Generalized Hyers–Ulam stability of ρ-functional inequalities

Abstract

Abstract In our research work generalized Hyers-Ulam stability of the following functional inequalities is analyzed by using fixed point approach: 0.1 ∥ f ( 2 x + y ) + f ( 2 x − y ) − 2 f ( x + y ) − 2 f ( x − y ) − 12 f ( x ) − ρ ( 4 f ( x + y 2 ) + 4 ( f ( x − y 2 ) − f ( x + y ) − f ( x − y ) ) − 6 f ( x ) , r ) ∥ ≥ r r + φ ( x , y ) $$\begin{aligned}& \biggl\Vert f(2x+y)+f(2x-y)-2f(x+y)-2f(x-y)-12f(x) \\& \quad {}-\rho \biggl(4f\biggl(x+\frac{y}{2}\biggr)+4\biggl(f\biggl(x- \frac{y}{2}\biggr)-f(x+y)-f(x-y)\biggr)-6f(x),r\biggr) \biggr\Vert \geq \frac{r}{r+\varphi (x, y)} \end{aligned}$$ and 0.2 ∥ f ( 2 x + y ) + f ( 2 x − y ) − 4 f ( x + y ) − 4 f ( x − y ) − 24 f ( x ) + 6 f ( y ) − ρ ( 8 f ( x + y 2 ) + 8 ( f ( x − y 2 ) − 2 f ( x + y ) − 2 f ( x − y ) ) − 12 f ( x ) + 3 f ( y ) , r ) ∥ ≥ r r + φ ( x , y ) $$\begin{aligned}& \biggl\Vert f(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y) \\& \qquad {}-\rho \biggl(8f\biggl(x+\frac{y}{2}\biggr)+8\biggl(f\biggl(x- \frac{y}{2}\biggr)-2f(x+y)-2f(x-y)\biggr)-12f(x)+3f(y),r\biggr) \biggr\Vert \\& \quad \geq \frac{r}{r+\varphi (x, y)} \end{aligned}$$ in the setting of fuzzy matrix, where ρ ≠ 2 $\rho \neq 2$ is a real number. We also discussed Hyers-Ulam stability from the application point of view.