Convex regular domains of tridiagonal matrices (Q1077493)

Let f and h be given real valued functions, g an arbitrary function and \({\mathcal A}(g)\) the tridiagonal matrix with upper diagonal h(1),...,h(n-1), lower diagonal f(1),...,f(n-1) and diagonal g(1),...,g(n). The authors prove that if there exists a positive sequence c(k) such that \(| g(1)| \geq c(1)\), \(| g(k)| \geq (| f(k-1)h(k-1)| /c(k- 1))+c(k),\) for \(2\leq k\leq n-1\) and \(| g(n)| >| f(n-1)k(n- 1)| /c(n-1),\) then \({\mathcal A}(g)\) is nonsingular. G is the set of vectors \(g=(g(1),g(2),...,g(n))\) which satisfy the above inequalities for some positive sequence c(k). If all components of a given vector g are nonzero O(g) denotes the set of vectors u such that \(g(k)u(k)>0\), \(1\leq k\leq n\). The authors prove the intersection of G and O(g) is convex. They describe a set of vectors W such that, for each \(g\in W\), the convex domain \(G\cap O(g)\) is bounded below by a singular surface (i.e. each vector in the surface is associated with a singular matrix). The results are applied to derive best possible conditions for the existence and uniqueness of solutions for boundary value problems of certain difference equations.