On Green’s function of an $n$-point boundary value problem

Abstract:

The Green’s function ${g_n}(x,s)$ for an n-point boundary value problem, ${y^{(n)}}(x) = 0,y({a_1}) = y({a_2}) = \cdots = y({a_n}) = 0$ is explicitly given. As a tool for discussing $\operatorname {sgn} g_n(x,s)$ on the square $[{a_1},{a_n}] \times [{a_1},{a_n}]$, some results about polynomials with coefficients as symmetric functions of a’s are obtained. It is shown that \[ \int _{{a_1}}^{{a_n}} {|{g_n}(x,s)|ds} \] is a suitable polynomial in x. Applications to n-point boundary value problems and lower bounds for ${a_m}\;(m \geq n)$ are included.