Pexiderized additive s-(alpha, beta)- functional inequalities and their stability

Wuttichai Suriyacharoen; Wutiphol Sintunavarat

Wuttichai Suriyacharoen Faculty of Science and Technology, Thammasat University, Pathum Thani, 12120, Thailand
Wutiphol Sintunavarat Faculty of Science and Technology\\ Thammasat University, Pathum Thani, 12120, Thailand.

Abstract

The present work aims to find the general solutions of the following three functional inequalities:
$$||f_1(x+y)+\overline{\alpha_1} f_2(\alpha_1 z)- \beta_1^{-1} f_3(\beta_1 (x+y+z))|| $$
$$\leq ||s\big(f_1(x-y)+\overline{\alpha_1} f_2(\alpha_1 z)
- \beta_1^{-1} f_3(\beta_1 (x-y+z))\big)||,$$
$$||g_1(x)+g_2(y)+2g_3(z)-\alpha_2 g_4(\beta_2 (x+y+2z))|| $$ $$\leq ||s\big(g_1(x)+g_2(-y)+2g_3(z)-\alpha_2 g_4(\beta_2 (x-y+2z))\big)||,$$
$$||h_1(x+y)-h_2(x)-h_3(y)|| \leq ||s\big(h_1(x-y)-h_2(x) - h_3(-y)\big)||$$
over complex vector spaces, where $s, \alpha_1, \alpha_2, \beta_1, \beta_2 \in \mathbb{C} \backslash \{0\}$ with $|s|<1$ and $|\alpha_1|=1$ are fixed complex numbers, without any regularity assumptions on the unknown functions. An analysis of their stability is here also carried out.