Fixed Point Theorems Of Multi-valued ?-contraction In Fuzzy Metric Space

In this thesis,we research on fixed points theorems in fuzzy metric space and Menger probabilistic normed spaces,such as fixed points theorems?the approximate property of fixed point and well-posedness of fixed point problem of multi-valued ?-contraction in fuzzy metric space,together with topological degree theory and its applications in Menger probabilistic normed spaces.Main contents are as follow:In the first chapter,we introduce the historical background and the development status of fuzzy metric space.We introduce the basic notions?topology and some properties of fuzzy metric space.We prove some properties of Hausdorff fuzzy metric.In the second chapter,we give the definition of multi-valued ?-contraction in fuzzy metric space and prove that there exists fixed points for such multi-valued operator.More if suppose the set of strict fixed point is not empty,the fixed point is unique.We prove the existence and uniqueness of an attractor.We deal with the well-posedness of fixed point problem of ?-contraction.In the third chapter,we introduce the topological degree theorey of compact continuous mapping in Menger probabilistic normed spaces.We prove the product theorem of topological degree and index product theorem of isolated zero point.We give out a fixed point theorem applying the topological degree:? is a bounded open set and T:(?) ? X is a compact continuous mapping,if for(?)x ?(?)?,t>0,FTx-x(t)? FTx-e(t),then T has at least one fixed point.