A FIXED POINT APPROACH TO THE INTUITIONISTIC FUZZY STABILITY OF QUINTIC AND SEXTIC FUNCTIONAL EQUATIONS

Document Type: Research Paper

Authors

1 School of Mathematics, Beijing Institute of Technology, Beijing 100081, People's Republic of China

2 Department of Statistical, University College London, Science 1-19 Torrington Place, London WC1E 7HB, United Kingdom

3 Department of Electrical and Computer Engineering, College of En- gineering, University of Kentucky, Lexington 40506, Usa and School of Communica- tion and Information Engineering, University of Electronic Science and Technology of China

Abstract

The fixed point alternative methods are implemented to give
Hyers-Ulam  stability for  the quintic functional equation $ f(x+3y)
- 5f(x+2y) + 10 f(x+y)- 10f(x)+ 5f(x-y) - f(x-2y) = 120f(y)$ and the
sextic functional equation $f(x+3y) - 6f(x+2y) + 15 f(x+y)- 20f(x)+
15f(x-y) - 6f(x-2y)+f(x-3y) = 720f(y)$   in the setting of
intuitionistic fuzzy normed spaces (IFN-spaces).  This method
introduces a metrical context and shows that the stability is
related to some fixed point of a suitable operator. Furthermore, the
interdisciplinary relation among the fuzzy set theory,  the theory
of intuitionistic spaces and the theory of functional equations are
also presented in the paper.

References

bibitem{Altun} I. Altun, {it Some fixed point theorems for single and multivalued mappings on ordered non-archimedean
fuzzy metric spaces}, Iranian Journal of Fuzzy Systems, {bf 7} (2010), 91-96.

bibitem{Aoki} T. Aoki, {it On the stability of the linear transformation in
Banach spaces}, J. Math. Soc. Japan, {bf 2} (1950), 64-66.

bibitem{Atanassov1}  K. T. Atanassov, {it Intuitionistic fuzzy sets}, Fuzzy Sets and
Systems, {bf 20} (1986), 87-96.


bibitem{Atanassov2}  K. T. Atanassov, {it New operations defined over the
intuitionistic fuzzy sets}, Fuzzy Sets and Systems, {bf 61} (1994),
137-42.

 


bibitem{Baktash}  E. Baktash, Y. J. Cho, M. Jalili, R. Saadati and S. M.
Vaezpour, {it On the stability of cubic mappings and quadratic
mappings in random normed spaces}, Journal of Inequalities and
Applications,  Article ID 902187, 11 pages, {bf 2008} (2008).

bibitem{cadariu}  L. Cu{a}dariu and V. Radu, {it Fixed points and stability
for functional equations in probabilistic metric and random normed
spaces}, Fixed Point Theory and Applications, Article ID 589143, 18 pages, {bf 2009} (2009).

 

 

bibitem{coker}  D. c{C}oker, {it An introduction to intuitionistic fuzzy
topological spaces}, Fuzzy Sets and Systems, {bf 88} (1997),
81-89.


bibitem{Deschrijver}  G. Deschrijver, C. Cornelis and E. E. Kerre, {it On the
representation of intuitionistic fuzzy $t$-norms and $t$-conorms},
IEEE Transaction on Fuzzy Systems, {bf 12} (2004), 45-61.

 

bibitem{Garia} J. G. Garc'{i}a and S. E. Rodabaugh, {it Order-theoretic,
topological, categorical redundancies of interval-valued sets, grey
sets, vague sets, interval-valued ``intuitionistic'' sets,
``intuitionistic'' fuzzy sets and topologies}, Fuzzy Sets and
Systems, {bf 156} (2005),  445-484.

 

bibitem{Gavruta}  P. Gu{a}vruc{t}a, {it A  generalization of the
Hyers-Ulam-Rassias stability of approximately additive mappings},
Journal of Mathematical Analysis and Applications, {bf
184} (1994), 431-436.

 

 

bibitem{Hosseini} S. B. Hosseini, D. O'Regan and R. Saadati, {it Some
results on intuitionistic fuzzy spaces}, Iranian Journal of Fuzzy Systems,
{bf 4} (2007),  53-64.


bibitem{Hyers} D. H. Hyers, {it On the stability of the linear functional
equation}, Proc. Nat. Acad. Sci. USA, {bf 27} (1941), 222-224.

 

bibitem{Isac}  G. Isac and T. M. Rassias, {it Stability of $psi$-additive
mappings: applications to nonlinear analysis}, International Journal
of Mathematics and Mathematical Sciences, {bf 19} (1996),
219-228.

 

bibitem{Junkim1} K. W. Jun and H. M. Kim, {it The generalized
Hyers-Ulam-Rassias stability of a cubic functional equation},
Journal of Mathematical Analysis and Applications, {bf 274} (2002),
867-878.

 

bibitem{Junkimchang} K. W. Jun, H. M. Kim and I. S. Chang, {it On the Hyers-Ulam
stability of an Euler-Lagrange type cubic functional equation},
Journal of Computational Analysis and Applications, {bf 7} (2005),
21-33.

 


bibitem{Merghadi} F. Merghadi and A. Aliouche, {it A related fixed point theorem in
$n$ fuzzy metric spaces}, Iranian Journal of Fuzzy Systems, {bf 7} (2010),
73-86.


bibitem{Mihet} D. Mihec{t}, {it The fixed point method for fuzzy stability
of the Jensen functional equation}, Fuzzy Sets and Systems, {bf
160} (2009), 1663-1667.


bibitem{Mihetsaadati}  D. Mihec{t}, R. Saadati and S. M. Vaezpour, {it The
stability of the quartic functional equation in random normed
spaces}, Acta Appl.  Math., {bf 110} (2010), 797-803.


bibitem{Mirmostafaee} A. K. Mirmostafaee and  M. S. Moslehian, {it Fuzzy versions
of Hyers-Ulam-Rassias theorem}, Fuzzy Sets and Systems, {bf
159} (2008), 720-729.


bibitem{Mohiuddine}  S. A. Mohiuddine and H. c{S}evli, {it Stability of Pexiderized
quadratic functional equation in intuitionistic fuzzy normed space},
Journal of Computational and Applied Mathematics, {bf 235} (2011),
2137-2146.


bibitem{Moslehian}  M. S. Moslehian and G. Sadeghi, {it Stability of two
types of cubic functional equations in non-archimedean spaces}, Real
Anal. Exchange, {bf 33} (2008), 375-383.

 

bibitem{Moszner} Z. Moszner, {it On the stability of functional equations},
Aequationes Math., {bf 77} (2009), 33-88.

 


bibitem{Mursaleen}  M. Mursaleen and S. A. Mohiuddine,  {it On stability of a
cubic functional equation in intuitionistic fuzzy normed spaces},
Chaos, Solitons and Fractals, {bf 42} (2009), 2997-3005.

 


 bibitem{Nozari} K. Nozari and B. Fazlpour, {it Some consequences of space-time
 fuzziness}, Chaos, Solitons and Fractals, {bf 34} (2007), 224-234.

 

bibitem{Paneah} B. Paneah, {it A new approach to the stability of linear
functional operators}, Aequationes Math., {bf 78} (2009), 45-61.

 

bibitem{Park} J. H. Park, {it  Intuitionistic fuzzy metric spaces}, Chaos,
Solitons and Fractals, {bf 22} (2004), 1039-1046.

 

bibitem{Radu} V. Radu, {it The fixed point alternative and the stability
of functional equations}, Fixed Point Theory, {bf 4} (2003),
91-96.


bibitem{Rafi} M. Rafi and M. S. M. Noorani, {it Fixed point theorem on intuitionistic
fuzzy metric spaces}, Iranian Journal of Fuzzy Systems, {bf 3} (2006), 23-29.

 

 bibitem{JMRassias} J. M. Rassias, {it Solution of the Ulam stability problem for quartic mappings},
  Glasnik Matemativ{c}ki, {bf 34} (1999), 243-252.

 

 bibitem{ThRassias} T. M. Rassias, {it On the stability of the linear mapping in Banach spaces}, Proc. Amer.
Math. Soc., {bf 72} (1978), 297-300.

 


bibitem{Saadati} R.  Saadati, {it  A note on ``Some results on the IF-normed
spaces''}, Chaos, Solitons and Fractals, {bf 41} (2009), 206-213.


bibitem{Saadaticho} R. Saadati, Y. J. Cho and J. Vahidi, {it The stability of the
quartic functional equation in various spaces}, Computers and
Mathematics with Applications, {bf 60} (2010), 1994-2002.

 

 bibitem{Saadatipark} R. Saadati and C. Park, {it Non-archimedean $mathscr{L}$-fuzzy normed spaces and stability of functional
equations}, Computers and Mathematics with Applications, {bf
60} (2010), 2488-2496.

 

bibitem{Saadatirazani} R. Saadati, A. Razani and H. Adibi, {it A common fixed point
theorem in $mathscr{L}$-fuzzy metric spaces}, Chaos Solitons
Fractals, {bf 33} (2007), 358-363.

 

 

bibitem{Saadatisedghi} R. Saadati, S. Sedghi and N. Shobe, {it Modified intuitionistic
fuzzy metric spaces and some fixed point theorems}, Chaos, Solitons
and Fractals, {bf 38} (2008), 36-47.

 

bibitem{Saadatisedghi} R. Saadati, S. Sedghi and H. Zhou, {it A common fixed point theorem for
$psi$-weakly commuting maps in $mathcal {L}$-fuzzy metric spaces},
Iranian Journal of Fuzzy Systems, {bf 5} (2008), 47-53.

 


bibitem{Saadativaezpour} R. Saadati, S. M. Vaezpour and Y. J. Cho, {it Quicksort
algorithm: application of a fixed point theorem in intuitionistic
fuzzy quasi-metric spaces at a domain of words},
 Journal of Computational and Applied Mathematics, {bf 228} (2009),
219-225.

 


bibitem{Sedghi} S. Sedghi, K. P. R. Rao and N. Shobe, {it A common fixed point theorem
for six weakly compatible mappings in $M$-fuzzy metric spaces},
Iranian Journal of Fuzzy Systems, {bf 5} (2008), 49-62.

 


bibitem{Ulam} S. M. Ulam, {it A collection of the mathematical problems},
Interscience, New York, 1960.

 

bibitem{xu1}  T. Z.  Xu, J. M.  Rassias, M. J. Rassias  and  W. X.
Xu, {it A fixed point approach to the stability of  quintic and
sextic functional equations in quasi-$beta$-normed spaces}, Journal
of Inequalities and Applications, Article ID
423231, 23 pages, {bf 2010} (2010).

 

bibitem{xu2} T. Z. Xu, J. M. Rassias and W. X. Xu, {it Intuitionistic fuzzy stability of a general mixed
additive-cubic equation}, Journal of Mathematical Physics, 063519, 21 pages, {bf
51} (2010).

 

bibitem{xu3}  T. Z. Xu, J. M. Rassias and W. X. Xu, {it Stability of a
general mixed additive-cubic functional equation in non-archimedean
fuzzy normed spaces}, Journal of Mathematical Physics, 093508, 19 pages, {bf
51} (2010).


bibitem{xu4} T. Z. Xu, J. M. Rassias and W. X. Xu, {it On the stability
of a general mixed additive-cubic functional equation in random
normed spaces}, Journal of Inequalities and Applications, Article ID 328473, 16 pages, {bf
2010} (2010).

 

 

 bibitem{xu5}  T. Z. Xu, J. M. Rassias and W. X. Xu, {it A fixed point approach to the stability of a general mixed
AQCQ-functional equation in non-archimedean normed spaces}, Discrete
Dynamics in Nature and Society, Article ID
812545,  24 pages, {bf 2010} (2010).

 

 

bibitem{xu6}  T. Z.  Xu, J. M.  Rassias and W. X.
Xu, {it A generalized mixed additive-cubic functional equation},
Journal of Computational Analysis and Applications, {bf 13} (2011),
1273-1282.


bibitem{Zhang} S. S. Zhang, J. M. Rassias and  R. Saadati, {it  Stability
of a cubic functional equation  in intuitionistic random normed
spaces}, Appl. Math. Mech. -Engl. Ed., {bf 31} (2010), 21-26.

Iranian Journal of Fuzzy Systems

Volume 9, Issue 5
November and December 2012
Pages 21-40

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