Abstract
Abstract Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some ℓp−type fractional difference sets via the gamma function. Although, we characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is ℓ∞. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.
Highlights
The gamma function of a real number x is defined by an improper integral: ∞∫︁ Γ (x) = e−t tx−1dt.It is known that for any natural number n, Γ(n + 1) = n!, and Γ(n + 1) = nΓ(n) hold for any real number n ∈/ {0, −1, −2, ...}
This paper considers some lp−type fractional difference sets via the gamma function
We characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is l∞
Summary
The gamma function of a real number x (except zero and the negative integers) is defined by an improper integral:. We can write the fractional difference operator defined in Eq (1.1) as an infinite matrix:. In the papers [11,12,13,14,15,16,17,18,19] different difference sets of sequences have been studied based on some newly defined infinite matrices. Some new results on the visualization and animations of the topologies of certain sets were illustrated in [20,21,22] The authors applied their software package for this purpose. We use the results of [31, 32] to obtain necessary and sufficient conditions for the classes of compact matrix operators from lp(∆(α)) spaces into the sets of bounded sequences and for the classes (l1(∆(α)), l∞) and (l∞(∆(α)), l∞)
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