Several Classes Of Nonlinear Differential Equation Boundary Value Problem For The Existence Of Positive Solutions Is Studied

Nonlinear functional analysis is a research field of mathematics which hasprofound theories and extensive applications. It takes the nonlinear problemsappearing in mathematics and the natural sciences as background to establishsome general theories and methods to handle nonlinear problems. Its rich theoryand advanced method have provided the efective theory tool for solving manykinds of nonlinear diferential equations,nonlinear integral equations and someother types of equations, and handling many nonlinear problems in computa-tional mathematics, cybernetics, optimized theory, dynamic system, economicalmathematics and so on. At present, the contents of nonlinear functional analysismainly have to topology degree theory, critical point theory, partial order method,analysis method, monotone mapping theory and so on. In recent years nonlinearproblems have received highly attention of the domestic and foreign mathematicsand natural science field, so the research on nonlinear functional analysis and itsapplications is very important in both theory and applications.The boundary value problems of nonlinear diferential equations are impor-tant subjects in the theory of diferential equations. Owing to the important inboth theory and in applications, boundary value problems for diferential equa-tions have been attracted many researchers, and a large number of results havebeen obtained. Under the impetus of functional analysis and practical problems,the development of the research on boundary value problems for nonlinear dif-ferential equations is rapid. However, boundary value problems for nonlinearsemi-positone diferential equation system are the hot spot which have discussedin recent years, and higher order singular diferential equation boundary valueproblem is one of most hot field of nonlinear functional analysis at present.The present paper employs the nonlinear functional analysis theory andmethod, such as cone theory, fixed point theory, fixed point index theory, to in-vestigate the existence for positive solutions to several kinds of singular boundaryvalue problems of nonlinear diferential equations(system), including higher or-der diferential equations, semi-positone problems and singular boundary valueproblems. By deep study, we obtain some new interesting results. The thesisis divided into three chapters. In Chapter1, we study the existence of positive solutions for the following tourth-order nonlinear singular semi-positone system: where are constants such that f,g may be singular at t=0and/or t=1,in which R+=[0,+∞),R-=(-∞,0]. By choosing a proper cone and the Guo—Krasnosel’skii fixed point theorem,the existence of the positive solutions is established. So this chapter improve the main resuls of[6],[13](see Remark1.3.1). In Chapter2,we study the existence of positive solutions of singular semi-positone boundary value problem where a,b,c,d are nonnegative constants such that1.are continuous. Lebesgue integrable.∫,g may be singular at ι=0and/or ι=1,and q can have finitely many singularities in [0,1]. And by the fixed point theory for strict set contraction operators and Ascoli—Arzela theorem,we obtain the existence of positive solution.And the main results improve[31],[32](see Remark2.3.1). In Chapter3,we considered the existence of positive solutions for the following higher order multi-point singular semi-positone boundary value problem and the multi-point boundary condition where λ>0is a parameter,m≥1andn≥2are integers,a:(0,1)â†'[0,∞)and a(l)may be singular at l=0and/or l=1,∫:[0,∞)â†'[0,∞)is continuous, there exists M>0such that.f(x)>-M,x∈[0,∞), ξj∈(0,1)and0<ξ1<…<ξ∈.<1,αj,βj∈[0,∞),j=1,…,m.By Guo—Krasnosel’skii fixed point theorem,we obtain sufficient conditions the existence of positive solution when λ is small or larger. And the main results improve[38],[39],[41](see Remark3.3.1).