Two-point simple-type self-adjoint boundary value problems for bending a beam -- dependency of Green functions on an interval length (Q1764342)

This paper deals with Green functions of the boundary value problem given by the linear equation \[ u^{(4)}-pu''+qu=f(x), \quad 0<x<L, \] and the boundary conditions \[ u^{(m_{j})}(0)=\alpha _{j}, u^{(n_{j})}(L)=\beta _{j}, \quad j=0,1. \] Here, \(p,q\) are constants satisfying \((\frac{p}{2}) ^{2}>q>0,\) \(p>0,\) and \(0\leq m_{0}<m_{1}\leq 3,\) \(0\leq n_{0}<n_{1}\leq 3,\) \(m=(m_{0},m_{1}),\) \( n=(n_{0},n_{1}).\) For nine choices of the parameters \((m,n)\), the problem is selfadjoint. The expression of the Green functions \(g(m,n,L;x,y) \) corresponding to the self-adjoint cases was obtained by \textit{Y. Kametaka, K. Takemura, Y. Suzuki} and \textit{A. Nagai} [Japan J. Ind. Appl. Math. 18, 543--566 (2001; Zbl 0991.34024)]. Now, the authors prove the dependence of \(g(m,n,L;x,y)\) on the interval length \(L.\) More precisely, they show that the Green function is monotone with respect to \(L\) in seven cases, and for \((m,n)\in \{(0,2,1,3),(0,1,1,3)\}\) three critical values \(0<L_{1}<L_{2}<L_{3}\) depending on \(p,q\) determine different zones of monotonicity of \(g\) with respect to \(L.\)