Positive Solutions For Several Kinds Of Boundary Value Problems Of Differential Equations

Nonlinear functional analysis which is an important and powerful tool for the non-linear problem is an important branch of the analysis of mathematics and has become an important direction of modern mathematics. As known to. us, when dealing with nonlinear problems of differential equation,especially the boundary value problems are often involved which origin from mathematics, engineering, biology, sybernetics and so on. With solving these problems, many important methods and theories such as upper and lower solutions method and fixed point theory of the cone have been developed effectively. This paper mainly investigates the existence of solutions for several kinds of boundery value problems of nonlinear differential equations by using fixed point theory, the theory of cone, nonlin-ear alternative theory and three-solution theorem. The existence and uniqueness of positive solutions for differential equations have been considered extensively since twenty years ago (see [1]-[40]).Based on these this paper discusses the existence of solutions for several kinds of boundary value problems of nonlinear differential equations. There are three chapters in this paper. Chapter 1 investigates the existence of unique positive and three positive solutions for the following second-order two-point boundary value problem with two parameters where f∈C([0,1]×[0,+∞),(0,+∞)),g∈C([0,1]×(-∞,+∞),(0,+∞)),λ≥0,μ≥0, andλ,μare not zero at the same time. In [1] the authors considered the existence of one positive solution by using Leray-Schauder fixed point theorem for the following second-order two-point boundery value problem with one parameterIn [2]the authors considered the existence of two positive solutions by using the fixed pointtheorem on cone for the following singular third-order three-point boundary value problem.However,there are few papers to investigate boundary value problom with two parameters asthis paper does. This paper firstly obtains one solution by using Leray-Schauder nonlinearalternative theory forλ> 0,μ> 0, and secondly gets an unique positive solution andthree positive solutions by using 0-concave operator fixed point theory and three-solutiontheorem forλ> 0,μ= 0. In chapter 2, we investigate the existence of positive solutions of the following fourth-order two-point BVP by using the fixed point theorem on cone In chapter 3 through constructing a special cone, we investigate the existence of twopositive solutions of the following second-order three-point BVP by using the fixed pointtheorem on conewhere the nonlinearity f(t,u) may be singular at t = 0, t = 1 and u = 0.