Boundary Value Problems For Differential Equations And Fixed Points For Nonlinear Operators

In this paper, we mainly apply theory of ordering in nonlinear functional analysis, cone theory, topological degree theory, cone expansion and compression theory, and the method of upper and lower solutions, partial ordering method, iterative method to study some nonlinear differential equations BVP and fixed points problems of mixed monotone operators, and obtain some new results. This paper is composed of four chapters.In chapter 1, we make it as introduction of this paper, which introduces the main contents of this paper.In chapter 2, we mainly deals with the following BVP for Sturm-liouville differential equationswhereα₀,β₀,α₁,β₁ are non-negative real number, and let J = [ξ,η],p(t)∈C¹(ξ,η), p(t) > 0, f(t,u)∈C[(ï¿¡,η)×R, R], R = (-∞, +∞),We have such conclutions that (SL.ρ)BVP(2.1.1) has at least one or three positive solutions under some conditions (Theorem 2.3.1,. Corollary 2.3.2).In chapter 3, we mainly discuss a second-order three-point boundary value problems under Nagumo conditions.and we obtain two theorems(Thorem 3.2.1, Thorem 3.2.2) about one or three positive solutions mainly by the method of upper and lower solutions and Nagumo condition(see difinition[3.1.2]).In chapter 4, we discuss two classes of fixed points problems of mixed monotone operators in cone, and obtain two fixed points theorems of mixed monotone operators under the conditions of compactness and complete continuity or not, respectivly (Thorem 4.1.2), Thorem 4.2.4). We mainly make use of cone expansion and compression theory, some properties of nonlinear operators(αconcave,βconvex,φconcave-(—ψ) convex operator), and iterative method to get existence conclutions of fixed points under different conditions.