The Existence Of Positive Solutions To Some Kinds Of Differential Equation

Along with science’s and technology’s development, various non-linear prob-lem has come up from the fields of physics, chemisty, mathematics, biology,medicine, economics, engineering, cybernetics, and these problems has arousedpeople’s widespread attention day by day. However, the nonlinear functionalanalysis ofers efective theoretic tools for these problems, and it is a subjectof profound theories and broad applications. The nonlinear functional analysisbases on nonlinear problems of math and science, constructs general theories andmethods, and plays an important role in dealing with all kinds of nonlinear inte-gral equations, nonlinear diferential equations and partial diferential equations.So the nonlinear analysis has become one important research directions in mod-ern mathematics. The nonlinear functional analysis is an important branch innonlinear analysis, because it can explain well various the natural phenomenon.The boundary value problem of nonlinear diferential equation stems from theapplied mathematics, the physics, the cybernetics and several kinds of applica-tion discipline. It is one of most active domains of functional analysis at present.The singular diferential equations is also the hot spot which have been discussedin recent years. So they become a very important domain of diferential equationresearch at present.In this paper, we use fixed point theorem, fixed point index and M¨onch Fixedpoint theorem, to study some kinds of singular value problems and non-singularboundary value problems. We obtain the existence and uniqueness of solutions forthese boundary value problems and apply the main results to the boundary valueproblem for the diferential equation. The thesis is divided into three chapters.In Chapter1, we investigate the the existence result of positive solutionsnonlinear singular fourth-order integral BVP for an increasing homeomorphismand homomorphism where λ>0, φ: Râ†'R is an increasing homeomorphism and homomorphism,φ(0)=0, w(t) can be singular at t=0, t=1, f (t, u) may also be singular atu=0.In this interval, the existence of at least one positive solution tothe boundary value problem is guaranteed. Our results extend and improve mangknown results including singular and nonsingular cases.(see Remark1.1.2)In Chapter2, we obtains the existence of multiple positive solutions for aboundary value problem of second nonlinear impulsive integro-diferential equa-tions in a Banach space (E,)Fixed point index theory are used to obtain the existence of multiple positivesolutions for boundary value problem (2.1.1)-(2.1.2) and generalize and improvethe results. This chapter makes up for this shortfall. This chapter promotethe research space from real space to abstract space, promote the diferentialequations to the integro-diferential equations and obtain the existence of multiplepositive solutions.(see Remark2.1.1)In Chapter3, we consider the following BVP for multi-point singular difer-ential equations of mixed type on an unbounded domain in Banach space andthe cone theory and M¨onch fixed point theorem combined with a monotoneiterative technique are used to investigate the positive solutions of a class ofboundary problems for second-order nonlinear singular diferential equations withmulti-point boundary value conditions on an infinite interval in Banach spaces.To demonstrate the applications of our results, a examples are also given in thepaper.(see Remark3.1.2)...