Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results
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Research Article Ordered Non-Archimedean Fuzzy Metric Spaces and Some Fixed Point Results Ishak Altun1 and Dorel Mihet¸2 1
Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey 2 Departament of Mathematics, Faculty of Mathematics and Computer Science, West University of Timis¸oara, Bv. V. Parvan 4, 300223 Timis¸oara, Romania Correspondence should be addressed to Ishak Altun, [email protected] Received 2 July 2009; Accepted 9 February 2010 Academic Editor: Mohamed A. Khamsi Copyright q 2010 I. Altun and D. Mihet¸. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In the present paper we provide two different kinds of fixed point theorems on ordered nonArchimedean fuzzy metric spaces. First, two fixed point theorems are proved for fuzzy order ψ-contractive type mappings. Then a common fixed point theorem is given for noncontractive type mappings. Kirk’s problem on an extension of Caristi’s theorem is also discussed.
1. Introduction and Preliminaries After the definition of the concept of fuzzy metric space by some authors 1–3, the fixed point theory on these spaces has been developing see, e.g., 4–9. Generally, this theory on fuzzy metric space is done for contractive or contractive-type mappings see 2, 10–13 and references therein. In this paper we introduce the concept of fuzzy order ψ-contractive mappings and give two fixed point theorems on ordered non-Archimedean fuzzy metric spaces for fuzzy order ψ-contractive type mappings. Then, using an idea in 14, we will provide a common fixed point theorem for weakly increasing single-valued mappings in a complete fuzzy metric space endowed with a partial order induced by an appropriate function. Some fixed point results on ordered probabilistic metric spaces can be found in 15. For the sake of completeness, we briefly recall some notions from the theory of fuzzy metric spaces used in this paper. Definition 1.1 see 16. A binary operation ∗ : 0, 1 × 0, 1 → 0, 1 is called a continuous t-norm if 0, 1, ∗ is an Abelian topological monoid with the unit 1 such that a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ 0, 1.
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Fixed Point Theory and Applications
A continuous t-norm ∗ is of Hadˇzi´c-type if there exists a strictly increasing sequence {bn } ⊂ 0, 1 such that bn ∗ bn bn for all n ∈ N. Definition 1.2 see 3. A fuzzy metric space in the sense of Kramosil and Mich´alek is a triple X, M, ∗, where X is a nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X 2 × 0, ∞, satisfying the following properties: KM-1 Mx, y, 0 0, for all x, y ∈ X, KM-2 Mx, y, t 1, for all t > 0 if and only if x y, KM-3 Mx, y, t My, x, t, for all x, y ∈ X and t > 0, KM-4 Mx, y, · : 0, ∞ → 0, 1 is left continuous, for all x, y ∈ X, KM-5 Mx, z, t s ≥ Mx, y, t ∗ My, z, s, for
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