Browse > Article
http://dx.doi.org/10.11568/kjm.2012.20.1.019

ON THE STABILITY OF THE QUADRATIC-ADDITIVE TYPE FUNCTIONAL EQUATION IN RANDOM NORMED SPACES VIA FIXED POINT METHOD  

Jin, Sun Sook (Department of Mathematics Education Gongju National University of Education)
Lee, Yang-Hi (Department of Mathematics Education Gongju National University of Education)
Publication Information
Korean Journal of Mathematics / v.20, no.1, 2012 , pp. 19-31 More about this Journal
Abstract
In this paper, we prove the stability in random normed spaces via fixed point method for the functional equation $$2f(x+y)+f(x-y)+f(y-x)-f(2x)-f(2y)=0$$.
Keywords
stability; additive mapping; random normed space; quadratic-additive type functional equation; fixed point theory;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
연도 인용수 순위
1 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66.   DOI
2 P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86.   DOI
3 Z. Gajda, On the stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991), 431-434.   DOI   ScienceOn
4 P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.   DOI   ScienceOn
5 D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224.   DOI   ScienceOn
6 S.-S. Jin and Y.-H. Lee, A fixed point approach to the stability of the Cauchy additive and quadratic type functional equation, J. Appl. Math. 2011 (2011), Article ID 817079, 16 pages.
7 S.-S. Jin and Y.-H. Lee, A fixed point approach to the stability of the quadratic-additive functional equation, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 18 (2011), 313-328.
8 S.-S. Jin and Y.-H. Lee, On the stability of the generalized quadratic and additive functional equation in random normed spaces via fixed point method, Korean J. Math. 19 (2011), 1-15.   DOI   ScienceOn
9 S.-M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137.   DOI   ScienceOn
10 H.-M. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, J. Math. Anal. Appl. 324 (2006), 358-372.   DOI   ScienceOn
11 Y.-H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc. 45 (2008), 397-403.   DOI   ScienceOn
12 Y.-H. Lee and K.W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), 305-315.   DOI   ScienceOn
13 Y.-H. Lee and K.W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Pexider equation, J. Math. Anal. Appl. 246 (2000), 627-638.   DOI   ScienceOn
14 Y.-H. Lee and K. W. Jun, A note on the Hyers-Ulam-Rassias stability of Pexider equation, J. Korean Math. Soc. 37 (2000), 111-124.
15 Y.-H. Lee and K.-W. Jun, On the stability of approximately additive mappings, Proc. Amer. Math. Soc. 128 (2000), 1361-1369.   DOI   ScienceOn
16 B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309.   DOI
17 D. Mihet and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572.   DOI   ScienceOn
18 V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91-96.
19 Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.   DOI   ScienceOn
20 I.A. Rus, Principles and applications of fixed point theory, Editura. Dacia, Cluj-Napoca, 1979 (in Romanian).
21 B. Schweizer and A. Sklar, Probabilistic metric spaces, Elsevier, North Holand, New York, 1983.
22 A.N. Serstnev, On the motion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280-283.
23 S.M. Ulam, A collection of mathematical problems, Interscience, New York (1968), 63.