Abstract
In this paper, we prove some common fixed point theorems for order contractive mappings on a σ ‐complete vector lattice. We apply new results to study the well‐posedness of a common fixed point problem for two contractive mappings. Our proofs are simple and purely order‐theoretic in nature. Consequently, the shadowing property and Ulam–Hyers stability of a common fixed point problem are studied. The existence of the solution of an integral equation is obtained as an application of our main result.