The Existence Of Positive Solutions For Second Order Singular Differential Equations Boundary Value Problems With Integral Boundary Conditions

The boundary value problems of ordinary differential equations arise in a variety of areas of applied science, such as applied mathematics, physics and control theory etc., and therefore the study for boundary value problems is very meaningful in application and theory. With the great development of science and technology, various kinds of nonlinear problems arise in many fields of natural science, engineering technology and social science (such as physics, ecology and economics etc). It is known that integral boundary value problems can cover two-point and multi-point boundary value problems as special cases under certain conditions. As a consequence, we study the problem on the existence of positive solutions for a class of second-order singular differential equations with three-point integral boundary value problems and a class of second-order singular differential equations with nonlinear integral boundary conditions.First, we study the problems on the existence of positive solutions for a kind of second-order singular differential equations with three-point integral boundary value conditions where the parameter λ>0,δ>ζ,ζis in[0,1],η(s),η2(s) are Riemann-Stieltjes integrals on [0,1],f:[0,1] x (0,+∞)â†'[0,+∞) is continuous, h(t):(0,1)â†'[0,+∞) is continuous, and our nonlinear term h(t) may be singular at t=0,1and f(t,x) may be singular at x=0. By means of the cone fixed point theorem and some analysis skills, we get some new results on the existence of positive solutions for the boundary value problem.Next, we study the problem on the existence of positive solutions for a class of second-order singular nonlinear integral boundary value problems where the parameter λ>0;f:(0,1)×(0,+∞)∞[0,+∞) is continuous and f(t,x)≠0; ζ(s),η(s) are nondecreasing on [0,1], and the integrals for above are Riemann-Stieltjes integrals; hi:[0,1]×[0,+∞)â†'[0,+∞)(i=1,2) are continuous; f(t,x) may be singular at t=0,1and x=0. By using Schauder’s fixed point theorem and some analysis skills, we present some new sufficient conditions which guarantee the existence of positive solutions for these boundary value problems. And our results generalize and improve relative results in the paper [12].