Quadratic \(\rho\)-functional inequalities in \(\beta\)-homogeneous normed spaces

Authors

Yuanfeng Park - Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Korea.
Yinhua Lu - Department of Mathematics, School of Science, ShenYang University of Technology, Shenyang 110870, P. R. China.
Gang Cui - Department of Mathematics, Yanbian University, Yanji 133001, P. R. China.
Choonkil Jin - Department of Mathematics, Yanbian University, Yanji 133001, P. R. China.

Abstract

In this paper, we solve the quadratic \(\rho\)-functional inequalities \[\|f(x+y)+f(x-y)-2f(x)-2f(y)\|\leq\|\rho(4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y))\|,\] where \(\rho\) is a fixed complex number with \(|\rho|<1\), and\[\|4f(\frac{x+y}{2})+f(x-y)-2f(x)-2f(y)\|\leq\|\rho(f(x+y)+f(x-y)-2f(x)-2f(y))\|,\] where \(\rho\) is a fixed complex number with \(|\rho|<1\). Using the direct method, we prove the Hyers-Ulam stability of the quadratic \(\rho\)-functional inequalities (1) and (2) in \(\beta\)- homogeneous complex Banach spaces.

Keywords

Hyers-Ulam stability, \(\beta\)-homogeneous space, quadratic \(\rho\)-functional inequality.

References

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