Eigenvalue intervals of two-point boundary value problems for general n-th order differential equations

Abstract

We consider the general $n^{th}$ order differential equation, $$y^{(n)}(x)+\lambda f(x,~y,~y',~y'',~.~.~.,~y^{(n-1)} )=0,~~x\in[a, b],$$ subject to the two-point boundary conditions $$y^{(i)}(a)=0,~~0\leq i\leq n-2,$$ $$ y^{(n-1)}(b)=0.$$ Values of the parameter $\lambda$ are determined for which the two-point boundary value problem has a positive solution by utilizing a fixed point theorem on cone.