Sequentially Right-like properties on Banach spaces

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