The Existence Of Positive Solutions For Fourth-Order Singular Boundary Value Problems

Singular nonlinear boundary value problems (SBVP. for short) have resulted from nuclear physics,gas dynamics. Newtonian fluid machanics. the theorey of boundary layer. nonlinear optics and so on. From 1980s,such problems have received a great deal of attention by many researchers. Therefore they become a new study field, and there are many excellent results. For example, see the results of reference [1-3. 5-7. 20. 27. 30].In recent years,the fourth-order boundary value problems can describe the deformation of an elastic beam equilibrium state and have comprehensive application in elasticity mechanics and engineering physics. Because either the function or the variable itself, which is mathematical model resulted from sonic important actual problems. may be singular at endpoints. the study of high order SBVP becomes very active.for example, see [2. 3. 19. 20-23. 27-30]. However, few papers have been reported on the same problems for multi-point SBVP. So the content that we studied has important theoretical and applicable vaue.This paper discusses the fourth-order SBVP more generally.and obtains some useful results on the basis of above discussions. At the same time.we study the corresponding multipoints problems and obtain the existence of at least one positive solution and multiple positive solutions to the problem.There are four chapters in the dissertation.In the first chapter,by using the Leggett-Williams fixed point theorem.we deal with the existence of multiple positive solutions and infinitely many positive solutions for the following fourth-order SBVPwhere f(t.x.y) maybe singular at t=0. and t=1. At present time.almost all papers have studied the existence of single or double solutions of the high order SBVP. To our knowledge.there is no paper to consider the existence of more solutions of the above problem. This chapter generalizes the previous results and an example is worked out to indicate our conditions are reasonable.In the second chapter, we investigates the existence of positive solutions of the following SBVPwhere the nonlinearity f(t.u) maybe singular at t=0.1, and u=0. 0<ξ,η<1,0< aξ,bη<1. In this chapter. under certain conditions.by constructing a special cone and using cone compression and expansion fixed point theorem.we obtain an excellent result and work out an example to indicate the application.The third chapter deals with the following multi-point SBVP with positive parameterλwhereξ_i.η_i∈(0.1).0<ξ₁<ξ₂<...<ξ_m-2<1,0<η₁<η₂<...<η_m2<1,a_i,b_i∈[0. +∞).sum from i=1 to m-2 a_i<1, sum from i=1 to m-2 b_i<1. and f(t.y) maybe singular at t-0.1 andy=0. By using fixed point theory in cones.an explicit interval forλis derived such that for anyλin this interval,the existence of at least one positive solution or double positive solutions to the above SBVP is guaranteed.Chapter 4 investigates the existence of at least one and multiple positive solutions of the following three-point SBVP in abstract space.where 00,f(t.x,y) maybe singular at t=0.1.x=θand y=θ,f∈C[(0.1)×P\{θ}×(-P)\{θ}. P].At last.two examples in finite-dimensional space or in infinite-dimensional space are worked out to indicate our conditions are reasonable.