Positivity conditions and bounds for Green's functions for higher order two-point BVP
The author studies the higher order two-point boundary value problem \[ \sum_{k=0}^n a_{n-k}(x) D^k u(x) = f(x), \quad 0<x<1, \] \[ u^{(2k)}(0) = u^{(2k)}(1) = 0, \quad k=0,1,\cdots,n-1. \] Here \(a_k(x)\), \(k=1,2,\cdots,n\), are real continuous functions, \(a_0(x)\equiv 1\), and \(D=-d^2/dx^2\). The author proves some sufficient conditions for the positivity of Green's function of the problem. He also obtains a set of upper and lower bounds for Green's function. At the end of the paper, an existence result for at least one positive solution for the problem is proved.
scientific article; zbMATH DE number 5997988
scientific article