Fixed Point Of Multivalued Mappings In Cone Metric Spaces

In this paper, we first note that the infimum of a set in general partially ordered topo-logical vector spaces dose not have the similar properties of the infimum of a real number set, and for this, we can not define Hausdorff cone metric on the family of all bounded and closed sets of a cone metric space. We define weakly continuous cone in ordered topological vector space, and find the above problem can be solved if the cone is weakly continuous. Then we get the fixed point theorems of strictly nonexpansive multivalued mappings un-der weakly continuous cone and non-weakly continuous cone respectively, applying our result, we generalize fixed point theorems for strictly nonexpansive single valued mappings of Huang and Zhang without the regular condition. We also discuss the fixed points of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in TVS-cone metric spaces, and give an example to explain that in the case of normal cone, the main result of Ing-Jer lin is wrong, we change conditions and prove it again. Finally, we obtain the coupled coincidence point theorem for set-valued mappings and single valued mappings in TV S-cone metric spaces.