Using weakly p-summable and Dunford-Pettis (resp. weakly p-summable and almost Dunford-Pettis) sequences, some geometric properties on Banach lattices are studied. Moreover, by the concept of relatively compact Dunford-Pettis property (briefly, DPrcP) and strong DPrcP, Banach lattices in which some of these properties coincide are characterized. As an application, Banach lattices with the Right fixed point property of order p are considered. In particular, it is established that for a Banach space X and a suitable Banach lattice F , a Banach lattice M ⊂K(X; F ) has the Right fixed point property of order p (resp. strong Right fixed point property of order p) if each evaluation operator y∗ on M is Dunford-Pettis p-convergent (resp. almost Dunford-Pettis p-convergent), where y∗ : M ! X∗ is defined by y∗(T ) = T ∗y∗ for y∗ 2 F ∗ and T 2 M.
