Fixed Point Theorems for Set Valued Mappings in Partially Ordered Metric Spaces

Let (X, ) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Assume that X satisfies; if a non-decreasing sequence xn → x in X , then xn x, for all n. Let F be a set valued mapping from X into X with nonempty closed bounded values satisfying; (i) there exists κ ∈ (0, 1) with D(F (x), F (y)) ≤ κd(x, y), for all x y, (ii) if d(x, y) < ε < 1 for some y ∈ F (x) then x y, (iii) there exists x0 ∈ X, and some x1 ∈ F (x0) with x0 x1 such that d(x0, x1) < 1. It is shown that F has a fixed point. Several consequences are also obtained. Keywords—Fixed point, partially ordered set, metric space, set valued mapping.