On finding fixed point of monotone mapping of partially ordered topological space

The problem of finding a fixed point of a continuous monotone mapping of a topological space into itself is considered. The solution of the problem is based on the fixed-point iteration method. A theorem on necessary and sufficient conditions for the convergence of the iterative process to one of the fixed points of the mapping is proved. Unlike other works devoted to fixed points of monotone mappings, the proposed theorem does not require the existence of a least upper bound for any partially ordered subset of a topological space.