Approximate Drygas mappings on a set of measure zero
DOI:
https://doi.org/10.4067/S0716-09172016000200007Keywords:
Drygas functional equation, stability, Baire category theorem, First category, Lebesgue measure, ecuación funcional de Drygas, estabilidad, teorema de categorías de Baire, primera categoría, medida de Lebesgue.Abstract
Let R be the set of real numbers, Y be a Banach space and f : R →Y. We prove the Hyers-Ulam stability for the Drygas functional equation
f (x + y) + f (x - y) = 2f (x) + f (y) + f (-y) for all (x, y) ∈ Ω, where Ω⊂ R2 is of Lebesgue measure 0.
References
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[13] B. R. Ebanks, PL. Kannappan and P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-innerproduct spaces, Canad. Math. Bull. 35, pp. 321-327, (1992).
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[15] G. L. Forti, J. Sikorska, Variations on the Drygas equations and its stability, Nonlinear Analysis, 74, pp. 343-350, (2011).
[16] R. Ger, J. Sikorska, On the Cauchy equation on spheres, Ann. Math. Sil., 11, pp. 89-99, (1997).
[17] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222, pp. 126-137, (1998).
[18] S.-M. Jung, P. K. Sahoo, Stability of functional equation of Drygas, Aequationes Math. 64, pp. 263-273, (2002).
[19] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, (2011).
[20] M. Kuczma, Functional equations on restricted domains, Aequationes Math. 18, pp. 1-34, (1978).
[21] Y.-H. Lee, Hyers-Ulam-Rassias stability of a quadratic-additive type functional equation on a restricted domain, Int. Journal of Math. Analysis, Vol. 7, no. 55, pp. 2745-2752, (2013).
[22] J. C. Oxtoby, Measure and Category, Springer, New York, (1980).
[23] J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 281, pp. 747-762, (2002).
[24] J. M. Rassias, M. J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281, pp. 516-524, (2003).
[25] J. Sikorska, On two conditional Pexider functional equations and their stabilities, Nonlinear Anal. 70, pp. 2673-2684, (2009).
[26] J. Sikorska, On a direct method for proving the Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 372, pp. 99-109, (2010).
[27] D. Yang, Remarks on the stability of Drygas equation and the Pexiderquadratic equation, Aequationes Math. 68, pp. 108-116, (2004).
[2] A. Bahyrycz, J. Brzdek, On solutions of the d’Alembert equation on a restricted domain, Aequationes Math. 85, pp. 169-183, (2013).
[3] B. Batko, Stability of an alternative functional equation, J. Math. Anal. Appl. 339, pp. 303-311, (2008).
[4] B.Batko, On approximation of approximate solutions of Dhombres equation, J. Math. Anal. Appl. 340, pp. 424-432, (2008).
[5] J. Brzdek, On the quotient stability of a family of functional equations, Nonlinear Anal. 71, pp. 4396-4404, (2009).
[6] J. Brzdek, On a method of proving the Hyers-Ulam stability of functional equations on restricted domains, Aust. J. Math. Anal. Appl. 6, pp. 1-10, (2009).
[7] J. Brzdek, J. Sikorska, A conditional exponential functional equation and its stability, Nonlinear Anal. 72, 2929-2934, (2010).
[8] J. Chung, Stability of functional equations on restricted domains in a group and their asymptotic behaviors, Comput. Math. Appl. 60, pp. 2653-2665, (2010).
[9] J. Chung, Stability of a conditional Cauchy equation on a set of measure zero, Aequationes Math. (2013), http://dx.doi.org/10.1007/s00010-013-0235-5.
[10] J. Chung and J. M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, J. Math. Anal. Appl. (in press).
[11] S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Inc., Palm Harbor, Florida, (2003).
[12] H. Drygas, Quasi-inner products and their applications, In: A. K. Gupta (ed.), Advances in Multivariate Statistical Analysis, 13-30, Reidel Publ. Co., (1987).
[13] B. R. Ebanks, PL. Kannappan and P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-innerproduct spaces, Canad. Math. Bull. 35, pp. 321-327, (1992).
[14] M. Fochi, An alternative functional equation on restricted domain, Aequationes Math. 70, pp. 201-212, (2005).
[15] G. L. Forti, J. Sikorska, Variations on the Drygas equations and its stability, Nonlinear Analysis, 74, pp. 343-350, (2011).
[16] R. Ger, J. Sikorska, On the Cauchy equation on spheres, Ann. Math. Sil., 11, pp. 89-99, (1997).
[17] S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222, pp. 126-137, (1998).
[18] S.-M. Jung, P. K. Sahoo, Stability of functional equation of Drygas, Aequationes Math. 64, pp. 263-273, (2002).
[19] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, (2011).
[20] M. Kuczma, Functional equations on restricted domains, Aequationes Math. 18, pp. 1-34, (1978).
[21] Y.-H. Lee, Hyers-Ulam-Rassias stability of a quadratic-additive type functional equation on a restricted domain, Int. Journal of Math. Analysis, Vol. 7, no. 55, pp. 2745-2752, (2013).
[22] J. C. Oxtoby, Measure and Category, Springer, New York, (1980).
[23] J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 281, pp. 747-762, (2002).
[24] J. M. Rassias, M. J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281, pp. 516-524, (2003).
[25] J. Sikorska, On two conditional Pexider functional equations and their stabilities, Nonlinear Anal. 70, pp. 2673-2684, (2009).
[26] J. Sikorska, On a direct method for proving the Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 372, pp. 99-109, (2010).
[27] D. Yang, Remarks on the stability of Drygas equation and the Pexiderquadratic equation, Aequationes Math. 68, pp. 108-116, (2004).
Published
2017-03-23
How to Cite
[1]
M. Almahalebi, “Approximate Drygas mappings on a set of measure zero”, Proyecciones (Antofagasta, On line), vol. 35, no. 2, pp. 225-233, Mar. 2017.
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