- Research Article
- Open Access
Fixed Point Theorems for Monotone Mappings on Partial Metric Spaces
© I. Altun and A. Erduran. 2011
- Received: 12 November 2010
- Accepted: 24 December 2010
- Published: 28 December 2010
- Partial Order
- Point Theorem
- Differential Geometry
- Matrix Equation
- Monotone Mapping
There are a lot of fixed and common fixed point results in different types of spaces. For example, metric spaces, fuzzy metric spaces, and uniform spaces. One of the most interesting is partial metric space, which is defined by Matthews . In partial metric spaces, the distance of a point in the self may not be zero. After the definition of partial metric space, Matthews proved the partial metric version of Banach fixed point theorem. Then, Valero , Oltra and Valero , and Altun et al.  gave some generalizations of the result of Matthews. Again, Romaguera  proved the Caristi type fixed point theorem on this space.
A partial metric space is a pair such that is a nonempty set and is a partial metric on . It is clear that if , then from (p1) and (p2) . But if , may not be 0. A basic example of a partial metric space is the pair , where for all . Other examples of partial metric spaces, which are interesting from a computational point of view, may be found in [1, 8].
On the other hand, existence of fixed points in partially ordered sets has been considered recently in , and some generalizations of the result of  are given in [10–15] in a partial ordered metric spaces. Also, in , some applications to matrix equations are presented; in [14, 15], some applications to ordinary differential equations are given. Also, we can find some results on partial ordered fuzzy metric spaces and partial ordered uniform spaces in [16–18], respectively.
The aim of this paper is to combine the above ideas, that is, to give some fixed point theorems on ordered partial metric spaces.
This shows that the contractive condition of Theorem 1 of  is not satisfied.
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