In this paper, we introduce a new class of subsets of class bounded linear operators between Banach spaces which is called p-(DPL) sets. Then, the relationship between these sets with equicompact sets is investigated. Moreover, we define p-version of Right sequentially continuous differentiable mappings and get some characterizations of these mappings. Finally, we prove that a mapping f : X → Y between real Banach spaces is Fréchet differentiable and f ′ takes bounded sets into p-(DPL) sets if and only if f may be written in the form f = • S where the intermediate space is normed, S is a Dunford-Pettis p-convergent operator, and g is a Gáteaux differentiable mapping with some additional properties.