On The Existence Of Positive Solutions Of Boundary Value Problems Of Differential Equations

Boundary value problems of ordinary diferential equations, as one of the impor-tant parts of the subject of ordinary diferential equation, exist in classical mechan-ics and eletrics pervasively. Ordinary diferential equations two-point boundary valueproblems(e.g. Dirichlet Boundary value problems, Neumann Boundary value problems,Robin Boundary value problems, Strum-Liouville Boundary value problems) have beenwidely studied. In fact, since Picard made researches on the existence and uniquenessof second order two-point boundary value problems of nonlinear ordinary diferentialequations by employing iterative methods in1893, the study on two-point boundaryvalue problems of ordinary diferential has been thriving.Since20th century, functional analysis has become the important theoretical basisof boundary value problems of ordinary diferential equations. In the mid-thirties ofthe last century, French mathematicians J. Leray and J. Schaulder set up the Leray-Schaulder theory. By employing the theory, researches made great success in linearordinary equations, integral equations and functional diferential equations. In partic-ular, the application of the theory to boundary value problems of ordinary diferentialequations forms the topological degree methods and functional methods. The core ofthe theory is the foundation and application of fixed point theorem.In this paper, we use the fixed point index theory, the fixed point theorem of gen-eralized cone expansion and compression as well as upper and lower solutions method,to study the positive solution of the several kinds of boundary value problems fornonlinear diferential equation.The thesis is divided into four chapters according to contents.The first chapter chiefly narrates the background of the nonlinear analysis andinnovation of the paper.In chapter2, we use the fixed point theorem of generalized cone expansion andcompression to investigate the following fourth order singular m-point boundary value problem where f∈C((0,1)×[0,+∞),[0,+∞)),f(t,u)is allowed to be singular at t=0,t=1, moreover0<αi<1,0<βi<1,i=1,2,…,m-2,0<η1<η2<…<ηm-2<1, and we obtain the existence of positive solution for the singular super-linear boundary value problem.In chapter3,By using upper and lower solutions method and Schaulder fixed point theorem,we investigate the second order boundary value problem with delay where f∈C([0,1]×[0,+∞),[0,+∞)),0<η1<<η2<…<ηm-2<1are constants, we obtain the existence of positive solution for the second order boundary value problem with delay.In chapter4,By using the fixed point index theory,we investigate the fourth order boundary value problem with p-Laplacian operator where φp(s)=|s|p-2s,p>1,(φp)-1=φq,1/p+1/q=1.we obtain the existence of at least one or at least two positive solutions for the fourth order boundary value problem with p-Laplacian.