C-Semigroups, Subordination Principle And The Levy Alpha-Stable Distribution On Discrete Time

In this paper, we introduce the notion of Lévy α-stable distribution within the discrete setting. Using this notion, a subordination principle is proved, which relates a sequence of solution operators – given by a discrete C-semigroup – for the abstract Cauchy problem of first order in discrete-time, with a sequence of solution operators for the abstract Cauchy problem of fractional order 0 < α < 1 in discrete-time. As an application, we establish the explicit solution of the abstract Cauchy problem in discrete-time that involves the Hilfer fractional difference operator and prove that, in some cases, such solution converges to zero. Our findings give new insights on the theory, provide original concepts and extend as well as improve recent results of relevant references on the subject.