×

Fixed points and the stability of an AQCQ-functional equation in non-Archimedean normed spaces. (English) Zbl 1192.39024

Summary: Using a fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation \(f(x+2y)+f(x - 2y)=4f(x+y)+4f(x - y) - 6f(x)+f(2y)+f( - 2y) - 4f(y) - 4f( - y)\) in non-Archimedean Banach spaces.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] M. S. Moslehian and G. Sadeghi, “A Mazur-Ulam theorem in non-Archimedean normed spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3405-3408, 2008. · Zbl 1160.46049 · doi:10.1016/j.na.2007.09.023
[2] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. · Zbl 0086.24101
[3] D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[4] T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. · Zbl 0040.35501 · doi:10.2969/jmsj/00210064
[5] Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 · doi:10.2307/2042795
[6] P. G\uavru\cta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431-436, 1994. · Zbl 0818.46043 · doi:10.1006/jmaa.1994.1211
[7] F. Skof, “Proprietà locali e approssimazione di operatori,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113-129, 1983. · Zbl 0599.39007 · doi:10.1007/BF02924890
[8] P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76-86, 1984. · Zbl 0549.39006 · doi:10.1007/BF02192660
[9] S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59-64, 1992. · Zbl 0779.39003 · doi:10.1007/BF02941618
[10] K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 267-278, 2002. · Zbl 1021.39014 · doi:10.1016/S0022-247X(02)00415-8
[11] S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 387-394, 2005. · Zbl 1072.39024 · doi:10.1016/j.jmaa.2004.12.062
[12] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. · Zbl 1011.39019
[13] Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431-434, 1991. · Zbl 0739.39013 · doi:10.1155/S016117129100056X
[14] D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1998. · Zbl 0907.39025
[15] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. · Zbl 0980.39024
[16] C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,” Bulletin des Sciences Mathématiques, vol. 132, no. 2, pp. 87-96, 2008. · Zbl 1144.39025 · doi:10.1016/j.bulsci.2006.07.004
[17] C. Park, “Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between C\ast -algebras,” Mathematische Nachrichten, vol. 281, no. 3, pp. 402-411, 2008. · Zbl 1142.39023 · doi:10.1002/mana.200510611
[18] C. Park and J. Cui, “Generalized stability of C\ast -ternary quadratic mappings,” Abstract and Applied Analysis, vol. 2007, Article ID 23282, 6 pages, 2007. · Zbl 1158.39020 · doi:10.1155/2007/23282
[19] C. Park and A. Najati, “Homomorphisms and derivations in C\ast -algebras,” Abstract and Applied Analysis, vol. 2007, Article ID 80630, 12 pages, 2007. · Zbl 1157.39017 · doi:10.1155/2007/80630
[20] Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae, vol. 39, pp. 292-293; 309, 1990.
[21] Th. M. Rassias, “On the stability of the quadratic functional equation and its applications,” Studia Universitatis Babe\cs-Bolyai. Mathematica, vol. 43, no. 3, pp. 89-124, 1998. · Zbl 1009.39025
[22] Th. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 352-378, 2000. · Zbl 0958.46022 · doi:10.1006/jmaa.2000.6788
[23] Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264-284, 2000. · Zbl 0964.39026 · doi:10.1006/jmaa.2000.7046
[24] Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23-130, 2000. · Zbl 0981.39014 · doi:10.1023/A:1006499223572
[25] Th. M. Rassias and P. \vSemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989-993, 1992. · Zbl 0761.47004 · doi:10.2307/2159617
[26] Th. M. Rassias and P. \vSemrl, “On the Hyers-Ulam stability of linear mappings,” Journal of Mathematical Analysis and Applications, vol. 173, no. 2, pp. 325-338, 1993. · Zbl 0789.46037 · doi:10.1006/jmaa.1993.1070
[27] Th. M. Rassias and K. Shibata, “Variational problem of some quadratic functionals in complex analysis,” Journal of Mathematical Analysis and Applications, vol. 228, no. 1, pp. 234-253, 1998. · Zbl 0945.30023 · doi:10.1006/jmaa.1998.6129
[28] L. C\uadariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 2003. · Zbl 1043.39010
[29] J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305-309, 1968. · Zbl 0157.29904 · doi:10.1090/S0002-9904-1968-11933-0
[30] G. Isac and Th. M. Rassias, “Stability of \psi -additive mappings: applications to nonlinear analysis,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, pp. 219-228, 1996. · Zbl 0843.47036 · doi:10.1155/S0161171296000324
[31] L. C\uadariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in Iteration Theory (ECIT ’02), vol. 346 of Grazer Mathematische Berichte, pp. 43-52, Karl-Franzens-Universitaet Graz, Graz, Austria, 2004. · Zbl 1060.39028
[32] L. C\uadariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,” Fixed Point Theory and Applications, vol. 2008, Article ID 749392, 15 pages, 2008. · Zbl 1146.39040 · doi:10.1155/2008/749392
[33] M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361-376, 2006. · Zbl 1118.39015 · doi:10.1007/s00574-006-0016-z
[34] C. Park, “Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,” Fixed Point Theory and Applications, vol. 2007, Article ID 50175, 15 pages, 2007. · Zbl 1167.39018 · doi:10.1155/2007/50175
[35] C. Park, “Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach,” Fixed Point Theory and Applications, vol. 2008, Article ID 493751, 9 pages, 2008. · Zbl 1146.39048 · doi:10.1155/2008/493751
[36] V. Radu, “The fixed point alternative and the stability of functional equations,” Fixed Point Theory, vol. 4, no. 1, pp. 91-96, 2003. · Zbl 1051.39031
[37] M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, C. Park, and S. Zolfaghari, “Stability of an additive-cubic-quartic functional equation,” Advances in Difference Equations, vol. 2009, Article ID 395693, 20 pages, 2009. · Zbl 1196.39015 · doi:10.1155/2009/395693
[38] M. Eshaghi Gordji, S. Abbaszadeh, and C. Park, “On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces,” Journal of Inequalities and Applications, vol. 2009, Article ID 153084, 26 pages, 2009. · Zbl 1187.39035 · doi:10.1155/2009/153084
[39] D. Mihe\ct and V. Radu, “On the stability of the additive Cauchy functional equation in random normed spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 567-572, 2008. · Zbl 1139.39040 · doi:10.1016/j.jmaa.2008.01.100
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.