Fixed Fuzzy Point Theorems for Fuzzy Mappings on Complete Metric Spaces

The Banach contraction principle appeared in explicit form in Banach’s thesis [5] in 1922 where it was used to establish the existence of a solution for an integral equation. Since then, it has become a very popular tool in solving existence problems in many branches of mathematics. Extensions of this principle were obtained either by generalizing the domain of mappings or by extending the contractive condition on the mappings ( see for example [1, 2, 4, 6, 7, 8, 10, 13, 14, 16, 18, 19] )s. Nadler [20] proved multivalued version of Banach contraction principle. In mathematical modeling of the real world problems, there are many inconveniences including the complexity of models and imprecision in differentiating the events exactly in real situations. Advances in computer science industry developed and modified many areas of research. There is still a major shortcoming of computers to deal with the uncertain and imprecise situations. To deal with this uncertainty Zadeh [23] in 1965, initiated the concept of fuzzy sets. Since then, many authors have employed this concept extensively in topology and analysis to develop this theory further and obtained several interesting applications. Now it is well recognized theory to handle uncertainties arising in various real life situations. Heilpern [12] introduced fuzzy mappings on a metric space and proved a fixed point theorem for fuzzy contraction mappings as a generalization of Nadler’s theorem [20]. For more results on fuzzy mappings we refer to [9, 21, 22].