A generalization of Drygas functional equation
DOI:
https://doi.org/10.4067/S0716-09172016000200002Keywords:
Automorphism group, difference operator, Drygas functional equation, automorfismo de grupos, operador diferencial, ecuación funcional de Drygas.Abstract
We obtain the Solutions of the following Drygas functional equation
∑ λ ∈Φ f (x + λy + aλ ) = κf(x)+ ∑ λ ∈Φ f(λy), x, y ∈ S
where S is an abelian semigroup, G is an abelian group, f ∈ GS, Φ is a finite automorphism group of S with order k, and aλ ∈ S, λ∈Φ.
References
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[5] A. Charifi, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias stability of a generalized Pexider functional equation, Banach J. Math. Anal, 1 (2), pp. 176-185, (2007).
[6] A. Charifi,B. Bouikhalene, E. Elqorachi, A. Redouani, Hyers-UlamRassias stability of a generalized Jensen functional equation, Aust. J. Math. Anal. Appl, 6 (1), pp. 1-16, (2009).
[7] AB. Chahbi, A. Charifi, B. Bouikhalene, S. Kabbaj, Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation, Arab Journal of Mathematical Sciences, 21 (1), pp. 67-83, (2015).
[8] D. Z. Djokovic, A representation theorem for (X1—1)(X2—1)...(Xn—1) and its applications, In Annales Polonici Mathematici 22 (2), pp. 189-198, (1969).
[9] H. Drygas, Quasi-inner products and their applications, Springer Netherlands., pp. 13-30, (1987).
[10] B. R. Ebanks, P. L. Kannappan, P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull, 35 (3), pp. 321-327, (1992).
[11] V. A. Faiziev, P. K. Sahoo, On Drygas functional equation on groups, Int. J. Appl. Math. Stat. 7, pp. 59-69, (2007).
[12] M. Frechet, Une d´ efinition fonctionnelles des polynˆ omes, Nouv. Ann. 9, pp. 145-162, (1909).
[13] A. Gianyi, A characterization of monomial functions, Aequationes Math. 54, pp. 343-361, (1997).
[14] D. H. Hyers, Transformations with bounded n-th differences, Pacific J. Math., 11, pp. 591-602, (1961).
[15] S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg, 70, pp. 175-190, (2000).
[16] S.-M. Jung, P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, J. Korean Math. Soc., 38 (3), pp. 645-656, (2001).
[17] S.-M. Jung, P. K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math., 64 (3), pp. 263-273, (2002).
[18] R. Ã Lukasik, Some generalization of Cauchy’s and the quadratic functional equations, Aequat. Math., 83, pp. 75-86, (2012).
[19] S. Mazur, W. Orlicz, Grundlegende Eigenschaften der Polynomischen Operationen, Erst Mitteilung, Studia Math., 5, pp. 50-68, (1934).
[20] A. K. Mirmostafaee, Non-Archimedean stability of quadratic equations, Fixed Point Theory, 11 (1), pp. 67-75, (2010).
[21] P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, Florida, (2011).
[22] P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59, pp.255-261, (2000).
[23] W. Smajdor, On set-valued solutions of a functional equation of Drygas, Aequ. Math. 77, pp. 89-97, (2009).
[24] H. Stetkær, Functional equations on abelian groups with involution. II, Aequationes Math. 55, pp. 227-240, (1998).
[25] H. Stetkær, Functional equations involving means of functions on the complex plane, Aequationes Math. 55, pp. 47-62, (1998).
[26] Gy. Szabo, Some functional equations related to quadratic functions, Glasnik Math. 38, pp. 107-118, (1983).
[27] D. Yang, Remarks on the stability of Drygas equation and the Pexiderquadratic equation, Aequationes Math. 68, pp. 108-116, (2004).
[2] L. M. Arriola, W. A. Beyer, Stability of the Cauchy functional equation over p-adic fields, Real Analysis Exchange, 31 (1), pp. 125-132, (2005).
[3] J. Baker, A general functional equation and its stability, Proceedings of the American Mathematical Society, 133(6), pp. 1657-1664, (2005).
[4] B. Bouikhalene and E. Elqorachi, Hyers-Ulam-Rassias stability of the Cauchy linear functional equation, Tamsui Oxford Journal of Mathematical Sciences 23 (4), pp. 449-459, (2007).
[5] A. Charifi, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias stability of a generalized Pexider functional equation, Banach J. Math. Anal, 1 (2), pp. 176-185, (2007).
[6] A. Charifi,B. Bouikhalene, E. Elqorachi, A. Redouani, Hyers-UlamRassias stability of a generalized Jensen functional equation, Aust. J. Math. Anal. Appl, 6 (1), pp. 1-16, (2009).
[7] AB. Chahbi, A. Charifi, B. Bouikhalene, S. Kabbaj, Operatorial approach to the non-Archimedean stability of a Pexider K-quadratic functional equation, Arab Journal of Mathematical Sciences, 21 (1), pp. 67-83, (2015).
[8] D. Z. Djokovic, A representation theorem for (X1—1)(X2—1)...(Xn—1) and its applications, In Annales Polonici Mathematici 22 (2), pp. 189-198, (1969).
[9] H. Drygas, Quasi-inner products and their applications, Springer Netherlands., pp. 13-30, (1987).
[10] B. R. Ebanks, P. L. Kannappan, P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull, 35 (3), pp. 321-327, (1992).
[11] V. A. Faiziev, P. K. Sahoo, On Drygas functional equation on groups, Int. J. Appl. Math. Stat. 7, pp. 59-69, (2007).
[12] M. Frechet, Une d´ efinition fonctionnelles des polynˆ omes, Nouv. Ann. 9, pp. 145-162, (1909).
[13] A. Gianyi, A characterization of monomial functions, Aequationes Math. 54, pp. 343-361, (1997).
[14] D. H. Hyers, Transformations with bounded n-th differences, Pacific J. Math., 11, pp. 591-602, (1961).
[15] S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg, 70, pp. 175-190, (2000).
[16] S.-M. Jung, P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type, J. Korean Math. Soc., 38 (3), pp. 645-656, (2001).
[17] S.-M. Jung, P. K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math., 64 (3), pp. 263-273, (2002).
[18] R. Ã Lukasik, Some generalization of Cauchy’s and the quadratic functional equations, Aequat. Math., 83, pp. 75-86, (2012).
[19] S. Mazur, W. Orlicz, Grundlegende Eigenschaften der Polynomischen Operationen, Erst Mitteilung, Studia Math., 5, pp. 50-68, (1934).
[20] A. K. Mirmostafaee, Non-Archimedean stability of quadratic equations, Fixed Point Theory, 11 (1), pp. 67-75, (2010).
[21] P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, Florida, (2011).
[22] P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59, pp.255-261, (2000).
[23] W. Smajdor, On set-valued solutions of a functional equation of Drygas, Aequ. Math. 77, pp. 89-97, (2009).
[24] H. Stetkær, Functional equations on abelian groups with involution. II, Aequationes Math. 55, pp. 227-240, (1998).
[25] H. Stetkær, Functional equations involving means of functions on the complex plane, Aequationes Math. 55, pp. 47-62, (1998).
[26] Gy. Szabo, Some functional equations related to quadratic functions, Glasnik Math. 38, pp. 107-118, (1983).
[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]
A. Charifi, M. Almahalebi, and S. Kabbaj, “A generalization of Drygas functional equation”, Proyecciones (Antofagasta, On line), vol. 35, no. 2, pp. 159-176, Mar. 2017.
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