## Abstract and Applied Analysis

### Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

#### Abstract

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equation $f(x+ky)+f(x-ky)=f(x+y)+f(x-y)+$ $(2(k+1)/k)f(ky)-2(k+1)f(y)$ for fixed integers $k$ with $k\ne 0,{\pm}1$ in fuzzy Banach spaces.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 953938, 22 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495689

Digital Object Identifier
doi:10.1155/2012/953938

Mathematical Reviews number (MathSciNet)
MR2910730

Zentralblatt MATH identifier
1242.39032

#### Citation

Eshaghi Gordji, M.; Azadi Kenary, H.; Rezaei, H.; Lee, Y. W.; Kim, G. H. Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces. Abstr. Appl. Anal. 2012 (2012), Article ID 953938, 22 pages. doi:10.1155/2012/953938. https://projecteuclid.org/euclid.aaa/1355495689

#### References

• S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, Inc, New York, NY, USA, 1964.
• 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.
• T. 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.
• Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991.
• S. Abbaszadeh, “Intuitionistic fuzzy stability of a quadratic and quartic functional equation,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 2, pp. 100–124, 2010.
• J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31, Cambridge University Press, Cambridge, UK, 1989.
• 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.
• D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951.
• M. B. Savadkouhi, M. E. Gordji, J. M. Rassias, and N. Ghobadipour, “Approximate ternary Jordan derivations on Banach ternary algebras,” Journal of Mathematical Physics, vol. 50, no. 4, article 042303, p. 9, 2009.
• P. Găvruţa, “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.
• P. Găvruţa and L. Găvruţa, “A new method for the generalized Hyers-Ulam-Rassias stability,” Journal of Mathematical Analysis and Applications, vol. 1, no. 2, pp. 11–18, 2010.
• D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhäuser Boston Inc., Boston, Mass, USA, 1998.
• G. Isac and T. M. Rassias, “On the Hyers-Ulam stability of $\psi$ -additive mappings,” Journal of Approximation Theory, vol. 72, no. 2, pp. 131–137, 1993.
• C. Park and M. E. Gordji, “Comment on “Approximate ternary Jordan derivations on Banach ternary algebras” [B. Savadkouhi Journal of Mathematical Physics vol. 50, article 042303, 2009],” Journal of Mathematical Physics, vol. 51, no. 4, 2010.
• C. Park and A. Najati, “Generalized additive functional inequalities in Banach algebras,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 2, pp. 54–62, 2010.
• C. Park and T. M. Rassias, “Isomorphisms in unital ${C}^{\ast\,\!}$-algebras,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 2, pp. 1–10, 2010.
• T. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.
• P. Kannappan, “Quadratic functional equation and inner product spaces,” Results in Mathematics, vol. 27, no. 3-4, pp. 368–372, 1995.
• F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.
• P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.
• 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.
• A. Grabiec, “The generalized Hyers-Ulam stability of a class of functional equations,” Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 217–235, 1996.
• C. Borelli and G. L. Forti, “On a general Hyers-Ulam stability result,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 2, pp. 229–236, 1995.
• C.-G. Park, “Generalized quadratic mappings in several variables,” Nonlinear Analysis. Theory, Methods & Applications, vol. 57, no. 5-6, pp. 713–722, 2004.
• C.-G. Park, “On the stability of the quadratic mapping in Banach modules,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 135–144, 2002.
• A. Ebadian, A. Najati, and M. Eshaghi Gordji, “On approximate additive-quartic and quadratic-cubic functional equations in two variables on abelian groups,” Results in Mathematics, vol. 58, no. 1-2, pp. 39–53, 2010.
• M. Eshaghi Gordji, “Stability of a functional equation deriving from quartic and additive functions,” Bulletin of the Korean Mathematical Society, vol. 47, no. 3, pp. 491–502, 2010.
• M. Eshaghi Gordji, “Stability of an additive-quadratic functional equation of two variables in F-spaces,” Journal of Nonlinear Science and its Applications, vol. 2, no. 4, pp. 251–259, 2009.
• M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of cubic and quartic functional equations in non-Archimedean spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1321–1329, 2010.
• M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces,” Applied Mathematics Letters, vol. 23, no. 10, pp. 1198–1202, 2010.
• M. Eshaghi Gordji and M. B. Savadkouhi, “Approximation of generalized homomorphisms in quasi-Banach algebras,” Mathematical Journal of the Ovidius University of Constantza, vol. 17, no. 2, pp. 203–213, 2009.
• M. Eshaghi Gordji and M. Bavand Savadkouhi, “On approximate cubic homomorphisms,” Advances in Difference Equations, vol. 2009, Article ID 618463, 11 pages, 2009.
• M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, “Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces,” Abstract and Applied Analysis, vol. 2009, Article ID 417473, 14 pages, 2009.
• M. Eshaghi Gordji and M. B. Savadkouhi, “Stability of mixed type cubic and quartic functional equations in random normed spaces,” Journal of Inequalities and Applications, vol. 2009, Article ID 527462, 9 pages, 2009.
• M. Eshaghi Gordji, A. Ebadian, and S. Zolfaghari, “Stability of a functional equation deriving from cubic and quartic functions,” Abstract and Applied Analysis, vol. 2008, Article ID 801904, 17 pages, 2008.
• M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, “Solution and stability of a mixed type additive, quadratic, and cubic functional equation,” Advances in Difference Equations, vol. 2009, Article ID 826130, 17 pages, 2009.
• 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.
• M. Eshaghi Gordji, T. Karimi, and S. Kaboli Gharetapeh, “Approximately n-Jordan homomorphisms on Banach algebras,” Journal of Inequalities and Applications, vol. 2009, Article ID 870843, 8 pages, 2009.
• M. Eshaghi Gordji and H. Khodaei, “Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5629–5643, 2009.
• M. Eshaghi Gordji, H. Khodaei, and R. Khodabakhsh, “General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces,” “Politehnica” University of Bucharest Scientific Bulletin. Series A, vol. 72, no. 3, pp. 69–84, 2010.
• M. Eshaghi Gordji and A. Najati, “Approximately ${J}^{\ast\,\!}$-homomorphisms: a fixed point approach,” Journal of Geometry and Physics, vol. 60, no. 5, pp. 809–814, 2010.
• M. Eshaghi Gordji, S. Zolfaghari, S. Kaboli Gharetapeh, A. Ebadian, and C. Park, “Solution and stability of generalized mixed type additive and quadratic functional equation in non-Archimedean spaces,” to appear in Annali dell'Universita di Ferrara.
• R. Farokhzad and S. A. R. Hosseinioun, “Perturbations of Jordan higher derivations in Banach ternary algebras: an alternative fixed point approach,” International Journal of Nonlinear Analysis and Applications, vol. 1, no. 1, pp. 42–53, 2010.
• M. Eshaghi Gordji and H. Khodaei, “On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations,” Abstract and Applied Analysis, vol. 2009, Article ID 923476, 11 pages, 2009.
• M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, “Solution and stability of a mixed type additive, quadratic, and cubic functional equation,” Advances in Difference Equations, vol. 2009, Article ID 826130, 17 pages, 2009.
• M. Eshaghi Gordji, S. Kaboli Gharetapeh, E. Rashidi, T. Karimi, and M. Aghaei, “Ternary Jordan-derivations in ${C}^{\ast\,\!}$-ternary algebras,” Journal of Computational Analysis and Applications, vol. 12, no. 2, pp. 463–470, 2010.
• M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour, “Generalized Hyers-Ulam stability of generalized $(N,K)$-derivations,” Abstract and Applied Analysis, vol. 2009, Article ID 437931, 8 pages, 2009.
• T. Bag and S. K. Samanta, “Finite dimensional fuzzy normed linear spaces,” Journal of Fuzzy Mathematics, vol. 11, no. 3, pp. 687–705, 2003.
• T. Bag and S. K. Samanta, “Fuzzy bounded linear operators,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 513–547, 2005.
• D. Miheţ 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.