## Stability problem for Jensen-type functional equations of cubic mappings.(English)Zbl 1118.39013

The authors introduce the following Jensen type functional equations of cubic mappings between real linear spaces: \begin{aligned} 4f\biggl(\frac{3x+y}{4}\biggr)&+ 4f\biggl(\frac{x+3y}{4}\biggr) =6f\biggl(\frac{x+y}{2}\biggr)+f(x)+f(y),\\ 9f\biggl(\frac{2x+y}{3}\biggr)&+ 9f\biggl(\frac{x+2y}{3}\biggr) =16f\biggl(\frac{x+y}{2}\biggr)+ f(x)+f(y). \end{aligned} The general solution of both functional equations is $$f(x) = C(x)+Q(x)+A(x)+f(0)$$, where $$C$$ is cubic, $$Q$$ is quadratic and $$A$$ is additive, see K.-W. Jun and H.-M. Kim [Math. Inequal. Appl. 6, No. 2, 289–302 (2003; Zbl 1032.39015)]. The authors also prove the generalized Hyers-Ulam-Rassias stability of both equations.

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges

Zbl 1032.39015
Full Text:

### References:

  Ulam, S. M.: A collection of Mathematical Problems, Interscience Publ., New York, 1960 · Zbl 0086.24101  Gruber, P. M.: Stability of isometries. Trans. Amer. Math. Soc., 245, 263–277 (1978) · Zbl 0393.41020  Zhou, D. X.: On a conjecture of Z. Ditzian. J. Approx. Theory, 69, 167–172 (1992) · Zbl 0755.41029  Hyers, D. H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci., 27, 222–224 (1941) · Zbl 0061.26403  Rassias, Th. M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc., 72, 297–300 (1978) · Zbl 0398.47040  Gajda, Z.: On stability of additive mappings. Internat. J. Math. and Math. Sci., 14, 431–434 (1991) · Zbl 0739.39013  Găvruta, P.: A generalization of the Hyers–Ulam–Rassias Stability of approximately additive mappings. J. Math. Anal. Appl., 184, 431–436 (1994) · Zbl 0818.46043  Hyers, D. H., Isac, G., Rassias, Th. M.: Stability of Functional Equations in Several Variables, Birkh”auser, Basel, 1998 · Zbl 0907.39025  Hyers, D. H., Rassias, Th. M.: Approximate homomorphisms. Aequationes Math., 44, 125–153 (1992) · Zbl 0806.47056  Rassias, Th. M.: On the stability of the quadratic functional equations and its applications, Studia Univ. ”Babes Bolyai”, Mathematica, XLIII, 89–124, 1998 · Zbl 1009.39025  Lee, Y. H., Jun, K. W.: A generalization of the Hyers–Ulam–Rassias stability of Jensen’s equation. J. Math. Anal. Appl., 238, 305–315 (1999) · Zbl 0933.39053  Trif, T.: Hyers–Ulam–Rassias stability of a Jensen type functional equation. J. Math. Anal. Appl., 250, 579–588 (2000) · Zbl 0964.39027  Trif, T.: On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions. J. Math. Anal. Appl., 272, 604–616 (2002) · Zbl 1036.39021  Park, C. G.: Universal Jensen’s equations in Banach modules over a C*–algebra and its unitary group. Acta Mathematica Sinica, English Series, 20(6), 1047–1056 (2004) · Zbl 1087.39514  Aczél, J., Dhombres, J.: Functional Equations in Several Variables, Cambridge Univ. Press, 1989 · Zbl 0685.39006  Czerwik, S.: The stability of the quadratic functional equation, in ’Stability of Mappings of Hyers–Ulam Type’ (edited by Th. M. Rassias and J. Tabor), Hadronic Press, Florida, 81–91, 1994 · Zbl 0844.39008  Jun, K. W., Lee, Y. H.: On the Hyers–Ulam–Rassias stability of a pexiderized quadratic inequality. Math. Ineq. Appl., 4(1), 93–118 (2001) · Zbl 0976.39031  Rassias, Th. M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl., 251, 264–284 (2000) · Zbl 0964.39026  Lee, Y. W.: On the stability of a quadratic Jensen type functional equation. J. Math. Anal. Appl., 270, 590–601 (2002) · Zbl 1007.39026  Trif, T.: Hyers–Ulam–Rassias stability of a quadratic functional equation. Bull. Korean Math. Soc., 40, 253–267 (2003) · Zbl 1049.39031  Rassias, J. M.: On the stability of the Euler–Lagrange functional equation. Chinese J. Math., 20, 185–190 (1992) · Zbl 0753.39003  Rassias, J. M.: Solution of the Ulam stability problem for Euler–Lagrange quadratic mappings. J. Math. Anal. Appl., 220, 613–639 (1998) · Zbl 0928.39014  Jun, K. W., Kim, H. M.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl., 274, 867–878 (2002) · Zbl 1021.39014  Jun, K. W., Kim, H. M.: Ulam stability problem for a mixed type of cubic and additive functional equation. Bull. Belgian Math. Soc., to appear · Zbl 1132.39022  Jun, K. W., Kim, H. M.: On the Hyers–Ulam–Rassias stability of a general cubic functional equation. Math. Ineq. Appl., 6(1), 289–302 (2003) · Zbl 1032.39015  Jun, K. W., Kim, H. M., Chang, Ick–Soon: On the Hyers–Ulam stability of an Euler–Lagrange type cubic functional equation. J. Comput. Anal. Appl., 7(1), 21–33 (2005) · Zbl 1087.39029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.