The natural duality between “topological” and “regular,” both considered as convergence space properties, extends naturally top-regular convergence spaces, resulting in the new concept of a p-topological convergence space. Taking advantage of this duality, the behavior ofp-topological andp-regular convergence spaces is explored, with particular emphasis on the former, since they have not been p...

In this paper, we are interested in the numerical solutions of stochastic functional differential equations (SFDEs) with jumps. Under the global Lipschitz condition, we show that the pth moment convergence of the Euler-Maruyama (EM) numerical solutions to SFDEs with jumps has order 1/p for any p ≥ 2. This is significantly different from the case of SFDEs without jumps where the order is 1/2 for...

Theorem 2 is the main result in this note. Lemma 1 is key to our main result. It is different from the result in [2] because it does not restrict the number of terms in the partial sum to an even one. Also, our inequalities (2) should be compared with the corresponding result in [2] for k = 2. Lemma 1. Let sn(p) be the nth partial sum of the p-series ∑∞ i=1(1/i), and let k be any integer greate...

We provide some new results on the weak convergence of sequences or nets lying in L,((T, & r), X) = L,(p, X), 1 <p < co, i.e., the space of equivalence classes of X-valued (X is a Banach space) Bochner integrable functions on the finite measure space ( r, z, I”). Our theorems generalize in several directions recent resuls on weak sequential convergence in L,(p, X) obtained by M. A. Khan and M. ...