Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces
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Research Article Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces Xin-Qi Hu School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Correspondence should be addressed to Xin-Qi Hu, [email protected] Received 23 November 2010; Accepted 27 January 2011 Academic Editor: Ljubomir B. Ciric Copyright q 2011 Xin-Qi Hu. 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. We prove a common fixed point theorem for mappings under φ-contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of S.Sedghi et al. 2010
1. Introduction Since Zadeh 1 introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani 2, 3 gave the concept of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space which have very important applications in quantum particle physics particularly in connection with both string and E-infinity theory. ´ c 5 discussed the Bhaskar and Lakshmikantham 4, Lakshmikantham and Ciri´ mixed monotone mappings and gave some coupled fixed point theorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al. 6 gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang 7 gave some common fixed point theorems under φ-contractions for compatible and weakly compatible mappings in Menger probabilistic metric spaces. Many authors 8– 23 have proved fixed point theorems in intuitionistic fuzzy metric spaces or probabilistic metric spaces. In this paper, using similar proof as in 7, we give a new common fixed point theorem under weaker conditions than in 6 and give an example which shows that the result is a genuine generalization of the corresponding result in 6.
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Fixed Point Theory and Applications
2. Preliminaries First we give some definitions. Definition 1 see 2. A binary operation ∗ : 0, 1 × 0, 1 → 0, 1 is continuous t-norm if ∗ is satisfying the following conditions: 1 ∗ is commutative and associative; 2 ∗ is continuous; 3 a ∗ 1 a for all a ∈ 0, 1; 4 a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ 0, 1. Definition 2 see 24. Let sup0 1 − δ. Definition 3 see 2. A 3-tuple X, M, ∗ is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous t-norm, and M is a fuzzy set on X 2 × 0, ∞ satisfying the following conditions, for each x, y, z ∈ X and t, s > 0: FM-1 Mx, y, t > 0; FM-2 Mx, y, t 1 if and only if x y; FM-3 Mx, y, t My, x, t; FM-4 Mx, y, t ∗ My, z, s ≤ Mx, z, t s; FM-5 Mx, y, · : 0, ∞ → 0, 1 is continuous. Let X, M, ∗ be a fuzzy metric space. For t > 0, the open ball Bx, r,
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