On the Ulam stability of a quadratic set-valued functional equation
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Authors
Faxing Wang
- Tongda College of Nanjing University of Posts and Telecommunications, Nanjing 210046, P. R. China.
Yonghong Shen
- School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, P. R. China.
Abstract
In this paper, we prove the Ulam stability of the following set-valued functional equation by employing the
direct method and the fixed point method, respectively,
\[f ( x -\frac{ y + z}{ 2}) \oplus f (x +\frac{ y - z}{ 2})\oplus f(x + z) = 3f(x) \oplus \frac{1}{ 2} f(y) \oplus \frac{3 }{2 }f(z).\]
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ISRP Style
Faxing Wang, Yonghong Shen, On the Ulam stability of a quadratic set-valued functional equation, Journal of Nonlinear Sciences and Applications, 7 (2014), no. 5, 359--367
AMA Style
Wang Faxing, Shen Yonghong, On the Ulam stability of a quadratic set-valued functional equation. J. Nonlinear Sci. Appl. (2014); 7(5):359--367
Chicago/Turabian Style
Wang, Faxing, Shen, Yonghong. "On the Ulam stability of a quadratic set-valued functional equation." Journal of Nonlinear Sciences and Applications, 7, no. 5 (2014): 359--367
Keywords
- Ulam stability
- Quadratic set-valued functional equation
- Hausdorff distance
- fixed point.
MSC
References
-
[1]
T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66.
-
[2]
C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, in: Lec. Notes in Math., vol. 580, Springer, Berlin (1977)
-
[3]
H. Y. Chu, A. Kim, S. Y. Yoo, On the stability of the generalized cubic set-valued functional equation, Appl. Math. Lett., 37 (2014), 7-14.
-
[4]
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.
-
[5]
G. L. Forti , Hyers-Ulam stability of functional equations in several variables, Aequat. Math., 50 (1995), 143-190.
-
[6]
P. Gavruca, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
-
[7]
D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222-224.
-
[8]
S. Y. Jang, C. Park, Y. Cho, Hyers-Ulam stability of a generalized additive set-valued functional equation, J. Inequal. Appl., 2013 (2013), 101
-
[9]
S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, (2011)
-
[10]
H. A. Kenary, H. Rezaei, On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory Appl., 2012 (2012), 81
-
[11]
G. Lu, C. Park , Hyers-Ulam stability of additive set-valued functional equations , Appl. Math. Lett., 24 (2011), 1312-1316.
-
[12]
C. Park, D. O'Regan, R. Saadati, Stability of some set-valued functional equations, Appl. Math. Lett., 24 (2011), 1910-1914.
-
[13]
M. Piszczek, The properties of functional inclusions and Hyers-Ulam stability, Aequat. Math., 85 (2013), 111-118.
-
[14]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
-
[15]
Th. M. Rassias , On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130.
-
[16]
H. Rådström, An embedding theorem for spaces of convex sets , Proc. Amer. Math. Soc., 3 (1952), 165-169.
-
[17]
P. K. Sahoo, P. Kannappan, Introduction to Functional Equations, CRC Press , Boca Raton (2011)
-
[18]
Y. H. Shen, Y. Y. Lan, On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces, J. Nonlinear Sci. Appl., 7 (2014), 368-378.
-
[19]
S. M. Ulam, Problems in Modern Mathematics, Wiley, New York (1960)