# On the stability of higher ring left derivations

Research paper by Yong-Soo Jung

Indexed on: 24 Oct '16Published on: 01 Sep '16Published in: Indian Journal of Pure and Applied Mathematics

#### Abstract

Abstract In this note, we investigate the Hyers-Ulam, the Isac and Rassias-type stability and the Bourgin-type superstability of a functional inequality corresponding to the following functional equation: $${h_n}\left( {xy} \right) = \sum\limits_{\begin{array}{*{20}{c}} {i + j = n} \\ {i \leqslant j} \end{array}} {\left[ {{h_i}\left( x \right){h_j}\left( y \right) + {c_{ij}}{h_i}\left( y \right){h_j}\left( x \right)} \right]}$$ , where $${c_{ij}} = \left\{ {\begin{array}{*{20}{c}} 1&{if\;i \ne j,} \\ 0&{if\;i = j.} \end{array}} \right.$$ AbstractIn this note, we investigate the Hyers-Ulam, the Isac and Rassias-type stability and the Bourgin-type superstability of a functional inequality corresponding to the following functional equation: $${h_n}\left( {xy} \right) = \sum\limits_{\begin{array}{*{20}{c}} {i + j = n} \\ {i \leqslant j} \end{array}} {\left[ {{h_i}\left( x \right){h_j}\left( y \right) + {c_{ij}}{h_i}\left( y \right){h_j}\left( x \right)} \right]}$$ , where $${c_{ij}} = \left\{ {\begin{array}{*{20}{c}} 1&{if\;i \ne j,} \\ 0&{if\;i = j.} \end{array}} \right.$$ $${h_n}\left( {xy} \right) = \sum\limits_{\begin{array}{*{20}{c}} {i + j = n} \\ {i \leqslant j} \end{array}} {\left[ {{h_i}\left( x \right){h_j}\left( y \right) + {c_{ij}}{h_i}\left( y \right){h_j}\left( x \right)} \right]}$$ $${h_n}\left( {xy} \right) = \sum\limits_{\begin{array}{*{20}{c}} {i + j = n} \\ {i \leqslant j} \end{array}} {\left[ {{h_i}\left( x \right){h_j}\left( y \right) + {c_{ij}}{h_i}\left( y \right){h_j}\left( x \right)} \right]}$$ $${c_{ij}} = \left\{ {\begin{array}{*{20}{c}} 1&{if\;i \ne j,} \\ 0&{if\;i = j.} \end{array}} \right.$$ $${c_{ij}} = \left\{ {\begin{array}{*{20}{c}} 1&{if\;i \ne j,} \\ 0&{if\;i = j.} \end{array}} \right.$$