Several Types Of The Differential Equation Boundary Value Problem For Positive Solutions And Applications

With the development of science and technology, various nonlinear problem hasaroused people,s widespread interest day by day, the nonlinear analysis has becomeone of the important research directions of modern mathematics. It mainly studiesall kinds of nonlinear diferential equations. Boundary value problem of diferentialequations has important applications in mathematics, physics, economics and so on.Fractional diferential equations is a hot topic in recent years, research in this area is avery important area. Since20th century, functional analysis has become the importanttheoretical basis of boundary value problems of ordinary diferential equations. In thispaper, using the cone theory, fixed point theorem for nonlinear functional methodssuch as the nonlinear fractional diferential equation(for Systems) integral existence ofpositive solutions for boundary value problems, and obtained some new results. Thethesis is divided into five sections according to contents.Chapter1Preference, we introduce some background materials from boundaryvalue problem of diferential equations theory and main works of this paper.Chapter2without any compact or continuous assumption, we obtain some newexistence and uniqueness theorems of positive fixed point of e-concave-convex mixedmonotone operators in Banach spaces partially ordered by a cone.Chapter3In this chapter, we deal with the existence of solutions for boundaryvalue problem for fractional order diferential equation of the formwherecDα0+denote α-order caputo-fractional order diferential,1<α <2, θ imply thezero element of Banach space E, f(t, u(t), u (t)):[0,1]×E×E→E. by using thefixed theorem of sadovski we obtain the existence of solutions for the equation.Chapter4In the cases of nonlinear term f being variable, we discuss the followingfourth-order boundary value problem with integral boundary conditions among φ: R→R is an increasing homeomorphism,Homomorphismφ(0)=0.we obtainthe existence of at least two positive solutions for the boundary value problem.