The Study Of Nonlinear Mixed Monotone Operators And Their Applications

In this thesis, some classes of fixed point theorems of mixed monotone operators are discussed by using cone theory and altering distance function as well as the properties of linear operator. Besides, the existence and uniqueness of positive solution for second order nonlinear differential equations are considered by using these theorems. The findings extend and improve some results in the relevant literatures. The overall structure of this thesis as follows:In chapter1, the background of discussed problems in this thesis is introduced briefly. Meanwhile, the main works are presented in detail.In chapter2, the existence of fixed points for mixed monotone mappings is proved by using altering distance function and some new reusults are obtained. Theorem2.3.1discusses the existence and uniqueness of fixed points of mixed monotone mappings without continuity in partially ordered metric space; Theorem2.3.2discusses the existence and uniqueness of fixed points for mixed monotone operators without the condition of existence of upper-lower solutions in real Banach space. For application purpose, related example is given for illustration.In chapter3, we discuss a class of0convex-φ concave mixed monotone operators without the continuity and compactness. At the same time, the existence theorem of positive fixed point is obtained. In particular, we do not demand the existence of upper-lower solutions condition. The relevant conclusions are developed and exteded.Theorem3.3.1discusses the φ convex-φ concave mixed monotone operator without including the condition of upper-lower solutions. The existence and uniqueness of fixed point of the operator are derived by using the cone theory and linear operator theory. Theorem3.3.3discusses the φ convex-φ concave mixed monotone operator and proves the existence of fixed point of the operator in a complete ordered closed subset. Some new results for this kind of operators on weaker condition are obtained.In chapter4, we consider three-point boundary value problems for second order nonlinear differential equations with mixed monotone term, whereη∈(0,1),β>0,1-βη>0.The research focus on the existence and uniqueness of positive solution of BVP (4.1.1), where the existence of upper-lower solutions is not demanded. It is proved by using a new fixed point theorem of generalized α-concave-convex operators. As application, we give related example to illustrate our results.