Positive Solutions And Applications Of A Class Of Mixed Monotone Operator Equations

Nonlinear operator theory as the nonlinear science theory and tools, already more and more attention by scholars. As a kind of nonlinear operator, the mixed monotone operator is very important, the existence and uniqueness of solutions for it also becomes particularly important.In this paper, on the basis of predecessors’research,we mainly be concerned with the existence and uniqueness of positive solution of a mixed monotone operator equation x=λA(x,x)+B(x,x) in Banach Space by the properties of normal cones and partially ordered methods. Using these two theorems, Elastic beam equation and Lane-Emden-Fowler singular problems are analyzed respectively. The organization of this article is as follows:In chapter one, we introduces the historical background of this issue and the significance of this research subject. And we analyze the present research in both domestic and international, and propose the structure of this article. Finally, we will give some preliminaries. Including the definitions of cone and the mixed monotone operator,fixed point.Moreover,Lemma1.4.2.1is given which will be used for later.In chapter two, we will give some preliminaries definitions and lemmas about altering distance function.In chapter three, we mainly give two theorems and the detailed proofs are also given. The two theorems are both for the operator equation x=λ(x,x)+B(x,x) whose the existence and uniqueness of positive solutions, while the difference comes from the limiting of A operator. In Theorem3.1, we will let the real numbers β>1, to prove the existence and uniqueness of equation; Theorem3.2will change the scope ofβto O<γ<β≤1, got the same conclusion, the existence and uniqueness of solutions for operator equations.In chapter four, the main purpose is useing elastic beam equation and Lane-Emden-Fowler equation with singular problem as an example to validate the conclusion in the third chapter. Through the contents of the third chapter, we can obtain the existence and uniqueness solution of the elastic beam equation with three parameters and general Lane-Emden-Fowler equation of singular problem solution.And give the detailed proof.