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

Abstract

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 ∗ $C^{*}$ -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: μ f ( ð , x + y 2 ) + μ f ( ð , x − y 2 ) = f ( ð , μ x ) . $$ \mu f \biggl(\eth , \frac{x+y}{2} \biggr) + \mu f \biggl(\eth , \frac{x-y}{2} \biggr) = f(\eth , \mu x) . $$ In particular, for μ = 1 $\mu = 1$ , the above equation turns out to be Jensen’s random operator equation.

Keywords

• Multi control functions
• Mittag–Leffler function
• H $\mathbb{H}$ -Fox function
• Hypergeometric function
• Wright function
• C ∗ $C^{*}$ -algebras