Stability of Pexiderized quadratic functional equation on a set of measure zero

Volume 9, Issue 6, pp 4554--4562 http://dx.doi.org/10.22436/jnsa.009.06.93

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Authors

Iz-iddine EL-Fassi - Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco. Abdellatif Chahbi - Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco. Samir Kabbaj - Department of Mathematics, Faculty of Sciences, University of Ibn Tofail, Kenitra, Morocco. Choonkil Park - Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea.

Abstract

Let \(\mathbb{R}\) be the set of real numbers and \(Y\) a Banach space. We prove the Hyers-Ulam stability theorem when \(f; h : \mathbb{R}\rightarrow Y\) satisfy the following Pexider quadratic inequality \[\|f(x + y) + f(x - y) - 2f(x) - 2h(y)\| \leq\epsilon ;\] in a set \(\Omega\subset \mathbb{R}^2\) of Lebesgue measure \(m(\Omega) = 0\).

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Keywords

• Pexider quadratic functional equation
• Hyers-Ulam stability
• first category Lebesgue measure
• Baire category theorem.

MSC

•  39B52
•  39B82

References

• [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.


• [2] L. R. Berrone, P-means and the solution of a functional equation involving Cauchy differences, Results Math., 68 (2015), 375-393.


• [3] D. G. Bourgin, Classes of transformations and bordering transformations , Bull. Amer. Math. Soc., 57 (1951), 223-237.


• [4] P. W. Cholewa, Remarks on the stability of functional equations , Aequationes Math., 27 (1984), 76-86.


• [5] J.-Y. Chung , Stability of a conditional Cauchy equation on a set of measure zero, Aequationes Math., 87 (2014), 391-400.


• [6] J.-Y. Chung, J. M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, J. Math. Anal. Appl., 419 (2014), 1065-1075.


• [7] S. Czerwik , On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59-64.


• [8] H. Drygas, Quasi-inner products and their applications, in: Advances in multivariate statistical analysis: Pillai Memorial Volume, Springer Netherlands, (1987), 13-30.


• [9] B. R. Ebanks, Pl. Kannappan, P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull., 35 (1992), 321-327.


• [10] Iz. El-Fassi, S. Kabbaj , Hyers-Ulam-Rassias stability of orthogonal quadratic functional equation in modular space, General Math. Notes, 26 (2015), 61-73.


• [11] Iz. El-Fassi, S. Kabbaj, On the generalized orthogonal stability of the Pexiderized quadratic functional equation in modular space, Math. Slovaca, (to appear),


• [12] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431-434.


• [13] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.


• [14] K.-W. Jun, H.-M. Kim , On the stability of an n-dimensional quadratic and additive functional equation, Math. Inequal. Appl., 9 (2006), 153-165.


• [15] S.-M. Jung, Z.-H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl., 2008 (2008), 11 pages.


• [16] S.-M. Jung, P. K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math., 64 (2002), 263-273.


• [17] H. Khadaei, On the stability of additive, quadratic, cubic and quartic set-valued functional equations, Results Math., 68 (2015), 1-10.


• [18] M. Mirzavaziri, M. S. Moslehian, A fixed point approach to stability of quadratic equation, Bull. Brazil. Math. Soc., 37 (2006), 361-376.


• [19] S. A. Mohiuddine, H. Sevli , Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comput. Appl. Math., 235 (2011), 2137-2146.


• [20] J. C. Oxtoby , Measure and Category, A survey of the analogies between topological and measure spaces: Second edition, Springer-Verlag, New York-Berlin (1980)


• [21] C. Park, T. M. Rassias , Fixed points and generalized Hyers-Ulam stability of quadratic functional equations, J. Math. Inequal., 1 (2007), 515-528.


• [22] M. Piszczek, J. Szczawińska, Stability of the Drygas functional equation on restricted domain, Results Math., 68 (2015), 11-24.


• [23] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.


• [24] T. M. Rassias , On a modified HyersUlam sequence, J. Math. Anal. Appl., 158 (1991), 106-113.


• [25] T. M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babe-Bolyai Math., 43 (1998), 89-124.


• [26] T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130.


• [27] T. M. Rassias, P. Semrl , On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114 (1992), 989-993.


• [28] F. Skof , Propriet locali e approssimazione di operatori , Rend. Semin. Mat. Fis. Milano, 53 (1983), 113-129.


• [29] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York (1961)