Theory Of Patially Ordered F-type Topological Space And Its Applications

In this paper, we mainly discuss the fixed point theorems of monotone mappings and the solvability of a class of nonlinear operator equations in partially ordered F-type topological spaces. The main contents are as follows:In chapter 1, the definition of F-type topological space and its characterization are presented. In F-type topological space, we introduce the concepts of φ-accessorial order, upper (lower) order-completeness, upper (lower) order-continuity, ordered upper (lower) semicontinuity etc. On this base, we research the basal properties of partially ordered F-type topological spaces.In chapter 2, by virtue of the properties of partially ordered F-type topological space, we establish some fixed point theorems of increasing mappings in this kind of space. At the same time, we investigate the existence of maximum fixed point and minimum fixed point of increasing mappings on an order interval. Making use of these results, we prove the couple fixed point theorems of mixed monotone mappings in F-type topological spaces.In chapter 3, as the direct applications of those theorems in chapter 2, we get several corresponding fixed point theorems of increasing mappings and the couple fixed point theorems of mixed monotone mappings in Menger probablistic metric spaces and fuzzy metric spaces.In chapter 4, in the frame of partially ordered F-type topological space, we research the existence of the solution of a nonlinear operator equation Lx = Nx, and the convergence of the iterative sequence {u_n} which satisfies Lu_n+1 = Nu_n for a solution of the equation. Then, we give the conditions for multiplicity of the solutions of this operator equation.The above results are the improvement and generalization of the corresponding results in partially ordered metric spaces which had been given by Zhang xian [2] , Feng yuqiang and Liu sanyang [14,20].