Research Of Positive Solutions To Boundary Value Problems For Nonlinear Singular Differential Equations

Based on the cone expansion and compression fixed point theorem of convex functional type, this paper mainly discusses the existence of positive solutions for two kinds of the fourth order singular nonlinear differential equations.First, we discuss the existence of positive solution for the fourth-order two-point singular boundary value problems where h:(0,1)?[0,+?) is continuous, and h(t)>0, f(t,x,y) is allowed to be singular at t=0, t=1 and x=y=0. By the function [jr1r2]n (t) and fn (t, u, v), we can prove that the operator T is completely continuous operator in C1 [0,1]. On the basis of a proper cone and a uniformly convex functional, we can prove that operator T has at least one fixed point in some particular set by the cone expansion and compression fixed point theorem of convex functional type, and the main results obtained generalize the existing results.After that, we study the existence of positive solution for the fourth order with integrals boundary value problem where ??C((0,1),[0,+00)),?(t) is allowed to be singular at t=0,1.f?C([0,1]×[0,+?) ×(-?,0],[0,+?)),ki,hi?L1[0,1] (i=1,2) is nonnegative. It discusses the properties for the Green function of the boundary value problem, and an equivalent integral equation with boundary vale problem is given. On the basis of completely continuous operator, we can prove that the fixed point of the operator is equivalent to the solution of the integral equation. By constructing a proper cone and a uniformly convex functional, the sufficient condition of existing one or two fixed points is given by the cone expansion and compression fixed point theorem of convex functional type, and the main results obtained generalize the existing results.Finally, some actual examples are given to illustrate the validity of the results of above.