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
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.
2. Main Result
This shows that the contractive condition of Theorem 1 of  is not satisfied.
- Matthews SG: Partial metric topology. In Proceedings of the 8th Summer Conference on General Topology and Applications, 1994. Volume 728. Annals of the New York Academy of Sciences; 183–197.Google Scholar
- Valero O: On Banach fixed point theorems for partial metric spaces. Applied General Topology 2005,6(2):229–240.MATHMathSciNetView ArticleGoogle Scholar
- Oltra S, Valero O: Banach's fixed point theorem for partial metric spaces. Rendiconti dell'Istituto di Matematica dell'Università di Trieste 2004,36(1–2):17–26.MATHMathSciNetGoogle Scholar
- Altun I, Sola F, Simsek H: Generalized contractions on partial metric spaces. Topology and Its Applications 2010,157(18):2778–2785. 10.1016/j.topol.2010.08.017MATHMathSciNetView ArticleGoogle Scholar
- Romaguera S: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory and Applications 2010, 2010:-6.MathSciNetView ArticleMATHGoogle Scholar
- Altun I, Simsek H: Some fixed point theorems on dualistic partial metric spaces. Journal of Advanced Mathematical Studies 2008,1(1–2):1–8.MATHMathSciNetGoogle Scholar
- Heckmann R: Approximation of metric spaces by partial metric spaces. Applied Categorical Structures 1999,7(1–2):71–83.MATHMathSciNetView ArticleGoogle Scholar
- Escardó MH: PCF extended with real numbers. Theoretical Computer Science 1996,162(1):79–115. 10.1016/0304-3975(95)00250-2MATHMathSciNetView ArticleGoogle Scholar
- Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society 2004,132(5):1435–1443. 10.1090/S0002-9939-03-07220-4MATHMathSciNetView ArticleGoogle Scholar
- Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Applicable Analysis 2008,87(1):109–116. 10.1080/00036810701556151MATHMathSciNetView ArticleGoogle Scholar
- Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory and Applications 2010, 2010:-17.Google Scholar
- Beg I, Butt AR: Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(9):3699–3704. 10.1016/j.na.2009.02.027MATHMathSciNetView ArticleGoogle Scholar
- Ciric L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008, 2008: 11.View ArticleMathSciNetMATHGoogle Scholar
- Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications 2010,72(3–4):1188–1197. 10.1016/j.na.2009.08.003MATHMathSciNetView ArticleGoogle Scholar
- Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005,22(3):223–239. 10.1007/s11083-005-9018-5MATHMathSciNetView ArticleGoogle Scholar
- Altun I: Some fixed point theorems for single and multi valued mappings on ordered non-Archimedean fuzzy metric spaces. Iranian Journal of Fuzzy Systems 2010,7(1):91–96.MATHMathSciNetGoogle Scholar
- Altun I, Miheţ D: Ordered non-Archimedean fuzzy metric spaces and some fixed point results. Fixed Point Theory and Applications 2010, 2010:-11.Google Scholar
- Altun I, Imdad M: Some fixed point theorems on ordered uniform spaces. Filomat 2009, 23: 15–22. 10.2298/FIL0903015AMATHView ArticleMathSciNetGoogle Scholar
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