Existence Of Positive Solutions For Nonlinear Fourth-order Boundary Problem

Nonlinear differential equations of fourth-order boundary problem in fluids mechanics and elastic mechanics of physics, have practical meanings, some scholars studied such problems, and received some conclusions. In this paper, we study three kinds of different boundary value conditions of the existence of positive solutions for fourth-order boundary problem, one class of nonlinear fourth-order two point boundary value problem, one class of nonlinear fourth-order two point boundary value problem with parameter and one class of nonlinear fourth-order three point boundary value problem with parameter. With apposite assumptions, prove the existence of positive solutions for three types of boundary value problems. Full paper is divided into four chapters:At first, elaborate the research background and the research status at home and abroad of the nonlinear fourth-order differential equations boundary value problem, basic knowledge, basic assumptions and mathematical theorems used in this article, include non-homogeneous equation of general solution structure and cone fixed point theory. Then, study the following conditions nonlinear fourth-order two-point boundary value problem(4) 4()()(,())(0)(1)(0)(1) 0u t u t f t u t u u u uì -r =í=￠ =￠ =￠￠ =?use the fourth-order boundary value problem into its equivalent of two second-order boundary value problem method, employ variation of constants, get operator equations, use the cone fixed point theorem, prove the existence of positive solutions to the problem. Afterward, study nonlinear fourth-order two-point boundary value problem with parameters(4) 4()()(,())(0) 0,(1) 0(0) 0,(1)u t u t f t u t u u u urlì - =?í= =?￠ =￠ = -?structure Green’s function of the nonlinear fourth-order two-point BVP with parameters l, find the right0l ?(0,￥), when 0l ?(0,l], and then use the cone fixed point theorem prove the existence of positive solutions. At last, study nonlinear fourth-order three-point boundary value problem with parameters(4) 4()()(,()),(0) 0,(1)() 0,(0) 0,(1)().u t u t f t u t u u u u u ura ha h lì - =?í= - =?￠ =￠ -￠ = -?by appropriate transformation, and use the results of the third chapter of the Green function, use the structure of non-homogeneous equations general solution, give the Green function of the boundary value problems, and establish upper and lower bounds estimate, apply conefixed point theorem, proved that when parameters0l ?(0,l], where0l ?(0,￥),the existence of positive solution to this problem.