M. Arunkumar, P. Agilan, C. Devi Shyamala Mary, Permanence of A Generalized AQ Functional Equation In Quasi-Beta Normed Spaces, International Journal of Pure and Applied Mathematics, Vol. 101, No. 6 (2015), 10131025.

M. Arunkumar, G.Shobana, S. Hemalatha, Ulam - Hyers, Ulam - Trassias, Ulam-Grassias, Ulam - Jrassias Stabilities of A Additive - Quadratic Mixed Type Functional Equation In Banach Spaces, International Journal of Pure and Applied Mathematics, Vol. 101, No. 6 (2015), 10271040.

M. Arunkumar, A. Bodaghi, T. Namachivayam and E. Sathya, A new type of the additive functional equations on intuitionistic fuzzy normed spaces, Commun. Korean Math. Soc. 32 (2017), No. 4, 915-932.

M. Arunkumar, C. Devi Shyamala Mary, Generalized Hyers - Ulam stability of additive - quadratic - cubic quartic functional equation in fuzzy normed spaces: $A$ direct method, International Journal of Mathematics And its Applications, Volume 4, Issue 4 (2016), 1-16.

M. Arunkumar, C. Devi Shyamala Mary, Generalized Hyers - Ulam stability of additive - quadratic - cubic quartic functional equation in fuzzy normed spaces: $A$ fixed point approach, International Journal of Mathematics And its Applications, Volume 4, Issue 4 (2016), 16-32.

K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.

T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11 (3) (2003) 687-705.

T. Bag and S.K. Samanta, Fuzzy bounded linear operators, Fuzzy Sets and Systems 151 (2005) 513-547.

A. Bodaghi, Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, $mathrm{J}$. Intel. Fuzzy Syst. 30 (2016), 2309-2317.

A. Bodaghi, C. Park and J. M. Rassias, Fundamental stabilities of the nonic functional equation in intuitionistic fuzzy normed spaces, Commun. Korean Math. Soc. 31 (2016), No. 4, 729-743.

L. Cadariu and V. Radu, Fixed points and stability for functional equations in probabilistic metric and random normed spaces, Fixed Point Theory and Applications. Article ID 589143,1 8 pages, 2009 (2009).

L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara, Ser. Mat. Inform. 41 (2003), 25-48.

L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber. 346 (2004), 43-52.

E. Castillo, A. Iglesias and R. Ruiz-coho, Functional Equations in Applied Sciences, Elsevier, B.V.Amslerdam, 2005 .

S.C. Cheng and J.N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994) 429-436.

P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86.

S. Czerwik, On the stability of the quadratic mappings in normed spaces, Abh. Math. Sem. Univ Hamburg., 62 (1992), 59-64.

S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.

G. L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl., 295 (2004), 127-133.

$mathrm{Z}$. Gajda, On the stability of additive mappings, Inter. J. Math. Math. Sci., 14 (1991), 431-434.

P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.

M. E. Gordji and M. B. Savadkouhi, Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces, Journal of Inequalities and Applications, doi: $10.1155 / 2009 / 527462$.

H.Azadi Kenary, S. Y. Jang and C. Park, A fixed point approach to the Hyers-Ulam stability of a functional equation in various normed spaces, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2011-67.

D. H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A., 27 (1941) 222-224.

D. H. Hyers, G. Isac, Th. M. Rassias, Stability of functional equations in several variables, Birkhauser, Basel, 1998.

K.W. Jun, H.M. Kim, On the stability of an n-dimensional quadratic and additive type functional equation, Math. Ineq. Appl, 9(1) (2006), 153-165.

S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.

L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functionsa question of priority, Aequationes Math., 75 (2008), 289-296.

B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968), 305309.

A. K. Mirmostafaee and M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, Vol. 159, no. 6, (2008), 720-729.

A. K. Mirmostafaee, M. Mirzavaziri and M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159, no. 6, (2008), 730-738.

A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Information Sciences, Vol. 178, no. $19,(2008), 3791-3798$.

A. K. Mirmostafaee and M. S. Moslehian, Fuzzy almost auadratic functions. Results in Mathematics. 52. no. 1-2.

A. Najati, Th.M. Rassias, Stability of a mixed functional equation in several variables on Banach modules, Nonlinear Analysis.-TMA, in press.

P. Narasimman and A. Bodaghi, Solution and stability of a mixed type functional equation, Filomat. 31 (2017), No. $5,1229-1239$.

M.J. Rassias, M. Arunkumar and S. Ramamoorthi, Stability of the Leibniz additive-quadratic functional equation in quasi- $beta$ normed spaces: direct and fixed point methods, Journal of Concrete and Applicable Mathematics, $mathbf{1 4}$ $(2014), 22-46$

J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130.

J. M. Rassias, Solution of the Ulam problem for cubic mappings, An. Univ. Timisoara Ser. [Mat. Inform. 38 (2000), 121-132.

J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Mathematicki., 34(54) no.2, (1999), 243-252.

J. M. Rassias, M. Arunkumar, E. Sathya and N. Mahesh Kumar, Solution And Stability Of A ACQ Functional Equation In Generalized 2-Normed Spaces, Intern. J. Fuzzy Mathematical Archive, Vol. 7, No. 2, (2015), 213-224.

J. M. Rassias, M. Arunkumar and T. Namachivayam, Stability Of The Leibniz Additive-Quadratic Functional Equation In Felbin's And Random Normed Spaces: A Direct Method, Jamal Academic Research Journal an Interdisciplinary, (2015), 102-110.

J. M. Rassias, M. Arunkumar, E.sathya, N. Mahesh Kumar, Generalized Ulam - Hyers Stability Of A (AQQ): Additive - Quadratic - Quartic Functional Equation, Malaya Journal of Matematik, 5(1) (2017), 122-142.

Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), 297300 .

Th. M. Rassias, On a modified Hyers-Ulam sequence, $mathrm{J}$. Math. Anal. Appl. 158no. 1, (1991), 106-113.

Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 no. 4, (1992), 989-993.

Th. M. Rassias and P.Semrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173no. 2, (1993), $325-338$.

Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., $246(2000), 352-378$

Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003.

K. Ravi, J. M. Rassias, M. Arunkumar and R. Kodandan, Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Article 114, 29 pp.

F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129.

S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964.

J. Z. Xiao and X. H. Zhu, Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133 (2003) 389-399.

T. Z. Xu and J. M. Rassias, M. J. Rassias and W. X. $mathrm{Xu}$, A fixed point approach to the stability of quintic and sextic functional equations in quasi- $beta$-normed spaces, $mathrm{J}$. Inequal. Appl. 2010, Art. ID 423231, 23 pp.

T. Z. Xu and J. M. Rassias, Approximate Septic and Octic mappings in quasi- $beta$-normed spaces, $mathrm{J}$. Computational Analysis and Applications, Vol.15, No. 6, 1110 - 1119, 2013 , copyright 2013 Eudoxus Press, LLC.

S. Zolfaghari, Approximation Of Mixed Type Functional Equations In p-Banach Spaces, J. Nonlinear Sci. Appl. $3(2010)$, no. $2,110-122$.