Stability of an ACQ-functional equation in various matrix normed spaces

Volume 8, Issue 1, pp 64--85 http://dx.doi.org/10.22436/jnsa.008.01.08

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)

• 1639 Downloads
• 2808 Views

Authors

Zhihua Wang - School of Science, Hubei University of Technology, Wuhan, Hubei 430068, P. R. China. Prasanna K. Sahoo - Department of Mathematics, University of Louisville, Louisville, KY 40292, USA.

Abstract

Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the following additive-cubic-quartic (ACQ ) functional equation \[11[f(x + 2y) + f(x - 2y)] = 44[f(x + y) + f(x - y)] + 12f(3y) - 48f(2y) + 60f(y) - 66f(x)\] in matrix Banach spaces. Furthermore, using the fixed point method, we also prove the Hyers-Ulam stability of the above functional equation in matrix fuzzy normed spaces.

Share and Cite

Keywords

• Fixed point method
• Hyers-Ulam stability
• matrix Banach space
• matrix fuzzy normed space
• additive-cubic-quartic functional equation.

MSC

•  47L25
•  39B82
•  39B72
•  46L07

References

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


• [2] T. Bag, S. K. Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11 (2003), 687-705.


• [3] L. Cădariu, V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber., 346 (2004), 43-52.


• [4] 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.


• [5] P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.


• [6] M. E. Gordji, S. K. Gharetapeh, C. Park, S. Zolfaghari, Stability of an additive-cubic-quartic functional equation, Adv. Diff. Equ., Article ID 395693, 2009 (2009), 20 pages.


• [7] O. Hadžić, E. Pap, V. Radu, Generalized contraction mapping principles in probabilistic metric spaces, Acta Math. Hungar., 101 (2003), 131-148.


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


• [9] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of functional equations in several variables, Birkhäuser, Basel (1998)


• [10] S. M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Science, New York (2011)


• [11] Pl. Kannappan, Functional equations and inequalities with applications, Springer Science, New York (2009)


• [12] H. Kenary, Nonlinear fuzzy approximation of a mixed type ACQ functional equation via fixed point alternative, Math. Sci., Article ID 54, 6 (2012), 10 pages.


• [13] J. R. Lee, C. Park, D. Shin, An AQCQ-functional equation in Matrix normed spaces, Result. Math., 64 (2013), 305-318.


• [14] J. R. Lee, D. Shin, C. Park, Hyers-Ulam stability of functional equations in matrix normed spaces, J. Ineq. Appl., Article ID 22, 2013 (2013), 11 pages.


• [15] D. Miheţ, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl., 343 (2008), 567-572.


• [16] A. K. Mirmostafaee, M. Mirzavaziri, M. S. Moslehian, Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159 (2008), 730-738.


• [17] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159 (2008), 720-729.


• [18] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy almost quadratic functions, Result. Math., 52 (2008), 161-177.


• [19] A. K. Mirmostafaee, M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci., 178 (2008), 3791-3798.


• [20] C. Park, J. R. Lee, D. Shin, An AQCQ-functional equation in matrix Banach spaces, Adv. Diff. Equ., Article ID 146, 2013 (2013), 15 pages.


• [21] C. Park, J. R. Lee, D. Shin, Functional equations and inequalities in matrix paranormed spaces, J. Ineq. Appl., Article ID 547, 2013 (2013), 13 pages.


• [22] C. Park, D. Shin, J. R. Lee, Fuzzy stability of functional inequalities in matrix fuzzy normed spaces, J. Ineq. Appl., Article ID 224, 2013 (2013), 28 pages.


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


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


• [25] Th. M. Rassias, Functional equations, inequalities and applications, Kluwer Academic, Dordrecht (2003)


• [26] P. K. Sahoo, Pl. Kannappan, Introduction to functional equations, CRC Press, Boca Raton (2011)


• [27] Z. Wang, X. Li, Th. M. Rassias, Stability of an additive-cubic-quartic functional equation in mutil-Banach spaces, Abst. Appl. Anal., Article ID 536520, 2011 (2011), 11 pages.


• [28] S. M. Ulam, Problems in modern mathematics, Chapter VI, Science Editions, Wiley, New York (1964)