The general solution of a quadratic functional equation and Ulam stability

Volume 8, Issue 5, pp 640--649 http://dx.doi.org/10.22436/jnsa.008.05.16

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1420 Downloads
• 2132 Views

Authors

Yaoyao Lan - College of Computer Science, Chongqing University, Chongqing 401331, China. - Key Laboratory of Chongqing University of Arts and Sciences, Chongqing 402160, China. Yonghong Shen - School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China.

Abstract

In this paper, we investigate the general solution of a new quadratic functional equation. We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed point method, respectively.

Share and Cite

Keywords

• functional equation
• Ulam stability
• quadratic mapping.

MSC

•  39A30
•  97I70

References

• [1] T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.


• [2] D. Barbu, C. Buse, A. Tabassum, Hyers-Ulam stability and discrete dichotomy, J. Math. Anal. Appl., 423 (2015), 1738-1752.


• [3] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, Singapore (2000)


• [4] K. Cieplinski, On the generalized Hyers-Ulam stability of multi-quadratic mappings, Comput. Math. Appl., 62 (2011), 3418-3426.


• [5] J. B. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305-309.


• [6] D. H. Hyers , On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224.


• [7] D. H. Hyers, G. Isac, T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser , Boston (1998)


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


• [9] S. M. Jung, On the Hyers-Ulam-Rassias stability of a quadratic functional equation , J. Math. Anal. Appl., 232 (1999), 384-393.


• [10] 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.


• [11] H. A. Kenary, H. Rezaei, Y. Gheisari, C. Park, On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory Appl., 2012 (2012), 17 pages.


• [12] Y. W. Lee, On the stability of a quadratic Jensen type functional equation, J. Math. Anal. Appl., 270 (2002), 590-601.


• [13] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Set. Sys., 159 (2008), 720-729.


• [14] M. S. Moslehian, T. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math., 1 (2007), 325-334.


• [15] K. Nikodem, D. Popa, On single-valuedness of set-valued maps satisfying linear inclusions, Banach J. Math. Anal., 3 (2009), 44-51.


• [16] C. G. Park, On the Hyers-Ulam-Rassias stability of generalized quadratic mappings in Banach modules , J. Math. Anal. Appl., 291 (2004), 214-223.


• [17] C. Park, H. A. Kenary, T. M. Rassias , Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non- Archimedean Banach spaces, J. Inequal. Appl., 2012 (2012), 18 pages.


• [18] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.


• [19] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96.


• [20] H. Y. Shen, Y. Y. Lan, On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces, J. Nonlinear Sci. Appl., 7 (2014), 368-378.


• [21] S. M. Ulam, Problems in Modern Mathematics, Wiley, New York (1964)