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

Abstract

AbstractIn 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 $C^{*}$ C ∗ -algebras and Lie $C^{*}$ C ∗ -algebras, multi-stability of derivations in $C^{*}$ C ∗ -algebras, and Lie $C^{*}$ C ∗ -algebras for the following random operator equation via fixed point methods: $$ \mu f \biggl(\eth , \frac{x+y}{2} \biggr) + \mu f \biggl(\eth , \frac{x-y}{2} \biggr) = f(\eth , \mu x) . $$ μ f ( ð , x + y 2 ) + μ f ( ð , x − y 2 ) = f ( ð , μ x ) . In particular, for $\mu = 1$ μ = 1 , the above equation turns out to be Jensen’s random operator equation.