Splines and anti-periodic boundary-value problems

Abstract

Aftabizadeh, Pavel and Huang showed in 1994 that some second-order differential equations on (0, π) with anti-periodic conditions y(0)+y(π)=0, y′(0)+y′(π)=0 have a unique solution. In the present paper, the authors consider a differential equation g(t, x(t), x′(t), x′′(t))=0 on (a, b) (t∈[a, b] and g continuous) having a solution satisfying the anti-periodic conditions x (i)(a)+x (i)(b)=0 (i=0, 1). They show that for every ϵ>0 there exist a positive integer k and a linear combination v k of spline functions such that and v k satisfies the exact anti-periodic boundary conditions.