Stability of generalized Jensen functional equation on a set of measure zero

 

Hajira Dimou

Youssef Aribou 

Abdellatif Chahbi 

Samir Kabbaj 

University of Ibn Tofail

Morocco 

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ABSTRACT

Let E is a complex vector space and F is real (or complex ) Banach space. In this paper, we prove the Hyers-Ulam stability for the generalized Jensen functional equation

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Subjclass [2010] : Primary 39B82; Secondary 39B52.

Keywords : K- Jensenfunctional equation, Hyers-Ulam stability.

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REFERENCES

[1]    T. Aoki, On the stability ofthe linear transformation in Banach spaces, Journal of the Mathematical Society of Japan, 2, pp. 64-66, (1950).

[2]    J. A. Baker, A generalfunctional equation and its stability, Proceeding of the American Mathematical Society, 133, pp. 1657-1664, (2005).

[3]    N. Brillouet-Belluot, J. Brzdek, K. Cieplinski, On some recent developments in Ulam’s type stability, Abstract and Applied Analysis, (2012).

[4]    J. Brzd§k, On a method ofproving the Hyers-Ulam stability offunc-tional equations on restricted domains, The Australian Journal of Mathematical Analysis and Applications, 6, pp. 1-10, (2009).

[5]    A. B. Chahbi, A. Charifi, B. Bouikhalene and S. Kabbaj, Nonarchimedean stability of a Pexider K-quadratic functional equation, Arab Journal of Mathematical Sciences, 21, pp. 67-83, (2015).

[6]    A. Chahbi, M. Almahalebi, A. Charifi and S.Kabbaj Generalized Jensen functional equation on restricted domain, Annals of West University of Timisoara-Mathematics, 52, pp. 29-39, (2014).

[7]    P. W. Cholewa, Remarks on the stability offunctional equations, Ae-quationes mathematicae, 27, pp. 76-86, (1984).

[8]    J. Chung , Stability of a conditional Cauchy equation on a set of measure zero, Aequationes mathematicae, 87, pp. 391-400, (2014).

[9]    J. Chung and J. M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, Journal of Mathematical Analysis and Applications, 419, pp. 1065-1075, (2014).

[10]    J. Chung and J. M. Rassias, On a measure zero Stability problem of a cyclic equation, Bulletin of the Australian Mathematical Society, 93,

pp. 1-11, (2016).

[11]    S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Univer-sitat Hamburg, 62, pp. 59-64, (1992).

[12]    P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184, pp. 431-436, (1994).

[13]    D. H. Hyers, On the stability ofthe linearfunctional equation, Proceedings of the National Academy of Sciences, 27, pp. 222-224, (1941).

[14]    D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Ae-quationes Mathematicae, 44, pp. 125-153, (1992).

[15]    D. H. Hyers, Transformations with bounded n-th differences, Pacific JournalofMathematics, 11, pp. 591-602, (1961).

[16]    K. W. Jun and Y. H. Lee, A generalization ofthe Hyers-Ulam-Rassias stability of Jensen's equation, Journal of Mathematical Analysis and Applications, 238, pp. 305-315, (1999).

[17]    S. M. Jung, On the HyersUlam stability of the functional equations that have the quadratic property, Journal of Mathematical Analysis and Applications, 222, pp. 126-137, (1998).

[18]    C. F. K. Jung, On generalized complete metric spaces, Bulletin of the American Mathematical Society, 75, pp. 113-116, (1969).

[19]    R. Lukasik, Some generalization of Cauch^s and the quadratic functional equations, Aequationes Mathematicae, 83, pp. 75-86, (2012).

[20]    A. Najati, S. M. Jung, Approximately quadratic mappings on restricted domains, Journal ofInequalities and Applications, (2010).

[21]    J. C. Oxtoby, Measure and Category, Springer, New-York (1980).

[22]    Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, pp. 297-300, (1978).

[23]    Th. M. Rassias, On the stability of the functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62, pp. 23-130, (2000).

[24]    Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proceedings of the American Mathematical Society, 114, pp. 989-993, (1992).

[25]    Th. M. Rassias, and J. Tabor, Stability ofMappings ofHyers-Ulam Type, Hardronic Press, (1994).

[26]    J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, Journal of Mathematical Analysis and Applications, 276, pp. 747-762, (2002).

[27]    F. Skof, Local properties and approximations of operators, Rendiconti del Seminario Matematico e Fisico di Milano, 53, pp. 113-129, (1983).

[28]    H. Stetkmr, Functional equations involving means offunctions on the complex plane, Aequationes Mathematicae, 56, pp. 47-62, (1998).

[29]    S. M. Ulam, A Collection ofMathematical Problems, Interscience Pub-lishers, 8(1960).

Hajira Dimou

Department of Mathematics, Faculty of Sciences,

IBN Tofail University,

BP: 14000, Kenitra,

Morocco

e-mail : dimouhajira@gmail.com

Youssef Aribou

Department of Mathematics, Faculty of Sciences,

IBN Tofail University,

BP: 14000, Kenitra,

Morocco

e-mail : aribouyoussef3@gmail.com

Abdellatif Chahbi

Department of Mathematics, Faculty of Sciences,

IBN Tofail University,

BP: 14000, Kenitra,

Morocco

e-mail : ab-1980@live.fr

Samir Kabbaj

Department of Mathematics, Faculty of Sciences,

IBN Tofail University,

BP: 14000, Kenitra,

Morocco

e-mail : samkabbaj@yahoo.fr

Received : June 2016. Accepted : August 2016