The Fixed Points Of Some Operators Of Operators In Ordered Passive Spaces

The fixed point theory plays an important role in nonlinear functional anal-ysis. In order to study the fixed point of some operators, the paper uses the properties of ordering difference and the sequence gaps, partial order method, lattice structure etc. Then we obtain some new fixed point theorems. The main research results are as follows:First, in this paper, by using the properties of ordering difference and the monotone iterative method as well as partial order method, we obtain existence and uniqueness of the fixed point for anti-mixed monotone operator and the mixed monotone operator in partially ordered Banach space. Finally the result is applied to initial value problem of the first order ordinary differential equation group and the existence and uniqueness of positive solutions are obtained.Second, by using the method of lattice structure combined with partial order, we study the operator A = CB in (ruo) complete Archimedean vector lattice and existence of the fixed point is obtained.Third, the concepts of order difference pair and sequence gaps are given. Then we use the properties of the sequence gaps, the monotone iterative method and mathematical induction to obtain the existence and uniqueness of fixed point for monotone operator without compactness, continuity, and any convex conditions. Meanwhile the conclusion is applied to the problem of Volterra integral equation group and the existence and uniqueness of positive solutions are obtained.