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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.

In this article, the class of all Dunford-Pettis p-convergent operators and p-Dunford-Pettis relatively compact property on Banach spaces are investigated. Moreover, we give some conditions on Banach spaces X and Y such that the class of bounded linear operators from X to Y and some its subspaces have the p-Dunford-Pettis relatively compact property. In addition, if Ω is a compact Hausdorff space, then we prove that dominated operators from the space of all continuous functions from K to Banach space X (in short C(Ω, X)) taking values in a Banach space with the p-(DP rcP ) are p-convergent when X has the Dunford-Pettis property of order p. Furthermore, we show that if T : C(Ω, X) → Y is a strongly bounded operator with representing measure m : Σ → L(X, Y ) andT : B(Ω, X) → Y is its extension, then T is Dunford-Pettis p-convergent if and only ifT is Dunford-Pettis p-convergent.Mathematics Subject Classification (2010). 46B20, 46B25,46B28.

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In this article, we introduce the notion of p-(DP L) sets. Also, a factorization result for differentiable mappings through Dunford-Pettis p-convergent operators is investigated. Namely, if X, Y are real Banach spaces and U is an open convex subset of X, then we obtain that, given a differentiable mapping f : U → Y its derivative f ′ takes U -bounded sets into p-(DP L) sets if and only if it happens f = g • S, where S is a Dunford-Pettis p-convergent operator from X into a suitable Banach space Z and g : S(U ) → Y is a Gâteaux differentiable mapping with some additional properties.