Quadratic $\rho$-functional inequalities in complex matrix normed spaces

Wang, Zhihua¹ ; Park, Choonkil²

¹ School of Science, Hubei University of Technology, Wuhan, Hubei 430068, P. R. China
² Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 9, p. 5344-5352.

Résumé

In this paper, we solve the following quadratic $\rho$ -functional inequalities

$\|f(x + y)+f(x - y) - 2f(x) - 2f(y)\| \leq\|\rho(2f(\frac{x + y}{2}) + 2f(\frac{x - y}{2}) - f(x) - f(y))\|,$

where $\rho$ is a fixed complex number with $|\rho| 1$, and

$\|2f(\frac{x + y}{2}) + 2f(\frac{x - y}{2}) - f(x) - f(y)\| \leq\|\rho(f(x + y)+f(x - y) - 2f(x) - 2f(y)\|,$

where $\rho$ is a fixed complex number with $|\rho| \frac{ 1}{2}$ . By using the direct method, we prove the Hyers-Ulam stability of these inequalities in complex matrix normed spaces, and prove the Hyers-Ulam stability of quadratic $\rho$-functional equations associated with these inequalities in complex matrix normed spaces.

Détail

DOI : 10.22436/jnsa.009.09.03

Classification : 39B62, 39B82, 39B52
Keywords: Hyers-Ulam stability, matrix normed space, quadratic \(\rho\)-functional equation, quadratic \(\rho\)-functional inequality.

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