Xu, T., Rassias, M., Xin Xu, W. (2012). A FIXED POINT APPROACH TO THE INTUITIONISTIC
FUZZY STABILITY OF QUINTIC AND SEXTIC
FUNCTIONAL EQUATIONS. Iranian Journal of Fuzzy Systems, 9(5), 21-40. doi: 10.22111/ijfs.2012.102
Tian Zhou Xu; Matina John Rassias; Wan Xin Xu. "A FIXED POINT APPROACH TO THE INTUITIONISTIC
FUZZY STABILITY OF QUINTIC AND SEXTIC
FUNCTIONAL EQUATIONS". Iranian Journal of Fuzzy Systems, 9, 5, 2012, 21-40. doi: 10.22111/ijfs.2012.102
Xu, T., Rassias, M., Xin Xu, W. (2012). 'A FIXED POINT APPROACH TO THE INTUITIONISTIC
FUZZY STABILITY OF QUINTIC AND SEXTIC
FUNCTIONAL EQUATIONS', Iranian Journal of Fuzzy Systems, 9(5), pp. 21-40. doi: 10.22111/ijfs.2012.102
Xu, T., Rassias, M., Xin Xu, W. A FIXED POINT APPROACH TO THE INTUITIONISTIC
FUZZY STABILITY OF QUINTIC AND SEXTIC
FUNCTIONAL EQUATIONS. Iranian Journal of Fuzzy Systems, 2012; 9(5): 21-40. doi: 10.22111/ijfs.2012.102
A FIXED POINT APPROACH TO THE INTUITIONISTIC
FUZZY STABILITY OF QUINTIC AND SEXTIC
FUNCTIONAL EQUATIONS
1School of Mathematics, Beijing Institute of Technology, Beijing
100081, People's Republic of China
2Department of Statistical, University College London, Science
1-19 Torrington Place, London WC1E 7HB, United Kingdom
3Department 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.