A generalization of Drygas functional equation

A. Charifi, Muaadh Almahalebi, S. Kabbaj

Resumen


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.


Palabras clave


Automorphism group; difference operator; Drygas functional equation; automorfismo de grupos; operador diferencial; ecuación funcional de Drygas.

Texto completo:

PDF

Referencias


M. Ait Sibaha, B. Bouikhalene, E. Elqorachi, Hyers-Ulam-Rassias stability of the K-quadratic functional equation, J. Ineq. Pure and appl.‏ Math 8, (2007).‏

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).‏

J. Baker, A general functional equation and its stability, Proceedings‏ of the American Mathematical Society, 133(6), pp. 1657-1664, (2005).‏

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).‏

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).‏

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).‏

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).‏

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).‏

H. Drygas, Quasi-inner products and their applications, Springer‏ Netherlands., pp. 13-30, (1987).‏

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).‏

V. A. Faiziev, P. K. Sahoo, On Drygas functional equation on groups,‏ Int. J. Appl. Math. Stat. 7, pp. 59-69, (2007).‏

M. Frechet, Une d´ efinition fonctionnelles des polynˆ omes, Nouv. Ann.‏ 9, pp. 145-162, (1909).‏

A. Gianyi, A characterization of monomial functions, Aequationes‏ Math. 54, pp. 343-361, (1997).‏

D. H. Hyers, Transformations with bounded n-th differences, Pacific J.‏ Math., 11, pp. 591-602, (1961).‏

S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh.‏ Math. Sem. Univ. Hamburg, 70, pp. 175-190, (2000).‏

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).‏

S.-M. Jung, P. K. Sahoo, Stability of a functional equation of Drygas,‏ Aequationes Math., 64 (3), pp. 263-273, (2002).‏

R. Ã Lukasik, Some generalization of Cauchy’s and the quadratic functional equations, Aequat. Math., 83, pp. 75-86, (2012).‏

S. Mazur, W. Orlicz, Grundlegende Eigenschaften der Polynomischen‏ Operationen, Erst Mitteilung, Studia Math., 5, pp. 50-68, (1934).‏

A. K. Mirmostafaee, Non-Archimedean stability of quadratic equations,‏ Fixed Point Theory, 11 (1), pp. 67-75, (2010).‏

P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations,‏ CRC Press, Boca Raton, Florida, (2011).‏

P. Sinopoulos, Functional equations on semigroups, Aequationes Math.‏ 59, pp.255-261, (2000).‏

W. Smajdor, On set-valued solutions of a functional equation of Drygas, Aequ. Math. 77, pp. 89-97, (2009).‏

H. Stetkær, Functional equations on abelian groups with involution. II,‏ Aequationes Math. 55, pp. 227-240, (1998).‏

H. Stetkær, Functional equations involving means of functions on the‏ complex plane, Aequationes Math. 55, pp. 47-62, (1998).‏

Gy. Szabo, Some functional equations related to quadratic functions,‏ Glasnik Math. 38, pp. 107-118, (1983).‏

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/S0716-09172016000200002

Enlaces refback

  • No hay ningún enlace refback.