In this paper, we study first the concept of p-sequentially Right property, which is p-version of the sequentially Right property. Also, we introduce a new class of subsets of Banach spaces which is called p-Right∗ set and obtain the relationship between p-Right subsets and p-Right∗ subsets of dual spaces. Furthermore, for 1 ≤ p < q ≤ ∞,we introduce the concepts of properties (SR)p,q and (SR∗)p,q in order to find a condition such that every Dunford-Pettis q-convergent operator is Dunford-Pettis p-convergent. Finally, we apply these concepts and obtain some characterizations of the p-Dunford-Pettis relatively compact property of Banach spaces and their dual spaces,