Some Problems Of Monotone Operators Associate With Condition

In this thesis we study a new kind of monotone operators,called T-monotone operators. Let X be a Banach space,T:X(?)X and A:X(?)X~* are multiple-valued operators. A is called T-monotone if (?)x,y∈X,(?)u∈T(x),v∈T(y) and (?)p∈A(x),q∈A(y) we have ≥0 The concept of T-monotone operators is natural and essential generalization of monotone operators. We describe some basic properties of T-monotone operators and get following main results: Firstly,we talk about the relations of T-monotone operators and monotone operators by theorem 3.2: Let H be a Hilbert space,A,T : H(?)H. Then A is T-monotonicity if and only if there exist monotone operator F : H(?)H such that A= F o T. Secondly,we proved the T-monotonicity and monotonicity of some special composite operators by theorem 4.7: Let H is a real Hilbert space,S,T : H(?)H. If S is a monotone operator,F is a linear relation and R(F) (?) Dom(S) and T is continue selection of F. Then SF is T-monotone.Thirdly,we get some properties and relations of T-monotone operators between T-monotone sets and non-extension operators by theorem 6.5: Let H is a real Hilbert space,S,T : H(?)H.Then S is T-monotone operator if and only if Graph(S) is a T-monotone set.We finished the article after many results which we expect on chapter 7.