In this paper, we introduce the notion of Lévy [Formula: see text]-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 [Formula: see text]-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 [Formula: see text] 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.
In this paper we introduce the class of (N, λ)-periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of (N, λ)-periodic solutions to the following class of Volterra difference equations with infinite delay: u(n + 1) = α n j=-∞ a(n-j)u(j) + f (n, u(n)), n ∈ Z, α ∈ C, where the kernel a and the nonlinear term f satisfy suitable conditions.
The main objective of this work is to deduce some interesting algebraic relationships that connect the degenerated generalized Apostol–Bernoulli, Apostol–Euler and Apostol– Genocchi polynomials and other families of polynomials such as the generalized Bernoulli polynomials of level m and the Genocchi polynomials. Futher, find new recurrence formulas for these three families of polynomials to study.
In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some examples.
We establish sufficient conditions for the existence and uniqueness of (N, λ)-periodic solutions for the following abstract model:where 0 < α ≤ 1, A is a closed linear operator defined in a Banach space X, Δ α denotes the fractional difference operator in the Weyl-like sense, and f satisfies appropriate conditions.
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