We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M φ(x, y) = (f ∗ φy)(x), (x, y) ∈ R n × R+, with kernel φy(t) = yφ(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quas...

We consider scalar equation ut(t, x) = uxx(t, x)− u(t, x) + Z R K(x− y)g(u(t − h, y))dy, (∗) where g : R+ → R+ has exactly two fixed points: x1 = 0, x2 = κ > 0, and K is non-negative and such that R

Two families A and B of sets are said to be cross-intersecting if each set in A intersects each set in B. For any two integers n and k with 1 6 k 6 n, let ([n] 6k ) denote the family of subsets of {1, . . . , n} of size at most k, and let Sn,k denote the family of sets in ([n] 6k ) that contain 1. The author recently showed that if A ⊆ ( [m] 6r ) , B ⊆ ( [n] 6s ) , and A and B are cross-interse...

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result of Guo and Qi [Taiwanese J. Math. 7 (2003), no. 2, 239–247] and others. As applications, several inequalities involving the volume of the unit ball in R are derived, which refine, generalize a...

(1) where n ≥ 2, f is a continuous function defined on R+ × R , R+ = [0,∞), R = (−∞,∞), τi ∈ C (R+) and τi(t) ≤ t for t ∈ R+ and i = 0, 1, . . . , n− 1 . We will suppose for the simplicity that inf t∈R+ τi(t) > −∞ for i = 0, 1, . . . , n − 1. Note, that C(I), s ∈ {0, 1, . . .}, I ⊂ R+ is the set of continuous functions on I that have continuous derivatives up to the order s. A special case of e...