Stability of functional inequality in digital metric space

Abstract

Abstract In the present article, the Hyers–Ulam stability of the following inequality is analyzed: 0.1 { d ( f ( ı + ȷ ) , ( f ( ı ) + f ( ȷ ) ) ) ≤ d ( ρ 1 ( ( f ( ı + ȷ ) + f ( ı − ȷ ) , 2 f ( ı ) ) ) d ( f ( ı + ȷ ) , ( f ( ı ) + f ( ȷ ) ) ) ≤ + d ( ρ 2 ( 2 f ( ı + ȷ 2 ) , ( f ( ı ) + f ( ȷ ) ) ) ) $$ \textstyle\begin{cases} d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq d (\rho _{1}((f(\imath +\jmath )+ f(\imath - \jmath ),\ 2f(\imath )) ) \\ \hphantom{ d (f(\imath +\jmath ), \ (f(\imath )+ \ f(\jmath )) )\leq}{}+ d (\rho _{2} (2f (\frac{\imath +\jmath}{2} ), \ (f(\imath )+ f(\jmath )) ) ) \end{cases} $$ in the setting of digital metric space, where ρ 1 $\rho _{1}$ and ρ 2 $\rho _{2}$ are fixed nonzero complex numbers with 1 > 2 | ρ 1 | + | ρ 2 | $1>\sqrt{2}|\rho _{1}|+|\rho _{2}|$ by using fixed point and direct approach.

Keywords

• Mathematical analysis
• Stability
• Metric spaces