Approximately Orthogonal Additive Set-valued Mappings

• Journal title : Kyungpook mathematical journal
• Volume 53, Issue 4,  2013, pp.639-646
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2013.53.4.646
Title & Authors
Approximately Orthogonal Additive Set-valued Mappings
Mirmostafaee, Alireza Kamel; Mahdavi, Mostafa;

Abstract
We investigate the stability of orthogonally additive set-valued functional equation $\small{F(x+y)=F(x)+F(y)(x{\perp}y)}$ in Hausdorff topology on closed convex subsets of a Banach space.
Keywords
Set-valued mappings;orthogonal space;Hausdorff metric;Hyers-Ulam stability;
Language
English
Cited by
1.
A Singular Behaviour of a Set-Valued Approximate Orthogonal Additivity, Results in Mathematics, 2016, 70, 1-2, 163
References
1.
G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1(1935), 169-172.

2.
J. Brzdek, D. Popa, B. Xu, Selection of set-valued maps satisfying a linear inclusion in single variable, Nonlinear Anal. 74(2011), 324-330.

3.
C. Casting and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Note in Math. 580(1977).

4.
S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publishing Co. Pte. Ltd (2002).

5.
M. Eshaghi Gordji, S. Abbaszadeh and C. Park, On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces, J. Ineq. Appl., 2009(2009), Article ID 153084, 26 pages.

6.
M. Eshaghi and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71(2009), 5629?5643.

7.
Z. Gajda and R. Ger, Subadditive mulifunctions and Hyers-Ulam stability, Numer. Math. 80(1987), 281-291.

8.
R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math. 43(1995), 143?151.

9.
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27(1941), 222-224.

10.
R. C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61(1947), 265?292.

11.
A. K. Mirmostafaee, Approximately additive mappings in non-Archimedean normed spaces, Bull. Korean Math. Soc. 46(2009), No. 2, 387-400.

12.
A. K. Mirmostafaee, Hyers-Ulam stability of cubic mappings in non-Archimedean normed spaces, Kyungpook Math. J. 50(2)(2010), 315-327.

13.
M. S. Moslehian, On the stability of the orthogonal Pexiderized Cauchy equation, J. Math. Anal. Appl., 318(1)(2006), 221-223.

14.
K. Nikodem, On quadratic set-valued functions, Publ. Math. Debrecen 30(1983), 297-301.

15.
C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl., 275(2002), 711?720.

16.
C. Park, On the stability of the orthogonally quartic functional equation, Bull. Iran. Math. Soc. 31(1)(2005), 63-70.

17.
D. Popa, A property of a functional inclusion connected with Hyers-Ulam stability, J. Math. Inequal. 4(2009), 591-598.

18.
H. Radstrom, An embedding theorem for space of convex sets, Proc. Amer. Math. Soc., 3(1952), 165-169.

19.
J. M. Rassias, The Ulam stability problem in approximation of approximately quadratic mappings by quartic mappings, Journal of Inequalities in Pure and Applied Mathematics, Issue 3, Article 52, 5(2004).

20.
Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300.

21.
J. Ratz, On orthogonally additive mappings, Aequationes Math. 28(1985), 35-49.

22.
J. Sikorska, Generalized orthogonal stability of some functional equations, J. Inequal. Appl. (2006), Art. ID 12404, 23 pp.

23.
A. Smajdor, Additive selections of superadditive set-valued functions, Aequations Math. 39(1990), 121-128.

24.
S. M. Ulam, Problems in Modern Mathematics, Science ed., John Wiley & Sons, New York, 1964 (Chapter VI, Some Questions in Analysis: Section 1,