The concept of weak orthogonality of order p (1 ≤ p ≤ ∞) in Banach lattices is introduced in order to obtain spaces with the weak fixed point property of order p. Moreover, various connections between a number of Banach space properties to imply the weak fixed point property, such as Opial condition, weak normal structure and property (M) are investigated. 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 of order p, if each evaluation operator ψy∗ onMis a p-convergent operator for y∗ ∈ F∗.
