Coupled Fixed Point Theorem And Its Applications Based On Partial Ordered Set

In this paper, by introducing the concepts of relatively well-ordered complete set and well-ordered complete set, we obtain some existence theorems of coupled fixed point of mixed monotone mapping. Under this background, we study the existence of coupled solutions and extremal coupled solutions of operator equations on partial set. Finally, we introduce the applications of coupled fixed theorems and existence theorems of solutions of abstract operator equations. The whole includes four chapters.In the first chapter, we introduce the development history and research status of coupled fixed theory. Then, we briefly introduce the symbols and the main results of this article.In the second chapter, we introduce the concepts of supremum and infimum center and mixed monotone mapping. After that, we introduce the concepts of relatively well-order complete set and well-order complete set, we obtain the new coupled fixed theorems and extremal coupled fixed theorems.In the third chapter, we study the existence of coupled solutions of abstract operator equations which are based on the second chapter. Through monotone iterative method, We obtain the needed well-ordered set which is well-ordered chain. Then we obtain the existence theorems of coupled solutions and extremal coupled solutions of abstract operator equations.In the forth chapter, we introduce the applications of coupled fixed theorems and existence theorems of solutions of abstract operator equations.