Bounds for solutions of multipoint boundary value problems (Q1614673)

The author considers the multipoint BVP \[ L_ny\geq 0,\quad 0\leq t\leq 1,\qquad y(a_i)=0,\quad i=1,2,\dots, n, \tag{1} \] where \(0\leq a_1\leq a_2 \leq \dots \leq a_n \leq 1\) and \(L_n\) is a linear disconjugate differential operator. Solutions to (1) are compared with solutions to the equation \[ L_ny=0. \tag{2} \] It is proved that, if \(y\) is a solution to (1), then there are linearly independent solutions \(p_1,\dots,p_{n-1}\) to (2) such that \[ p_i(a_j)=0,\quad p_i(a_{i+1})\not=0, \quad i=1,\dots,n-1, j=1,\dots, i, \] with \[ yp_i\geq 0,\quad 0\leq t\leq a_{i+1},\qquad W(p_i,y)\leq 0, \quad 0\leq t\leq a_{i+2},\;i=1,\dots,n-1. \] And also, there are linearly independent solutions \(q_1,\dots,q_{n-1}\) to (2) such that \[ q_i(a_{n+1-j})=0,\quad q_i(a_{n-i})\not=0, \quad i=1,\dots,n-1, j=1,\dots, i, \] with \[ yq_i\geq 0,\quad a_{n-i}\leq t\leq 1,\qquad W(q_i,y)\geq 0, \quad a_{n-i-1}\leq t\leq 1,\;i=1,\dots,n-1. \] The proofs are similar to those by \textit{U. Elias} [Proc. Am. Math. Soc. 128, 475--484 (2000; Zbl 0932.34004)] and they involve induction on the order of \(L_n\) and the derivative of the quotient of Wronskians. The results complete earlier ones by \textit{P. W. Eloe} and \textit{J. Henderson} [Rocky Mt. J. Math. 29, 821--829 (1999; Zbl 0954.34010)].