Stability of system of additive functional equations in various Banach spaces: Classical Hyers methods
Downloads
DOI:
https://doi.org/10.26637/MJM0601/0014Abstract
In this paper, authors proved the generalized Ulam - Hyers stability of system of additive functional equations
$$
\begin{aligned}
& f\left(\sum_{a=1}^n a x_a\right)=\sum_{a=1}^n\left(a f\left(x_a\right)\right) ; \quad n \geq 1 \\
& g\left(\sum_{a=1}^n 2 a y_{2 a}\right)=\sum_{a=1}^n\left(2 a g\left(y_{2 a}\right)\right) ; \quad n \geq 1 \\
& h\left(\sum_{a=1}^n(2 a-1) z_{2 a-1}\right)=\sum_{a=1}^n\left((2 a-1) h\left(z_{2 a-1}\right)\right) ; \quad n \geq 1
\end{aligned}
$$
where $n$ is a positive integer, which is originating from sum of first $n$, natural numbers, even natural numbers and odd natural numbers, respectively in various Banach spaces.
Keywords:
Additive functional equations, Generalized Ulam - Hyers stability, Banach spaces, 2-Banach space, Quasi - Beta - 2- Banach space, Fuzzy Quasi - Beta - 2 - Banach space, Random Quasi - Beta - 2 - Banach space.Mathematics Subject Classification:
Mathematics- Pages: 91-112
- Date Published: 01-01-2018
- Vol. 6 No. 01 (2018): Malaya Journal of Matematik (MJM)
J. Aczel Lectures on Functional Equations and Their Applications, Academic Press, New York (1966). MR:348020 (1967).
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, Solution and stability of Arun-additive functional equations, International Journal Mathematical Sciences and Engineering Applications, 4, No.3, August 2010 .
M. Arunkumar, C. Leela Sabari, Solution and stability of a functional equation originating from a chemical equation, International Journal Mathematical Sciences and Engineering Applications, f 5 (2) (2011), 1-8..
M. Arunkumar, S. Karthikeyan, Solution and Stability of n-dimensional Additive functional equation, International Journal of Applied Mathematics, 25 (2), (2012), 163174.
M. Arunkumar, S. Hema latha, C. Devi Shaymala Mary, Functional equation originating from arithmetic Mean of consecutive terms of an arithmetic Progression are stable in banach space: Direct and fixed point method, JP Journal of Mathematical Sciences, 3, Issue 1, 2012, Pages 27-43.
M. Arunkumar, G. Vijayanandhraj, S. Karthikeyan, Solution and Stability of a Functional Equation Originating From $n$ Consecutive Terms of an Arithmetic Progression, Universal Journal of Mathematics and Mathematical Sciences, 2, No. 2, (2012), 161-171.
M. Arunkumar, Solution and stability of modified additive and quadratic functional equation in generalized 2-normed spaces, International Journal Mathematical Sciences and Engineering Applications, 7 No. I (2013), 383-391.
M. Arunkumar, Functional equation originating from sum of First natural numbers is stable in Cone Banach Spaces: Direct and Fixed Point Methods, International Journal of Information Science and Intelligent System (IJISIS ISSN:2307-9142), 2 No. 4 (2013), 89 - 104.
M. Arunkumar, Stability of n-dimensional Additive functional equation in Generalized 2 - normed space, Demonstrato Mathematica $49(3)$, (2016), 319 - 33 .
M. Arunkumar, P. Agilan, Additive functional equation and inequality are Stable in Banach space and its applications, Malaya Journal of Matematik (MJM), Vol 1, Issue 1, (2013), 10-17.
A. Malceski, R. Malceski, K. Anevska and S. Malceski, $A$ Remark about Quasi 2-Normed Space, Applied Mathematical Sciences, 9, 2015, no. 55, 2717-2727.
S. Czerwik, Functional Equations and Inequalities in SeveralVariables, World Scientific, River Edge, NJ, 2002.
S. Gahler 2 Metrische Raume und ihre Topologische Struktur, Math. Nachr., 26 (1963), 115-148.
S. Gahler, Lineare 2-Normierte Raume, Math. Nachr., 28 (1964) $1-43$.
S. Gahler, Uber 2-Banach-Raume, Math. Nachr. 42 (1969), 335-347.
P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA. 27 (1941), 222224.
D. H. Hyers, G. Isac and Th.M. Rassias, Stability of functional equations in several variables, Birkhauser, Basel, 1998.
O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 2001.
O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 2001.
O. Hadzic, E. Pap and M. Budincevic, Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces, Kybernetika. 38 , no. 3 (2002), 363-382.
S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, 2009.
A. K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets and Systems, 12 (1984), 143-154.
M. Kir and M. Acikgoz, A study involving the completion of quasi 2-normed space, Int. J. Anal. (2013). http://dx.doi.org/10.1155/2013/512372.
D.O. Lee, Hyers-Ulam stability of an addtiive type functional equation, J. Appl. Math. and Computing, 13 (2003) no.1-2, 471-477.
Z. Lewandowska, Generalized 2-normed spaces, Stuspske Prace Matematyczno-Fizyczne, 1 (2001), no.4, $33-40$
L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions- a question of priority, Aequationes Math., 75 (2008), 289-296.
M.J.Rassias, M. Arunkumar, E.Sathya, ai Type n- Variable Multi n-Dimensional Additive Functional Equation, Malaya Journal of Matematik, 5(2) (2017), 278-292.
J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130.
Th. M. Rassias, On the stability of the linear mapping inBanach spaces, Proc. Amer. Math. Soc., 72 (2) (1978) 297-300.
K. Ravi, M. Arunkumar, On a n-dimensional additive Functional Equation with fixed point Alternative, Proceedings of ICMS 2007,314-330.
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, Autumn 2008 Vol.3, No. 08, 36-47.
S. Rolewicz, Metric Linear Spaces, Reidel, Dordrecht, 1984.
P. K. Sahoo, Pl. Kannappan, Introduction to Functional Equations, Chapman and Hall/CRC Taylor and Francis Group, 2011.
B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing, New York, NY, USA, 1983.
A. N. Sherstnev, On the notion of a random normed space, Doklady Akademii Nauk SSSR. 149 (1963), 280-283 (Russian).
J. Tabor, Stability of the Cauchy functional equation in quasi-Banach spaces, Ann. Polon. Math. 83 (2004) 243255
S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Ed., Wiley, New York, 1940.
A. White, 2-Banach spaces, Doctorial Diss., St. Louis Univ., 1968.
A. White, 2-Banach spaces, Math. Nachr. 42 (1969), $43-60$.
- A
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2018 MJM
This work is licensed under a Creative Commons Attribution 4.0 International License.
