Stability of functional inequality in digital metric space

Abstract

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