According to the three concepts, weak orthogonality, WORTH property and nonstrictly Opial condition in Banach spaces, the disjoint (positive) version of them in Banach lattices is introduced. Also, Banach lattices in which these three properties are equivalent to the order continuity of the norm are characterized. As an application, some conditions for a Banach lattice under which any of these three properties implies the weak fixed point property are provided. Next, the disjoint (positive) weak fixed point property for some operator spaces is studied. In particular, it is established that for each Banach space X and a suitable Banach lattice F, a Banach lattice M ⊂ K(X, F) has the weak fixed point property (resp. disjoint (positive) weak fixed point property) if and only if each evaluation operator ψy∗ on M is completely continuous (resp. almost Dunford-Pettis) operator, where ψy∗ : M → X∗ is defined by ψy∗(T ) = T ∗y∗ for y∗ ∈ Y ∗ and T ∈ M.
