A Fixed Point Theorem in Orbitally Complete Partially Ordered Metric Spaces

Let be a partially ordered set and be a mapping. We prove a fixed point theorem for the map satisfying a contractive condition in orbits, when is -orbitally complete. Our result extends and generalizes the results of Samet et al. (2013) to partially ordered sets. Also, we generalize the results of Ran and Reurings (2004). 1. Introduction In 1922, Banach [1] established a very fundamental result in fixed point theory, namely, the Banach contraction principle. Large number of generalizations have been made on this principle (we refer to Berinde and Vetro [2], Chatterjea [3], ？iri？ [4], and Kannan [5, 6]). The Banach contraction principle forces the map to be continuous. In 1968, Kannan [5] established a fixed point theorem for a map which need not be continuous. Later in 1971, Reich [7] proved a fixed point theorem through which the Banach contraction principle and Kannan's fixed point theorem are generalized. Recently, Samet et al. [8] established new fixed point theorems in complete metric spaces that generalize the Banach contraction principle and Reich’s and Kannan’s fixed point theorems. Theorem 1 (see [8]). Let be a complete metric space, a lower semicontinuous function, and a given mapping. Suppose that, for any , there exists such that implies for all . Then, has a unique fixed point . Moreover, one has . Theorem 2 (see [8]). Let be a complete metric space, a lower semi-continuous function and a given mapping. Suppose that there exists a constant such that for all . Then, has a unique fixed point . Moreover, one has . Theorem 3 (see [8]). Let be a complete metric space, a lower semi-continuous function and a given mapping. Suppose that, there exist with such that for all , Then, has a unique fixed point . Moreover, one has . Establishing the existence of fixed points in partially ordered sets is of recent interest in fixed point theory. In 2004, Ran and Reurings [9] established the existence of fixed points for contraction maps in partially ordered sets. Definition 4. Let be a partially ordered set. A map is said to be non-decreasing if, for any with , one has . Theorem 5 (see [9]). Let be a partially ordered set and be a metric on such that is a complete metric space. Suppose that is a continuous map and there exists such that for each with . If there exists such that , then has a fixed point in . Definition 6. Let be a nonempty set and . Let . The orbit of is defined by . Definition 7. Let be a metric space and . is said to be T-orbitally complete if every Cauchy sequence in , , converges to a point in . Definition 8. Let be a metric space and .