The (2,2)-disconjugacy of a fourth order difference equation

Abstract

Let a fourth order linear difference operator L be defined by for t in a discrete interval [a + 2,b + 2] where b - a is a nonegative integer. Solutions of Ly = 0 are then determined on the discrete interval [a,b + 4]. Defining generalized zeros appropriately, the equation Ly = 0 is (2,2)-disconjugate on [a,b + 4] provided no nontrivial solution of Ly = 0 has two distinct generalized zeros, each of order two or more, in [a,b + 4]. Letting q–(t) = max{−q(t),0}, the main theorem gives a condition on q(t), in the form of a strict inequality that q–(t) summed over the interval [a + 2b + 2] satisfies, which guarantees that Ly = 0 is (2,2)-disconjugate on [a,b + 4]. An example is given showing that the inequality is sharp. The proof of the main result utilizes an appropriately defined quadratic form.