On the Ulam stability of a quadratic set-valued functional equation

Wang, Faxing¹ ; Shen, Yonghong²

¹ Tongda College of Nanjing University of Posts and Telecommunications, Nanjing 210046, P. R. China
² School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, P. R. China

Journal of nonlinear sciences and its applications, Tome 7 (2014) no. 5, p. 359-367.

Résumé

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,

$f ( x -\frac{ y + z}{ 2}) \oplus f (x +\frac{ y - z}{ 2})\oplus f(x + z) = 3f(x) \oplus \frac{1}{ 2} f(y) \oplus \frac{3 }{2 }f(z).$

Détail

DOI : 10.22436/jnsa.007.05.06

Classification : 39B72, 54H25, 54C60
Keywords: Ulam stability, Quadratic set-valued functional equation, Hausdorff distance, fixed point.

Bibliographie

[1] T. Aoki On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, Volume 2 (1950), pp. 64-66

[2] Castaing, C.; Valadier, M. Convex analysis and measurable multifunctions, in: Lec. Notes in Math., vol. 580, Springer, Berlin, 1977

[3] Chu, H. Y.; Kim, A.; Yoo, S. Y. On the stability of the generalized cubic set-valued functional equation, Appl. Math. Lett., Volume 37 (2014), pp. 7-14

[4] Diaz, J. B.; Margolis, B. A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., Volume 74 (1968), pp. 305-309

[5] G. L. Forti Hyers-Ulam stability of functional equations in several variables, Aequat. Math., Volume 50 (1995), pp. 143-190

[6] Gavruca, P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., Volume 184 (1994), pp. 431-436

[7] Hyers, D. H. On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, Volume 27 (1941), pp. 222-224

[8] Jang, S. Y.; Park, C.; Cho, Y. Hyers-Ulam stability of a generalized additive set-valued functional equation, J. Inequal. Appl., Volume 2013 (2013), pp. 1-101

[9] Jung, S. M. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, , 2011

[10] Kenary, H. A.; Rezaei, H. On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory Appl., Volume 2012 (2012), pp. 1-81

[11] Lu, G.; C. Park Hyers-Ulam stability of additive set-valued functional equations , Appl. Math. Lett., Volume 24 (2011), pp. 1312-1316

[12] Park, C.; O'Regan, D.; Saadati, R. Stability of some set-valued functional equations, Appl. Math. Lett., Volume 24 (2011), pp. 1910-1914

[13] Piszczek, M. The properties of functional inclusions and Hyers-Ulam stability, Aequat. Math., Volume 85 (2013), pp. 111-118

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

[15] Th. M. Rassias On the stability of functional equations and a problem of Ulam, Acta Appl. Math., Volume 62 (2000), pp. 23-130

[16] Rådström, H. An embedding theorem for spaces of convex sets , Proc. Amer. Math. Soc., Volume 3 (1952), pp. 165-169

[17] Sahoo, P. K.; Kannappan, P. Introduction to Functional Equations, CRC Press , Boca Raton, 2011

[18] Shen, Y. H.; Lan, Y. Y. On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces, J. Nonlinear Sci. Appl., Volume 7 (2014), pp. 368-378

[19] Ulam, S. M. Problems in Modern Mathematics, Wiley, New York, 1960