Existence And Uniqueness Of Solutions For A Second Order Differential Equation With Integral Boundary Conditions

The ordinary differential equations has many different forms,complex branches and wide range of applications,so the research of the equations earns more and more concern.After centuries of development,the theory of ordinary differential equations become more and more perfect,and lots of methods are used in the research.For example, Nagumo conditions, upper and lower solution methods, fixed point theorem, Leray-Schauder degree theory, the regular perturbation method, the matching method, and so on. All of these are the most important methods.In the past two years, the boundary value problems of the differential equations with integral boundary conditions developed rapidly as a new research topic, including two, three, multi-point and nonlinear boundary conditions, etc. But the uniqueness of this kind of problems is still almost a blank.So the uniqueness of solution of the second order differential equations boundary value problem with integral boundary conditions is still a relatively new subject.In this paper, we consider the following second order differential equation with the upper and lower solution method, Nagumo conditions and Leray-Schauder degree theory:(φ(u’(t)))’+f(t,u(t),u’(t))=0,0<t<1subject to the following integral conditions: where f:[0,1]x R2â†'R,hi:Râ†'R(i=1,2),and fi are continuous, ki(i=1,2)are nonnegative constant, φ(u) is continuous and strictly increasing function with φ(0)=0,φ(R)=R.In the paper, we prove the uniqueness of the solutions firstly.This work Contributes a lot to the research of the boundary value problems of the differential equations with integral boundary conditions.