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{d(f(ı+ȷ),(f(ı)+f(ȷ)))≤d(ρ1((f(ı+ȷ)+f(ı−ȷ),2f(ı)))d(f(ı+ȷ),(f(ı)+f(ȷ)))≤+d(ρ2(2f(ı+ȷ2),(f(ı)+f(ȷ))))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} d (f(\\imath +\\jmath ), \\ (f(\\imath )+ \\ f(\\jmath )) )\\leq d (\\rho _{1}((f(\\imath +\\jmath )+ f(\\imath - \\jmath ),\\ 2f(\\imath )) ) \\\\ \\hphantom{ d (f(\\imath +\\jmath ), \\ (f(\\imath )+ \\ f(\\jmath )) )\\leq}{}+ d (\\rho _{2} (2f (\\frac{\\imath +\\jmath}{2} ), \\ (f(\\imath )+ f(\\jmath )) ) ) \\end{cases} $$\\end{document} in the setting of digital metric space, where ρ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho _{1}$\\end{document} and ρ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho _{2}$\\end{document} are fixed nonzero complex numbers with 1>2|ρ1|+|ρ2|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$1>\\sqrt{2}|\\rho _{1}|+|\\rho _{2}|$\\end{document} by using fixed point and direct approach.