Class Of Differential Boundary Value Problem

Nonlinear Functional Analysis is one of the most important research fields in modern anal-ysis mathematics. Its mainly theory include cone theory, topological degree theory, fixed pointtheory and so on. It has various methods to research nonlinear problems such as partial ordermethod, variational method and topological degree method et.al. All those of both rich theoriesand methods provide a much effective theoretical tool for solving nonlinear problems arising inthe fields of science and technology. Especially, it plays an important role to handle all kinds ofmathematical models corresponding to nonlinear differential equations, nonlinear integral equa-tions and partial differential equations in actual problems. Both domestic and foreign well-knownmathematics, such as H.Amann[51], K.Deimling[52], Gongqing Zhang[55], Wenyuan Chen[53],Dajun Guo[45]-[47], [54] and so on, has obtain excellent achievements in variations fields ofnonlinear functional analysis.The present paper mainly investigates a class of higher-order differential equations boundaryvalue problems, especially higher-order nonlinear singular boundary value problem. Nonlinearboundary value problem appears in the fields of nonlinear optics, newtonian ?uid mechanics andso on. It has attracted many attentions of mathematics and other technicians. The existence,uniqueness and multiple positive solutions for nonlinear singular boundary value problems havebeen studied extensively in recent years(see [12]-[14], [16]-[28] and the references therein).By using partial order method, topological degree theory, fixed point index theory and so on,the aim of the present paper further deeply discusses higher-order nonlinear singular boundaryvalue problems on the basis of known references.The paper is divided into four chapters. In chapter one, we mainly introduce background ofnonlinear singular boundary value problems and the main work of the present paper. The chap-ter two, we discuss the existence of nonlinear fourth-order differential equation Sturm-Liouvilleboundary value problems. By applying topological degree theory, we obtain new sufficient condi-tions of the bounded solutions for fourth-order Sturm-Liouville boundary value problems. The in-teresting point is that the nonlinear term not only contain second-order and third-order derivatives,but also may be change sign. In chapter three, we consider the positive solutions of fourth-ordernonlinear singular Sturm-Liouville boundary value problems by making use of fixed point indextheory. Under some weaker conditions, we establish some new results of at least one positivesolutions, at least two positive solutions for the fourth-order singular boundary value problems.The fourth chapter focus on the study of the existence of solutions for nth-order differential equa-tions with integral boundary conditions and with derivative boundary conditions. By constructing valid integral operator together with employing fixed point index theory, we obtain the existenceof at least one solutions and multiple solutions for nth-order ordinary differential equations withintegral boundary conditions and with derivative boundary conditions. The main results extendand improve the corresponding results of webb[1]. However, our new results cover a wide rangeof functions.