Special functions and multi-stability of the Jensen type random operator equation in C^{*}-algebras via fixed point

Abstract

In this paper, we apply some special functions to introduce a new class of control functions that help us define the concept of multi-stability. Further, we investigate the multi-stability of homomorphisms in C^{*}-algebras and Lie C^{*}-algebras, multi-stability of derivations in C^{*}-algebras, and Lie C^{*}-algebras for the following random operator equation via fixed point methods: μf(ð,x+y2)+μf(ð,x−y2)=f(ð,μx).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mu f \\biggl(\\eth , \\frac{x+y}{2} \\biggr) + \\mu f \\biggl(\\eth , \\frac{x-y}{2} \\biggr) = f(\\eth , \\mu x) . $$\\end{document} In particular, for mu = 1, the above equation turns out to be Jensen’s random operator equation.