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Erschienen in:
Simple Proofs of Some Bernstein–Mordell Type Inequalities Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

A Fixed Point Approach to Stability of the Quadratic Equation

verfasst von : M. Almahalebi, A. Charifi, S. Kabbaj, E. Elqorachi

Erschienen in: Topics in Mathematical Analysis and Applications

Verlag: Springer International Publishing

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Abstract

In this paper, by using the fixed point method in Banach spaces, we prove the Hyers–Ulam–Rassias stability for the quadratic functional equation
$$\displaystyle{f\left (\sum _{i=1}^{m}x_{ i}\right ) =\sum _{ i=1}^{m}f(x_{ i}) + \frac{1} {2}\sum _{1\leq i<j\leq m}\{f(x_{i} + x_{j}) - f(x_{i} - x_{j})\}.}$$
The concept of the Hyers–Ulam–Rassias stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978).

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Vorheriges Kapitel Hilbert-Type Inequalities Including Some Operators, the Best Possible Constants and Applications: A Survey
Nächstes Kapitel Aspects of Global Analysis of Circle-Valued Mappings
Literatur
1.
Ait Sibaha, M., Bouikhalene, B., Elqorachi, E.: Hyers-Ulam-Rassias stability of the K-quadratic functional equation. J. Inequal. Pure Appl. Math. 8 (2007). Article 89
2.
3.
Akkouchi, M.: Hyers-Ulam-Rassias stability of Nonlinear Volterra integral equation via a fixed point approach. Acta Univ. Apulensis Math. Inform. 26, 257–266 (2011)MathSciNetMATH
4.
Almahalebi, M.: A fixed point approach of quadratic functional equations. Int. J. Math. Anal. 7, 1471–1477 (2013)MathSciNetMATH
5.
Almahalebi, M., Kabbaj, S.: A fixed point approach to the orthogonal stability of an additive - quadratic functional equation. Adv. Fixed Point Theory 3, 464–475 (2013)
6.
Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)MATHCrossRef
7.
Baker, J.A.: The stability of certain functional equations. Proc. Am. Math. Soc. 112, 729–732 (1991)MATHCrossRef
8.
Bouikhalene, B., Elqorachi, E., Rassias, Th.M.: On the Hyers-Ulam stability of approximately Pexider mappings. Math. Inequal. Appl. 11, 805–818 (2008)MathSciNetMATH
9.
Brzdȩk, J.: On a method of proving the Hyers-Ulam stability of functional equations on restricted domains. Aust. J. Math. Anal. Appl. 6(1), 1–10 (2009). Article 4
10.
Cǎdariu, L., Radu, V.: Fixed points and the stability of Jensens functional equation. J. Inequal. Pure Appl. Math. 4(1), 7 (2003). Article 4
11.
Cǎdariu, L., Radu, V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Berichte 346, 43–52 (2004)
12.
Charifi, A., Bouikhalene, B., Elqorachi, E.: Hyers-Ulam-Rassias stability of a generalized Pexider functional equation. Banach J. Math. Anal. 1, 176–185 (2007)MathSciNetMATHCrossRef
13.
Charifi, A., Bouikhalene, B., Elqorachi, E., Redouani, A.: Hyers-Ulam-Rassias stability of a generalized Jensen functional equation. Aust. J. Math. Anal. Appl. 19, 1–16 (2009)MathSciNet
14.
Cho, Y.J., Gordji, M.E., Zolfaghari, S.: Solutions and stability of generalized mixed type QC functional equations in random normed spaces. J. Inequal. Appl. 2010 (2010). Article ID 403101
15.
16.
Cieplisński, K.: Applications of fixed point theorems to the Hyers-Ulam stability of functional equations, a survey. Ann. Funct. Anal. 3, 151–164 (2012)MathSciNetCrossRef
17.
18.
19.
Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)MathSciNetMATHCrossRef
20.
21.
Forti, G.-L., Sikorska, J.: Variations on the Drygas equation and its stability. Nonlinear Anal. Theory Methods Appl. 74, 343–350 (2011)MathSciNetMATHCrossRef
22.
23.
Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)MathSciNetCrossRef
24.
Gordji, M.E., Rassias, J.M., Savadkouhi, M.B.: Approximation of the quadratic and cubic functional equations in RN-spaces. Eur. J. Pure Appl. Math. 2, 494–507 (2009)MathSciNetMATH
25.
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)MathSciNetCrossRef
26.
27.
Hyers, D.H., Isac, G., Rassias, Th.M.: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proc. Am. Math. Soc. 126, 425–430 (1998)MathSciNetMATHCrossRef
28.
Hyers, D.H., Isac, G.I., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)MATHCrossRef
29.
Janfada, M., Sadeghi, G.: Generalized Hyers-Ulam stability of a quadratic functional equation with involution in quasi-β-normed spaces. J. Appl. Math. Inform. 29, 1421–1433 (2011)MathSciNetMATH
30.
Jun, K.-W., Lee, Y.-H.: A generalization of the Hyers-Ulam-Rassias stability of Jensen’s equation. J. Math. Anal. Appl. 238, 305–315 (1999)MathSciNetMATHCrossRef
31.
32.
Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011)MATHCrossRef
33.
Jung, S.-M., Kim, B.: Local stability of the additive functional equation and its applications. IJMMS. 2003(1), 15–26 (2003)MathSciNetMATH
34.
Jung, S.-M., Lee, Z.-H.: A fixed point approach to the stability of quadratic functional equation with involution. Fixed Point Theory Appl. 5 (2008). Article ID 732086
35.
Jung, S.-M., Sahoo, P.K.: Hyers-Ulam stability of the quadratic equation of Pexider type. J. Korean Math. Soc. 38(3), 645–656 (2001)MathSciNetMATH
36.
Jung, S.-M., Moslehian, M.S., Sahoo, P.K.: Stability of generalized Jensen equation on restricted domains. J. Math Inequal. 4, 191–206 (2010)MathSciNetMATHCrossRef
37.
Kannappan, Pl.: Functional Equations and Inequalities with Applications. Springer, New York (2009)MATHCrossRef
38.
Lee, Y.H., Jung, K.W.: A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation. J. Math. Anal. Appl. 246, 627–638 (2000)MathSciNetMATHCrossRef
39.
Manar, Y., Elqorachi, E., Bouikhalene, B.: Fixed point and Hyers-Ulam-Rassias stability of the quadratic and Jensen functional equations. Nonlinear Funct. Anal. Appl. 15(2), 273–284 (2010)MathSciNetMATH
40.
41.
Moslehian, M.S., Najati, A.: Application of a fixed point theorem to a functional inequality. Fixed Point Theory 10, 141–149 (2009)MathSciNetMATH
42.
Moslehian, M.S., Sadeghi, Gh.: Stability of linear mappings in quasi-Banach modules. Math. Inequal. Appl. 11, 549–557 (2008)MathSciNetMATH
43.
44.
Najati, A., Moghimi, M.B.: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J. Math. Anal. Appl. 337, 399–415 (2008)MathSciNetMATHCrossRef
45.
Najati, A., Park, C.: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. J. Math. Anal. Appl. 335, 763–778 (2007)MathSciNetMATHCrossRef
46.
47.
48.
Pourpasha, M.M., Rassias, J.M., Saadati, R., Vaezpour, S.M.: A fixed point approach to the stability of Pexider quadratic functional equation with involution. J. Inequal. Appl. (2010). doi:10.1155/2010/839639. Article ID 839639MathSciNetMATH
49.
Radosław, ł.: The solution and the stability of the Pexiderized K-quadratic functional equation. In: 12th Debrecen-Katowice Winter Seminar on Functional Equations and Inequalities, Hajdúszoboszló, 25–28 January 2012
50.
Radosław, ł.: Some generalization of Cauchys and the quadratic functional equations. Aequationes Math. 83, 75–86 (2012)
51.
Rassias, Th.M.: On the stability of linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)MATHCrossRef
52.
Rassias, Th.M.: The problem of S. M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352–378 (2000)MATHCrossRef
53.
Rassias, Th.M., Brzdȩk, J.: Functional Equations in Mathematical Analysis. Springer, New York (2011)MATH
54.
Schwaiger, J.: The functional equation of homogeneity and its stability properties. Österreich. Akad. Wiss. Math.-Natur, Kl, Sitzungsber. Abt. II 205, 3–12 (1996)MathSciNet
55.
Skof, F.: Sull’approssimazione delle applicazioni localmente δ-additive. Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. 117, 377–389 (1983)MathSciNetMATH
56.
Skof, F.: Approssimazione di funzioni δ-quadratic su dominio restretto. Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. 118, 58–70 (1984)MathSciNetMATH
57.
58.
59.
60.
Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1961). Problems in Modern Mathematics. Wiley, New York (1964)
61.
Yang, D.: Remarks on the stability of Drygas equation and the Pexider-quadratic equation. Aequationes Math. 68, 108–116 (2004)MathSciNetMATH
Metadaten
Titel
A Fixed Point Approach to Stability of the Quadratic Equation
verfasst von
M. Almahalebi
A. Charifi
S. Kabbaj
E. Elqorachi
Copyright-Jahr
2014
Buch
Topics in Mathematical Analysis and Applications

Print ISBN: 978-3-319-06553-3

Electronic ISBN: 978-3-319-06554-0

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-06554-0_3

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