Approximate Drygas mappings on a set of measure zero

Muaadh Almahalebi

Resumen


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.


Palabras clave


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.

Texto completo:

PDF

Referencias


C. Alsina, J. L. Garcia-Roig, On a conditional Cauchy equation on‏ rhombuses, in: J.M. Rassias (Ed.), Functional Analysis, Approximation Theory and Numerical Analysis, World Scientific, (1994).‏

A. Bahyrycz, J. Brzd¸ ek,On solutions of the d’Alembert equation on a‏ restricted domain, Aequationes Math. 85, pp. 169-183, (2013).‏

B. Batko, Stability of an alternative functional equation, J. Math.‏ Anal. Appl. 339, pp. 303-311, (2008).‏

B.Batko, On approximation of approximate solutions of Dhombres‏ equation, J. Math. Anal. Appl. 340, pp. 424-432, (2008).‏

J. Brzd¸ ek, On the quotient stability of a family of functional equations,‏ Nonlinear Anal. 71, pp. 4396-4404, (2009).‏

J. Brzd¸ ek, 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).‏

J. Brzd¸ ek, J. Sikorska, A conditional exponential functional equation‏ and its stability, Nonlinear Anal. 72, 2929-2934, (2010).‏

J. Chung, Stability of functional equations on restricted domains in a‏ group and their asymptotic behaviors, Comput. Math. Appl. 60, pp. 2653-2665, (2010).

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.‏

J. Chung and J. M. Rassias, Quadratic functional equations in a set‏ of Lebesgue measure zero, J. Math. Anal. Appl. (in press).‏

S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias‏ Type, Hadronic Press, Inc., Palm Harbor, Florida, (2003).‏

H. Drygas, Quasi-inner products and their applications, In: A. K.‏ Gupta (ed.), Advances in Multivariate Statistical Analysis, 13-30, Reidel Publ. Co., (1987).

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).‏

M. Fochi, An alternative functional equation on restricted domain,‏ Aequationes Math. 70, pp. 201-212, (2005).‏

G. L. Forti, J. Sikorska, Variations on the Drygas equations and its‏ stability, Nonlinear Analysis, 74, pp. 343-350, (2011).‏

R. Ger, J. Sikorska, On the Cauchy equation on spheres, Ann. Math.‏ Sil., 11, pp. 89-99, (1997).‏

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).

S.-M. Jung, P. K. Sahoo, Stability of functional equation of Drygas,‏ Aequationes Math. 64, pp. 263-273, (2002).‏

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in‏ Nonlinear Analysis, Springer, New York, (2011).‏

M. Kuczma, Functional equations on restricted domains, Aequationes‏ Math. 18, pp. 1-34, (1978).‏

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).

J. C. Oxtoby, Measure and Category, Springer, New York, (1980).‏

J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl. 281, pp. 747-762, (2002).‏

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).‏

J. Sikorska, On two conditional Pexider functional equations and their‏ stabilities, Nonlinear Anal. 70, pp. 2673-2684, (2009).‏

J. Sikorska, On a direct method for proving the Hyers-Ulam stability‏ of functional equations, J. Math. Anal. Appl. 372, pp. 99-109, (2010).‏

D. Yang, Remarks on the stability of Drygas equation and the Pexiderquadratic equation, Aequationes Math. 68, pp. 108-116, (2004).‏




DOI: http://dx.doi.org/10.4067/S0716-09172016000200007

Enlaces refback

  • No hay ningún enlace refback.