Functional inequalities for a generalized quadratic functional equation in various Banach spaces
Downloads
Abstract
In this paper, we study the stability of a generalized quadratic functional equation in sense of Ulam, Hyers and Rassias in Banach spaces, Quasi- $\beta$-2-Banach spaces and intuitionistic fuzzy-2-Banach spaces via two alternate methods.
Keywords:
Quadratic functional equation, Generalized Hyers-Ulam stability, Banach space, Quasi- $beta$-2 -Banach space ```, intuitionistic fuzzy-2-Banach space, Hyers method, fixed point methodMathematics Subject Classification:
Mathematics- Pages: 966-980
- Date Published: 01-01-2021
- Vol. 9 No. 01 (2021): Malaya Journal of Matematik (MJM)
J. Aczél, Lectures on functional equations and sheir applications, translated by Scripta Technica, Inc. Supplemented by the author. Edited by Hansjorg Oser, Mathematics in Science and Engineering, Vol. 19, Academic Press, New York, 1966.
J. Acze] and J. Dhombres, Funcrional equations in several variables, Encyclopedia of Mathematics and its Applications, 31, Cambridge Univ. Press, Cambridge, 1989.
T. Aoki, On the stabiliry of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.
M. Arunkumar, Stabiliry of n-dimensional Additive fienctonal equation in Generalized 2 - normed space, Demonstrato Mathematica 49 (3), (2016), 319 - 33 .
M.Arunkumar, E.Sathya, S.Karthikeyan, G. Ganapathy, T. Namachivayam, Stability of System of Additive Futerional Equations in Various Banach Spaces: Classical Hyers Methods, Malaya Journal of Matematik, Volume 6, Issue 1, 2018, 91-1 12.
$161 mathrm{~K}$. Atanassoy, Intuitionisnic fuczy sers, Fuzry sets and Systems. 20 (1986), No. 1, 87-96.
J.-H. Bae and K.-W. Jun, On the generalized HyersUan-Rassias stability of a quadraric functional equation, Bull. Korean Math. Soc. 38 (2001), no. 2, 325-336.
A. Bodaghi, Stabiliry of a quartic functional equation, The Scientific World Journal. 2014, Art. ID 752146, 9 pages, doà: $10.1155 / 2014 / 752146$.
A. Bodaghi, Intuitionistic firzy stabiliry of the generallied forms of cubic and quaric functional equations, J. Intel. Fuzzy Syst. 30 (2016), 2309-2317.
A. Bodaghi, I. A. Alias and M. Eshaghi Gordji, On the ssabiliry of quadratic double cenralizers and quadratic multipliers: A fixed point approach, J. Inequal. Appl. 2011, Article ID 957541 (2011).
A. Bodaghi, C. Park and J. M. Rassaas, Fuendanental stabilities of the novic funcrional equarion in intuitionisric fuzz nomed spaces, Commun. Korean Math. Soc., 31 (2016), No. $4,729-743$.
E. Castillo, A. Iglesias and R. Ruíz-Cobo, Functional equations in applied sciences, Mathematics in Science and Engineering, 199, Elsevier B. V., Amsterdam, 2005.
I.-S. Chang and H.-M. Kim, Hyers-Ulam-Rassias stability of a quadratic finctional equation, Kyungpook Math. J. $42(2002)$, no. 1, 71-86.
I.-S. Chang, E. H. Lee and H.-M. Kim, On Hyers-UlamRassias stability of a quadratic functional equation, Math. Inequal. Appl. 6 (2003), no. 1, 87-95.
P. W. Cholewa, Remarks on the stability of functional equarions, Aequationes Math. 27 (1984), no. 1-2, 76-86. [16] $mathrm{S}$. Czerwik, On the stability of the quadratic mapping in nomed spaces, Abh. Math. Sem. Univ. Hamburg 62 $(1992), 59-64$.
S. Czerwik, Srability of Functional Equarions of Ulam - Hyers - Rassias Type, Hadronic Press, Plam Harbor, Florida, 2003.
M. Eshaghi Gordji and A. Bodaghi, On the Hyers-UlamRassias stabiliry problem for quadratic functional equations, East J. Approx., 16 (2010), no. 2, 123-130.
M. Eshaghi Gordji, A. Bodaghi and C. Park, A fixed point approuch to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras, U. P. B. Sci. Bull., Series A. 73, Iss. 2 (2011), 65-73.
S. Gahler, Lineare 2-Nomierie Raume, Math. Nachr., $28(1964) 1-43$.
S. Gahler, Uber 2-Banach-Raime, Math. Nachr. 42 (1969), 335-347.
P. Gãvrutĕ, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436.
D. H. Hyers, On the sability of the linear functional equarion, Proc. Nat. Acad. Sci. U. S. A. 27 (1941), $222-$ 224.
D. H. Hyers and Th. M. Rassias, Appooximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153.
D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser Boston, Boston, MA, 1998.
D. H. Hyers, G. Isac and Th. M. Rassias, On the asymptoticiry aspect of Hyers-Ulam stability of mappings, Proc. Amer. Math. Soc. 126 (1998), no. 2, 425-430.
${ }^{277]} mathrm{G}$. Isac and Th. M. Rassias, Stability of $Psi$-additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219-228.
S.-M. Jung, On the Hyers-Ulam stability of the funcsional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), no. 1, 126-137.
S.-M. Jung, On the Hyers-Ulam-Rassias stabiliry of a quadratic finctional equation, J. Math. Anal. Appl. 232 (1999), no. 2, 384-393.
S.-M. Jung and B. Kim, On the stability of the quadratic functional equation on bounded domains, Abh. Math. Sem. Univ. Hamburg 69 (1999), 293-308.
S.-M. Jung and P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, J. Korean Math. Soc. 38 (2001), no. 3, $645-656$.
Pl. Kannappan, Quedratic functional equation and innter product spaces, Results Math. 27 (1995), no. 3-4, 368372.
Pl. Kannappan, Functional equations and inequaliries with applications, Springer Monographs in Mathematics, Springer, New York, 2009.
A. K. Katsaras, Fuzzy topological veclor spaces $I$, Fuzzy Sets and Systems, 12 (1984), 143-154.
M. Kir and M. Acikgoz, A study imolving the connpletion of quasi 2-normed space, Int. J. Anal. (2013). httpe//dx.doi.org/10.1155/2013/512372.
Z. Lewandowska, Generalized 2-nomed spaces, Stuspske Prace Matematyczno-Fizyczne, 1 (2001), no.4. $33-40$.
A. Malceski, R. Malceski, K. Anevska and S. Malceski, A Remark abour Qnasi 2-Normed Space, Applied Mathematical Sciences, 9, 2015, no. 55, 2717-2727.
L. Maligranda, A result of Tosio Aoki about a generalisation of Hyers-Ulam stabiliry of additive functions - a quession of prioriry, Aequationes Math. 75 (2008), no. 3, $289-296$.
B. Margolis and J. B. Diaz, A fixed point theorem of the alternative, for contractions on a generalized complete merric space, Bull. Amer. Math. Soc. 74 (1968), 305309.
V. Radu, The fixed point allemative and the stability of functional equitionts, Fixed Point Theory 4 (2003), no. 1 , $91-96$.
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. $1,126-130$.
J. M. Rassias, Solution of the Ulam stabiliry problem for Euler-Lagrange quadratic mappings, J. Math. Anal. Appl. 220 (1998), no. 2, 613-639.
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2 , 297-300.
Th. M. Rassias, On the stability of mappings, Rend. Sem. Mat. Fis. Milano 58 (1988), 91-99 (1990).
Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), no. 1, 106-113.
S. Rolewicz, Metric Linear Spaces, Reidel, Dordrecht, 1984.
R. Saadati, J. H. Park, On the intaitionistic fitzry topological spuces, Chaos, Solitons and Fractals. 27 (2006), $331-344$
R. Saadati, S. Sedghi and N. Shobe, Modified intiritionistic ficzy metric spaces and sonte fited point theorems, Chaos, Solitons and Fractals, 38 (2008), 36-47.
F. Skof, Lacal properties and approximation of operarors, Rend. Sem. Mat. Fis. Mīlano 53 (1983), 113-129 (1986).
S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964.
A. White, 2-Banach spaces, Doctorial Diss., St. Louis Univ., 1968.
A. White, 2-Banach spaces, Math. Nachr. 42 (1969), $43-60$.
Similar Articles
- S. Gomathi, M. Sakthi, M. Shangeetha, Effect of mass transfer on unsteady three-dimensional MHD dusty Couette flow , Malaya Journal of Matematik: Vol. 9 No. 01 (2021): Malaya Journal of Matematik (MJM)
You may also start an advanced similarity search for this article.
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 MJM
This work is licensed under a Creative Commons Attribution 4.0 International License.
