Some Fixed Point Theorems For Nonlinear Contractions In Fuzzy Metric Spaces

In this paper,two classes of fixed point theorems for Boyd-Wong's type nonlinear cyclic ?contractive mappings are established in fuzzy metric spaces.These results can deduce the corresponding nonlinear form of the results,but also can deduce the fixed point theorems for Alber-Guerre Delabriere type and Geraghty type nonlinear cyclic contractive mappings.At the same time,we give two examples to illustrate the reliability of our theorems.Because the metric space is a special fuzzy metric space,we can apply our results to metric spaces and obtain the results of He et alThe Menger probabilistic metric space is also a special fuzzy metric space,but the establishing of fixed point theorems needs the condition(R-2)in fuzzy metric spaces In the setting of Menger probabilistic metric spaces,it usually only needs t-norm of H-type.We give a counterexample to show that there is no inclusion relation between the probabilistic metric space with the t-norm of H-type and the fuzzy metric space satisfying(R-2).So,the fixed point theorems in fuzzy metric spaces can not be directly applied to Menger probability metric spaces.