## Annals of Functional Analysis

### Stability of a functional equation of Whitehead on semigroups

#### Abstract

‎Let $S$ be a semigroup and $X$ a Banach space‎. ‎The functional‎ ‎equation $\varphi (xyz)‎+ ‎\varphi (x)‎ + ‎\varphi (y)‎ + ‎\varphi (z) =‎ ‎\varphi (xy)‎ + ‎\varphi (yz)‎ + ‎\varphi (xz)$ is said to be stable for‎ ‎the pair $(X‎, ‎S)$ if and only if $f‎: ‎S\to X$ satisfying $\|‎ ‎f(xyz)+f(x)‎ + ‎f(y)‎ + ‎f(z)‎ - ‎f(xy)‎- ‎f(yz)-f(xz)\| \leq \delta$ for‎ ‎some positive real number $\delta$ and all $x‎, ‎y‎, ‎z \in S$‎, ‎there is‎ ‎a solution $\varphi‎ : ‎S \to X$ such that $f-\varphi$ is bounded‎. ‎In‎ ‎this paper‎, ‎among others‎, ‎we prove the following results‎: ‎1) this‎ ‎functional equation‎, ‎in general‎, ‎is not stable on an arbitrary‎ ‎semigroup; 2) this equation is stable on periodic semigroups; 3)‎ ‎this equation is stable on abelian semigroups; 4) any semigroup with‎ ‎left (or right) law of reduction can be embedded into a semigroup‎ ‎with left (or right) law of reduction where this equation is stable‎. ‎The main results of this paper generalize the works of Jung [J‎. ‎Math‎. ‎Anal‎. ‎Appl‎. ‎222 (1998)‎, ‎126--137]‎, ‎Kannappan [Results Math‎. ‎27‎ ‎(1995)‎, ‎368--372] and Fechner [J‎. ‎Math‎. ‎Anal‎. ‎Appl‎. ‎322 (2006)‎, ‎774--786]‎.

#### Article information

Source
Ann. Funct. Anal., Volume 3, Number 2 (2012), 32-57.

Dates
First available in Project Euclid: 12 May 2014

https://projecteuclid.org/euclid.afa/1399899931

Digital Object Identifier
doi:10.15352/afa/1399899931

Mathematical Reviews number (MathSciNet)
MR2948387

#### Citation

A‎. ‎Fa\u iziev, Valeriy; K‎. ‎Sahoo, Prasanna. Stability of a functional equation of Whitehead on semigroups. Ann. Funct. Anal. 3 (2012), no. 2, 32--57. doi:10.15352/afa/1399899931. https://projecteuclid.org/euclid.afa/1399899931

#### References

• I.-S. Chang and H.-M. Kim, Hyers-Ulam-Rassias stability of a quadratic functional equation, Kyungpook Math. J. 42 (2002), 71–86.
• P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86.
• S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.
• S. Czerwik, The stability of the quadratic functional equation, In: Stability of Mappings of Hyers-Ulam Type (ed. Th. M. Rassias & J. Tabor), Hadronic Press, Florida, 1994, 81–91.
• S. Czerwik and K. Dlutek, Quadratic difference operators in $L_p$ spaces, Aequationes Math. 67 (2004), 1–11.
• S. Czerwik and K. Dlutek, Stability of the quadratic functional equation in Lipschitz spaces, J. Math. Anal. Appli. 293 (2004), 79–88.
• B.R. Ebanks, PL. Kannappan and P.K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner-product spaces, Canad. Math. Bull. 35 (1992), 321–327.
• V. A. Faiziev and P.K. Sahoo, On the stability of Drygas functional equation on groups, Banach J. Math. Anal., 1 (2007), 43–55.
• V. A. Faiziev and P.K. Sahoo, On Drygas functional equation on groups, Internat. J. Appl. Math. Stat. 7 (2007), 59–69.
• V.A. Faiziev and P.K. Sahoo, On the stability of the quadratic functional equation on groups, Bull. Belgian Math. Soc. Simon Stevin 15 (2008), 135–151.
• V.A. Faiziev and P.K. Sahoo, Stability of Drygas functional equation on $T( 3, \mathbb{R} )$, Internat. J. Appl. Math. Stat. 7 (2007), 70–81.
• W. Fechner, On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings, J. Math. Anal. Appl. 322 (2006), 774–786.
• I. Fenyő, On an inequality of P. W. Cholewa, In: General Inequalities 5 (ed. W. Walter), Birkhauser, Basel, 1987, 277–280.
• P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 169–176.
• R. Ger, Functional inequalities stemming from stability questions, In: General Inequalities 6 (ed. W. Walter), Birkhauser, Basel, 1992, 227–240.
• D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
• S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126–137.
• S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg 70 (2000), 175–190.
• S.-M. Jung and P.K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math. 64 (2002), 263–273.
• Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), 368-372.
• G.-H. Kim, On the stability of the quadratic mapping in normed spaces, Internat. J. Math. & Math. Sci. 25 (2001), 217-229.
• Th.M. Rassias, On the stability of the functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 284–124.
• F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129.
• S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964.
• J.H.C. Whitehead, A certain exact sequence, Ann. of Math. 52 (1950), 51–110.
• D. Yang, Remarks on the stability of Drygas' equation and the Pexider-quadratic equation, Aequationes Math. 68 (2004), 108–116.
• D. Yang, The stability of the quadratic functional equation on amenable groups, J. Math. Anal. Appl. 291 (2004), 666–672.