The Attractor Of Operator Semigroup On The Uniform Spaces

The concept of semigroups of operators is defined in separate uniform spaces.The relations between totally bounded sets,basic bounded sets and relative compact sets are discussed.It is concluded that the basic bounded sets are equivalent to the relative compact sets and the relative compact sets are the bounded sets.In the process of discussing totally bounded sets and ralative compact sets,we prove the completion theorem of uniform space.The totally bounded set,the relative compact set and the basic bounded set in the complete uniform space are equivalent.Further,the TK class operator semigroup is defined by means of a totally bounded set in a separate uniform space,and TAK is defined in conjunction with the concept of a uniform space network.At the same time,we also defines a global TB-attractor that attracts a totally bounded set.The relations between ?-limit set and orbit,the minimum closed global attractor and the minimum closed global TB-attractor of the semigroup of operators are discussed in separate uniform spaces.The ?-limit set of the set of TK,TAK class operator semigroups attracts its own nonempty invariant minimum compact set and global(or TB)Sufficient conditions exist for attrac-tors.For example,the TAK class operator semigroup in a separate unifor-m space(X,U?).if B and the orbit r+(B)are a totally bounded set,?(B)is a nonempty minimal compact invariant set that attracts B.And if B is connected and semigroup {Vt} is continuous,?-(B)is also connected.Let {Vt} be a totally bounded TK,TAK class semigroup of operators defined in a separate uni-form space.Non-empty minimum closed global attractors M = Ux?X?(x)and non-empty minimal closed global TB-attractors M UB?B?(B),and M and M are all positive.