Discretization of C0$$ {C}_0 $$‐semigroups and discrete semigroups of operators in Banach spaces

摘要

In this paper, we introduce the notion of a τ$$ tau $$‐discrete semigroup {Tτn}n∈ℕ0$$ {left{{T}_{tau}^nright}}_{nin {mathbb{N}}_0} $$ generated by a closed linear operator A$$ A $$ in a Banach space X$$ X $$. We show that {Tτn}n∈ℕ0$$ {left{{T}_{tau}^nright}}_{nin {mathbb{N}}_0} $$ allows us to write the solution to an abstract discrete difference equation of first order as a discrete variation of parameters formula. Moreover, we study the main properties of {Tτn}n∈ℕ0$$ {left{{T}_{tau}^nright}}_{nin {mathbb{N}}_0} $$ and its relation with the well‐known notion of discrete semigroups. Finally, we characterize uniform exponential stability of a C0$$ {C}_0 $$‐semigroup {T(t)}t≥0$$ {left{T(t)right}}_{tge 0} $$ in terms of the τ$$ tau $$‐discrete semigroup {Tτn}n∈ℕ0$$ {left{{T}_{tau}^nright}}_{nin {mathbb{N}}_0} $$.