General solution and two methods of generalized Ulam - Hyers stability of n-dimensional AQCQ functional equation

Sandra Pinelas; M. Arunkumar; T. Namachivayam; E. Sathya

doi:10.26637/mjm502/003

Abstract

In this paper, we achieve the general solution and generalized Ulam - Hyers stability of a $n$-dimensional additive-quadratic-cubic-quartic (AQCQ) functional equation
$$
\begin{aligned}
&f\left(\sum_{i=1}^{n-1} v_i+2 v_n\right)+f\left(\sum_{i=1}^{n-1} v_i-2 v_n\right)\\= & 4 f\left(\sum_{i=1}^n v_i\right)+4 f\left(\sum_{i=1}^{n-1} v_i-v_n\right)-6 f\left(\sum_{i=1}^{n-1} v_i\right) \\
& +f\left(2 v_n\right)+f\left(-2 v_n\right)-4 f\left(v_n\right)-4 f\left(-v_n\right)
\end{aligned}
$$
where $n$ is a positive integer with $n \geq 3$ in Banach Space (BS) via direct and fixed point methods. The stability results are discussed in two different ways by assuming $n$ is an odd positive integer and $n$ is an even positive integer.

Keywords:

AQCQ functional equation, generalized Ulam - Hyers stability, Banach space, fixed point

Mathematics Subject Classification:

39B52, 32B72, 32B82

Authors Imprint References Funding Related Articles Downloads

Sandra Pinelas Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, Athens 15342, Greece. https://orcid.org/0000-0002-0984-0159
M. Arunkumar Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, Tamil Nadu, India.
T. Namachivayam Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, Tamil Nadu, India.
E. Sathya Department of Mathematics, Government Arts College, Tiruvannamalai - 606 603, Tamil Nadu, India.

Pages: 202-240
Date Published: 01-04-2017
Vol. 5 No. 02 (2017): Malaya Journal of Matematik (MJM)

J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989. DOI: https://doi.org/10.1017/CBO9781139086578

C. Alsina, On the stability of a functional equation arising in probabilistic normed spaces, General Inequal., Oberwolfach 5, 263-271 (1986). Birkhuser, Basel (1987). DOI: https://doi.org/10.1007/978-3-0348-7192-1_20

T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. DOI: https://doi.org/10.2969/jmsj/00210064

M. Arunkumar, Matina J. Rassias, Y. Zhang, Ulam-Hyers stability of a 2-variable AC-mixed type functional equation: direct and fixed point methods, J. Mod. Math. Front., 1 (3) (2012), 10-26.

M. Arunkumar, Solution and stability of modified additive and quadratic functional equation in generalized 2-normed spaces, International J. Math. Sci. Eng. Appl., 7, No. I (2013), 383-391.

M. Arunkumar, G. Britto Antony Xavier, Functional equation Originating from sum of higher Powers of Arithmetic Progression using Difference Operator is stable in Banach space: Direct and Fixed point Methods, Malaya J. Mat., 1. Iss. 1 (2014), 49-60. DOI: https://doi.org/10.26637/mjm201/007

M. Arunkumar, S. Karthikeyan, Solution and Intuitionistic Fuzzy stability of n-dimensional quadratic functional equation: Direct and Fixed Point Methods, Int. J. Adv. Math. Sci., 2 (1) (2014), 21-33. DOI: https://doi.org/10.14419/ijams.v2i1.1498

M. Arunkumar, Perturbation of $n$ Dimensional AQ - 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. DOI: https://doi.org/10.14419/ijams.v2i1.1499

M. Arunkumar, T. Namachivayam, Stability of a n-dimensional additive functional equation in random normed space, Int. J. Math. . Anal., 4, No. 2 (2012), 179-186.

M. Arunkumar, T. Namachivayam, Stability of a n-dimensional additive functional equation in random normed space: A fixed point approach, International Journal of Physical, Chemical and Mathematical Sciences (IJPCMS) - ISSN 2278 - 683X, Vol. 2; No. 1, (2013), 100 - 108.

M. Arunkumar, S. Hemalatha, E. Sathya, 2-Variable AQCQ - functional equation, Int. J. Adv. Math. Sci., 3 (1) (2015), 65-86 . DOI: https://doi.org/10.14419/ijams.v3i1.4401

M. Arunkumar, S. Hema latha, Additive -Quartic Functional Equations Are Stable in Random Normed Space, Jamal Academic Research Journal an Interdisciplinary, (2015), 31-38.

M. Arunkumar, S. Murthy, S. Hema latha, M. Arulselvan, Additive - quartic functional equations are stable in random normed space: a fixed point method, Malaya Journal of Mathematics, (2015), 215 - 227.

A. Bodaghi, Stability of a quartic functional equation, The Scientific World Journal. 2014, Art. ID 752146, 9 pages, doi:10.1155/2014/752146. DOI: https://doi.org/10.1155/2014/752146

A. Bodaghi, I. A. Alias and M. H. Ghahramani, Ulam stability of a quartic functional equation, Abs. Appl. Anal., 2012, Art. ID 232630 (2012). DOI: https://doi.org/10.1155/2012/232630

D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), 223-237. DOI: https://doi.org/10.1090/S0002-9904-1951-09511-7

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

L. Cădariu, 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.

Y. J. Cho, M. Eshaghi Gordji and S. Zolfaghari, Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces, J. Ineq. Appl, doi:10.1155/2010/403101 DOI: https://doi.org/10.1155/2010/403101

S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. DOI: https://doi.org/10.1142/4875

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. DOI: https://doi.org/10.1016/j.jmaa.2004.03.011

Z. Gajda, On the stability of additive mappings, Inter. J. Math. Math. Sci., 14 (1991), 431-434. DOI: https://doi.org/10.1155/S016117129100056X

P. Gǎvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184 (1994), 431-436. DOI: https://doi.org/10.1006/jmaa.1994.1211

M. Eshaghi Gordji, H. Khodaie, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, arxiv: 0812. 2939v1 Math FA, 15 Dec 2008.

M. Eshaghi Gordji and M. B. Savadkouhi, Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces, J. Ineq. Appl., doi:10.1155/2009/527462. DOI: https://doi.org/10.1155/2009/527462

M. Eshaghi Gordji, M. Bavand Savadkouhi, Choonkil Park, Quadratic-Quartic Functional Equations in RN-Spaces, Journal of Inequalities and Applications, doi:10.1155/2009/868423. DOI: https://doi.org/10.1155/2009/868423

M. Eshaghi Gordji, S. Zolfaghari , J. M. Rassias and M. B. Savadkouhi, Solution and Stability of a Mixed type Cubic and Quartic functional equation in Quasi-Banach spaces, Abstract and Applied Analysis, Volume 2009, Art. ID 417473, 1-14, Doi:10.1155/2009/417473. DOI: https://doi.org/10.1155/2009/417473

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 The. and Appl., doi:10.1186/1687-1812-2011-67. DOI: https://doi.org/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. DOI: https://doi.org/10.1073/pnas.27.4.222

D. H. Hyers, G. Isac and Th.M. Rassias, Stability of functional equations in several variables,Birkhauser, Basel, 1998. DOI: https://doi.org/10.1007/978-1-4612-1790-9

K. W. Jun and H. M. Kim, On the stability of an n-dimensional quadratic and additive type functional equation, Math. Ineq. Appl 9(1) (2006), 153-165. DOI: https://doi.org/10.7153/mia-09-16

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

P. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, 2009. DOI: https://doi.org/10.1007/978-0-387-89492-8

L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions-a question of priority, Aequ. Math., 75 (2008), 289-296. DOI: https://doi.org/10.1007/s00010-007-2892-8

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), 305-309. DOI: https://doi.org/10.1090/S0002-9904-1968-11933-0

D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J Math Anal Appl. 343 (2008), 567-572 . DOI: https://doi.org/10.1016/j.jmaa.2008.01.100

D. Mihet, The probabilistic stability for a functional equation in a single variable, Acta Math Hungar. 123, 249256 (2009). doi:10.1007/s10474-008-8101-y DOI: https://doi.org/10.1007/s10474-008-8101-y

M. Mohamadi, Y. Cho, C. Park, P. Vetro, R. Saadati, Random stability of an additive-quadratic-quartic functional equation J Inequal Appl 2010, 18 (2010). Article ID 754210. DOI: https://doi.org/10.1155/2010/754210

S. Murthy, M. Arunkumar, V. Govindan, General Solution and Generalized Ulam-Hyers Stability of a Generalized n- Type Additive Quadratic Functional Equation in Banach Space and Banach Algebra: Direct and Fixed Point Methods, International Journal of Advanced Mathematical Sciences, 3 (1) (2015), 25-64. DOI: https://doi.org/10.14419/ijams.v3i1.4402

C. Park and A. Bodaghi, On the stability of *-derivations on Banach *-algebras, Adv. Difference Equ. 2012, Article No. 138, 10 pages (2012). DOI: https://doi.org/10.1186/1687-1847-2012-138

C. Park, J. R. Lee, An AQCQ-functional equation in paranormed spaces, Advances in Difference Equations, doi: 10.1186/1687-1847-2012-63. DOI: https://doi.org/10.1186/1687-1847-2012-63

J. Matina, Rassias, M. Arunkumar, 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, 14 (2014), 22-46.

J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, $46(1982) 126-130$. DOI: https://doi.org/10.1016/0022-1236(82)90048-9

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 and H. M. Kim, Generalized Hyers-Ulam stability for general additive functional equations in quasi- $beta$-normed spaces J. Math. Anal. Appl. 356 (2009), no. 1, 302-309. DOI: https://doi.org/10.1016/j.jmaa.2009.03.005

J. M. Rassias, K.Ravi, M.Arunkumar and B.V.Senthil Kumar, Ulam Stability of Mixed type Cubic and Additive functional equation, Functional Ulam Notions (F.U.N) Nova Science Publishers, 2010, Chapter $13,149-175$.

J.M. Rassias, R. Saadati, G. Sadeghi, J. Vahidi, On nonlinear stability in various random normed spaces Journal of Inequalities and Applications 2011 doi:10.1186/1029-242X-2011-62. DOI: https://doi.org/10.1186/1029-242X-2011-62

John M. Rassias, Matina J. Rassias, M.Arunkumar, T. Namachivayam, Ulam - Hyers stability of a 2variable AC - mixed Type functional equation in Felbin's type spaces: Fixed point methods, International Mathematical Forum, Vol. 8, No. 27, (2013), 1307 - 1322. DOI: https://doi.org/10.12988/imf.2013.35108

J. M. Rassias, M. Arunkumar, E. Sathya, N. Mahesh Kumar Solution And Stability Of A ACQ Functional Equation In Generalized 2-Normed Spaces, Intern. J. Fuzzy Math. Arch., 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.

Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), $297-300$. DOI: https://doi.org/10.1090/S0002-9939-1978-0507327-1

Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 no. 1, (1991), 106-113. DOI: https://doi.org/10.1016/0022-247X(91)90270-A

T. 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. DOI: https://doi.org/10.1090/S0002-9939-1992-1059634-1

Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl.,246 (2000), 352-378. DOI: https://doi.org/10.1006/jmaa.2000.6788

Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003. DOI: https://doi.org/10.1007/978-94-017-0225-6

K. Ravi, M. Arunkumar and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, 3, No. 08 (2008), 36-47.

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, Art. 114, 29 pages.

F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113-129. DOI: https://doi.org/10.1007/BF02924890

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

T.Z. Xu, J.M. Rassias, W.X Xu, Generalized Ulam-Hyers stability of a general mixed AQCQ-functional equation in multi-Banach spaces: a fixed point approach, Eur. J. Pure Appl. Math. 3 (2010), 1032-1047.

T.Z. Xu, J. M. Rassias, W.X. Xu, A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-Archimedean normed spaces, Discrete Dyn. Nat. Soc. 2010, Art. ID 812545, 24 pages. DOI: https://doi.org/10.1155/2010/812545