Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

Youming Zhou,Yan Xu

doi:10.1016/j.jmaa.2005.07.014

Abstract

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form { − u ″ = f ( t , v ) , t ∈ ( 0 , 1 ) , − v ″ = g ( t , u ) , t ∈ ( 0 , 1 ) , u ( 0 ) = v ( 0 ) = 0 , α u ( η ) = u ( 1 ) , α v ( η ) = v ( 1 ) . Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones.

Highlights

In this paper, we study a higher-order quasilinear equation with p-Laplacian φp u(n−1) + g(t) f t, u, u, . . . , u(n−2) = 0, 0 < t < 1, n ≥ 3, (1.1)
All of the above-mentioned references dealt with the case of the nonlinearity without singularities
For the singular case of multipoint boundary value problems, to our acknowledge, no one has studied the existence of positive solutions in this case

Summary

Introduction

On the existence of positive solutions of multipoint boundary value problems for second-order ordinary differential equation, some authors have obtained the existence results (see [5,6,7,8, 10]). For the singular case of multipoint boundary value problems, to our acknowledge, no one has studied the existence of positive solutions in this case.

Results

Conclusion