# A fixed point approach to the stability of a general quartic functional equation

Volume 20, Issue 3, pp 207--215
Publication Date: December 09, 2019 Submission Date: September 20, 2019 Revision Date: October 11, 2019 Accteptance Date: October 15, 2019
• 177 Views

### Authors

Yang-Hi Lee - Department of Mathematics Education, Gongju National University of Education, Gongju 32553, Republic of Korea. Soon-Mo Jung - Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea.

### Abstract

In this paper, we study the generalized Hyers-Ulam stability of the quartic functional equation $f(x+3y) - 5f(x+2y) + 10f(x+y) - 10f(x) + 5f(x-y) - f(x-2y) = 0,$ by applying the fixed point method.

### Keywords

• Fixed point method
• fixed point
• stability
• generalized Hyers-Ulam stability
• quartic functional equation

•  39B82
•  39B52
•  47H10
•  47N99

### References

• [1] M. Almahalebi, On the stability of a generalization of Jensen functional equation, Acta Math. Hungar., 154 (2018), 187--198

• [2] S. Alshybania, S. M. Vaezpoura, R. Saadati, Generalized Hyers-Ulam stability of sextic functional equation in random normed spaces, J. Comput. Anal. Appl., 24 (2018), 370--381

• [3] J. Brzdęk, E. Karapinar, A. Petruşel, A fixed point theorem and the Ulam stability in generalized dq-metric spaces, J. Math. Anal. Appl., 467 (2018), 501--520

• [4] L. Cădariu, V. Radu, Fixed points and the stability of Jensen's functional equation, JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), 7 pages

• [5] L. Cădariu, V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform., 41 (2003), 25--48

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

• [7] 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

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

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

• [10] D. H. Hyers, G. Isac, T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston (1998)

• [11] D. H. Hyers, T. M. Rassias, Approximate homomorphisms, Aequationes Math., 44 (1992), 125--153

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

• [13] S.-M. Jung, T.-S. Kim, A fixed point approach to the stability of the cubic functional equation, Bol. Soc. Mat. Mexicana (3), 12 (2006), 51--57

• [14] Y.-H. Lee, Hyers-Ulam-Rassias stability of a quadratic-additive type functional equation on a restricted domain, Int. J. Math. Anal. (Ruse), 7 (2013), 2745--2752

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

• [16] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York-London (1960)