Positive Solutions Of Fourth Order Boundaryvalue Problems For Nonlinear Ordinary Differential Equations

Nonlinear functional analysis has been one of the most signifcant branches with a widerange of application in modern mathematics, which mainly includes topological degree the-ory,the variational method, monotone operator theory,cone theory.etc. The most importantproblem is nonlinear integral equations and diferential equations. People at home and abroadhad profound works in the research of nonlinear functional analysis. L.E.J.Brouwer had estab-lished the concept of topological degree for fnite dimensional space(Broawer degree) in1912.J.Leray had extended the completely continuous feld of Banach space, he established the con-cept of Leray-schauder degree in1943. E.Rothe, M.A.Krasnosel’skii, P.H.Rabinowitz, H.Amann,K.Deimling had carried on nonlinear functional analysis and applications. Many famous math-ematicians in China, say Zhang Gongging, Guo Dajun, Sun Jingxian etc., played an importantrole in nonlinear functional analysis, and the results of their research are also widely used incontrol theory, optimization theory,computational mathematics, economics mathematics.The fourth-order boundary value problems of nonlinear diferential equations is amongthe most important parts in the theory of diferential equations.It plays an important role inboth applications and practical signifcance, boundary value problems for ordinary diferentialequations have been studied widely, many fruitful results are established in recent years. Mostof these results are obtained via transforming the fourth-order boundary value problem intoa second-order boundary value problem, and then we apply the fxed point index theory, thetopological degree theory, the cone theory, the lower and upper solutions method to obtainpositive solutions.Although there are many scholars studing the existence of fourth-order boundary valueproblems especially on the existence of positive solutions, but because of those most commonfxed point index theorem have many conditions, so that the applicability of them has certainlimitations. Therefore, there are still many challenging problems to be solved. This paper is onthe basis of the previous studies to improve and broaden some restrictions. So that, this articlegeneralized their results.In this thesis, we mainly use fxed point index theory to the system of fourth-order non-linear diferential equations boundary value problems. It divides into four chapters.In Chapter One, we study the existence of positive solutions of a fourth-order integral boundary value problem:where f∈C([0,1]×R+,R+), αi≥0,βi≥0, αi and βi are nondecreasing functions on [0,1]. The highlights of this article are that directly obtained the Green function of the equations (1.1.1) are difficult. We establish the Green function of the equations (1.1.1) by using and transforming the Green function of the equations (1.1.2). Based on a priori estimates achieved by utilizing some integral identities and inequalities, we use fixed point index theory to prove the existence of positive solutions for the equations of (1.1.1).In the second chapter, we study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problems:where f, g∈C([0,1]×R8+,R+)(R+:=[0,∞)).. The highlights of this article are that the application of concave functions in obtaining a priori estimates, the application of nonnegative matrices in obtaining a priori estimates, the reduction of order. By the method of article [31], We can obtain the existence and multiplicity of positive solutions(1.2.1) by utilizing some integral identities and inequalities and R+2-monotone matrices.In the third chapter, we study the existence of positive solutions for the system of fourth-order boundary value problems:where f, g∈C([0,1]×R8+,R+)(R+:=[0,∞)).. The main difficulty in treating (1.3.1) comes from the presence of odd orders, in particular, in the nonlinearity f, g. To overcome this difficulty, we employ the method of order reduction to transform (1.3.1) to an initial value problem for a first-order integro-differential equation. By the method of [49], and based on a priori estimates achieved by developing spectral properties of associated parameterized linear integral operators, we use the fixed point index theory to prove the the existence and multiplicity of positive solutions(1.3.1).In Chapter Four, we study the existence of positive solutions for the system of fourth-order boundary value problems:where f, g∈C([0,1]×R8+,R+)(R+:=[0,∞)). Compared with the chapter three, the chapter contains different boundary value conditions and the presence of first order derivatives in the nonlinearities, we first use the method of order reduction with Chapter three, and transform (1.4.1) to an initial value problem for a first-order integro-differential equation. We can obtain some integral identities and inequalities and a priori estimates. In this chapter, that matrix theory also plays an important role in our proofs.