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The investigation on the oscillatory and asymptotic properties of differential equations plays an important role in many applications, such as engineering, physics and medicine. In this work, we establish some new sufficient conditions for both the oscillatory and asymptotic behavior of the solutions to a second-order neutral impulsive differential system (IDS, for short) with deviating argument and fixed moments of impulse effect under both canonical and non-canonical operators with various values of the neutral coefficient. Also, we obtain the condition for the existence of a positive solution of the system under consideration. Finally, two examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

Let Ω ∈ L s (S n − 1) , with s ≥ 1 , be a homogeneous function of degree zero. In this paper, we establish the boundedness of the homogeneous fractional integral operator on weighted Herz spaces with variable exponent K ˙ p (⋅) α , q (w). [ABSTRACT FROM AUTHOR]

We investigate relationships between the sign of the discrete fractional sequential difference ( Δ 1 + a − μ ν Δ a μ f) (t) and the convexity of the function t ↦ f (t). In particular, we consider the case in which the bound ( Δ 1 + a − μ ν Δ a μ f) (t) ≥ ε f (a) , for some ε > 0 and where f (a) < 0 , is satisfied. Thus, we allow for the case in which the sequential difference may be negative, and we show that even though the fractional difference can be negative, the convexity of the function f can be implied by the above inequality nonetheless. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. We use a combination of both hard analysis and numerical simulation. [ABSTRACT FROM AUTHOR]